Ladder

13 scenarios, 87 inputs, N from 1 to 10,000, every cell against 200-digit arithmetic.

Each scenario is measured twice: against the exact inputs, which is what a protocol sees, and against the inputs as each format rounds them, which isolates the arithmetic. The chart covers every input; the table below it is the reference input, cell by cell.

9,384 cells checked live on Sepolia at block 11,928,247, 0 mismatches. 2026-10-16, runner 0xfcdf2f18…. Re-run: uv run scripts/ladder/livecap.py

How this was measured

truth: For every scenario, input and N the same loop is run in exact real arithmetic with mpmath at 300 decimal digits (scripts/ladder/ladder.py), from the exact rational parameters in scenarios.json. Closed forms are used where the loop has one (compounding, the cumulative product, the geometric mean, the round trip, per-second and continuous accrual, the option walk and the decay); the bonding curve, the amortising loan and the Black-Scholes chain are stepped.
rounding: Each library receives the parameters floored into its own representation (2^-128, 2^-64, 10^-18 or 10^-27 units, or the nearest float for ABDKMathQuad and FP256; scripts/ladder/inputs.json). truthFormat is the exact loop started from those floored parameters, so digitsFormat measures the library's arithmetic alone, while digits measures what a protocol sees: the format's rounding of the inputs plus the arithmetic.
digits: digits = floor(-log10 |v - t| / |t|), the number of correct leading significant digits by relative error, capped at 100, which is reported as exact. It is not a string comparison: a value can be correct to 36 digits while its 37th printed digit happens to agree with the truth, or while a carry makes the printed prefix differ earlier.
agree: agree is the count of leading significant digits of the printed value (42 significant digits) that are literally equal to the printed truth. The demo page colours the rest of the value with it. It is a reading aid, not a measurement: a carry across 0.999… / 1.000… gives 0.
gas: gasTotal is measured inside the adapter around the scenario function (the parameter reads and the loop), after the runner has warmed the adapter and the FP127 object, so the call into the adapter and the ABI decoding are excluded. The FP127 (deployed) form includes one staticcall per op; FP127Lib and the other libraries are internal calls. Aave, Morpho and Maker answer only the accrual scenarios their models describe and report Unsupported elsewhere, which the tables show as a revert. gasPerStep = gasTotal / N. Reverted cells carry no gas.
live: LadderRunner.run(scenario, form, n, p) on Sepolia executes any cell with one eth_call and returns (ok, raw, reason, gasPerStep, gasTotal). scripts/ladder/livecap.py sends every cell's floored parameters p and compares the returned raw word with the committed one bit for bit (or the revert selector for a reverted cell). liveMaxN is the largest N per form the public RPC completed within its gas limit, measured by scripts/ladder/livecap.py.
reproduce: make ladder regenerates ladder.json and ladder.md from the Foundry harness (test/ladder/Ladder.t.sol) and this script; make ladder-check fails CI when the committed dataset differs. Versions of every library and of forge are recorded in the dataset. Generated 2026-10-15; versions: aave aave-v3-origin@8305565ae342 (MathUtils, transcribed), abdk v3.2, balancer balancer-v3-monorepo@2b9ce4b1b8ad (LogExpMath.sol, vendored), forge forge Version: 1.7.1, fp127 60e7ec3, huff sepolia:0xfae694D0c2c44181791F838c54Ed64C3151FfE30@11857616, maker dss pot.sol rpow via solady v0.1.26, morpho morpho-blue@8e26ca6a8dbc (MathLib, transcribed), prb-math v4.2.0, solady v0.1.26.

Daily compound interest

Step: b = b * (1 + r). Result: balance after N days.

Per-block or per-day accrual in every lending protocol: one mul per step, N steps.

Over all 10 inputs

vs exact inputs (what a protocol sees) vs inputs as each format rounds them (the arithmetic alone)

Line: median correct digits over the inputs; band: the range; capped at 80 on the axis. FP127Lib coincides with FP127 and is not drawn. A missing point means every input reverted at that N. FP127 emerald, Solady gold, ABDK sapphire, PRBMath red, Quad violet, Balancer teal, Aave orange, Morpho plum, Maker slate, FP256 black, labelled at the line end; Aave, Morpho and Maker appear only on the accrual scenarios their models describe. Hover a point for the numbers.

N	FP127	FP127Lib	ABDK	Solady	PRBMath	Quad	Balancer	Aave	Morpho	Maker	FP256
1	38 (35–39)	80 (35–80)	38 (35–39)	80 (35–80)	19 (16–20)	80 (16–80)	18 (15–19)	80 (15–80)	18 (15–19)	80 (15–80)	33 (33–35)	33 (33–35)	18 (15–19)	80 (15–80)	— 10 rev	— 10 rev	— 10 rev	— 10 rev	27 (24–28)	80 (24–80)	71 (71–71)	71 (71–71)
2	38 (35–38)	42 (35–47)	38 (35–38)	42 (35–47)	19 (15–19)	23 (16–27)	18 (14–19)	22 (14–26)	18 (14–19)	22 (14–26)	33 (33–34)	33 (33–34)	18 (14–19)	22 (14–26)	— 10 rev	— 10 rev	— 10 rev	— 10 rev	26 (25–27)	27 (25–27)	71 (71–71)	71 (71–71)
5	38 (35–38)	42 (35–46)	38 (35–38)	42 (35–46)	18.5 (15–19)	22.5 (15–27)	18 (14–19)	21 (14–26)	18 (14–19)	21 (14–26)	33 (32–33)	33 (32–33)	18 (14–19)	21 (14–26)	— 10 rev	— 10 rev	— 10 rev	— 10 rev	26 (24–27)	27 (24–27)	70 (70–71)	70 (70–71)
10	37 (34–38)	41 (34–45)	37 (34–38)	41 (34–45)	18 (15–19)	22 (15–26)	17 (14–18)	21 (14–25)	17 (14–18)	21 (14–25)	32 (32–33)	32 (32–33)	17 (14–18)	21 (14–25)	— 10 rev	— 10 rev	— 10 rev	— 10 rev	26 (24–26)	26 (24–26)	70 (70–71)	70 (70–71)
20	37 (34–37)	41 (34–45)	37 (34–37)	41 (34–45)	18 (15–18)	22 (15–26)	17 (13–18)	21 (13–25)	17 (13–18)	21 (13–25)	32 (32–32)	32 (32–32)	17 (13–18)	21 (13–25)	— 10 rev	— 10 rev	— 10 rev	— 10 rev	26 (24–26)	26 (24–26)	70 (70–70)	70 (70–70)
50	37 (34–37)	41 (34–45)	37 (34–37)	41 (34–45)	17.5 (14–18)	21 (14–25)	17 (13–18)	20 (13–24)	17 (13–18)	20 (13–24)	32 (31–32)	32 (31–32)	17 (13–18)	20 (13–24)	— 10 rev	— 10 rev	— 10 rev	— 10 rev	25 (24–26)	26 (24–26)	69 (69–70)	69 (69–70)
100	36 (33–37)	40 (33–44)	36 (33–37)	40 (33–44)	17 (14–18)	21 (14–25)	16 (13–17)	20 (13–24)	16 (13–17)	20 (13–24)	31 (31–32)	31 (31–32)	16 (13–17)	20 (13–24)	— 10 rev	— 10 rev	— 10 rev	— 10 rev	25 (24–25)	25 (24–26)	69 (69–69)	69 (69–69)
365	36 (33–36)	40 (33–44)	36 (33–36)	40 (33–44)	16.5 (14–17)	21 (14–25)	16 (12–17)	19 (12–23)	16 (12–17)	19 (12–23)	31 (31–31)	31 (31–31)	16 (12–17)	19 (12–23)	— 10 rev	— 10 rev	— 10 rev	— 10 rev	24 (24–25)	25 (24–25)	68 (68–69)	68 (68–69)
1000	35 (32–36)	39 (32–44)	35 (32–36)	39 (32–44)	16 (13–17)	20 (13–24)	15 (12–16)	19 (12–23)	15 (12–16)	19 (12–23)	30 (30–31)	30 (30–31)	15 (12–16)	19 (12–23)	— 10 rev	— 10 rev	— 10 rev	— 10 rev	24 (24–24)	24 (24–25)	68 (68–68)	68 (68–68)
10000	34 (32–35)	39 (32–43)	34 (32–35)	39 (32–43)	15 (13–16)	19.5 (13–24)	14 (11–15)	18 (11–23)	14 (11–15)	18 (11–23)	29 (29–30)	29 (29–30)	14 (11–15)	18 (11–23)	— 10 rev	— 10 rev	— 10 rev	— 10 rev	23 (23–23)	23 (23–24)	67 (67–67)	67 (67–67)

Median (min–max) correct digits over the inputs, capped at 80; "rev" counts inputs that reverted at that N and are excluded.

The inputs and the ops

input	parameters	why
reference reference	principal = 10000, rate = 0.1/365	$10,000 at 10% APR, the headline cell
tiny-low	principal = 1, rate = 0.01/365	one unit at 1%: the integer part is small, the fraction carries everything
tiny-high	principal = 1, rate = 0.5/365	one unit at 50%
mid-1pct	principal = 10000, rate = 0.01/365	a slow rate: the per-step factor is close to 1
mid-5pct	principal = 10000, rate = 0.05/365	a typical lending rate
mid-50pct	principal = 10000, rate = 0.5/365	a high rate: the balance grows 150x by N = 10,000
large	principal = 1000000, rate = 0.1/365	a million-unit position
huge	principal = 100000000, rate = 0.1/365	a hundred-million-unit pool
huge-high	principal = 100000000, rate = 0.5/365	the largest balance the ladder reaches, near 10^14
fraction	principal = 0.001, rate = 0.1/365	a sub-unit balance: leading zeros in WAD

Library calls per step: FP127 mul; FP127Lib mul; ABDK mul; Solady sMulWad; PRBMath mul; Quad mul; Balancer mulDown; Aave unsupported; Morpho unsupported; Maker rpow, rmul; FP256 mul. Parameters reach each adapter floored into its representation (ladder-inputs.json).

The reference input in full: principal = 10000, rate = 0.1/365

Each cell: the value with the digits that agree with the truth in ink and the rest in red; below it, correct digits against the exact truth, against the format-rounded truth, the signed error in units of the last place, and gas per step.

N	truth	FP127 (deployed, staticcall)	FP127Lib (inline)	ABDKMath64x64	Solady FixedPointMathLib	PRBMath SD59x18	ABDKMathQuad (binary128)	Balancer LogExpMath	Aave v3 MathUtils (RAY)	Morpho Blue MathLib	Maker rpow (RAY)	FP256 (float, inline)
1	10002.7397260273972602739726027397260273973	10002.7397260273972602739726027397260273831 38 digits · exact vs format · -4810.96 ULP · 2148 gas/step	10002.7397260273972602739726027397260273831 38 digits · exact vs format · -4810.96 ULP · 888 gas/step	10002.7397260273972600785191700012433102529 19 digits · exact vs format · -3605.48 ULP · 934 gas/step	10002.73972602739726 19 digits · exact vs format · -273.973 ULP · 744 gas/step	10002.73972602739726 19 digits · exact vs format · -273.973 ULP · 1104 gas/step	10002.7397260273972602739726027397248603154 33 digits · 33 vs format · ULP · 2933 gas/step	10002.73972602739726 19 digits · exact vs format · -273.973 ULP · 1015 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	10002.7397260273972602739726 27 digits · exact vs format · -2739.73 ULP · 610 gas/step	10002.7397260273972602739726027397260273973 71 digits · 71 vs format · ULP · 2901 gas/step
2	10005.4802026646650403452805404391067742541	10005.4802026646650403452805404391067742258 38 digits · 42 vs format · -9625.4 ULP · 1885 gas/step	10005.4802026646650403452805404391067742258 38 digits · 42 vs format · -9625.4 ULP · 623 gas/step	10005.4802026646650399542451230494588543252 19 digits · 23 vs format · -7213.33 ULP · 610 gas/step	10005.480202664665039797 19 digits · 22 vs format · -548.281 ULP · 488 gas/step	10005.480202664665039797 19 digits · 22 vs format · -548.281 ULP · 848 gas/step	10005.480202664665040345280540439104093104 33 digits · 33 vs format · ULP · 1974 gas/step	10005.480202664665039797 19 digits · 22 vs format · -548.281 ULP · 759 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	10005.48020266466504034528053 26 digits · 27 vs format · -10439.1 ULP · 420 gas/step	10005.4802026646650403452805404391067742541 71 digits · 71 vs format · ULP · 2067 gas/step
5	10013.7061382924386210524857642965155424808	10013.7061382924386210524857642965155424101 38 digits · 42 vs format · -24083.1 ULP · 1727 gas/step	10013.7061382924386210524857642965155424101 38 digits · 42 vs format · -24083.1 ULP · 464 gas/step	10013.7061382924386200740608440784562560566 19 digits · 23 vs format · -18048.8 ULP · 415 gas/step	10013.706138292438619679 18 digits · 21 vs format · -1373.49 ULP · 335 gas/step	10013.706138292438619679 18 digits · 21 vs format · -1373.49 ULP · 695 gas/step	10013.7061382924386210524857642965089797718 33 digits · 33 vs format · ULP · 1398 gas/step	10013.706138292438619679 18 digits · 21 vs format · -1373.49 ULP · 606 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	10013.70613829243862105248574 26 digits · 26 vs format · -24296.5 ULP · 197 gas/step	10013.7061382924386210524857642965155424808 70 digits · 70 vs format · ULP · 1566 gas/step
10	10027.4310624075663873328585783270357249574	10027.4310624075663873328585783270357248157 37 digits · 41 vs format · -48232.5 ULP · 1674 gas/step	10027.4310624075663873328585783270357248157 37 digits · 41 vs format · -48232.5 ULP · 411 gas/step	10027.4310624075663853733086060271428863189 18 digits · 22 vs format · -36147.3 ULP · 350 gas/step	10027.431062407566384582 18 digits · 21 vs format · -2750.86 ULP · 284 gas/step	10027.431062407566384582 18 digits · 21 vs format · -2750.86 ULP · 644 gas/step	10027.43106240756638733285857832702353373 32 digits · 32 vs format · ULP · 1206 gas/step	10027.431062407566384582 18 digits · 21 vs format · -2750.86 ULP · 555 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	10027.43106240756638733285853 26 digits · 26 vs format · -48327.0 ULP · 121 gas/step	10027.4310624075663873328585783270357249574 70 digits · 70 vs format · ULP · 1399 gas/step
20	10054.9373711336135548503369243222913751056	10054.9373711336135548503369243222913748213 37 digits · 41 vs format · -96731.2 ULP · 1648 gas/step	10054.9373711336135548503369243222913748213 37 digits · 41 vs format · -96731.2 ULP · 384 gas/step	10054.937371133613550920398935217825453492 18 digits · 22 vs format · -72494.6 ULP · 318 gas/step	10054.937371133613549333 18 digits · 21 vs format · -5517.34 ULP · 258 gas/step	10054.937371133613549333 18 digits · 21 vs format · -5517.34 ULP · 618 gas/step	10054.9373711336135548503369243222684323598 32 digits · 32 vs format · ULP · 1110 gas/step	10054.937371133613549333 18 digits · 21 vs format · -5517.34 ULP · 529 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	10054.93737113361355485033684 26 digits · 26 vs format · -84322.3 ULP · 68 gas/step	10054.9373711336135548503369243222913751056 70 digits · 70 vs format · ULP · 1316 gas/step
50	10137.9098421416552959332725454243516547326	10137.909842141655295933272545424351654016 37 digits · 41 vs format · -243824.0 ULP · 1632 gas/step	10137.909842141655295933272545424351654016 37 digits · 41 vs format · -243824.0 ULP · 368 gas/step	10137.9098421416552860271844023476450757926 18 digits · 21 vs format · -182735.0 ULP · 298 gas/step	10137.909842141655282022 17 digits · 20 vs format · -13911.3 ULP · 243 gas/step	10137.909842141655282022 17 digits · 20 vs format · -13911.3 ULP · 603 gas/step	10137.9098421416552959332725454242952441686 32 digits · 32 vs format · ULP · 1053 gas/step	10137.909842141655282022 17 digits · 20 vs format · -13911.3 ULP · 514 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	10137.90984214165529593327234 25 digits · 26 vs format · -205424.0 ULP · 31 gas/step	10137.9098421416552959332725454243516547326 69 digits · 69 vs format · ULP · 1266 gas/step
100	10277.721596739264220164675884991779643531	10277.7215967392642201646758849917796420782 36 digits · 40 vs format · -494375.0 ULP · 1627 gas/step	10277.7215967392642201646758849917796420782 36 digits · 40 vs format · -494375.0 ULP · 363 gas/step	10277.7215967392642000792805576092625585716 17 digits · 21 vs format · -370510.0 ULP · 292 gas/step	10277.721596739264191958 17 digits · 20 vs format · -28206.7 ULP · 238 gas/step	10277.721596739264191958 17 digits · 20 vs format · -28206.7 ULP · 598 gas/step	10277.7215967392642201646758849916676515892 31 digits · 31 vs format · ULP · 1034 gas/step	10277.721596739264191958 17 digits · 20 vs format · -28206.7 ULP · 509 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	10277.72159673926422016467546 25 digits · 25 vs format · -424992.0 ULP · 17 gas/step	10277.721596739264220164675884991779643531 69 digits · 69 vs format · ULP · 1249 gas/step
365	11051.5578161626437393801159668054461177946	11051.5578161626437393801159668054461120925 36 digits · 40 vs format · -1.94032e+6 ULP · 1624 gas/step	11051.5578161626437393801159668054461120925 36 digits · 40 vs format · -1.94032e+6 ULP · 359 gas/step	11051.5578161626436605492209384882240641446 17 digits · 21 vs format · -1.45417e+6 ULP · 287 gas/step	11051.557816162643628692 16 digits · 19 vs format · -110688.0 ULP · 234 gas/step	11051.557816162643628692 16 digits · 19 vs format · -110688.0 ULP · 594 gas/step	11051.5578161626437393801159668049905361029 31 digits · 31 vs format · ULP · 1020 gas/step	11051.557816162643628692 16 digits · 19 vs format · -110688.0 ULP · 505 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	11051.5578161626437393801143 24 digits · 25 vs format · -1.66681e+6 ULP · 6 gas/step	11051.5578161626437393801159668054461177946 68 digits · 68 vs format · ULP · 1237 gas/step
1000	13151.2942008121758666272801482974151894245	13151.2942008121758666272801482974151708344 35 digits · 39 vs format · -6.32587e+6 ULP · 1624 gas/step	13151.2942008121758666272801482974151708344 35 digits · 39 vs format · -6.32587e+6 ULP · 358 gas/step	13151.294200812175609621539323218497585799 16 digits · 20 vs format · -4.74092e+6 ULP · 286 gas/step	13151.294200812175505839 16 digits · 19 vs format · -360788.0 ULP · 233 gas/step	13151.294200812175505839 16 digits · 19 vs format · -360788.0 ULP · 593 gas/step	13151.2942008121758666272801482960083685618 30 digits · 30 vs format · ULP · 1016 gas/step	13151.294200812175505839 16 digits · 19 vs format · -360788.0 ULP · 504 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	13151.29420081217586662727473 24 digits · 24 vs format · -5.4183e+6 ULP · 2 gas/step	13151.2942008121758666272801482974151894245 68 digits · 68 vs format · ULP · 1234 gas/step
10000	154769.34069144112806190764064318746174622	154769.340691441128061907640643187459558592 34 digits · 39 vs format · -7.44411e+8 ULP · 1641 gas/step	154769.340691441128061907640643187459558592 34 digits · 39 vs format · -7.44411e+8 ULP · 358 gas/step	154769.340691441097818583337575315539424992 15 digits · 20 vs format · -5.57891e+8 ULP · 286 gas/step	154769.340691441085644663 15 digits · 18 vs format · -4.24172e+7 ULP · 233 gas/step	154769.340691441085644663 15 digits · 18 vs format · -4.24172e+7 ULP · 593 gas/step	154769.340691441128061907640643019251346999 29 digits · 29 vs format · ULP · 1015 gas/step	154769.340691441085644663 15 digits · 18 vs format · -4.24172e+7 ULP · 504 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	154769.34069144112806190700363 23 digits · 23 vs format · -6.37013e+8 ULP · 0 gas/step	154769.34069144112806190764064318746174622 67 digits · 67 vs format · ULP · 1233 gas/step

Compound interest in one pow

Step: b0 * (1 + r)^N. Result: the same balance, computed with one real-exponent pow.

The clever way to skip the loop. pow is exp(N ln(1+r)) in every library, so its precision is bounded by ln and exp, and the error grows with N without any loop.

Over all 10 inputs

vs exact inputs (what a protocol sees) vs inputs as each format rounds them (the arithmetic alone)

N	FP127	FP127Lib	ABDK	Solady	PRBMath	Quad	Balancer	Aave	Morpho	Maker	FP256
1	38 (35–39)	80 (35–80)	38 (35–39)	80 (35–80)	19 (16–19)	19 (16–19)	17 (15–17)	18 (15–18)	18 (15–19)	80 (15–80)	33 (33–33)	33 (33–33)	17 (15–17)	18 (15–18)	— 10 rev	— 10 rev	— 10 rev	— 10 rev	— 10 rev	— 10 rev	71 (71–71)	71 (71–71)
2	38 (35–38)	38 (35–38)	38 (35–38)	38 (35–38)	18 (16–18)	18.5 (16–19)	17 (15–17)	17 (15–18)	16 (15–16)	16 (15–16)	33 (33–34)	33 (33–34)	17 (15–17)	17 (15–18)	— 10 rev	— 10 rev	— 10 rev	— 10 rev	— 10 rev	— 10 rev	71 (71–72)	71 (71–72)
5	38 (35–38)	38 (35–39)	38 (35–38)	38 (35–39)	18 (16–18)	18 (16–19)	17 (15–17)	17 (15–17)	16 (15–16)	16 (15–16)	33 (32–33)	33 (32–33)	17 (15–17)	18 (15–18)	— 10 rev	— 10 rev	— 10 rev	— 10 rev	— 10 rev	— 10 rev	70 (70–71)	70 (70–71)
10	37 (35–38)	39 (35–40)	37 (35–38)	39 (35–40)	18 (16–18)	18 (16–18)	17 (15–17)	17 (15–18)	16 (15–16)	16 (15–16)	33 (32–34)	33 (32–34)	17 (15–18)	18 (15–18)	— 10 rev	— 10 rev	— 10 rev	— 10 rev	— 10 rev	— 10 rev	70 (70–70)	70 (70–70)
20	37 (35–37)	39 (35–39)	37 (35–37)	39 (35–39)	17 (16–18)	18 (16–18)	16 (15–17)	16 (15–17)	15 (15–15)	15 (15–15)	33 (32–33)	33 (32–33)	17 (15–17)	18 (15–18)	— 10 rev	— 10 rev	— 10 rev	— 10 rev	— 10 rev	— 10 rev	70 (70–70)	70 (70–70)
50	37 (35–37)	39 (37–39)	37 (35–37)	39 (37–39)	17 (16–17)	17 (16–18)	16 (15–17)	16 (15–17)	15 (15–15)	15 (15–15)	32 (32–33)	32 (32–33)	17 (15–17)	17 (15–18)	— 10 rev	— 10 rev	— 10 rev	— 10 rev	— 10 rev	— 10 rev	69 (69–70)	69 (69–70)
100	36 (35–37)	39 (37–39)	36 (35–37)	39 (37–39)	17 (16–17)	17 (16–18)	16 (15–17)	16 (15–17)	15 (14–15)	15 (14–15)	32 (31–33)	32 (31–33)	16 (15–17)	17 (15–18)	— 10 rev	— 10 rev	— 10 rev	— 10 rev	— 10 rev	— 10 rev	69 (69–69)	69 (69–69)
365	36 (35–36)	39 (35–42)	36 (35–36)	39 (35–42)	16 (16–16)	16.5 (16–17)	15 (15–16)	15 (15–16)	14 (14–14)	14 (14–14)	31 (31–32)	31 (31–32)	16 (15–17)	17.5 (15–18)	— 10 rev	— 10 rev	— 10 rev	— 10 rev	— 10 rev	— 10 rev	68 (68–69)	68 (68–69)
1000	35 (35–36)	39 (36–39)	35 (35–36)	39 (36–39)	16 (15–16)	16 (16–17)	15 (14–16)	15 (14–16)	14 (13–14)	14 (13–14)	31 (30–32)	31 (30–32)	15 (15–16)	18 (15–18)	— 10 rev	— 10 rev	— 10 rev	— 10 rev	— 10 rev	— 10 rev	68 (68–68)	68 (68–68)
10000	34 (34–35)	40 (36–42)	34 (34–35)	40 (36–42)	15 (15–15)	15 (15–16)	14 (13–15)	14 (14–15)	13 (12–13)	13 (12–13)	30 (29–31)	30 (29–31)	14.5 (14–15)	18 (16–18)	— 10 rev	— 10 rev	— 10 rev	— 10 rev	— 10 rev	— 10 rev	67 (67–67)	67 (67–67)

Median (min–max) correct digits over the inputs, capped at 80; "rev" counts inputs that reverted at that N and are excluded.

The inputs and the ops

input	parameters	why
reference reference	principal = 10000, rate = 0.1/365	$10,000 at 10% APR, the headline cell
tiny-low	principal = 1, rate = 0.01/365	one unit at 1%: the integer part is small, the fraction carries everything
tiny-high	principal = 1, rate = 0.5/365	one unit at 50%
mid-1pct	principal = 10000, rate = 0.01/365	a slow rate: the per-step factor is close to 1
mid-5pct	principal = 10000, rate = 0.05/365	a typical lending rate
mid-50pct	principal = 10000, rate = 0.5/365	a high rate: the balance grows 150x by N = 10,000
large	principal = 1000000, rate = 0.1/365	a million-unit position
huge	principal = 100000000, rate = 0.1/365	a hundred-million-unit pool
huge-high	principal = 100000000, rate = 0.5/365	the largest balance the ladder reaches, near 10^14
fraction	principal = 0.001, rate = 0.1/365	a sub-unit balance: leading zeros in WAD

Library calls per step: FP127 pow, mul; FP127Lib pow, mul; ABDK exp, mul, ln (pow = exp(y ln x)); Solady powWad, sMulWad; PRBMath pow, mul; Quad exp(y ln x), mul; Balancer pow, mulDown; Aave unsupported; Morpho unsupported; Maker unsupported; FP256 pow, mul. Parameters reach each adapter floored into its representation (ladder-inputs.json).

The reference input in full: principal = 10000, rate = 0.1/365

N	truth	FP127 (deployed, staticcall)	FP127Lib (inline)	ABDKMath64x64	Solady FixedPointMathLib	PRBMath SD59x18	ABDKMathQuad (binary128)	Balancer LogExpMath	Aave v3 MathUtils (RAY)	Morpho Blue MathLib	Maker rpow (RAY)	FP256 (float, inline)
1	10002.7397260273972602739726027397260273973	10002.7397260273972602739726027397260273831 38 digits · exact vs format · -4810.96 ULP · 3285 gas/step	10002.7397260273972602739726027397260273831 38 digits · exact vs format · -4810.96 ULP · 1130 gas/step	10002.7397260273972595364180837584910932492 19 digits · 19 vs format · -13605.5 ULP · 11498 gas/step	10002.73972602739725 17 digits · 18 vs format · -10274.0 ULP · 2153 gas/step	10002.73972602739726 19 digits · exact vs format · -273.973 ULP · 1436 gas/step	10002.7397260273972602739726027397232825936 33 digits · 33 vs format · ULP · 28822 gas/step	10002.73972602739725 17 digits · 18 vs format · -10274.0 ULP · 3096 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	10002.7397260273972602739726027397260273973 71 digits · 71 vs format · ULP · 4037 gas/step
2	10005.4802026646650403452805404391067742541	10005.4802026646650403452805404391067742095 38 digits · 38 vs format · -15179.4 ULP · 1747 gas/step	10005.4802026646650403452805404391067742095 38 digits · 38 vs format · -15179.4 ULP · 682 gas/step	10005.4802026646650383841577469645756082173 18 digits · 18 vs format · -36176.3 ULP · 5749 gas/step	10005.48020266466503 17 digits · 18 vs format · -10345.3 ULP · 1076 gas/step	10005.48020266466484 16 digits · 16 vs format · -200345.0 ULP · 5366 gas/step	10005.480202664665040345280540439104093104 33 digits · 33 vs format · ULP · 14413 gas/step	10005.48020266466503 17 digits · 18 vs format · -10345.3 ULP · 1548 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	10005.4802026646650403452805404391067742541 71 digits · 71 vs format · ULP · 2576 gas/step
5	10013.7061382924386210524857642965155424808	10013.7061382924386210524857642965155424218 38 digits · 38 vs format · -20077.1 ULP · 2136 gas/step	10013.7061382924386210524857642965155424218 38 digits · 38 vs format · -20077.1 ULP · 2117 gas/step	10013.7061382924386174642777946885985329573 18 digits · 18 vs format · -66190.8 ULP · 2282 gas/step	10013.70613829243859 17 digits · 17 vs format · -31052.5 ULP · 430 gas/step	10013.70613829243812 16 digits · 16 vs format · -501052.0 ULP · 2118 gas/step	10013.7061382924386210524857642965105574936 33 digits · 33 vs format · ULP · 5728 gas/step	10013.70613829243861 17 digits · 18 vs format · -11052.5 ULP · 619 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	10013.7061382924386210524857642965155424808 70 digits · 70 vs format · ULP · 3974 gas/step
10	10027.4310624075663873328585783270357249574	10027.4310624075663873328585783270357248201 37 digits · 39 vs format · -46727.5 ULP · 1068 gas/step	10027.4310624075663873328585783270357248201 37 digits · 39 vs format · -46727.5 ULP · 1058 gas/step	10027.4310624075663802559285620041862330254 18 digits · 18 vs format · -130546.0 ULP · 1141 gas/step	10027.43106240756634 17 digits · 17 vs format · -47332.9 ULP · 215 gas/step	10027.4310624075654 16 digits · 16 vs format · -987333.0 ULP · 1060 gas/step	10027.4310624075663873328585783270298446172 33 digits · 33 vs format · ULP · 2864 gas/step	10027.43106240756638 18 digits · 18 vs format · -7332.86 ULP · 309 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	10027.4310624075663873328585783270357249574 70 digits · 70 vs format · ULP · 1988 gas/step
20	10054.9373711336135548503369243222913751056	10054.9373711336135548503369243222913748266 37 digits · 39 vs format · -94936.2 ULP · 534 gas/step	10054.9373711336135548503369243222913748266 37 digits · 39 vs format · -94936.2 ULP · 529 gas/step	10054.9373711336135415973444540149728254619 17 digits · 18 vs format · -244475.0 ULP · 570 gas/step	10054.93737113361346 17 digits · 17 vs format · -94850.3 ULP · 107 gas/step	10054.93737113361157 15 digits · 15 vs format · -1.98485e+6 ULP · 530 gas/step	10054.937371133613554850336924322282631856 33 digits · 33 vs format · ULP · 1432 gas/step	10054.93737113361354 17 digits · 18 vs format · -14850.3 ULP · 154 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	10054.9373711336135548503369243222913751056 70 digits · 70 vs format · ULP · 993 gas/step
50	10137.9098421416552959332725454243516547326	10137.9098421416552959332725454243516540042 37 digits · 38 vs format · -247838.0 ULP · 213 gas/step	10137.9098421416552959332725454243516540042 37 digits · 38 vs format · -247838.0 ULP · 211 gas/step	10137.9098421416552629724383560727574149496 17 digits · 17 vs format · -608020.0 ULP · 230 gas/step	10137.90984214165505 16 digits · 16 vs format · -245933.0 ULP · 43 gas/step	10137.90984214165031 15 digits · 15 vs format · -4.98593e+6 ULP · 217 gas/step	10137.9098421416552959332725454243299540485 32 digits · 32 vs format · ULP · 573 gas/step	10137.90984214165528 17 digits · 18 vs format · -15933.3 ULP · 61 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	10137.9098421416552959332725454243516547326 69 digits · 69 vs format · ULP · 397 gas/step
100	10277.721596739264220164675884991779643531	10277.7215967392642201646758849917796420803 36 digits · 39 vs format · -493651.0 ULP · 106 gas/step	10277.7215967392642201646758849917796420803 36 digits · 39 vs format · -493651.0 ULP · 105 gas/step	10277.7215967392641536234942911676171206636 17 digits · 17 vs format · -1.22747e+6 ULP · 115 gas/step	10277.72159673926373 16 digits · 16 vs format · -490165.0 ULP · 21 gas/step	10277.72159673925412 15 digits · 15 vs format · -1.01002e+7 ULP · 109 gas/step	10277.7215967392642201646758849917370713488 32 digits · 32 vs format · ULP · 286 gas/step	10277.72159673926418 17 digits · 17 vs format · -40164.7 ULP · 30 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	10277.721596739264220164675884991779643531 69 digits · 69 vs format · ULP · 198 gas/step
365	11051.5578161626437393801159668054461177946	11051.5578161626437393801159668054461120828 36 digits · 39 vs format · -1.94363e+6 ULP · 29 gas/step	11051.5578161626437393801159668054461120828 36 digits · 39 vs format · -1.94363e+6 ULP · 29 gas/step	11051.5578161626434791067405405406987028982 16 digits · 16 vs format · -4.8012e+6 ULP · 31 gas/step	11051.55781616264184 15 digits · 15 vs format · -1.89938e+6 ULP · 5 gas/step	11051.55781616260411 14 digits · 14 vs format · -3.96294e+7 ULP · 29 gas/step	11051.5578161626437393801159668052855700814 31 digits · 31 vs format · ULP · 78 gas/step	11051.55781616264362 16 digits · 18 vs format · -119380.0 ULP · 8 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	11051.5578161626437393801159668054461177946 68 digits · 68 vs format · ULP · 54 gas/step
1000	13151.2942008121758666272801482974151894245	13151.2942008121758666272801482974151708376 35 digits · 39 vs format · -6.32477e+6 ULP · 10 gas/step	13151.2942008121758666272801482974151708376 35 digits · 39 vs format · -6.32477e+6 ULP · 10 gas/step	13151.294200812175020048715168297093214278 16 digits · 16 vs format · -1.56166e+7 ULP · 11 gas/step	13151.29420081216967 15 digits · 15 vs format · -6.19663e+6 ULP · 2 gas/step	13151.29420081204668 14 digits · 14 vs format · -1.29187e+8 ULP · 10 gas/step	13151.2942008121758666272801482968934704975 31 digits · 31 vs format · ULP · 28 gas/step	13151.2942008121755 16 digits · 18 vs format · -366627.0 ULP · 3 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	13151.2942008121758666272801482974151894245 68 digits · 68 vs format · ULP · 19 gas/step
10000	154769.34069144112806190764064318746174622	154769.340691441128061907640643187459558676 34 digits · 40 vs format · -7.44383e+8 ULP · 1 gas/step	154769.340691441128061907640643187459558676 34 digits · 40 vs format · -7.44383e+8 ULP · 1 gas/step	154769.340691441028485565711836358104847022 15 digits · 15 vs format · -1.83686e+9 ULP · 1 gas/step	154769.34069144039984 14 digits · 14 vs format · -7.28222e+8 ULP · 0 gas/step	154769.34069142592578 13 digits · 13 vs format · -1.52023e+10 ULP · 1 gas/step	154769.340691441128061907640643126283994619 30 digits · 30 vs format · ULP · 2 gas/step	154769.34069144108545 15 digits · 17 vs format · -4.26119e+7 ULP · 0 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	154769.34069144112806190764064318746174622 67 digits · 67 vs format · ULP · 2 gas/step

Square-root bonding curve

Step: cost += sqrt(s); s += 1. Result: total cost of N unit buys.

Bonding curves price each token at a root of the supply; the sum of N floors is where the digits go.

Over all 7 inputs

vs exact inputs (what a protocol sees) vs inputs as each format rounds them (the arithmetic alone)

N	FP127	FP127Lib	ABDK	Solady	PRBMath	Quad	Balancer	Aave	Morpho	Maker	FP256
1	43 (39–80)	43 (39–80)	43 (39–80)	43 (39–80)	23 (20–80)	23 (20–80)	22 (18–80)	22 (18–80)	22 (18–80)	22 (18–80)	34 (33–80)	34 (33–80)	18 (17–80)	18 (17–80)	— 7 rev	— 7 rev	— 7 rev	— 7 rev	— 7 rev	— 7 rev	72 (71–80)	72 (71–80)
2	40 (38–43)	40 (38–43)	40 (38–43)	40 (38–43)	20 (19–24)	20 (19–24)	19 (18–22)	19 (18–22)	19 (18–22)	19 (18–22)	34 (33–34)	34 (33–34)	18 (18–18)	18 (18–18)	— 7 rev	— 7 rev	— 7 rev	— 7 rev	— 7 rev	— 7 rev	71 (71–73)	71 (71–73)
5	40 (38–43)	40 (39–43)	40 (38–43)	40 (39–43)	21 (19–24)	21 (19–24)	19 (18–22)	19 (18–22)	19 (18–22)	19 (18–22)	33 (33–34)	33 (33–34)	18 (18–18)	18 (18–18)	— 7 rev	— 7 rev	— 7 rev	— 7 rev	— 7 rev	— 7 rev	71 (70–72)	71 (70–72)
10	40 (38–43)	40 (38–43)	40 (38–43)	40 (38–43)	20 (19–24)	20 (19–24)	19 (18–22)	19 (18–22)	19 (18–22)	19 (18–22)	33 (33–33)	33 (33–33)	18 (18–18)	18 (18–18)	— 7 rev	— 7 rev	— 7 rev	— 7 rev	— 7 rev	— 7 rev	71 (70–72)	71 (70–72)
20	40 (38–43)	40 (38–43)	40 (38–43)	40 (38–43)	21 (18–24)	21 (19–24)	19 (18–22)	19 (18–22)	19 (18–22)	19 (18–22)	33 (33–33)	33 (33–33)	18 (17–18)	18 (17–18)	— 7 rev	— 7 rev	— 7 rev	— 7 rev	— 7 rev	— 7 rev	71 (70–71)	71 (70–71)
50	40 (37–43)	40 (38–43)	40 (37–43)	40 (38–43)	21 (18–24)	21 (19–24)	19 (18–22)	19 (18–22)	19 (18–22)	19 (18–22)	32 (32–32)	32 (32–32)	18 (18–18)	18 (18–18)	— 7 rev	— 7 rev	— 7 rev	— 7 rev	— 7 rev	— 7 rev	70 (70–71)	70 (70–71)
100	40 (37–43)	40 (38–43)	40 (37–43)	40 (38–43)	21 (18–24)	21 (19–24)	19 (18–22)	19 (18–22)	19 (18–22)	19 (18–22)	32 (32–32)	32 (32–32)	18 (17–18)	18 (17–18)	— 7 rev	— 7 rev	— 7 rev	— 7 rev	— 7 rev	— 7 rev	70 (70–71)	70 (70–71)
365	40 (36–43)	40 (38–43)	40 (36–43)	40 (38–43)	21 (17–24)	21 (19–24)	19 (18–22)	19 (18–22)	19 (18–22)	19 (18–22)	31 (31–31)	31 (31–31)	18 (17–18)	18 (17–18)	— 7 rev	— 7 rev	— 7 rev	— 7 rev	— 7 rev	— 7 rev	70 (69–71)	70 (69–71)
1000	40 (36–43)	40 (38–43)	40 (36–43)	40 (38–43)	21 (17–24)	21 (19–24)	19 (18–22)	19 (18–22)	19 (18–22)	19 (18–22)	31 (31–31)	31 (31–31)	18 (17–18)	18 (17–18)	— 7 rev	— 7 rev	— 7 rev	— 7 rev	— 7 rev	— 7 rev	70 (69–70)	70 (69–70)
10000	40 (36–43)	40 (39–43)	40 (36–43)	40 (39–43)	21 (16–24)	21 (19–24)	20 (18–22)	20 (18–22)	20 (18–22)	20 (18–22)	30 (30–30)	30 (30–30)	18 (18–18)	18 (18–18)	— 7 rev	— 7 rev	— 7 rev	— 7 rev	— 7 rev	— 7 rev	70 (68–70)	70 (68–70)

Median (min–max) correct digits over the inputs, capped at 80; "rev" counts inputs that reverted at that N and are excluded.

The inputs and the ops

input	parameters	why
reference reference	supply = 1000, unit = 1	a thousand tokens out, unit buys
from-one	supply = 1, unit = 1	the curve from its first token
half-units	supply = 1000, unit = 0.5	buys that are not integers
deep	supply = 1000000, unit = 1	a million out: sqrt is large, the sum is huge
micro	supply = 1, unit = 0.001	thousandth-unit buys from the start
very-deep	supply = 1000000000, unit = 1000	a billion out in thousand-token lots
small-steps	supply = 10, unit = 0.01	hundredth-unit buys from ten

Library calls per step: FP127 sqrt, add; FP127Lib sqrt, add; ABDK sqrt, add; Solady sqrtWad, add; PRBMath sqrt, add; Quad sqrt, add; Balancer pow(x, 1/2), add; Aave unsupported; Morpho unsupported; Maker unsupported; FP256 sqrt, add. Parameters reach each adapter floored into its representation (ladder-inputs.json).

The reference input in full: supply = 1000, unit = 1

N	truth	FP127 (deployed, staticcall)	FP127Lib (inline)	ABDKMath64x64	Solady FixedPointMathLib	PRBMath SD59x18	ABDKMathQuad (binary128)	Balancer LogExpMath	Aave v3 MathUtils (RAY)	Morpho Blue MathLib	Maker rpow (RAY)	FP256 (float, inline)
1	31.6227766016837933199889354443271853371956	31.6227766016837933199889354443271853371954 41 digits · 41 vs format · -0.065532 ULP · 4491 gas/step	31.6227766016837933199889354443271853371954 41 digits · 41 vs format · -0.065532 ULP · 4198 gas/step	31.6227766016837933199458594846209535944581 20 digits · 20 vs format · -0.794611 ULP · 1814 gas/step	31.622776601683793319 19 digits · 19 vs format · -0.988935 ULP · 1262 gas/step	31.622776601683793319 19 digits · 19 vs format · -0.988935 ULP · 1753 gas/step	31.6227766016837933199889354443271822760817 34 digits · 34 vs format · ULP · 3234 gas/step	31.622776601683793313 18 digits · 18 vs format · -6.98894 ULP · 3206 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	31.6227766016837933199889354443271853371956 71 digits · 71 vs format · ULP · 5056 gas/step
2	63.2613606407965424630952270291281684242756	63.2613606407965424630952270291281684242738 40 digits · 40 vs format · -0.626369 ULP · 4276 gas/step	63.2613606407965424630952270291281684242738 40 digits · 40 vs format · -0.626369 ULP · 3986 gas/step	63.2613606407965424630393536098438289627666 21 digits · 21 vs format · -1.03068 ULP · 1574 gas/step	63.261360640796542462 19 digits · 19 vs format · -1.09523 ULP · 1050 gas/step	63.261360640796542462 19 digits · 19 vs format · -1.09523 ULP · 1545 gas/step	63.2613606407965424630952270291281601345083 33 digits · 33 vs format · ULP · 3118 gas/step	63.26136064079654243 18 digits · 18 vs format · -33.0952 ULP · 2994 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	63.2613606407965424630952270291281684242756 71 digits · 71 vs format · ULP · 5227 gas/step
5	158.271878503221320234300570350652186729491	158.271878503221320234300570350652186729484 40 digits · 40 vs format · -2.49809 ULP · 4147 gas/step	158.271878503221320234300570350652186729484 40 digits · 40 vs format · -2.49809 ULP · 3859 gas/step	158.271878503221320234157706852151648035942 21 digits · 21 vs format · -2.63537 ULP · 1424 gas/step	158.271878503221320232 19 digits · 19 vs format · -2.30057 ULP · 923 gas/step	158.271878503221320232 19 digits · 19 vs format · -2.30057 ULP · 1419 gas/step	158.271878503221320234300570350652157047843 33 digits · 33 vs format · ULP · 3023 gas/step	158.271878503221320149 18 digits · 18 vs format · -85.3006 ULP · 2867 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	158.271878503221320234300570350652186729491 71 digits · 71 vs format · ULP · 5312 gas/step
10	316.938155912380578582534163023823506795788	316.938155912380578582534163023823506795773 40 digits · 40 vs format · -5.22996 ULP · 4104 gas/step	316.938155912380578582534163023823506795773 40 digits · 40 vs format · -5.22996 ULP · 3817 gas/step	316.938155912380578582243466356782590764851 21 digits · 21 vs format · -5.36241 ULP · 1376 gas/step	316.938155912380578577 19 digits · 19 vs format · -5.53416 ULP · 881 gas/step	316.938155912380578577 19 digits · 19 vs format · -5.53416 ULP · 1375 gas/step	316.938155912380578582534163023823397966508 33 digits · 33 vs format · ULP · 2985 gas/step	316.938155912380578369 18 digits · 18 vs format · -213.534 ULP · 2825 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	316.938155912380578582534163023823506795788 72 digits · 72 vs format · ULP · 5339 gas/step
20	635.450002940208176168731350846033723899844	635.450002940208176168731350846033723899815 40 digits · 40 vs format · -9.84813 ULP · 4083 gas/step	635.450002940208176168731350846033723899815 40 digits · 40 vs format · -9.84813 ULP · 3796 gas/step	635.450002940208176168183050458448946073986 21 digits · 21 vs format · -10.1144 ULP · 1351 gas/step	635.450002940208176158 19 digits · 19 vs format · -10.7314 ULP · 860 gas/step	635.450002940208176158 19 digits · 19 vs format · -10.7314 ULP · 1354 gas/step	635.450002940208176168731350846033258920681 33 digits · 33 vs format · ULP · 2964 gas/step	635.450002940208175685 18 digits · 18 vs format · -483.731 ULP · 2804 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	635.450002940208176168731350846033723899844 71 digits · 71 vs format · ULP · 5352 gas/step
50	1600.35088143869734634844272805794430858922	1600.35088143869734634844272805794430858915 40 digits · 40 vs format · -22.7297 ULP · 4008 gas/step	1600.35088143869734634844272805794430858915 40 digits · 40 vs format · -22.7297 ULP · 3692 gas/step	1600.35088143869734634698901928895331536751 21 digits · 21 vs format · -26.8162 ULP · 1339 gas/step	1600.350881438697346324 19 digits · 19 vs format · -24.4427 ULP · 847 gas/step	1600.350881438697346324 19 digits · 19 vs format · -24.4427 ULP · 1342 gas/step	1600.35088143869734634844272805794137163294 32 digits · 32 vs format · ULP · 2952 gas/step	1600.350881438697344991 18 digits · 18 vs format · -1357.44 ULP · 2802 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	1600.35088143869734634844272805794430858922 71 digits · 71 vs format · ULP · 5360 gas/step
100	3239.29226451438240670571916593133108873454	3239.2922645143824067057191659313310887344 40 digits · 40 vs format · -47.8637 ULP · 3976 gas/step	3239.2922645143824067057191659313310887344 40 digits · 40 vs format · -47.8637 ULP · 3646 gas/step	3239.29226451438240670297179918390995112532 21 digits · 21 vs format · -50.68 ULP · 1336 gas/step	3239.292264514382406656 19 digits · 19 vs format · -49.7192 ULP · 843 gas/step	3239.292264514382406656 19 digits · 19 vs format · -49.7192 ULP · 1338 gas/step	3239.29226451438240670571916593132005776791 32 digits · 32 vs format · ULP · 2947 gas/step	3239.292264514382402598 17 digits · 17 vs format · -4107.72 ULP · 2805 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	3239.29226451438240670571916593133108873454 70 digits · 70 vs format · ULP · 5362 gas/step
365	12536.2620049472952153455284521422191448628	12536.2620049472952153455284521422191448622 40 digits · 40 vs format · -177.874 ULP · 3952 gas/step	12536.2620049472952153455284521422191448622 40 digits · 40 vs format · -177.874 ULP · 3612 gas/step	12536.2620049472952153357631008279149398277 21 digits · 21 vs format · -180.139 ULP · 1334 gas/step	12536.262004947295215169 19 digits · 19 vs format · -176.528 ULP · 840 gas/step	12536.262004947295215169 19 digits · 19 vs format · -176.528 ULP · 1335 gas/step	12536.2620049472952153455284521420603831495 31 digits · 31 vs format · ULP · 2943 gas/step	12536.262004947295205758 18 digits · 18 vs format · -9587.53 ULP · 2763 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	12536.2620049472952153455284521422191448628 71 digits · 71 vs format · ULP · 5364 gas/step
1000	38540.0786548103528040149168003592993412283	38540.0786548103528040149168003592993412268 40 digits · 40 vs format · -482.75 ULP · 3948 gas/step	38540.0786548103528040149168003592993412268 40 digits · 40 vs format · -482.75 ULP · 3604 gas/step	38540.0786548103528039883598592385549608252 21 digits · 21 vs format · -489.889 ULP · 1333 gas/step	38540.078654810352803518 19 digits · 19 vs format · -496.917 ULP · 839 gas/step	38540.078654810352803518 19 digits · 19 vs format · -496.917 ULP · 1345 gas/step	38540.078654810352804014916800358010846558 31 digits · 31 vs format · ULP · 2942 gas/step	38540.078654810352774145 18 digits · 18 vs format · -29869.9 ULP · 2779 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	38540.0786548103528040149168003592993412283 70 digits · 70 vs format · ULP · 5364 gas/step
10000	748008.007615874676777707582886478609217336	748008.007615874676777707582886478609217321 40 digits · 40 vs format · -4994.81 ULP · 4035 gas/step	748008.007615874676777707582886478609217321 40 digits · 40 vs format · -4994.81 ULP · 3694 gas/step	748008.007615874676777438468874836807742668 21 digits · 21 vs format · -4964.28 ULP · 1345 gas/step	748008.007615874676772695 20 digits · 20 vs format · -5012.58 ULP · 839 gas/step	748008.007615874676772695 20 digits · 20 vs format · -5012.58 ULP · 1354 gas/step	748008.007615874676777707582886253357595001 30 digits · 30 vs format · ULP · 2941 gas/step	748008.00761587467625955 18 digits · 18 vs format · -518158.0 ULP · 2702 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	748008.007615874676777707582886478609217336 70 digits · 70 vs format · ULP · 5364 gas/step

exp(ln(x)) round trip

Step: x = exp(ln(x)). Result: x after N round trips; the answer is the input.

Isolates transcendental error. Every step should be the identity; what remains is the library's ln and exp error, compounded.

Over all 10 inputs

vs exact inputs (what a protocol sees) vs inputs as each format rounds them (the arithmetic alone)

N	FP127	FP127Lib	ABDK	Solady	PRBMath	Quad	Balancer	Aave	Morpho	Maker	FP256
1	38 (36–39)	39 (38–80)	38 (36–39)	39 (38–80)	19 (17–19)	19 (17–19)	18 (17–80)	18 (17–80)	17 (16–80)	17 (16–80)	32.5 (32–34)	32.5 (32–33)	18 (17–80)	18 (17–80)	— 10 rev	— 10 rev	— 10 rev	— 10 rev	— 10 rev	— 10 rev	71.5 (70–80)	80 (70–80)
2	38 (36–39)	39 (38–80)	38 (36–39)	39 (38–80)	18 (16–18)	18 (16–18)	17 (16–80)	17 (16–80)	16 (16–80)	16 (16–80)	32 (31–33)	32 (31–33)	17 (17–80)	17 (17–80)	— 10 rev	— 10 rev	— 10 rev	— 10 rev	— 10 rev	— 10 rev	71.5 (70–80)	80 (70–80)
5	38 (36–39)	39 (38–80)	38 (36–39)	39 (38–80)	18 (16–18)	18 (16–18)	17 (16–80)	17 (16–80)	16 (16–80)	16 (16–80)	32 (31–33)	32 (31–33)	17 (17–80)	17 (17–80)	— 10 rev	— 10 rev	— 10 rev	— 10 rev	— 10 rev	— 10 rev	71.5 (70–80)	80 (70–80)
10	38 (36–39)	39 (38–80)	38 (36–39)	39 (38–80)	17.5 (16–18)	17.5 (16–18)	17 (15–80)	17 (15–80)	16 (15–80)	16 (15–80)	31.5 (31–33)	31.5 (31–33)	17 (16–80)	17 (16–80)	— 10 rev	— 10 rev	— 10 rev	— 10 rev	— 10 rev	— 10 rev	71.5 (70–80)	80 (70–80)
20	38 (36–39)	39 (38–80)	38 (36–39)	39 (38–80)	17 (15–17)	17 (15–17)	16 (15–80)	16 (15–80)	15 (15–80)	15 (15–80)	31 (30–33)	31 (30–33)	16 (16–80)	16 (16–80)	— 10 rev	— 10 rev	— 10 rev	— 10 rev	— 10 rev	— 10 rev	71.5 (70–80)	80 (70–80)
50	38 (36–39)	39 (38–80)	38 (36–39)	39 (38–80)	17 (15–17)	17 (15–17)	16 (15–80)	16 (15–80)	15 (15–80)	15 (15–80)	31 (30–33)	31 (30–33)	16 (16–80)	16 (16–80)	— 10 rev	— 10 rev	— 10 rev	— 10 rev	— 10 rev	— 10 rev	71.5 (70–80)	80 (70–80)
100	38 (36–39)	39 (38–80)	38 (36–39)	39 (38–80)	17 (15–17)	17 (15–17)	16 (14–80)	16 (14–80)	15 (14–80)	15 (14–80)	30.5 (30–33)	30.5 (30–33)	16 (15–80)	16 (15–80)	— 10 rev	— 10 rev	— 10 rev	— 10 rev	— 10 rev	— 10 rev	71.5 (70–80)	80 (70–80)
365	38 (36–39)	39 (38–80)	38 (36–39)	39 (38–80)	16 (14–16)	16 (14–16)	15 (14–80)	15 (14–80)	14 (14–80)	14 (14–80)	30 (29–33)	30 (29–33)	15 (15–80)	15 (15–80)	— 10 rev	— 10 rev	— 10 rev	— 10 rev	— 10 rev	— 10 rev	71.5 (70–80)	80 (70–80)
1000	38 (36–39)	39 (38–80)	38 (36–39)	39 (38–80)	16 (14–16)	16 (14–16)	15 (13–80)	15 (13–80)	14 (13–80)	14 (13–80)	29.5 (29–33)	29.5 (29–33)	15 (14–80)	15 (14–80)	— 10 rev	— 10 rev	— 10 rev	— 10 rev	— 10 rev	— 10 rev	71.5 (70–80)	80 (70–80)
10000	38 (36–39)	39 (38–80)	38 (36–39)	39 (38–80)	15 (13–15)	15 (13–15)	14 (12–80)	14 (12–80)	13 (12–80)	13 (12–80)	28.5 (28–33)	28.5 (28–33)	14 (13–80)	14 (13–80)	— 10 rev	— 10 rev	— 10 rev	— 10 rev	— 10 rev	— 10 rev	71.5 (70–80)	80 (70–80)

Median (min–max) correct digits over the inputs, capped at 80; "rev" counts inputs that reverted at that N and are excluded.

The inputs and the ops

input	parameters	why
reference reference	x0 = 1.5	1.5 is representable in every format, so a perfect library returns it exactly
hundredth	x0 = 0.01	ln is large and negative
tenth	x0 = 0.1	a typical price ratio
near-one-below	x0 = 0.7	ln is small: cancellation territory
just-above-one	x0 = 1.0001	ln is tiny: the log1p regime
two	x0 = 2	the simplest non-trivial value
ten	x0 = 10	a round decade
thousand	x0 = 1000	three decades
million	x0 = 1000000	six decades, near ABDK's comfortable range
awkward	x0 = 123456.789	a value with no special structure

Library calls per step: FP127 ln, exp; FP127Lib ln, exp; ABDK ln, exp; Solady lnWad, expWad; PRBMath ln, exp; Quad ln, exp; Balancer ln, exp; Aave unsupported; Morpho unsupported; Maker unsupported; FP256 ln, exp. Parameters reach each adapter floored into its representation (ladder-inputs.json).

The reference input in full: x0 = 1.5

N	truth	FP127 (deployed, staticcall)	FP127Lib (inline)	ABDKMath64x64	Solady FixedPointMathLib	PRBMath SD59x18	ABDKMathQuad (binary128)	Balancer LogExpMath	Aave v3 MathUtils (RAY)	Morpho Blue MathLib	Maker rpow (RAY)	FP256 (float, inline)
1	1.5	1.49999999999999999999999999999999999999706 38 digits · 38 vs format · -1.0 ULP · 10190 gas/step	1.49999999999999999999999999999999999999706 38 digits · 38 vs format · -1.0 ULP · 10151 gas/step	1.49999999999999999989157978275144955659925 19 digits · 19 vs format · -2.0 ULP · 10892 gas/step	1.5 exact · exact vs format · 0 ULP · 1441 gas/step	1.499999999999999982 16 digits · 16 vs format · -18.0 ULP · 10093 gas/step	1.49999999999999999999999999999999961481401 33 digits · 33 vs format · ULP · 23587 gas/step	1.499999999999999998 17 digits · 17 vs format · -2.0 ULP · 2617 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	1.5 exact · exact vs format · ULP · 15907 gas/step
2	1.5	1.49999999999999999999999999999999999999706 38 digits · 38 vs format · -1.0 ULP · 10018 gas/step	1.49999999999999999999999999999999999999706 38 digits · 38 vs format · -1.0 ULP · 9982 gas/step	1.49999999999999999978315956550289911319851 18 digits · 18 vs format · -4.0 ULP · 10698 gas/step	1.5 exact · exact vs format · 0 ULP · 1272 gas/step	1.499999999999999966 16 digits · 16 vs format · -34.0 ULP · 9902 gas/step	1.49999999999999999999999999999999922962802 33 digits · 33 vs format · ULP · 23376 gas/step	1.499999999999999996 17 digits · 17 vs format · -4.0 ULP · 2448 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	1.5 exact · exact vs format · ULP · 15738 gas/step
5	1.5	1.49999999999999999999999999999999999999706 38 digits · 38 vs format · -1.0 ULP · 9914 gas/step	1.49999999999999999999999999999999999999706 38 digits · 38 vs format · -1.0 ULP · 9880 gas/step	1.49999999999999999934947869650869733959553 18 digits · 18 vs format · -12.0 ULP · 10578 gas/step	1.5 exact · exact vs format · 0 ULP · 1170 gas/step	1.499999999999999919 16 digits · 16 vs format · -81.0 ULP · 9772 gas/step	1.49999999999999999999999999999999807407006 32 digits · 32 vs format · ULP · 23252 gas/step	1.499999999999999992 17 digits · 17 vs format · -8.0 ULP · 2346 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	1.5 exact · exact vs format · ULP · 15636 gas/step
10	1.5	1.49999999999999999999999999999999999999706 38 digits · 38 vs format · -1.0 ULP · 9880 gas/step	1.49999999999999999999999999999999999999706 38 digits · 38 vs format · -1.0 ULP · 9846 gas/step	1.49999999999999999864474728439311945749068 18 digits · 18 vs format · -25.0 ULP · 10538 gas/step	1.5 exact · exact vs format · 0 ULP · 1136 gas/step	1.499999999999999834 15 digits · 15 vs format · -166.0 ULP · 9794 gas/step	1.49999999999999999999999999999999614814011 32 digits · 32 vs format · ULP · 23224 gas/step	1.499999999999999984 16 digits · 16 vs format · -16.0 ULP · 2312 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	1.5 exact · exact vs format · ULP · 15602 gas/step
20	1.5	1.49999999999999999999999999999999999999706 38 digits · 38 vs format · -1.0 ULP · 9863 gas/step	1.49999999999999999999999999999999999999706 38 digits · 38 vs format · -1.0 ULP · 9829 gas/step	1.49999999999999999718107435153768847158062 17 digits · 17 vs format · -52.0 ULP · 10522 gas/step	1.5 exact · exact vs format · 0 ULP · 1119 gas/step	1.499999999999999678 15 digits · 15 vs format · -322.0 ULP · 9801 gas/step	1.49999999999999999999999999999999229628022 32 digits · 32 vs format · ULP · 23211 gas/step	1.499999999999999969 16 digits · 16 vs format · -31.0 ULP · 2295 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	1.5 exact · exact vs format · ULP · 15585 gas/step
50	1.5	1.49999999999999999999999999999999999999706 38 digits · 38 vs format · -1.0 ULP · 9853 gas/step	1.49999999999999999999999999999999999999706 38 digits · 38 vs format · -1.0 ULP · 9819 gas/step	1.49999999999999999279005555297139551385044 17 digits · 17 vs format · -133.0 ULP · 10507 gas/step	1.5 exact · exact vs format · 0 ULP · 1109 gas/step	1.499999999999999206 15 digits · 15 vs format · -794.0 ULP · 9789 gas/step	1.49999999999999999999999999999998074070056 31 digits · 31 vs format · ULP · 23192 gas/step	1.499999999999999924 16 digits · 16 vs format · -76.0 ULP · 2285 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	1.5 exact · exact vs format · ULP · 15575 gas/step
100	1.5	1.49999999999999999999999999999999999999706 38 digits · 38 vs format · -1.0 ULP · 9850 gas/step	1.49999999999999999999999999999999999999706 38 digits · 38 vs format · -1.0 ULP · 9816 gas/step	1.49999999999999998552590099731851580600051 17 digits · 17 vs format · -267.0 ULP · 10502 gas/step	1.5 exact · exact vs format · 0 ULP · 1106 gas/step	1.499999999999998394 14 digits · 14 vs format · -1606.0 ULP · 9773 gas/step	1.49999999999999999999999999999996148140111 31 digits · 31 vs format · ULP · 23189 gas/step	1.499999999999999849 15 digits · 15 vs format · -151.0 ULP · 2282 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	1.5 exact · exact vs format · ULP · 15572 gas/step
365	1.5	1.49999999999999999999999999999999999999706 38 digits · 38 vs format · -1.0 ULP · 9849 gas/step	1.49999999999999999999999999999999999999706 38 digits · 38 vs format · -1.0 ULP · 9813 gas/step	1.49999999999999994681988343958600751193444 16 digits · 16 vs format · -981.0 ULP · 10488 gas/step	1.5 exact · exact vs format · 0 ULP · 1103 gas/step	1.499999999999994003 14 digits · 14 vs format · -5997.0 ULP · 9778 gas/step	1.49999999999999999999999999999985940711406 31 digits · 31 vs format · ULP · 23181 gas/step	1.499999999999999452 15 digits · 15 vs format · -548.0 ULP · 2279 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	1.5 exact · exact vs format · ULP · 15569 gas/step
1000	1.5	1.49999999999999999999999999999999999999706 38 digits · 38 vs format · -1.0 ULP · 9854 gas/step	1.49999999999999999999999999999999999999706 38 digits · 38 vs format · -1.0 ULP · 9813 gas/step	1.4999999999999998541748078006996536259976 16 digits · 16 vs format · -2690.0 ULP · 10495 gas/step	1.5 exact · exact vs format · 0 ULP · 1103 gas/step	1.49999999999998402 13 digits · 13 vs format · -15980.0 ULP · 9761 gas/step	1.49999999999999999999999999999961481401112 30 digits · 30 vs format · ULP · 23187 gas/step	1.499999999999998499 14 digits · 14 vs format · -1501.0 ULP · 2279 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	1.5 exact · exact vs format · ULP · 15569 gas/step
10000	1.5	1.49999999999999999999999999999999999999706 38 digits · 38 vs format · -1.0 ULP · 9924 gas/step	1.49999999999999999999999999999999999999706 38 digits · 38 vs format · -1.0 ULP · 9813 gas/step	1.49999999999999854104334659488095837787114 15 digits · 15 vs format · -26913.0 ULP · 10500 gas/step	1.5 exact · exact vs format · 0 ULP · 1103 gas/step	1.499999999999839112 12 digits · 12 vs format · -160888.0 ULP · 9758 gas/step	1.49999999999999999999999999999614814011123 29 digits · 29 vs format · ULP · 23224 gas/step	1.499999999999984999 13 digits · 13 vs format · -15001.0 ULP · 2279 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	1.5 exact · exact vs format · ULP · 15569 gas/step

Amortising loan

Step: b = b * (1 + r) - payment. Result: outstanding balance after N daily payments.

Subtracting nearly equal numbers. The payment retires the loan at step 366, so at N = 365 the balance is one payment and relative error is amplified several hundred times.

Over all 8 inputs

vs exact inputs (what a protocol sees) vs inputs as each format rounds them (the arithmetic alone)

N	FP127	FP127Lib	ABDK	Solady	PRBMath	Quad	Balancer	Aave	Morpho	Maker	FP256
1	39 (38–39)	80 (80–80)	39 (38–39)	80 (80–80)	20 (19–20)	80 (80–80)	19 (18–19)	80 (80–80)	19 (18–19)	80 (80–80)	33 (33–33)	33 (33–33)	19 (18–19)	80 (80–80)	— 8 rev	— 8 rev	— 8 rev	— 8 rev	— 8 rev	— 8 rev	71 (71–71)	71 (71–71)
2	38 (38–38)	44 (41–46)	38 (38–38)	44 (41–46)	19 (19–19)	24 (22–25)	19 (17–19)	23 (21–24)	19 (17–19)	23 (21–24)	33 (33–33)	33 (33–33)	19 (17–19)	23 (21–24)	— 8 rev	— 8 rev	— 8 rev	— 8 rev	— 8 rev	— 8 rev	71 (70–72)	71 (70–72)
5	38 (37–38)	43 (40–44)	38 (37–38)	43 (40–44)	19 (18–19)	24 (21–24)	19 (17–19)	23 (20–23)	19 (17–19)	23 (20–23)	32 (32–33)	32 (32–33)	19 (17–19)	23 (20–23)	— 8 rev	— 8 rev	— 8 rev	— 8 rev	— 8 rev	— 8 rev	70 (70–71)	70 (70–71)
10	38 (37–38)	43 (40–44)	38 (37–38)	43 (40–44)	19 (18–19)	24 (21–24)	18 (17–18)	22 (20–23)	18 (17–18)	22 (20–23)	32 (32–32)	32 (32–32)	18 (17–18)	22 (20–23)	— 8 rev	— 8 rev	— 8 rev	— 8 rev	— 8 rev	— 8 rev	70 (70–71)	70 (70–71)
20	37 (37–37)	42.5 (40–43)	37 (37–37)	42.5 (40–43)	18 (18–18)	23 (20–24)	18 (16–18)	22 (19–23)	18 (16–18)	22 (19–23)	32 (32–32)	32 (32–32)	18 (16–18)	22 (19–23)	— 8 rev	— 8 rev	— 8 rev	— 8 rev	— 8 rev	— 8 rev	70 (69–71)	70 (69–71)
50	37 (36–37)	42 (39–43)	37 (36–37)	42 (39–43)	18 (17–18)	23 (20–23)	18 (16–18)	22 (19–22)	18 (16–18)	22 (19–22)	31 (31–32)	31 (31–32)	18 (16–18)	22 (19–22)	— 8 rev	— 8 rev	— 8 rev	— 8 rev	— 8 rev	— 8 rev	69 (69–70)	69 (69–70)
100	37 (36–37)	42 (39–42)	37 (36–37)	42 (39–42)	17 (17–18)	22 (20–23)	17 (16–17)	21 (18–22)	17 (16–17)	21 (18–22)	31 (31–31)	31 (31–31)	17 (16–17)	21 (18–22)	— 8 rev	— 8 rev	— 8 rev	— 8 rev	— 8 rev	— 8 rev	69 (69–69)	69 (69–69)
365	34 (33–36)	39 (36–41)	34 (33–36)	39 (36–41)	15 (14–17)	19 (17–22)	14.5 (13–17)	18 (15–21)	14.5 (13–17)	18 (15–21)	28 (28–31)	28 (28–31)	14.5 (13–17)	18 (15–21)	— 8 rev	— 8 rev	— 8 rev	— 8 rev	— 8 rev	— 8 rev	66 (66–68)	66 (66–68)
1000	36 (36–36)	41 (38–42)	36 (36–36)	41 (38–42)	17 (16–17)	21.5 (19–22)	16.5 (15–17)	20.5 (18–22)	16.5 (15–17)	20.5 (18–22)	31 (30–31)	31 (30–31)	16.5 (15–17)	20.5 (18–22)	— 8 rev	— 8 rev	— 8 rev	— 8 rev	— 8 rev	— 8 rev	69 (68–70)	69 (68–70)
10000	35 (31–35)	40.5 (36–42)	35 (31–35)	40.5 (36–42)	15.5 (12–16)	21.5 (16–22)	15.5 (11–16)	21 (15–21)	15.5 (11–16)	21 (15–21)	29 (25–30)	29 (25–30)	15.5 (11–16)	21 (15–21)	— 8 rev	— 8 rev	— 8 rev	— 8 rev	— 8 rev	— 8 rev	67 (63–68)	67 (63–68)

Median (min–max) correct digits over the inputs, capped at 80; "rev" counts inputs that reverted at that N and are excluded.

The inputs and the ops

input	parameters	why
reference reference	principal = 300000, rate = 0.05/365, term = 366; payment = principal * rate / (1 - (1 + rate)^(-term))	a mortgage-sized loan retiring at step 366
small	principal = 10000, rate = 0.05/365, term = 366; payment = principal * rate / (1 - (1 + rate)^(-term))	a small loan
high-rate	principal = 300000, rate = 0.1/365, term = 366; payment = principal * rate / (1 - (1 + rate)^(-term))	double the rate
ten-year	principal = 300000, rate = 0.05/365, term = 3650; payment = principal * rate / (1 - (1 + rate)^(-term))	a ten-year term: at N = 365 nine tenths remain
large-low	principal = 1000000, rate = 0.03/365, term = 366; payment = principal * rate / (1 - (1 + rate)^(-term))	a large loan at a low rate
very-high	principal = 300000, rate = 0.2/365, term = 366; payment = principal * rate / (1 - (1 + rate)^(-term))	20% APR
micro	principal = 500, rate = 0.05/365, term = 366; payment = principal * rate / (1 - (1 + rate)^(-term))	a tiny loan: the payment is a few units
retires-at-10001	principal = 300000, rate = 0.05/365, term = 10001; payment = principal * rate / (1 - (1 + rate)^(-term))	retires one step past the ladder's end, so N = 10,000 is the one-payment cell

Library calls per step: FP127 mul, sub; FP127Lib mul, sub; ABDK mul, sub; Solady sMulWad, sub; PRBMath mul, sub; Quad mul, sub; Balancer mulDown, sub; Aave unsupported; Morpho unsupported; Maker unsupported; FP256 mul, sub. Parameters reach each adapter floored into its representation (ladder-inputs.json).

The reference input in full: principal = 300000, rate = 0.05/365, term = 366

N	truth	FP127 (deployed, staticcall)	FP127Lib (inline)	ABDKMath64x64	Solady FixedPointMathLib	PRBMath SD59x18	ABDKMathQuad (binary128)	Balancer LogExpMath	Aave v3 MathUtils (RAY)	Morpho Blue MathLib	Maker rpow (RAY)	FP256 (float, inline)
1	299200.647990355892260602435720377950399328	299200.647990355892260602435720377950399116 39 digits · exact vs format · -72164.0 ULP · 2551 gas/step	299200.647990355892260602435720377950399116 39 digits · exact vs format · -72164.0 ULP · 1318 gas/step	299200.647990355892257670659354490405590354 20 digits · exact vs format · -54081.7 ULP · 1434 gas/step	299200.647990355892256493 19 digits · exact vs format · -4109.44 ULP · 1147 gas/step	299200.647990355892256493 19 digits · exact vs format · -4109.44 ULP · 1507 gas/step	299200.647990355892260602435720377827909005 33 digits · 33 vs format · ULP · 4718 gas/step	299200.647990355892256493 19 digits · exact vs format · -4109.44 ULP · 1418 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	299200.647990355892260602435720377950399328 71 digits · 71 vs format · ULP · 4478 gas/step
2	298401.186480436490807815912870306637504191	298401.186480436490807815912870306637503767 38 digits · 44 vs format · -144146.0 ULP · 2163 gas/step	298401.186480436490807815912870306637503767 38 digits · 44 vs format · -144146.0 ULP · 928 gas/step	298401.186480436490801959744604415769586581 19 digits · 25 vs format · -108027.0 ULP · 967 gas/step	298401.186480436490799607 19 digits · 23 vs format · -8208.91 ULP · 766 gas/step	298401.186480436490799607 19 digits · 23 vs format · -8208.91 ULP · 1126 gas/step	298401.186480436490807815912870306394512417 33 digits · 33 vs format · ULP · 3604 gas/step	298401.186480436490799607 19 digits · 23 vs format · -8208.91 ULP · 1037 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	298401.186480436490807815912870306637504191 71 digits · 71 vs format · ULP · 3524 gas/step
5	296002.14479901587277453479645551265335414	296002.144799015872774534796455512653353085 38 digits · 43 vs format · -358998.0 ULP · 1930 gas/step	296002.144799015872774534796455512653353085 38 digits · 43 vs format · -358998.0 ULP · 694 gas/step	296002.144799015872759949917492061643997658 19 digits · 24 vs format · -269044.0 ULP · 686 gas/step	296002.14479901587275409 19 digits · 23 vs format · -20444.8 ULP · 538 gas/step	296002.14479901587275409 19 digits · 23 vs format · -20444.8 ULP · 898 gas/step	296002.14479901587277453479645551210093214 32 digits · 32 vs format · ULP · 2936 gas/step	296002.14479901587275409 19 digits · 23 vs format · -20444.8 ULP · 809 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	296002.14479901587277453479645551265335414 70 digits · 70 vs format · ULP · 2952 gas/step
10	292001.550590734601252336129721761763542342	292001.550590734601252336129721761763540245 38 digits · 43 vs format · -713431.0 ULP · 1852 gas/step	292001.550590734601252336129721761763540245 38 digits · 43 vs format · -713431.0 ULP · 616 gas/step	292001.55059073460122335180776456664553109 19 digits · 24 vs format · -534666.0 ULP · 593 gas/step	292001.550590734601211706 18 digits · 22 vs format · -40630.1 ULP · 462 gas/step	292001.550590734601211706 18 digits · 22 vs format · -40630.1 ULP · 822 gas/step	292001.550590734601252336129721760665355576 32 digits · 32 vs format · ULP · 2713 gas/step	292001.550590734601211706 18 digits · 22 vs format · -40630.1 ULP · 733 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	292001.550590734601252336129721761763542342 70 digits · 70 vs format · ULP · 2761 gas/step
20	283992.137644809178562284250368709832406964	283992.137644809178562284250368709832402825 37 digits · 42 vs format · -1.40858e+6 ULP · 1813 gas/step	283992.137644809178562284250368709832402825 37 digits · 42 vs format · -1.40858e+6 ULP · 577 gas/step	283992.137644809178505058517780201832181319 18 digits · 23 vs format · -1.05563e+6 ULP · 546 gas/step	283992.137644809178482065 18 digits · 22 vs format · -80219.3 ULP · 424 gas/step	283992.137644809178482065 18 digits · 22 vs format · -80219.3 ULP · 784 gas/step	283992.137644809178562284250368707754318788 32 digits · 32 vs format · ULP · 2602 gas/step	283992.137644809178482065 18 digits · 22 vs format · -80219.3 ULP · 695 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	283992.137644809178562284250368709832406964 70 digits · 70 vs format · ULP · 2666 gas/step
50	259897.967206003300994908944979016288932429	259897.967206003300994908944979016288922485 37 digits · 42 vs format · -3.38378e+6 ULP · 1790 gas/step	259897.967206003300994908944979016288922485 37 digits · 42 vs format · -3.38378e+6 ULP · 553 gas/step	259897.96720600330085743730115094285437749 18 digits · 23 vs format · -2.5359e+6 ULP · 518 gas/step	259897.967206003300802194 18 digits · 22 vs format · -192715.0 ULP · 401 gas/step	259897.967206003300802194 18 digits · 22 vs format · -192715.0 ULP · 761 gas/step	259897.967206003300994908944979011264557986 31 digits · 31 vs format · ULP · 2535 gas/step	259897.967206003300802194 18 digits · 22 vs format · -192715.0 ULP · 672 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	259897.967206003300994908944979016288932429 69 digits · 69 vs format · ULP · 2608 gas/step
100	219520.339088217564302677320754471630738482	219520.339088217564302677320754471630719953 37 digits · 42 vs format · -6.30525e+6 ULP · 1782 gas/step	219520.339088217564302677320754471630719953 37 digits · 42 vs format · -6.30525e+6 ULP · 545 gas/step	219520.339088217564046516289567173574326375 17 digits · 22 vs format · -4.72534e+6 ULP · 509 gas/step	219520.339088217563943577 17 digits · 21 vs format · -359100.0 ULP · 393 gas/step	219520.339088217563943577 17 digits · 21 vs format · -359100.0 ULP · 753 gas/step	219520.339088217564302677320754463555623545 31 digits · 31 vs format · ULP · 2513 gas/step	219520.339088217563943577 17 digits · 21 vs format · -359100.0 ULP · 664 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	219520.339088217564302677320754471630738482 69 digits · 69 vs format · ULP · 2589 gas/step
365	840.332785974796123490236849916581575798202	840.332785974796123490236849916581535234713 34 digits · 39 vs format · -1.3803e+7 ULP · 1777 gas/step	840.332785974796123490236849916581535234713 34 digits · 39 vs format · -1.3803e+7 ULP · 540 gas/step	840.332785974795562720213691620951834693187 15 digits · 19 vs format · -1.03444e+7 ULP · 502 gas/step	840.332785974795337317 15 digits · 18 vs format · -786173.0 ULP · 388 gas/step	840.332785974795337317 15 digits · 18 vs format · -786173.0 ULP · 748 gas/step	840.332785974796123490236849899778912354514 28 digits · 28 vs format · ULP · 2496 gas/step	840.332785974795337317 15 digits · 18 vs format · -786173.0 ULP · 659 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	840.332785974796123490236849916581575798202 66 digits · 66 vs format · ULP · 2575 gas/step
1000	-556627.388400841571518712768710676765307471	-556627.388400841571518712768710676765225396 36 digits · 41 vs format · 2.79286e+7 ULP · 1777 gas/step	-556627.388400841571518712768710676765225396 36 digits · 41 vs format · 2.79286e+7 ULP · 538 gas/step	-556627.388400841570384060383227392554417179 17 digits · 22 vs format · 2.09306e+7 ULP · 500 gas/step	-556627.388400841569927932 17 digits · 21 vs format · 1.59078e+6 ULP · 386 gas/step	-556627.388400841569927932 17 digits · 21 vs format · 1.59078e+6 ULP · 753 gas/step	-556627.388400841571518712768710656833355199 31 digits · 31 vs format · ULP · 2191 gas/step	-556627.388400841569927932 17 digits · 21 vs format · 1.59078e+6 ULP · 726 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	-556627.388400841571518712768710676765307471 69 digits · 69 vs format · ULP · 2536 gas/step
10000	-16823263.3532103472394141655896251577011087	-16823263.3532103472394141655896251576317416 35 digits · 41 vs format · 2.36044e+10 ULP · 1794 gas/step	-16823263.3532103472394141655896251576317416 35 digits · 41 vs format · 2.36044e+10 ULP · 538 gas/step	-16823263.3532103462804450766778299741766034 16 digits · 22 vs format · 1.76899e+10 ULP · 500 gas/step	-16823263.353210345895187359 16 digits · 21 vs format · 1.34423e+9 ULP · 386 gas/step	-16823263.353210345895187359 16 digits · 21 vs format · 1.34423e+9 ULP · 756 gas/step	-16823263.3532103472394141655896036031306915 29 digits · 29 vs format · ULP · 2032 gas/step	-16823263.353210345895187359 16 digits · 21 vs format · 1.34423e+9 ULP · 761 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	-16823263.3532103472394141655896251577011087 68 digits · 68 vs format · ULP · 2516 gas/step

Running geometric mean

Step: L += ln(x_i); g = exp(L / N), x_i = (i + 2) / (i + 1). Result: geometric mean of N terms; truth telescopes to ((N + 2) / 2)^(1 / N).

Index and basket maths: a sum of logs, one division, one exp. The division by N and the final exp expose every accumulated ULP.

Over all 1 input

vs exact inputs (what a protocol sees) vs inputs as each format rounds them (the arithmetic alone)

N	FP127	FP127Lib	ABDK	Solady	PRBMath	Quad	Balancer	Aave	Morpho	Maker	FP256
1	38 (38–38)	38 (38–38)	38 (38–38)	38 (38–38)	19 (19–19)	19 (19–19)	80 (80–80)	80 (80–80)	16 (16–16)	16 (16–16)	33 (33–33)	33 (33–33)	17 (17–17)	17 (17–17)	— 1 rev	— 1 rev	— 1 rev	— 1 rev	— 1 rev	— 1 rev	80 (80–80)	80 (80–80)
2	38 (38–38)	38 (38–38)	38 (38–38)	38 (38–38)	18 (18–18)	18 (18–18)	17 (17–17)	17 (17–17)	16 (16–16)	16 (16–16)	33 (33–33)	33 (33–33)	17 (17–17)	17 (17–17)	— 1 rev	— 1 rev	— 1 rev	— 1 rev	— 1 rev	— 1 rev	71 (71–71)	71 (71–71)
5	38 (38–38)	38 (38–38)	38 (38–38)	38 (38–38)	19 (19–19)	19 (19–19)	17 (17–17)	17 (17–17)	16 (16–16)	16 (16–16)	33 (33–33)	33 (33–33)	17 (17–17)	17 (17–17)	— 1 rev	— 1 rev	— 1 rev	— 1 rev	— 1 rev	— 1 rev	71 (71–71)	71 (71–71)
10	38 (38–38)	38 (38–38)	38 (38–38)	38 (38–38)	18 (18–18)	18 (18–18)	17 (17–17)	17 (17–17)	16 (16–16)	16 (16–16)	33 (33–33)	33 (33–33)	17 (17–17)	17 (17–17)	— 1 rev	— 1 rev	— 1 rev	— 1 rev	— 1 rev	— 1 rev	71 (71–71)	71 (71–71)
20	38 (38–38)	38 (38–38)	38 (38–38)	38 (38–38)	18 (18–18)	18 (18–18)	17 (17–17)	17 (17–17)	16 (16–16)	16 (16–16)	33 (33–33)	33 (33–33)	17 (17–17)	17 (17–17)	— 1 rev	— 1 rev	— 1 rev	— 1 rev	— 1 rev	— 1 rev	72 (72–72)	72 (72–72)
50	38 (38–38)	38 (38–38)	38 (38–38)	38 (38–38)	18 (18–18)	18 (18–18)	17 (17–17)	17 (17–17)	16 (16–16)	16 (16–16)	33 (33–33)	33 (33–33)	17 (17–17)	17 (17–17)	— 1 rev	— 1 rev	— 1 rev	— 1 rev	— 1 rev	— 1 rev	71 (71–71)	71 (71–71)
100	39 (39–39)	39 (39–39)	39 (39–39)	39 (39–39)	18 (18–18)	18 (18–18)	17 (17–17)	17 (17–17)	16 (16–16)	16 (16–16)	33 (33–33)	33 (33–33)	17 (17–17)	17 (17–17)	— 1 rev	— 1 rev	— 1 rev	— 1 rev	— 1 rev	— 1 rev	71 (71–71)	71 (71–71)
365	38 (38–38)	38 (38–38)	38 (38–38)	38 (38–38)	18 (18–18)	18 (18–18)	17 (17–17)	17 (17–17)	16 (16–16)	16 (16–16)	33 (33–33)	33 (33–33)	17 (17–17)	17 (17–17)	— 1 rev	— 1 rev	— 1 rev	— 1 rev	— 1 rev	— 1 rev	71 (71–71)	71 (71–71)
1000	39 (39–39)	39 (39–39)	39 (39–39)	39 (39–39)	18 (18–18)	18 (18–18)	17 (17–17)	17 (17–17)	16 (16–16)	16 (16–16)	33 (33–33)	33 (33–33)	17 (17–17)	17 (17–17)	— 1 rev	— 1 rev	— 1 rev	— 1 rev	— 1 rev	— 1 rev	72 (72–72)	72 (72–72)
10000	38 (38–38)	38 (38–38)	38 (38–38)	38 (38–38)	18 (18–18)	18 (18–18)	17 (17–17)	17 (17–17)	16 (16–16)	16 (16–16)	33 (33–33)	33 (33–33)	17 (17–17)	17 (17–17)	— 1 rev	— 1 rev	— 1 rev	— 1 rev	— 1 rev	— 1 rev	71 (71–71)	71 (71–71)

Median (min–max) correct digits over the inputs, capped at 80; "rev" counts inputs that reverted at that N and are excluded.

The inputs and the ops

input	parameters	why
reference reference		the scenario has no parameters; its inputs are the integers 1..N

Library calls per step: FP127 div, ln, add, exp; FP127Lib div, ln, add, exp; ABDK div, ln, add, exp; Solady sDivWad, lnWad, add, expWad; PRBMath div, ln, add, exp; Quad div, ln, add, exp; Balancer divDown, ln, add, exp; Aave unsupported; Morpho unsupported; Maker unsupported; FP256 div, ln, add, exp. Parameters reach each adapter floored into its representation (ladder-inputs.json).

The reference input in full:

N	truth	FP127 (deployed, staticcall)	FP127Lib (inline)	ABDKMath64x64	Solady FixedPointMathLib	PRBMath SD59x18	ABDKMathQuad (binary128)	Balancer LogExpMath	Aave v3 MathUtils (RAY)	Morpho Blue MathLib	Maker rpow (RAY)	FP256 (float, inline)
1	1.5	1.49999999999999999999999999999999999999706 38 digits · 38 vs format · -1.0 ULP · 12962 gas/step	1.49999999999999999999999999999999999999706 38 digits · 38 vs format · -1.0 ULP · 11962 gas/step	1.49999999999999999989157978275144955659925 19 digits · 19 vs format · -2.0 ULP · 12146 gas/step	1.5 exact · exact vs format · 0 ULP · 2580 gas/step	1.499999999999999982 16 digits · 16 vs format · -18.0 ULP · 12066 gas/step	1.49999999999999999999999999999999961481401 33 digits · 33 vs format · ULP · 28422 gas/step	1.499999999999999998 17 digits · 17 vs format · -2.0 ULP · 4358 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	1.5 exact · exact vs format · ULP · 22480 gas/step
2	1.41421356237309504880168872420969807856967	1.41421356237309504880168872420969807856606 38 digits · 38 vs format · -1.22996 ULP · 9601 gas/step	1.41421356237309504880168872420969807856606 38 digits · 38 vs format · -1.22996 ULP · 8929 gas/step	1.41421356237309504865536785578328249357583 18 digits · 18 vs format · -2.69914 ULP · 10322 gas/step	1.414213562373095047 17 digits · 17 vs format · -1.80169 ULP · 2042 gas/step	1.414213562373095034 16 digits · 16 vs format · -14.8017 ULP · 10383 gas/step	1.41421356237309504880168872420969779175425 33 digits · 33 vs format · ULP · 25241 gas/step	1.414213562373095047 17 digits · 17 vs format · -1.80169 ULP · 3360 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	1.41421356237309504880168872420969807856967 71 digits · 71 vs format · ULP · 17874 gas/step
5	1.28473515712343933867943306541867483181504	1.28473515712343933867943306541867483181363 38 digits · 38 vs format · -0.480657 ULP · 7459 gas/step	1.28473515712343933867943306541867483181363 38 digits · 38 vs format · -0.480657 ULP · 6950 gas/step	1.28473515712343933856945041305230859052244 19 digits · 19 vs format · -2.02882 ULP · 8889 gas/step	1.284735157123439337 17 digits · 17 vs format · -1.67943 ULP · 1720 gas/step	1.284735157123439324 16 digits · 16 vs format · -14.6794 ULP · 9067 gas/step	1.28473515712343933867943306541867435790427 33 digits · 33 vs format · ULP · 22612 gas/step	1.284735157123439337 17 digits · 17 vs format · -1.67943 ULP · 2767 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	1.28473515712343933867943306541867483181504 71 digits · 71 vs format · ULP · 15065 gas/step
10	1.19623119885131548973381914341377367153755	1.19623119885131548973381914341377367153437 38 digits · 38 vs format · -1.08179 ULP · 6709 gas/step	1.19623119885131548973381914341377367153437 38 digits · 38 vs format · -1.08179 ULP · 6243 gas/step	1.19623119885131548953026078829786627011345 18 digits · 18 vs format · -3.75499 ULP · 8503 gas/step	1.196231198851315488 17 digits · 17 vs format · -1.73382 ULP · 1612 gas/step	1.196231198851315477 16 digits · 16 vs format · -12.7338 ULP · 8686 gas/step	1.19623119885131548973381914341377331359046 33 digits · 33 vs format · ULP · 21921 gas/step	1.196231198851315488 17 digits · 17 vs format · -1.73382 ULP · 2535 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	1.19623119885131548973381914341377367153755 71 digits · 71 vs format · ULP · 14062 gas/step
20	1.12737820415783302414307801981343475345086	1.12737820415783302414307801981343475344916 38 digits · 38 vs format · -0.579813 ULP · 6341 gas/step	1.12737820415783302414307801981343475344916 38 digits · 38 vs format · -0.579813 ULP · 5900 gas/step	1.12737820415783302402173635115012473306706 18 digits · 18 vs format · -2.23836 ULP · 8308 gas/step	1.127378204157833022 17 digits · 17 vs format · -2.14308 ULP · 1558 gas/step	1.127378204157833012 16 digits · 16 vs format · -12.1431 ULP · 8538 gas/step	1.12737820415783302414307801981343445385091 33 digits · 33 vs format · ULP · 21654 gas/step	1.127378204157833022 17 digits · 17 vs format · -2.14308 ULP · 2322 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	1.12737820415783302414307801981343475345086 72 digits · 72 vs format · ULP · 13597 gas/step
50	1.06733184424855890455496508133611557432867	1.06733184424855890455496508133611557432543 38 digits · 38 vs format · -1.10411 ULP · 6071 gas/step	1.06733184424855890455496508133611557432543 38 digits · 38 vs format · -1.10411 ULP · 5631 gas/step	1.06733184424855890443106787301985605154186 18 digits · 18 vs format · -2.2855 ULP · 8184 gas/step	1.067331844248558903 17 digits · 17 vs format · -1.55497 ULP · 1526 gas/step	1.067331844248558893 16 digits · 16 vs format · -11.555 ULP · 8376 gas/step	1.06733184424855890455496508133611529688231 33 digits · 33 vs format · ULP · 21578 gas/step	1.067331844248558903 17 digits · 17 vs format · -1.55497 ULP · 2194 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	1.06733184424855890455496508133611557432867 71 digits · 71 vs format · ULP · 13242 gas/step
100	1.04010144984871012198554698020468702744746	1.04010144984871012198554698020468702744667 39 digits · 39 vs format · -0.267912 ULP · 5936 gas/step	1.04010144984871012198554698020468702744667 39 digits · 39 vs format · -0.267912 ULP · 5485 gas/step	1.04010144984871012182421276537347409885115 18 digits · 18 vs format · -2.97609 ULP · 8147 gas/step	1.04010144984871012 17 digits · 17 vs format · -1.98555 ULP · 1515 gas/step	1.040101449848710111 16 digits · 16 vs format · -10.9855 ULP · 8318 gas/step	1.04010144984871012198554698020468666440993 33 digits · 33 vs format · ULP · 21638 gas/step	1.04010144984871012 17 digits · 17 vs format · -1.98555 ULP · 2151 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	1.04010144984871012198554698020468702744746 71 digits · 71 vs format · ULP · 13076 gas/step
365	1.01438248702624022745474729116356250455137	1.01438248702624022745474729116356250454769 38 digits · 38 vs format · -1.25442 ULP · 5740 gas/step	1.01438248702624022745474729116356250454769 38 digits · 38 vs format · -1.25442 ULP · 5250 gas/step	1.0143824870262402273362115079358147795574 18 digits · 18 vs format · -2.1866 ULP · 8117 gas/step	1.014382487026240225 17 digits · 17 vs format · -2.45475 ULP · 1507 gas/step	1.014382487026240217 16 digits · 16 vs format · -10.4547 ULP · 8258 gas/step	1.01438248702624022745474729116356196091094 33 digits · 33 vs format · ULP · 21830 gas/step	1.014382487026240225 17 digits · 17 vs format · -2.45475 ULP · 2120 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	1.01438248702624022745474729116356250455137 71 digits · 71 vs format · ULP · 13498 gas/step
1000	1.00623596930045890152123699191085464878457	1.00623596930045890152123699191085464878384 39 digits · 39 vs format · -0.247777 ULP · 5666 gas/step	1.00623596930045890152123699191085464878384 39 digits · 39 vs format · -0.247777 ULP · 5155 gas/step	1.00623596930045890141761782965446059279202 18 digits · 18 vs format · -1.91144 ULP · 8110 gas/step	1.006235969300458899 17 digits · 17 vs format · -2.52124 ULP · 1506 gas/step	1.006235969300458891 16 digits · 16 vs format · -10.5212 ULP · 8241 gas/step	1.00623596930045890152123699191085417921205 33 digits · 33 vs format · ULP · 22011 gas/step	1.006235969300458899 17 digits · 17 vs format · -2.52124 ULP · 2113 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	1.00623596930045890152123699191085464878457 72 digits · 72 vs format · ULP · 14344 gas/step
10000	1.00085210215007978718759617264671502942782	1.00085210215007978718759617264671502942658 38 digits · 38 vs format · -0.419256 ULP · 5695 gas/step	1.00085210215007978718759617264671502942658 38 digits · 38 vs format · -0.419256 ULP · 5105 gas/step	1.00085210215007978706630228926677261824807 18 digits · 18 vs format · -2.23748 ULP · 8107 gas/step	1.000852102150079785 17 digits · 17 vs format · -2.1876 ULP · 1505 gas/step	1.000852102150079776 16 digits · 16 vs format · -11.1876 ULP · 8188 gas/step	1.00085210215007978718759617264671431263658 33 digits · 33 vs format · ULP · 22491 gas/step	1.000852102150079785 17 digits · 17 vs format · -2.1876 ULP · 2109 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	1.00085210215007978718759617264671502942782 71 digits · 71 vs format · ULP · 14801 gas/step

Black-Scholes d1 along a price path

Step: acc += d1(S); S += tick, with d1 = (ln(S / K) + (r + sigma^2 / 2) T) / (sigma sqrt(T)). Result: sum of d1 over N ticks.

The options-desk case: ln, div, sqrt and mul in one expression, re-evaluated as the spot moves and aggregated.

Over all 8 inputs

vs exact inputs (what a protocol sees) vs inputs as each format rounds them (the arithmetic alone)

N	FP127	FP127Lib	ABDK	Solady	PRBMath	Quad	Balancer	Aave	Morpho	Maker	FP256
1	37 (35–38)	37 (35–38)	37 (35–38)	37 (35–38)	18 (17–19)	18 (17–19)	17 (17–80)	17 (17–80)	17 (16–80)	17 (16–80)	34 (33–34)	34 (33–35)	17 (17–80)	17 (17–80)	— 8 rev	— 8 rev	— 8 rev	— 8 rev	— 8 rev	— 8 rev	71 (70–71)	71 (71–71)
2	37 (35–38)	37 (35–39)	37 (35–38)	37 (35–39)	17.5 (16–18)	18 (16–19)	17.5 (15–18)	17.5 (15–18)	16 (14–16)	16 (14–16)	32.5 (30–33)	32.5 (30–33)	17 (15–18)	17 (15–18)	— 8 rev	— 8 rev	— 8 rev	— 8 rev	— 8 rev	— 8 rev	70 (69–72)	70 (69–71)
5	37 (35–38)	37 (35–38)	37 (35–38)	37 (35–38)	17 (16–18)	18 (16–18)	17 (15–18)	17 (15–18)	15.5 (14–16)	15.5 (14–16)	32 (30–33)	32 (30–32)	17 (15–18)	17 (15–18)	— 8 rev	— 8 rev	— 8 rev	— 8 rev	— 8 rev	— 8 rev	70 (68–71)	70 (68–71)
10	37 (35–38)	37 (35–38)	37 (35–38)	37 (35–38)	17 (16–18)	18 (16–18)	17 (15–18)	17 (15–18)	15.5 (14–16)	15.5 (14–16)	32 (30–32)	32 (30–32)	17 (15–17)	17 (15–17)	— 8 rev	— 8 rev	— 8 rev	— 8 rev	— 8 rev	— 8 rev	70 (68–71)	70 (68–71)
20	37 (35–37)	37 (35–38)	37 (35–37)	37 (35–38)	17.5 (16–18)	18 (16–18)	17 (15–17)	17 (15–17)	16 (14–16)	16 (14–16)	32 (30–32)	32 (30–32)	17 (15–17)	17 (15–17)	— 8 rev	— 8 rev	— 8 rev	— 8 rev	— 8 rev	— 8 rev	70 (68–71)	70 (68–71)
50	37 (35–38)	37 (35–38)	37 (35–38)	37 (35–38)	17.5 (16–18)	18 (16–18)	17 (15–17)	17 (15–17)	16 (14–16)	16 (14–16)	31 (30–32)	31 (30–32)	17 (15–17)	17 (15–17)	— 8 rev	— 8 rev	— 8 rev	— 8 rev	— 8 rev	— 8 rev	69 (67–70)	69 (67–70)
100	37 (36–39)	37.5 (36–39)	37 (36–39)	37.5 (36–39)	17.5 (16–19)	18 (16–19)	17 (16–17)	17 (16–17)	16 (14–16)	16 (14–16)	31 (30–32)	31 (30–32)	17 (16–17)	17 (16–17)	— 8 rev	— 8 rev	— 8 rev	— 8 rev	— 8 rev	— 8 rev	69 (67–71)	69 (67–71)
365	37 (36–38)	37 (36–38)	37 (36–38)	37 (36–38)	17.5 (17–19)	18 (17–19)	17 (16–17)	17 (16–17)	16 (15–17)	16 (15–17)	30.5 (30–31)	30.5 (30–31)	17 (16–17)	17 (16–17)	— 8 rev	— 8 rev	— 8 rev	— 8 rev	— 8 rev	— 8 rev	68 (67–70)	68 (67–71)
1000	37 (36–38)	37.5 (37–38)	37 (36–38)	37.5 (37–38)	17 (17–18)	18 (17–19)	17 (16–18)	17 (16–18)	16 (15–17)	16 (15–17)	30 (30–31)	30 (30–31)	17 (16–18)	17 (16–18)	— 8 rev	— 8 rev	— 8 rev	— 8 rev	— 8 rev	— 8 rev	68 (67–70)	68 (67–70)
10000	37 (36–38)	38 (37–39)	37 (36–38)	38 (37–39)	18 (17–20)	18 (17–20)	17 (17–18)	17 (17–18)	16 (16–17)	16 (16–17)	30 (29–30)	30 (29–30)	17 (17–17)	17 (17–17)	— 8 rev	— 8 rev	— 8 rev	— 8 rev	— 8 rev	— 8 rev	67.5 (67–70)	67.5 (67–70)

Median (min–max) correct digits over the inputs, capped at 80; "rev" counts inputs that reverted at that N and are excluded.

The inputs and the ops

input	parameters	why
reference reference	T = 0.5, r = 0.05, sigma = 0.2, spot = 100, strike = 100, tick = 0.01	at the money, 20% vol, six months
low-vol	T = 0.5, r = 0.05, sigma = 0.05, spot = 100, strike = 100, tick = 0.01	5% vol: the denominator is small
high-vol	T = 0.5, r = 0.05, sigma = 0.8, spot = 100, strike = 100, tick = 0.01	80% vol
in-the-money	T = 0.5, r = 0.05, sigma = 0.2, spot = 120, strike = 100, tick = 0.01	ln(S/K) is positive and not small
out-of-the-money	T = 0.5, r = 0.05, sigma = 0.2, spot = 80, strike = 100, tick = 0.01	ln(S/K) is negative
short-dated	T = 0.01, r = 0.05, sigma = 0.2, spot = 100, strike = 100, tick = 0.01	sqrt(T) is small
long-dated	T = 2, r = 0.05, sigma = 0.2, spot = 100, strike = 100, tick = 0.01	two years
coarse-ticks	T = 0.5, r = 0.05, sigma = 0.2, spot = 100, strike = 100, tick = 1	whole-unit ticks: S doubles by N = 100 and reaches 10,100

Library calls per step: FP127 div, ln, add, sqrt, mul; FP127Lib div, ln, add, sqrt, mul; ABDK div, ln, add, sqrt, mul; Solady sDivWad, lnWad, add, sqrtWad, sMulWad; PRBMath div, ln, add, sqrt, mul; Quad div, ln, add, sqrt, mul; Balancer divDown, ln, add, pow(x, 1/2), mulDown; Aave unsupported; Morpho unsupported; Maker unsupported; FP256 div, ln, add, sqrt, mul. Parameters reach each adapter floored into its representation (ladder-inputs.json).

The reference input in full: T = 0.5, r = 0.05, sigma = 0.2, spot = 100, strike = 100, tick = 0.01

N	truth	FP127 (deployed, staticcall)	FP127Lib (inline)	ABDKMath64x64	Solady FixedPointMathLib	PRBMath SD59x18	ABDKMathQuad (binary128)	Balancer LogExpMath	Aave v3 MathUtils (RAY)	Morpho Blue MathLib	Maker rpow (RAY)	FP256 (float, inline)
1	0.247487373415291633540295526736697163749693	0.247487373415291633540295526736697163730619 37 digits · 37 vs format · -6.49024 ULP · 14759 gas/step	0.247487373415291633540295526736697163730619 37 digits · 37 vs format · -6.49024 ULP · 8335 gas/step	0.247487373415291633360190565182890054529707 18 digits · 18 vs format · -3.32235 ULP · 11764 gas/step	0.247487373415291635 17 digits · 17 vs format · 1.4597 ULP · 3822 gas/step	0.247487373415291635 17 digits · 17 vs format · 1.4597 ULP · 6998 gas/step	0.247487373415291633540295526736697176149717 34 digits · 34 vs format · ULP · 13162 gas/step	0.247487373415291633 17 digits · 17 vs format · -0.540296 ULP · 7708 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	0.247487373415291633540295526736697163749693 71 digits · 71 vs format · ULP · 17684 gas/step
2	0.495681818258787601119016496784504841607013	0.49568181825878760111901649678450484156673 37 digits · 37 vs format · -13.7077 ULP · 13481 gas/step	0.49568181825878760111901649678450484156673 37 digits · 37 vs format · -13.7077 ULP · 8968 gas/step	0.495681818258787600290932867652671234282025 17 digits · 17 vs format · -15.2754 ULP · 10635 gas/step	0.495681818258787601 18 digits · 18 vs format · -0.119016 ULP · 2921 gas/step	0.495681818258787559 16 digits · 16 vs format · -42.119 ULP · 8436 gas/step	0.495681818258787601119016496784503462509083 32 digits · 32 vs format · ULP · 19642 gas/step	0.495681818258787597 17 digits · 17 vs format · -4.11902 ULP · 6386 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	0.495681818258787601119016496784504841607013 70 digits · 70 vs format · ULP · 19656 gas/step
5	1.24450687446379156294859558697855008088958	1.24450687446379156294859558697855008077686 37 digits · 37 vs format · -38.3547 ULP · 12714 gas/step	1.24450687446379156294859558697855008077686 37 digits · 37 vs format · -38.3547 ULP · 9347 gas/step	1.24450687446379155986792702792698150915385 17 digits · 17 vs format · -56.8283 ULP · 9957 gas/step	1.244506874463791555 17 digits · 17 vs format · -7.9486 ULP · 2381 gas/step	1.244506874463791342 15 digits · 15 vs format · -220.949 ULP · 9431 gas/step	1.24450687446379156294859558697854168099986 32 digits · 32 vs format · ULP · 23377 gas/step	1.244506874463791545 16 digits · 16 vs format · -17.9486 ULP · 5592 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	1.24450687446379156294859558697855008088958 70 digits · 70 vs format · ULP · 20832 gas/step
10	2.50668346780494102921027404289032901502747	2.50668346780494102921027404289032901475369 36 digits · 37 vs format · -93.1626 ULP · 12459 gas/step	2.50668346780494102921027404289032901475369 36 digits · 37 vs format · -93.1626 ULP · 9474 gas/step	2.50668346780494102210437926125941032751143 17 digits · 17 vs format · -131.081 ULP · 9731 gas/step	2.50668346780494101 17 digits · 17 vs format · -19.2103 ULP · 2201 gas/step	2.506683467804940505 15 digits · 15 vs format · -524.21 ULP · 9734 gas/step	2.50668346780494102921027404289029780119392 31 digits · 31 vs format · ULP · 24511 gas/step	2.50668346780494099 16 digits · 16 vs format · -39.2103 ULP · 5328 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	2.50668346780494102921027404289032901502747 69 digits · 69 vs format · ULP · 21229 gas/step
20	5.08401051403297891352661479689167511150537	5.08401051403297891352661479689167511088225 36 digits · 37 vs format · -212.039 ULP · 12331 gas/step	5.08401051403297891352661479689167511088225 36 digits · 37 vs format · -212.039 ULP · 9537 gas/step	5.08401051403297889923552088653480041102739 17 digits · 17 vs format · -263.624 ULP · 9618 gas/step	5.084010514032978876 17 digits · 17 vs format · -37.5266 ULP · 2111 gas/step	5.084010514032977814 15 digits · 15 vs format · -1099.53 ULP · 9905 gas/step	5.0840105140329789135266147968915545422869 31 digits · 31 vs format · ULP · 25006 gas/step	5.084010514032978836 16 digits · 16 vs format · -77.5266 ULP · 5196 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	5.08401051403297891352661479689167511150537 69 digits · 69 vs format · ULP · 21425 gas/step
50	13.2391487646747260018911032773984733598061	13.2391487646747260018911032773984733581989 36 digits · 37 vs format · -546.895 ULP · 12256 gas/step	13.2391487646747260018911032773984733581989 36 digits · 37 vs format · -546.895 ULP · 9575 gas/step	13.23914876467472596450114172283996083479 17 digits · 17 vs format · -689.723 ULP · 9551 gas/step	13.239148764674725887 17 digits · 17 vs format · -114.891 ULP · 2057 gas/step	13.239148764674722973 15 digits · 15 vs format · -3028.89 ULP · 10052 gas/step	13.2391487646747260018911032773977398443755 31 digits · 31 vs format · ULP · 25202 gas/step	13.239148764674725785 16 digits · 16 vs format · -216.891 ULP · 5116 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	13.2391487646747260018911032773984733598061 69 digits · 69 vs format · ULP · 21133 gas/step
100	28.237364393266085066216191925904648830649	28.2373643932660850662161919259046488271773 36 digits · 37 vs format · -1181.35 ULP · 12304 gas/step	28.2373643932660850662161919259046488271773 36 digits · 37 vs format · -1181.35 ULP · 9677 gas/step	28.2373643932660849925468139753892415910741 17 digits · 17 vs format · -1358.96 ULP · 9528 gas/step	28.23736439326608483 17 digits · 17 vs format · -236.216 ULP · 2039 gas/step	28.237364393266079006 15 digits · 15 vs format · -6060.22 ULP · 10098 gas/step	28.2373643932660850662161919259017616314702 30 digits · 30 vs format · ULP · 25187 gas/step	28.237364393266084623 16 digits · 16 vs format · -443.216 ULP · 5090 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	28.237364393266085066216191925904648830649 68 digits · 68 vs format · ULP · 20371 gas/step
365	136.745450883799826897439024462329942560739	136.745450883799826897439024462329942542832 36 digits · 37 vs format · -6093.27 ULP · 12423 gas/step	136.745450883799826897439024462329942542832 36 digits · 37 vs format · -6093.27 ULP · 9845 gas/step	136.745450883799826610829741130315184705069 17 digits · 17 vs format · -5287.01 ULP · 9512 gas/step	136.745450883799826231 17 digits · 17 vs format · -666.439 ULP · 2025 gas/step	136.745450883799804954 15 digits · 15 vs format · -21943.4 ULP · 10135 gas/step	136.745450883799826897439024462292089304604 30 digits · 30 vs format · ULP · 25027 gas/step	136.745450883799825305 16 digits · 16 vs format · -1592.44 ULP · 5071 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	136.745450883799826897439024462329942560739 68 digits · 68 vs format · ULP · 19895 gas/step
1000	589.474773952864477005737983804430862752872	589.474773952864477005737983804430862672337 36 digits · 37 vs format · -27404.5 ULP · 12561 gas/step	589.474773952864477005737983804430862672337 36 digits · 37 vs format · -27404.5 ULP · 9989 gas/step	589.474773952864476086858664533041363142729 17 digits · 18 vs format · -16950.3 ULP · 9508 gas/step	589.474773952864476583 18 digits · 18 vs format · -422.738 ULP · 2022 gas/step	589.474773952864417545 15 digits · 15 vs format · -59460.7 ULP · 10171 gas/step	589.474773952864477005737983804151811806625 30 digits · 30 vs format · ULP · 24823 gas/step	589.474773952864472524 17 digits · 17 vs format · -4481.74 ULP · 5066 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	589.474773952864477005737983804430862752872 68 digits · 68 vs format · ULP · 19877 gas/step
10000	29787.5592875314291664765495822016511063025	29787.5592875314291664765495822016511027086 36 digits · 39 vs format · -1.22295e+6 ULP · 13263 gas/step	29787.5592875314291664765495822016511027086 36 digits · 39 vs format · -1.22295e+6 ULP · 10331 gas/step	29787.5592875314291460913610626315417562182 18 digits · 19 vs format · -376040.0 ULP · 9506 gas/step	29787.559287531429310994 17 digits · 17 vs format · 144517.0 ULP · 2021 gas/step	29787.559287531428723829 16 digits · 16 vs format · -442648.0 ULP · 10223 gas/step	29787.5592875314291664765495821771299686063 30 digits · 30 vs format · ULP · 24408 gas/step	29787.559287531429097595 17 digits · 17 vs format · -68881.5 ULP · 5313 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	29787.5592875314291664765495822016511063025 67 digits · 67 vs format · ULP · 20135 gas/step

Cumulative product of small factors

Step: p = p * (1 + eps_i), eps_i = +eps for odd i, -eps for even i. Result: product after N factors.

The general form of fee accrual, index levels and LP share accounting: many factors a basis point from one. The drift from 1 is where the digits are, and it is small.

Over all 5 inputs

vs exact inputs (what a protocol sees) vs inputs as each format rounds them (the arithmetic alone)

N	FP127	FP127Lib	ABDK	Solady	PRBMath	Quad	Balancer	Aave	Morpho	Maker	FP256
1	38 (38–39)	80 (80–80)	38 (38–39)	80 (80–80)	19 (19–20)	80 (80–80)	80 (80–80)	80 (80–80)	80 (80–80)	80 (80–80)	34 (33–35)	34 (33–35)	80 (80–80)	80 (80–80)	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	71 (71–71)	71 (71–71)
2	38 (38–39)	38 (38–39)	38 (38–39)	38 (38–39)	19 (19–20)	19 (19–20)	80 (80–80)	80 (80–80)	80 (80–80)	80 (80–80)	33 (33–34)	33 (33–34)	80 (80–80)	80 (80–80)	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	71 (71–71)	71 (71–71)
5	38 (37–38)	38 (38–38)	38 (37–38)	38 (38–38)	18 (18–19)	18 (18–19)	80 (20–80)	80 (20–80)	80 (20–80)	80 (20–80)	33 (33–33)	33 (33–33)	80 (20–80)	80 (20–80)	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	71 (70–71)	71 (70–71)
10	37 (37–37)	37 (37–37)	37 (37–37)	37 (37–37)	18 (18–19)	18 (18–18)	18 (17–22)	18 (17–22)	18 (17–22)	18 (17–22)	32 (32–33)	32 (32–33)	18 (17–22)	18 (17–22)	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	71 (70–72)	71 (70–72)
20	37 (37–37)	37 (37–37)	37 (37–37)	37 (37–37)	18 (18–18)	18 (18–18)	17 (17–22)	17 (17–22)	17 (17–22)	17 (17–22)	32 (32–32)	32 (32–32)	17 (17–22)	17 (17–22)	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	70 (70–71)	70 (70–71)
50	37 (37–37)	37 (37–37)	37 (37–37)	37 (37–37)	17 (17–18)	17 (17–17)	16 (16–21)	16 (16–21)	16 (16–21)	16 (16–21)	32 (32–32)	32 (32–32)	16 (16–21)	16 (16–21)	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	70 (69–71)	70 (69–71)
100	36 (36–36)	36 (36–36)	36 (36–36)	36 (36–36)	17 (17–17)	17 (17–17)	16 (16–20)	16 (16–20)	16 (16–20)	16 (16–20)	31 (31–31)	31 (31–31)	16 (16–20)	16 (16–20)	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	70 (69–70)	70 (69–70)
365	36 (36–36)	36 (36–36)	36 (36–36)	36 (36–36)	16 (16–17)	16 (16–17)	15 (15–19)	15 (15–19)	15 (15–19)	15 (15–19)	31 (31–31)	31 (31–31)	15 (15–19)	15 (15–19)	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	69 (69–71)	69 (69–71)
1000	35 (35–35)	35 (35–35)	35 (35–35)	35 (35–35)	16 (16–16)	16 (16–16)	15 (15–18)	15 (15–18)	15 (15–18)	15 (15–18)	30 (30–30)	30 (30–30)	15 (15–18)	15 (15–18)	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	69 (68–70)	70 (68–70)
10000	34 (30–34)	34 (30–34)	34 (30–34)	34 (30–34)	15 (11–15)	15 (11–15)	14 (9–16)	14 (9–16)	14 (9–16)	14 (9–16)	29 (29–29)	29 (29–29)	14 (9–16)	14 (9–16)	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	69 (67–70)	69 (67–70)

Median (min–max) correct digits over the inputs, capped at 80; "rev" counts inputs that reverted at that N and are excluded.

The inputs and the ops

input	parameters	why
reference reference	eps = 0.0001	a basis point
percent	eps = 0.01	one percent
tenth-percent	eps = 0.001	ten basis points
micro	eps = 0.000001	a hundredth of a basis point: the factors are within WAD's last digits of one
five-percent	eps = 0.05	large factors: the product falls to 4e-6 by N = 10,000

Library calls per step: FP127 mul; FP127Lib mul; ABDK mul; Solady sMulWad; PRBMath mul; Quad mul; Balancer mulDown; Aave unsupported; Morpho unsupported; Maker unsupported; FP256 mul. Parameters reach each adapter floored into its representation (ladder-inputs.json).

The reference input in full: eps = 0.0001

N	truth	FP127 (deployed, staticcall)	FP127Lib (inline)	ABDKMath64x64	Solady FixedPointMathLib	PRBMath SD59x18	ABDKMathQuad (binary128)	Balancer LogExpMath	Aave v3 MathUtils (RAY)	Morpho Blue MathLib	Maker rpow (RAY)	FP256 (float, inline)
1	1.0001	1.00009999999999999999999999999999999999957 39 digits · exact vs format · -0.1456 ULP · 2644 gas/step	1.00009999999999999999999999999999999999957 39 digits · exact vs format · -0.1456 ULP · 1411 gas/step	1.00009999999999999999123964644631712417322 20 digits · exact vs format · -0.1616 ULP · 1491 gas/step	1.0001 exact · exact vs format · 0 ULP · 1234 gas/step	1.0001 exact · exact vs format · 0 ULP · 1594 gas/step	1.00009999999999999999999999999999999815111 35 digits · 35 vs format · ULP · 5693 gas/step	1.0001 exact · exact vs format · 0 ULP · 1505 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	1.0001 71 digits · 71 vs format · ULP · 4637 gas/step
2	0.99999999	0.999999989999999999999999999999999999999066 39 digits · 39 vs format · -0.317885 ULP · 2188 gas/step	0.999999989999999999999999999999999999999066 39 digits · 39 vs format · -0.317885 ULP · 940 gas/step	0.999999989999999999950967832784698430259596 19 digits · 19 vs format · -0.904484 ULP · 944 gas/step	0.99999999 exact · exact vs format · 9.33264e-284 ULP · 789 gas/step	0.99999999 exact · exact vs format · 9.33264e-284 ULP · 1149 gas/step	0.999999989999999999999999999999999863682863 33 digits · 33 vs format · ULP · 3409 gas/step	0.99999999 exact · exact vs format · 9.33264e-284 ULP · 1060 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	0.99999999 71 digits · 71 vs format · ULP · 2990 gas/step
5	1.00009997999800010001	1.00009997999800010000999999999999999999409 38 digits · 38 vs format · -2.01232 ULP · 1915 gas/step	1.00009997999800010000999999999999999999409 38 digits · 38 vs format · -2.01232 ULP · 657 gas/step	1.00009997999800009986460722077872276258859 18 digits · 18 vs format · -2.68202 ULP · 616 gas/step	1.0000999799980001 20 digits · 20 vs format · -0.01 ULP · 522 gas/step	1.0000999799980001 20 digits · 20 vs format · -0.01 ULP · 882 gas/step	1.0000999799980001000099999999999995066009 33 digits · 33 vs format · ULP · 2023 gas/step	1.0000999799980001 20 digits · 20 vs format · -0.01 ULP · 793 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	1.00009997999800010001 71 digits · 71 vs format · ULP · 2005 gas/step
10	0.9999999500000009999999900000000499999999	0.999999950000000999999990000000049999988515 37 digits · 37 vs format · -3.87424 ULP · 1824 gas/step	0.999999950000000999999990000000049999988515 37 digits · 37 vs format · -3.87424 ULP · 563 gas/step	0.999999950000000999714502847304231636371696 18 digits · 18 vs format · -5.26631 ULP · 506 gas/step	0.999999950000000997 17 digits · 17 vs format · -2.99999 ULP · 433 gas/step	0.999999950000000997 17 digits · 17 vs format · -2.99999 ULP · 793 gas/step	0.999999950000000999999990000000048779719584 32 digits · 32 vs format · ULP · 1566 gas/step	0.999999950000000997 17 digits · 17 vs format · -2.99999 ULP · 704 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	0.9999999500000009999999900000000499999999 72 digits · 72 vs format · ULP · 1676 gas/step
20	0.9999999000000044999998800000020999999748	0.9999999000000044999998800000020999999476 37 digits · 37 vs format · -9.25555 ULP · 1779 gas/step	0.9999999000000044999998800000020999999476 37 digits · 37 vs format · -9.25555 ULP · 516 gas/step	0.999999900000004499382375011684587207128061 18 digits · 18 vs format · -11.391 ULP · 452 gas/step	0.999999900000004492 17 digits · 17 vs format · -7.99988 ULP · 388 gas/step	0.999999900000004492 17 digits · 17 vs format · -7.99988 ULP · 748 gas/step	0.999999900000004499999880000002097764127958 32 digits · 32 vs format · ULP · 1335 gas/step	0.999999900000004492 17 digits · 17 vs format · -7.99988 ULP · 659 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	0.9999999000000044999998800000020999999748 71 digits · 71 vs format · ULP · 1511 gas/step
50	0.999999750000029999997700000126499994687	0.99999975000002999999770000012649999461417 37 digits · 37 vs format · -24.7829 ULP · 1751 gas/step	0.99999975000002999999770000012649999461417 37 digits · 37 vs format · -24.7829 ULP · 488 gas/step	0.9999997500000299984314680590280488559074 17 digits · 17 vs format · -28.8919 ULP · 419 gas/step	0.999999750000029977 16 digits · 16 vs format · -22.9977 ULP · 362 gas/step	0.999999750000029977 16 digits · 16 vs format · -22.9977 ULP · 722 gas/step	0.99999975000002999999770000012649392197215 32 digits · 32 vs format · ULP · 1197 gas/step	0.999999750000029977 16 digits · 16 vs format · -22.9977 ULP · 633 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	0.999999750000029999997700000126499994687 70 digits · 70 vs format · ULP · 1413 gas/step
100	0.999999500000122499980400002302999788124016	0.999999500000122499980400002302999787978012 36 digits · 36 vs format · -49.6824 ULP · 1742 gas/step	0.999999500000122499980400002302999787978012 36 digits · 36 vs format · -49.6824 ULP · 478 gas/step	0.999999500000122497052421760566076613940822 17 digits · 17 vs format · -54.0117 ULP · 408 gas/step	0.999999500000122452 16 digits · 16 vs format · -47.9804 ULP · 353 gas/step	0.999999500000122452 16 digits · 16 vs format · -47.9804 ULP · 713 gas/step	0.999999500000122499980400002302988171046357 31 digits · 31 vs format · ULP · 1151 gas/step	0.999999500000122452 16 digits · 16 vs format · -47.9804 ULP · 624 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	0.999999500000122499980400002302999788124016 70 digits · 70 vs format · ULP · 1380 gas/step
365	1.00009817981964726372164161629041717960188	1.00009817981964726372164161629041717909031 36 digits · 36 vs format · -174.078 ULP · 1736 gas/step	1.00009817981964726372164161629041717909031 36 digits · 36 vs format · -174.078 ULP · 472 gas/step	1.00009817981964725354870546225605210111098 16 digits · 16 vs format · -187.658 ULP · 400 gas/step	1.000098179819647083 15 digits · 15 vs format · -180.722 ULP · 346 gas/step	1.000098179819647083 15 digits · 15 vs format · -180.722 ULP · 706 gas/step	1.00009817981964726372164161629037401189909 31 digits · 31 vs format · ULP · 1118 gas/step	1.000098179819647083 15 digits · 15 vs format · -180.722 ULP · 617 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	1.00009817981964726372164161629041717960188 71 digits · 71 vs format · ULP · 1356 gas/step
1000	0.999995000012474979291525730285725552297672	0.999995000012474979291525730285725550870416 35 digits · 35 vs format · -485.67 ULP · 1736 gas/step	0.999995000012474979291525730285725550870416 35 digits · 35 vs format · -485.67 ULP · 470 gas/step	0.999995000012474951939407674661453029330005 16 digits · 16 vs format · -504.558 ULP · 398 gas/step	0.999995000012474475 15 digits · 15 vs format · -504.292 ULP · 345 gas/step	0.999995000012474475 15 digits · 15 vs format · -504.292 ULP · 705 gas/step	0.999995000012474979291525730285603759470012 30 digits · 30 vs format · ULP · 1110 gas/step	0.999995000012474475 15 digits · 15 vs format · -504.292 ULP · 616 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	0.999995000012474979291525730285725552297672 70 digits · 70 vs format · ULP · 1350 gas/step
10000	0.999950001249729179425101682297158079008892	0.999950001249729179425101682297158064279421 34 digits · 34 vs format · -5012.18 ULP · 1753 gas/step	0.999950001249729179425101682297158064279421 34 digits · 34 vs format · -5012.18 ULP · 469 gas/step	0.999950001249728906329693883803244602859195 15 digits · 15 vs format · -5037.72 ULP · 397 gas/step	0.999950001249724158 14 digits · 14 vs format · -5021.43 ULP · 344 gas/step	0.999950001249724158 14 digits · 14 vs format · -5021.43 ULP · 704 gas/step	0.999950001249729179425101682295945149558818 29 digits · 29 vs format · ULP · 1105 gas/step	0.999950001249724158 14 digits · 14 vs format · -5021.43 ULP · 615 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	0.999950001249729179425101682297158079008892 70 digits · 70 vs format · ULP · 1347 gas/step

Annual compounding at 10% (README claim check)

Step: b = b * 1.1. Result: balance after N years.

The old README asserted $25,937.42 true against $25,932.38 for Solady at N = 10 with no source. This row regenerates that number.

Over all 5 inputs

vs exact inputs (what a protocol sees) vs inputs as each format rounds them (the arithmetic alone)

N	FP127	FP127Lib	ABDK	Solady	PRBMath	Quad	Balancer	Aave	Morpho	Maker	FP256
1	38 (38–80)	80 (80–80)	38 (38–80)	80 (80–80)	19 (19–80)	80 (80–80)	80 (80–80)	80 (80–80)	80 (80–80)	80 (80–80)	33 (33–80)	33 (33–80)	80 (80–80)	80 (80–80)	— 5 rev	— 5 rev	— 5 rev	— 5 rev	80 (80–80)	80 (80–80)	80 (71–80)	72 (71–80)
2	38 (38–80)	77 (38–80)	38 (38–80)	77 (38–80)	19 (19–80)	39 (20–80)	80 (80–80)	80 (80–80)	80 (80–80)	80 (80–80)	33 (33–80)	33 (33–80)	80 (80–80)	80 (80–80)	— 5 rev	— 5 rev	— 5 rev	— 5 rev	80 (80–80)	80 (80–80)	80 (71–80)	72 (71–80)
5	38 (37–80)	42 (38–80)	38 (37–80)	42 (38–80)	18 (18–80)	23 (19–80)	80 (80–80)	80 (80–80)	80 (80–80)	80 (80–80)	33 (32–80)	33 (32–80)	80 (80–80)	80 (80–80)	— 5 rev	— 5 rev	— 5 rev	— 5 rev	80 (80–80)	80 (80–80)	70 (70–80)	70 (70–80)
10	37 (37–80)	42 (38–80)	37 (37–80)	42 (38–80)	18 (18–80)	23 (18–80)	80 (80–80)	80 (80–80)	80 (80–80)	80 (80–80)	32 (32–80)	32 (32–80)	80 (80–80)	80 (80–80)	— 5 rev	— 5 rev	— 5 rev	— 5 rev	80 (80–80)	80 (80–80)	70 (70–80)	70 (70–80)
20	37 (37–80)	42 (37–80)	37 (37–80)	42 (37–80)	18 (18–80)	22 (18–80)	80 (18–80)	80 (18–80)	80 (18–80)	80 (18–80)	32 (32–80)	32 (32–80)	80 (18–80)	80 (18–80)	— 5 rev	— 5 rev	— 5 rev	— 5 rev	80 (28–80)	80 (28–80)	70 (70–80)	70 (70–80)
50	37 (36–80)	41 (37–80)	37 (36–80)	41 (37–80)	17 (17–80)	22 (18–80)	22 (18–25)	22 (18–25)	22 (18–25)	22 (18–25)	32 (31–80)	32 (31–80)	22 (18–25)	22 (18–25)	— 5 rev	— 5 rev	— 5 rev	— 5 rev	29 (28–33)	29 (28–33)	69 (69–80)	69 (69–80)
100	36 (36–80)	41 (37–80)	36 (36–80)	41 (37–80)	17 (17–17) 1 rev	22 (18–24) 1 rev	22 (18–25)	22 (18–25)	22 (18–25)	22 (18–25)	31 (31–32)	31 (31–32)	22 (18–25)	22 (18–25)	— 5 rev	— 5 rev	— 5 rev	— 5 rev	29 (27–33)	29 (27–33)	69 (69–80)	69 (69–80)
365	36 (36–36) 1 rev	41 (37–44) 1 rev	36 (36–36) 1 rev	41 (37–44) 1 rev	16 (16–16) 3 rev	20 (18–22) 3 rev	21.5 (18–24) 1 rev	21.5 (18–24) 1 rev	21.5 (18–24) 1 rev	21.5 (18–24) 1 rev	31 (31–31)	31 (31–31)	21.5 (18–24) 1 rev	21.5 (18–24) 1 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	28 (26–28) 1 rev	28 (26–28) 1 rev	68 (68–69)	68 (68–69)
1000	35 (35–35) 4 rev	41 (41–41) 4 rev	35 (35–35) 4 rev	41 (41–41) 4 rev	— 5 rev	— 5 rev	21 (21–21) 4 rev	21 (21–21) 4 rev	21.5 (18–24) 1 rev	21.5 (18–24) 1 rev	30 (30–31)	30 (30–31)	21 (21–21) 4 rev	21 (21–21) 4 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	68 (68–69)	68 (68–69)
10000	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	29 (29–30)	29 (29–30)	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	67 (67–67)	67 (67–67)

Median (min–max) correct digits over the inputs, capped at 80; "rev" counts inputs that reverted at that N and are excluded.

The inputs and the ops

input	parameters	why
reference reference	principal = 10000, rate = 0.1	the README claim
five-pct	principal = 10000, rate = 0.05	5% a year
unit	principal = 1, rate = 0.1	one unit
million	principal = 1000000, rate = 0.1	a million units: ABDK overflows sooner
fifty-pct	principal = 10000, rate = 0.5	50% a year: every library overflows before N = 100

The reference input in full: principal = 10000, rate = 0.1

N	truth	FP127 (deployed, staticcall)	FP127Lib (inline)	ABDKMath64x64	Solady FixedPointMathLib	PRBMath SD59x18	ABDKMathQuad (binary128)	Balancer LogExpMath	Aave v3 MathUtils (RAY)	Morpho Blue MathLib	Maker rpow (RAY)	FP256 (float, inline)
1	11000.0	10999.9999999999999999999999999999999999824 38 digits · exact vs format · -6000.0 ULP · 2403 gas/step	10999.9999999999999999999999999999999999824 38 digits · exact vs format · -6000.0 ULP · 1143 gas/step	10999.9999999999999996747393482543486697978 19 digits · exact vs format · -6000.0 ULP · 1189 gas/step	11000.0 exact · exact vs format · 0 ULP · 999 gas/step	11000.0 exact · exact vs format · 0 ULP · 1359 gas/step	10999.9999999999999999999999999999984222782 33 digits · 33 vs format · ULP · 3188 gas/step	11000.0 exact · exact vs format · 0 ULP · 1270 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	11000.0 exact · exact vs format · 0 ULP · 610 gas/step	11000.0 exact · 72 vs format · ULP · 3156 gas/step
2	12100.0	12099.9999999999999999999999999999999999612 38 digits · 77 vs format · -13200.0 ULP · 2012 gas/step	12099.9999999999999999999999999999999999612 38 digits · 77 vs format · -13200.0 ULP · 750 gas/step	12099.9999999999999992844265661595670735551 19 digits · 39 vs format · -13200.0 ULP · 737 gas/step	12100.0 exact · exact vs format · -1.52906e-279 ULP · 616 gas/step	12100.0 exact · exact vs format · -1.52906e-279 ULP · 976 gas/step	12099.9999999999999999999999999999968445564 33 digits · 33 vs format · ULP · 2101 gas/step	12100.0 exact · exact vs format · -1.52906e-279 ULP · 887 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	12100.0 exact · exact vs format · -1.52906e-270 ULP · 420 gas/step	12100.0 71 digits · 71 vs format · ULP · 2194 gas/step
5	16105.1	16105.0999999999999999999999999999999998709 38 digits · 42 vs format · -43923.6 ULP · 1778 gas/step	16105.0999999999999999999999999999999998709 38 digits · 42 vs format · -43923.6 ULP · 515 gas/step	16105.0999999999999976188968728307848721215 18 digits · 23 vs format · -43923.6 ULP · 466 gas/step	16105.1 exact · exact vs format · -3.05812e-279 ULP · 386 gas/step	16105.1 exact · exact vs format · -3.05812e-279 ULP · 746 gas/step	16105.0999999999999999999999999999876937699 33 digits · 33 vs format · ULP · 1449 gas/step	16105.1 exact · exact vs format · -3.05812e-279 ULP · 657 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	16105.1 exact · exact vs format · -3.05812e-270 ULP · 197 gas/step	16105.1 70 digits · 70 vs format · ULP · 1617 gas/step
10	25937.424601	25937.4246009999999999999999999999999995842 37 digits · 42 vs format · -141480.0 ULP · 1700 gas/step	25937.4246009999999999999999999999999995842 37 digits · 42 vs format · -141480.0 ULP · 436 gas/step	25937.4246009999999923303156020015780569565 18 digits · 23 vs format · -141481.0 ULP · 376 gas/step	25937.424601 exact · exact vs format · -9.17435e-279 ULP · 309 gas/step	25937.424601 exact · exact vs format · -9.17435e-279 ULP · 669 gas/step	25937.4246009999999999999999999999543604713 32 digits · 32 vs format · ULP · 1228 gas/step	25937.424601 exact · exact vs format · -9.17435e-279 ULP · 580 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	25937.424601 exact · exact vs format · -9.17435e-270 ULP · 121 gas/step	25937.424601 70 digits · 70 vs format · ULP · 1425 gas/step
20	67274.9994932560009201	67274.9994932560009200999999999999999978432 37 digits · 42 vs format · -733929.0 ULP · 1661 gas/step	67274.9994932560009200999999999999999978432 37 digits · 42 vs format · -733929.0 ULP · 397 gas/step	67274.9994932560008803138352877215577763081 18 digits · 22 vs format · -733925.0 ULP · 331 gas/step	67274.9994932560009201 exact · exact vs format · -4.89299e-278 ULP · 271 gas/step	67274.9994932560009201 exact · exact vs format · -4.89299e-278 ULP · 631 gas/step	67274.9994932560009200999999999997671187464 32 digits · 32 vs format · ULP · 1117 gas/step	67274.9994932560009201 exact · exact vs format · -4.89299e-278 ULP · 542 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	67274.9994932560009201 exact · exact vs format · -4.89299e-269 ULP · 68 gas/step	67274.9994932560009201 70 digits · 70 vs format · ULP · 1330 gas/step
50	1173908.52879695316506666495990358319938982	1173908.52879695316506666495990358319929573 37 digits · 41 vs format · -3.20161e+7 ULP · 1637 gas/step	1173908.52879695316506666495990358319929573 37 digits · 41 vs format · -3.20161e+7 ULP · 373 gas/step	1173908.52879695316333107199969348899415422 17 digits · 22 vs format · -3.2016e+7 ULP · 304 gas/step	1173908.528796953165066617 22 digits · 22 vs format · -47.9599 ULP · 248 gas/step	1173908.528796953165066617 22 digits · 22 vs format · -47.9599 ULP · 608 gas/step	1173908.52879695316506666495990357292852018 32 digits · 32 vs format · ULP · 1052 gas/step	1173908.528796953165066617 22 digits · 22 vs format · -47.9599 ULP · 519 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	1173908.5287969531650666649599 29 digits · 29 vs format · -3583.2 ULP · 31 gas/step	1173908.52879695316506666495990358319938982 69 digits · 69 vs format · ULP · 1272 gas/step
100	137806123.398222701841183371720896367762643	137806123.398222701841183371720896367740554 36 digits · 41 vs format · -7.51675e+9 ULP · 1630 gas/step	137806123.398222701841183371720896367740554 36 digits · 41 vs format · -7.51675e+9 ULP · 365 gas/step	137806123.398222701433700108702637421309589 17 digits · 22 vs format · -7.51674e+9 ULP · 295 gas/step	137806123.398222701841177177 22 digits · 22 vs format · -6194.72 ULP · 240 gas/step	137806123.398222701841177177 22 digits · 22 vs format · -6194.72 ULP · 600 gas/step	137806123.39822270184118337172089399648371 31 digits · 31 vs format · ULP · 1030 gas/step	137806123.398222701841177177 22 digits · 22 vs format · -6194.72 ULP · 511 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	137806123.39822270184118337172017 29 digits · 29 vs format · -726368.0 ULP · 17 gas/step	137806123.398222701841183371720896367762643 69 digits · 69 vs format · ULP · 1253 gas/step
365	12833055803133526968.9944800789796565545148	12833055803133526968.9944800789796565470065 36 digits · 41 vs format · -2.55495e+21 ULP · 1624 gas/step	12833055803133526968.9944800789796565470065 36 digits · 41 vs format · -2.55495e+21 ULP · 360 gas/step	reverts bare revert (require without a reason, or out of gas) 0x00000000	12833055803133526968.993902795476311067 22 digits · 22 vs format · -5.77284e+14 ULP · 235 gas/step	12833055803133526968.993902795476311067 22 digits · 22 vs format · -5.77284e+14 ULP · 595 gas/step	12833055803133526968.9944800789788530437363 31 digits · 31 vs format · ULP · 1015 gas/step	12833055803133526968.993902795476311067 22 digits · 22 vs format · -5.77284e+14 ULP · 506 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	12833055803133526968.9944800782719799783126 28 digits · 28 vs format · -7.07677e+17 ULP · 6 gas/step	12833055803133526968.9944800789796565545148 68 digits · 68 vs format · ULP · 1239 gas/step
1000	2.46993291800582633412408838508522147770973e+45	reverts Overflow() 0x35278d12	reverts Overflow() 0x35278d12	reverts bare revert (require without a reason, or out of gas) 0x00000000	reverts Solady ExpOverflow() 0xedcd4dd4	2.46993291800582633412397727736857096657984e+45 22 digits · 22 vs format · -1.11108e+41 ULP · 602 gas/step	2.46993291800582633412408838508478949012537e+45 30 digits · 30 vs format · ULP · 1011 gas/step	reverts Panic (checked arithmetic overflow or division by zero) 0x4e487b71	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Solady RPowOverflow() (Maker's rpow) 0x49f7642b	2.46993291800582633412408838508522147770973e+45 68 digits · 68 vs format · ULP · 1235 gas/step
10000	8.44990025120034795728149043293472294488798e+417	reverts Overflow() 0x35278d12	reverts Overflow() 0x35278d12	reverts bare revert (require without a reason, or out of gas) 0x00000000	reverts Solady ExpOverflow() 0xedcd4dd4	reverts PRBMath_SD59x18_Mul_Overflow 0x120b5b43	8.44990025120034795728149043291992031813128e+417 29 digits · 29 vs format · ULP · 1009 gas/step	reverts Panic (checked arithmetic overflow or division by zero) 0x4e487b71	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Solady RPowOverflow() (Maker's rpow) 0x49f7642b	8.44990025120034795728149043293472294488798e+417 67 digits · 67 vs format · ULP · 1234 gas/step

Per-second accrual

Step: b = b0 * (1 + r/31536000)^(N * 86400). Result: balance after N days of per-second compounding.

How Maker's jug and pot accrue: the rate per second, compounded every second, N days at a time. The rate is 1.6e-9 at 5% APR, which an 18-decimal grid holds to ten digits before any arithmetic; FP127 computes the power as one exact integer power (compound), the WAD and 64.64 libraries as exp(n ln(1 + r)), Maker as rpow.

Over all 6 inputs

vs exact inputs (what a protocol sees) vs inputs as each format rounds them (the arithmetic alone)

N	FP127	FP127Lib	ABDK	Solady	PRBMath	Quad	Balancer	Aave	Morpho	Maker	FP256
1	33 (33–35)	39 (38–39)	33 (33–35)	39 (38–39)	14 (14–14)	14 (14–14)	12.5 (12–13)	13 (13–13)	12 (12–12)	12 (12–12)	28 (28–29)	28 (28–29)	13 (13–13)	17.5 (17–18)	— 6 rev	— 6 rev	— 6 rev	— 6 rev	22 (22–23)	22 (22–23)	66 (66–67)	66 (66–67)
2	33 (33–34)	38 (38–39)	33 (33–34)	38 (38–39)	13 (13–14)	14 (13–14)	12 (12–12)	12 (12–13)	11 (11–11)	11 (11–11)	28 (28–29)	28 (28–29)	12.5 (12–13)	17 (17–18)	— 6 rev	— 6 rev	— 6 rev	— 6 rev	22 (21–23)	22 (22–22)	66 (66–66)	66 (66–66)
5	33 (32–34)	39 (39–39)	33 (32–34)	39 (39–39)	13 (13–13)	13 (13–13)	12 (12–12)	12 (12–12)	11 (11–11)	11 (11–11)	28 (28–28)	28 (28–28)	12.5 (12–13)	17.5 (17–18)	— 6 rev	— 6 rev	— 6 rev	— 6 rev	21 (21–22)	21.5 (21–22)	65 (65–66)	65 (65–66)
10	32 (32–34)	39 (38–40)	32 (32–34)	39 (38–40)	13 (13–13)	13 (13–13)	11.5 (11–12)	12 (12–12)	11 (11–11)	11 (11–11)	27 (27–28)	27 (27–28)	12 (12–12)	17.5 (17–18)	— 6 rev	— 6 rev	— 6 rev	— 6 rev	21 (21–22)	21 (21–22)	65 (65–66)	65 (65–66)
20	32 (32–33)	38 (38–39)	32 (32–33)	38 (38–39)	12 (12–13)	13 (12–13)	11 (11–11)	11 (11–12)	10 (10–10)	10 (10–10)	27 (27–28)	27 (27–28)	11.5 (11–12)	18 (17–18)	— 6 rev	— 6 rev	— 6 rev	— 6 rev	21 (20–22)	21 (21–21)	65 (65–65)	65 (65–65)
50	32 (31–33)	39 (38–39)	32 (31–33)	39 (38–39)	12 (12–12)	12 (12–12)	11 (11–11)	11 (11–11)	10 (10–10)	10 (10–10)	27 (27–27)	27 (27–27)	11.5 (11–12)	18 (17–18)	— 6 rev	— 6 rev	— 6 rev	— 6 rev	20 (20–21)	20.5 (20–21)	64 (64–65)	64 (64–65)
100	31 (31–33)	39 (38–39)	31 (31–33)	39 (38–39)	12 (12–12)	12 (12–12)	10.5 (10–11)	11 (11–11)	10 (10–10)	10 (10–10)	26 (26–27)	26 (26–27)	11 (11–11)	18 (17–18)	— 6 rev	— 6 rev	— 6 rev	— 6 rev	20 (20–21)	20 (20–21)	64 (64–65)	64 (64–65)
365	31 (31–32)	39 (39–40)	31 (31–32)	39 (39–40)	11 (11–11)	11 (11–12)	10 (10–10)	10 (10–11)	9 (9–9)	9 (9–9)	26 (26–26)	26 (26–26)	10.5 (10–11)	18 (17–18)	— 6 rev	— 6 rev	— 6 rev	— 6 rev	19 (19–21)	20 (20–20)	63.5 (63–64)	63.5 (63–64)
1000	30 (30–32)	38.5 (38–39)	30 (30–32)	38.5 (38–39)	11 (11–11)	11 (11–11)	9.5 (9–10)	10 (10–10)	9 (9–9)	9 (9–9)	25 (25–26)	25 (25–26)	10 (10–10)	18 (17–18)	— 6 rev	— 6 rev	— 6 rev	— 6 rev	19 (19–20)	19 (19–20)	63 (63–64)	63 (63–64)
10000	29 (29–31)	40 (39–44)	29 (29–31)	40 (39–44)	10 (10–10)	10 (10–10)	8.5 (8–9)	9 (9–9)	8 (8–8)	8 (8–8)	24 (24–25)	24 (24–25)	9 (9–9)	18 (17–18)	— 6 rev	— 6 rev	— 6 rev	— 6 rev	18 (18–19)	18 (18–19)	62 (62–63)	62 (62–63)

Median (min–max) correct digits over the inputs, capped at 80; "rev" counts inputs that reverted at that N and are excluded.

The inputs and the ops

input	parameters	why
reference reference	principal = 10000, rateSec = 0.05/31536000	$10,000 at 5% APR, the lending protocol's steady state
one-pct	principal = 10000, rateSec = 0.01/31536000	a slow rate: 1% APR is 3.2e-10 a second
twenty-pct	principal = 10000, rateSec = 0.2/31536000	20% APR
fifty-pct	principal = 10000, rateSec = 0.5/31536000	50% APR, where the three-term expansions strain
unit	principal = 1, rateSec = 0.05/31536000	one unit, the fraction carries everything
huge	principal = 100000000, rateSec = 0.05/31536000	a hundred-million-unit pool

Library calls per step: FP127 compound, mul; FP127Lib compound, mul; ABDK ln, mul, exp; Solady powWad, sMulWad; PRBMath pow, mul; Quad ln, mul, exp; Balancer pow, mulDown; Aave unsupported; Morpho unsupported; Maker rpow, rmul; FP256 pow, mul. Parameters reach each adapter floored into its representation (ladder-inputs.json).

The reference input in full: principal = 10000, rateSec = 0.05/31536000

N	truth	FP127 (deployed, staticcall)	FP127Lib (inline)	ABDKMath64x64	Solady FixedPointMathLib	PRBMath SD59x18	ABDKMathQuad (binary128)	Balancer LogExpMath	Aave v3 MathUtils (RAY)	Morpho Blue MathLib	Maker rpow (RAY)	FP256 (float, inline)
1	10001.3699568431307942024761884151451229546	10001.3699568431307942024761884151438388605 33 digits · 39 vs format · -4.36955e+8 ULP · 9390 gas/step	10001.3699568431307942024761884151438388605 33 digits · 39 vs format · -4.36955e+8 ULP · 7284 gas/step	10001.3699568430573376724354583977572019649 14 digits · 14 vs format · -1.35503e+9 ULP · 11908 gas/step	10001.369956842326 13 digits · 13 vs format · -8.04794e+8 ULP · 2517 gas/step	10001.36995683515467 12 digits · 12 vs format · -7.97612e+9 ULP · 10845 gas/step	10001.369956843130794202476188261990122001 28 digits · 28 vs format · ULP · 31502 gas/step	10001.36995684296813 13 digits · 17 vs format · -1.62664e+8 ULP · 3464 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	10001.36995684313079420207488 22 digits · 22 vs format · -4.01308e+8 ULP · 3471 gas/step	10001.3699568431307942024761884151451229546 66 digits · 66 vs format · ULP · 20209 gas/step
2	10002.7401013644367924940998199373753631246	10002.7401013644367924940998199373727946122 33 digits · 38 vs format · -8.74019e+8 ULP · 4825 gas/step	10002.7401013644367924940998199373727946122 33 digits · 38 vs format · -8.74019e+8 ULP · 3778 gas/step	10002.7401013642898598829042433600022832252 13 digits · 14 vs format · -2.71043e+9 ULP · 5961 gas/step	10002.74010136282699 12 digits · 12 vs format · -1.6098e+9 ULP · 1258 gas/step	10002.74010134848237 11 digits · 11 vs format · -1.59544e+10 ULP · 5429 gas/step	10002.740101364436792494099819631025612394 28 digits · 28 vs format · ULP · 15753 gas/step	10002.74010136411143 13 digits · 17 vs format · -3.25362e+8 ULP · 1732 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	10002.74010136443679249329706 22 digits · 22 vs format · -8.0276e+8 ULP · 1809 gas/step	10002.7401013644367924940998199373753631246 66 digits · 66 vs format · ULP · 10107 gas/step
5	10006.851661254534624362262641758190829368	10006.8516612545346243622626417581844054175 33 digits · 39 vs format · -2.18596e+9 ULP · 2100 gas/step	10006.8516612545346243622626417581844054175 33 digits · 39 vs format · -2.18596e+9 ULP · 1681 gas/step	10006.851661254167143054114030853440908686 13 digits · 13 vs format · -6.77883e+9 ULP · 2370 gas/step	10006.85166125050847 12 digits · 12 vs format · -4.02615e+9 ULP · 503 gas/step	10006.85166121463218 11 digits · 11 vs format · -3.99024e+10 ULP · 2152 gas/step	10006.8516612545346243622626409920059179602 28 digits · 28 vs format · ULP · 6278 gas/step	10006.85166125372091 13 digits · 18 vs format · -8.13714e+8 ULP · 692 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	10006.85166125453462436025495 21 digits · 21 vs format · -2.00769e+9 ULP · 803 gas/step	10006.851661254534624362262641758190829368 65 digits · 65 vs format · ULP · 4045 gas/step
10	10013.7080170352639378227825839791637869183	10013.7080170352639378227825839791509301991 32 digits · 39 vs format · -4.37491e+9 ULP · 1076 gas/step	10013.7080170352639378227825839791509301991 32 digits · 39 vs format · -4.37491e+9 ULP · 868 gas/step	10013.7080170345284718452436023561347155919 13 digits · 13 vs format · -1.3567e+10 ULP · 1185 gas/step	10013.70801702720612 12 digits · 12 vs format · -8.05782e+9 ULP · 251 gas/step	10013.70801695540438 11 digits · 11 vs format · -7.98596e+10 ULP · 1077 gas/step	10013.7080170352639378227825824457485972179 27 digits · 27 vs format · ULP · 3139 gas/step	10013.7080170336354 12 digits · 18 vs format · -1.62854e+9 ULP · 346 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	10013.70801703526393781876444 21 digits · 21 vs format · -4.01814e+9 ULP · 416 gas/step	10013.7080170352639378227825839791637869183 65 digits · 65 vs format · ULP · 2023 gas/step
20	10027.4348250436317842775218518191025424128	10027.4348250436317842775218518190767937232 32 digits · 38 vs format · -8.76183e+9 ULP · 551 gas/step	10027.4348250436317842775218518190767937232 32 digits · 38 vs format · -8.76183e+9 ULP · 447 gas/step	10027.4348250421588361327648875054308064136 12 digits · 13 vs format · -2.71711e+10 ULP · 593 gas/step	10027.43482502749406 11 digits · 11 vs format · -1.61377e+10 ULP · 125 gas/step	10027.43482488369373 10 digits · 10 vs format · -1.59938e+11 ULP · 549 gas/step	10027.4348250436317842775218487480669392567 27 digits · 27 vs format · ULP · 1569 gas/step	10027.43482504037025 12 digits · 18 vs format · -3.26153e+9 ULP · 173 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	10027.43482504363178426947455 21 digits · 21 vs format · -8.0473e+9 ULP · 215 gas/step	10027.4348250436317842775218518191025424128 65 digits · 65 vs format · ULP · 1011 gas/step
50	10068.7282526709369311954105617373559711168	10068.7282526709369311954105617372913343473 32 digits · 39 vs format · -2.19948e+10 ULP · 230 gas/step	10068.7282526709369311954105617372913343473 32 digits · 39 vs format · -2.19948e+10 ULP · 190 gas/step	10068.7282526672393980502265287135799098905 12 digits · 12 vs format · -6.82074e+10 ULP · 238 gas/step	10068.72825263042649 11 digits · 11 vs format · -4.05104e+10 ULP · 50 gas/step	10068.72825226944521 10 digits · 10 vs format · -4.01492e+11 ULP · 221 gas/step	10068.7282526709369311954105540281535213435 27 digits · 27 vs format · ULP · 632 gas/step	10068.72825266274953 12 digits · 18 vs format · -8.1874e+9 ULP · 69 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	10068.72825267093693117520939 20 digits · 20 vs format · -2.02012e+10 ULP · 92 gas/step	10068.7282526709369311954105617373559711168 64 digits · 64 vs format · ULP · 404 gas/step
100	10137.9288626133938773324977277050127626522	10137.928862613393877332497727704882600624 31 digits · 39 vs format · -4.42918e+10 ULP · 118 gas/step	10137.928862613393877332497727704882600624 31 digits · 39 vs format · -4.42918e+10 ULP · 97 gas/step	10137.9288626059479863986134695608143374557 12 digits · 12 vs format · -1.37352e+11 ULP · 119 gas/step	10137.92886253181615 11 digits · 11 vs format · -8.15777e+10 ULP · 25 gas/step	10137.92886180489168 10 digits · 10 vs format · -8.08502e+11 ULP · 110 gas/step	10137.9288626133938773324977121806418126368 26 digits · 26 vs format · ULP · 316 gas/step	10137.92886259690654 11 digits · 18 vs format · -1.64873e+10 ULP · 34 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	10137.92886261339387729181774 20 digits · 20 vs format · -4.068e+10 ULP · 47 gas/step	10137.9288626133938773324977277050127626522 64 digits · 64 vs format · ULP · 202 gas/step
365	10512.7109633435455501160300546893018100032	10512.7109633435455501160300546888091552906 31 digits · 39 vs format · -1.67642e+11 ULP · 34 gas/step	10512.7109633435455501160300546888091552906 31 digits · 39 vs format · -1.67642e+11 ULP · 28 gas/step	10512.7109633153633430060688858098671971675 11 digits · 11 vs format · -5.1987e+11 ULP · 32 gas/step	10512.71096303477923 10 digits · 10 vs format · -3.08766e+11 ULP · 6 gas/step	10512.71096028341784 9 digits · 9 vs format · -3.06013e+12 ULP · 30 gas/step	10512.7109633435455501160299959305845636861 26 digits · 26 vs format · ULP · 86 gas/step	10512.71096328114209 11 digits · 18 vs format · -6.24035e+10 ULP · 9 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	10512.71096334354554996205899 19 digits · 20 vs format · -1.53971e+11 ULP · 13 gas/step	10512.7109633435455501160300546893018100032 63 digits · 63 vs format · ULP · 55 gas/step
1000	11468.1243852882470306069512046958528107607	11468.1243852882470306069512046943804052294 30 digits · 38 vs format · -5.01034e+11 ULP · 12 gas/step	11468.1243852882470306069512046943804052294 30 digits · 38 vs format · -5.01034e+11 ULP · 10 gas/step	11468.1243852040183872067688475304692019563 11 digits · 11 vs format · -1.55374e+12 ULP · 11 gas/step	11468.12438436543188 10 digits · 10 vs format · -9.22815e+11 ULP · 2 gas/step	11468.12437614239109 9 digits · 9 vs format · -9.14586e+12 ULP · 11 gas/step	11468.1243852882470306069510290826738711704 25 digits · 25 vs format · ULP · 31 gas/step	11468.12438510174076 10 digits · 18 vs format · -1.86506e+11 ULP · 3 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	11468.12438528824703014677523 19 digits · 19 vs format · -4.60176e+11 ULP · 5 gas/step	11468.1243852882470306069512046958528107607 63 digits · 63 vs format · ULP · 20 gas/step
10000	39348.1163898331886255451916114813260386943	39348.1163898331886255451916114308065441121 29 digits · 40 vs format · -1.71909e+13 ULP · 1 gas/step	39348.1163898331886255451916114308065441121 29 digits · 40 vs format · -1.71909e+13 ULP · 1 gas/step	39348.1163869432317911623009365484904265031 10 digits · 10 vs format · -5.33103e+13 ULP · 1 gas/step	39348.11635817061095 9 digits · 9 vs format · -3.16626e+13 ULP · 0 gas/step	39348.11607603104394 8 digits · 8 vs format · -3.13802e+14 ULP · 1 gas/step	39348.1163898331886255451855860429041149103 24 digits · 24 vs format · ULP · 3 gas/step	39348.11638343399906 9 digits · 18 vs format · -6.39919e+12 ULP · 0 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	39348.11638983318860975615968 18 digits · 18 vs format · -1.5789e+13 ULP · 0 gas/step	39348.1163898331886255451916114813260386943 62 digits · 62 vs format · ULP · 2 gas/step

Continuous accrual

Step: b = b0 * e^(r * N / 365). Result: balance after N days of continuous accrual.

How Aave and Morpho accrue: the ideal is e^(rate * time), and both ship a three-term expansion of it (1 + x + x^2/2 + x^3/6), which Aave notes 'slightly underpays liquidity providers and undercharges borrowers'. Every library is judged against the exponential it approximates. The step here is one accrual spanning N days, which is what a market that nobody touched for N days computes on its next interaction; the expansion's error is x^4/24 with x = rate * years, so it is invisible at daily updates and the whole story at a year or more. Morpho holds the rate per second in WAD (1,585,489,599 wei at 5% APR), ten significant digits before any arithmetic; both digit counts show that loss, since the format truth is built from the annual rate the other libraries receive.

Over all 6 inputs

vs exact inputs (what a protocol sees) vs inputs as each format rounds them (the arithmetic alone)

N	FP127	FP127Lib	ABDK	Solady	PRBMath	Quad	Balancer	Aave	Morpho	Maker	FP256
1	38 (38–38)	38 (38–38)	38 (38–38)	38 (38–38)	19 (18–19)	19 (18–19)	18 (17–18)	18 (17–18)	17.5 (17–18)	17.5 (17–18)	33.5 (33–34)	33.5 (33–34)	18 (17–18)	18 (17–18)	16 (12–19)	16 (12–19)	13 (12–13)	13 (12–13)	— 6 rev	— 6 rev	71 (71–72)	71 (71–72)
2	38 (38–38)	38 (38–38)	38 (38–38)	38 (38–38)	19 (18–19)	19 (18–19)	18 (17–18)	18 (17–18)	17 (17–18)	17 (17–18)	33 (33–34)	33 (33–34)	18 (17–18)	18 (17–18)	15 (11–18)	15 (11–18)	12.5 (11–13)	12.5 (11–13)	— 6 rev	— 6 rev	71 (71–72)	71 (71–72)
5	39 (38–39)	39 (38–39)	39 (38–39)	39 (38–39)	19 (19–19)	19 (19–19)	18 (18–18)	18 (18–18)	17 (17–18)	17 (17–18)	34 (33–34)	34 (33–34)	18 (18–18)	18 (18–18)	14 (10–16)	14 (10–16)	12.5 (10–13)	12.5 (10–13)	— 6 rev	— 6 rev	71.5 (71–73)	71.5 (71–73)
10	38 (38–39)	38 (38–39)	38 (38–39)	38 (38–39)	18.5 (18–19)	18.5 (18–19)	18 (18–18)	18 (18–18)	17 (17–17)	17 (17–17)	33 (33–34)	33 (33–34)	18 (18–18)	18 (18–18)	12 (8–15)	12 (8–15)	12 (8–12)	12 (8–12)	— 6 rev	— 6 rev	71 (71–72)	71 (71–72)
20	38 (38–39)	38 (38–39)	38 (38–39)	38 (38–39)	19 (18–19)	19 (18–19)	18 (17–18)	18 (17–18)	17.5 (17–18)	17.5 (17–18)	34 (33–34)	34 (33–34)	18 (17–18)	18 (17–18)	11 (7–14)	11 (7–14)	11 (7–11)	11 (7–11)	— 6 rev	— 6 rev	71 (71–73)	71 (71–73)
50	38 (38–39)	38 (38–39)	38 (38–39)	38 (38–39)	18 (18–19)	18 (18–19)	18 (17–18)	18 (17–18)	17 (17–17)	17 (17–17)	33 (33–34)	33 (33–34)	18 (17–18)	18 (17–18)	10 (6–12)	10 (6–12)	10 (6–11)	10 (6–11)	— 6 rev	— 6 rev	71.5 (71–72)	71.5 (71–72)
100	38 (38–39)	38 (38–39)	38 (38–39)	38 (38–39)	19 (18–19)	19 (18–19)	18 (18–18)	18 (18–18)	17 (17–18)	17 (17–18)	33.5 (33–34)	33.5 (33–34)	18 (18–18)	18 (18–18)	8 (4–11)	8 (4–11)	8 (4–11)	8 (4–11)	— 6 rev	— 6 rev	71 (71–73)	71 (71–73)
365	38 (38–40)	39 (38–40)	38 (38–40)	39 (38–40)	19 (19–19)	19.5 (19–20)	18 (18–18)	18 (18–18)	18 (17–18)	18 (17–18)	33 (33–34)	33 (33–34)	18 (18–18)	18 (18–18)	6 (2–9)	6 (2–9)	6 (2–9)	6 (2–9)	— 6 rev	— 6 rev	71 (71–71)	71 (71–71)
1000	38 (38–38)	38 (38–38)	38 (38–38)	38 (38–38)	18 (18–19)	19 (18–19)	17.5 (17–18)	17.5 (17–18)	17 (17–17)	17 (17–17)	33 (33–34)	33 (33–34)	17.5 (17–18)	17.5 (17–18)	4 (1–7)	4 (1–7)	4 (1–7)	4 (1–7)	— 6 rev	— 6 rev	71 (71–71)	71 (71–71)
10000	37 (37–38)	38 (38–38)	37 (37–38)	38 (38–38)	17.5 (17–19)	19 (19–19)	17.5 (17–18)	17.5 (17–18)	17 (17–18)	17 (17–18)	33 (32–33)	33 (32–33)	17 (17–18)	17 (17–18)	1 (0–3)	1 (0–3)	1 (0–3)	1 (0–3)	— 6 rev	— 6 rev	71 (70–72)	71 (71–72)

Median (min–max) correct digits over the inputs, capped at 80; "rev" counts inputs that reverted at that N and are excluded.

The inputs and the ops

input	parameters	why
reference reference	principal = 10000, rateAnnual = 0.05; rateSec = rateAnnual / 31536000	$10,000 at 5% APR, the lending protocol's steady state
one-pct	principal = 10000, rateAnnual = 0.01; rateSec = rateAnnual / 31536000	a slow rate: 1% APR is 3.2e-10 a second
twenty-pct	principal = 10000, rateAnnual = 0.2; rateSec = rateAnnual / 31536000	20% APR
fifty-pct	principal = 10000, rateAnnual = 0.5; rateSec = rateAnnual / 31536000	50% APR, where the three-term expansions strain
unit	principal = 1, rateAnnual = 0.05; rateSec = rateAnnual / 31536000	one unit, the fraction carries everything
huge	principal = 100000000, rateAnnual = 0.05; rateSec = rateAnnual / 31536000	a hundred-million-unit pool

Library calls per step: FP127 mul, div, exp; FP127Lib mul, div, exp; ABDK mul, div, exp; Solady sMulWad, sDivWad, expWad; PRBMath mul, div, exp; Quad mul, div, exp; Balancer mulDown, divDown, exp; Aave calculateCompoundedInterest, rayMul; Morpho wTaylorCompounded, wMulDown; Maker unsupported; FP256 mul, div, exp. Parameters reach each adapter floored into its representation (ladder-inputs.json).

The reference input in full: principal = 10000, rateAnnual = 0.05

N	truth	FP127 (deployed, staticcall)	FP127Lib (inline)	ABDKMath64x64	Solady FixedPointMathLib	PRBMath SD59x18	ABDKMathQuad (binary128)	Balancer LogExpMath	Aave v3 MathUtils (RAY)	Morpho Blue MathLib	Maker rpow (RAY)	FP256 (float, inline)
1	10001.3699568442168947523014256696449249239	10001.3699568442168947523014256696449249036 38 digits · 38 vs format · -6905.4 ULP · 10489 gas/step	10001.3699568442168947523014256696449249036 38 digits · 38 vs format · -6905.4 ULP · 7338 gas/step	10001.3699568442168940643130503609370407503 19 digits · 19 vs format · -12691.1 ULP · 5082 gas/step	10001.36995684421689 18 digits · 18 vs format · -4752.3 ULP · 2062 gas/step	10001.36995684421689 18 digits · 18 vs format · -4752.3 ULP · 5114 gas/step	10001.3699568442168947523014256696434184119 33 digits · 33 vs format · ULP · 11226 gas/step	10001.36995684421689 18 digits · 18 vs format · -4752.3 ULP · 3453 gas/step	10001.369956844216748025579 16 digits · 16 vs format · -1.46727e+14 ULP · 1608 gas/step	10001.36995684405409 13 digits · 13 vs format · -1.62805e+8 ULP · 1442 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	10001.3699568442168947523014256696449249239 71 digits · 71 vs format · ULP · 13368 gas/step
2	10002.740101366609291175926524648192624907	10002.7401013666092911759265246481926248923 38 digits · 38 vs format · -5007.45 ULP · 5244 gas/step	10002.7401013666092911759265246481926248923 38 digits · 38 vs format · -5007.45 ULP · 3669 gas/step	10002.7401013666092903378959655391611249797 19 digits · 19 vs format · -15458.9 ULP · 2541 gas/step	10002.74010136660929 18 digits · 18 vs format · -1175.93 ULP · 1031 gas/step	10002.74010136660928 17 digits · 17 vs format · -11175.9 ULP · 2564 gas/step	10002.7401013666092911759265246481906849063 33 digits · 33 vs format · ULP · 5615 gas/step	10002.74010136660929 18 digits · 18 vs format · -1175.93 ULP · 1726 gas/step	10002.74010136660694348404654 15 digits · 15 vs format · -2.34769e+15 ULP · 804 gas/step	10002.74010136628159 13 digits · 13 vs format · -3.27701e+8 ULP · 721 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	10002.740101366609291175926524648192624907 72 digits · 72 vs format · ULP · 6687 gas/step
5	10006.8516612599681035447190899643916635711	10006.8516612599681035447190899643916635726 39 digits · 39 vs format · 491.306 ULP · 2097 gas/step	10006.8516612599681035447190899643916635726 39 digits · 39 vs format · 491.306 ULP · 1467 gas/step	10006.8516612599681028100695456828361784574 19 digits · 19 vs format · -13551.9 ULP · 1010 gas/step	10006.8516612599681 18 digits · 18 vs format · -3544.72 ULP · 412 gas/step	10006.85166125996809 17 digits · 17 vs format · -13544.7 ULP · 1025 gas/step	10006.8516612599681035447190899643908704196 34 digits · 34 vs format · ULP · 2234 gas/step	10006.8516612599681 18 digits · 18 vs format · -3544.72 ULP · 690 gas/step	10006.85166125987638929232741 14 digits · 14 vs format · -9.17143e+16 ULP · 321 gas/step	10006.85166125906267 13 digits · 13 vs format · -9.05434e+8 ULP · 288 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	10006.8516612599681035447190899643916635711 73 digits · 73 vs format · ULP · 2677 gas/step
10	10013.7080170461383418594567841237800085191	10013.7080170461383418594567841237800084967 38 digits · 38 vs format · -7629.65 ULP · 1048 gas/step	10013.7080170461383418594567841237800084967 38 digits · 38 vs format · -7629.65 ULP · 733 gas/step	10013.7080170461383407274832957334353977785 18 digits · 18 vs format · -20881.2 ULP · 506 gas/step	10013.70801704613834 18 digits · 18 vs format · -1859.46 ULP · 206 gas/step	10013.70801704613833 17 digits · 17 vs format · -11859.5 ULP · 515 gas/step	10013.7080170461383418594567841237776326456 33 digits · 33 vs format · ULP · 1117 gas/step	10013.70801704613834 18 digits · 18 vs format · -1859.46 ULP · 345 gas/step	10013.70801704467071276233857 12 digits · 12 vs format · -1.46763e+18 ULP · 160 gas/step	10013.70801704304217 12 digits · 12 vs format · -3.09617e+9 ULP · 144 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	10013.7080170461383418594567841237800085191 72 digits · 72 vs format · ULP · 1339 gas/step
20	10027.4348250654104056540386919480208485404	10027.4348250654104056540386919480208485182 38 digits · 38 vs format · -7542.53 ULP · 524 gas/step	10027.4348250654104056540386919480208485182 38 digits · 38 vs format · -7542.53 ULP · 366 gas/step	10027.4348250654104049888262761847101955937 19 digits · 19 vs format · -12271.0 ULP · 253 gas/step	10027.4348250654104 18 digits · 18 vs format · -5654.04 ULP · 103 gas/step	10027.4348250654104 18 digits · 18 vs format · -5654.04 ULP · 267 gas/step	10027.4348250654104056540386919480198713688 34 digits · 34 vs format · ULP · 558 gas/step	10027.4348250654104 18 digits · 18 vs format · -5654.04 ULP · 172 gas/step	10027.43482504192190401276379 11 digits · 11 vs format · -2.34885e+19 ULP · 80 gas/step	10027.43482503866036 11 digits · 11 vs format · -2.675e+10 ULP · 72 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	10027.4348250654104056540386919480208485404 71 digits · 71 vs format · ULP · 669 gas/step
50	10068.7282527256076979927807360904191194169	10068.7282527256076979927807360904191193953 38 digits · 38 vs format · -7337.64 ULP · 209 gas/step	10068.7282527256076979927807360904191193953 38 digits · 38 vs format · -7337.64 ULP · 146 gas/step	10068.7282527256076968512343849937451523147 18 digits · 18 vs format · -21057.8 ULP · 103 gas/step	10068.72825272560769 18 digits · 18 vs format · -7992.78 ULP · 41 gas/step	10068.72825272560768 17 digits · 17 vs format · -17992.8 ULP · 108 gas/step	10068.728252725607697992780736090417657742 33 digits · 33 vs format · ULP · 223 gas/step	10068.72825272560769 18 digits · 18 vs format · -7992.78 ULP · 69 gas/step	10068.72825180733318423959192 10 digits · 10 vs format · -9.18275e+20 ULP · 32 gas/step	10068.72825179914578 10 digits · 10 vs format · -9.26462e+11 ULP · 28 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	10068.7282527256076979927807360904191194169 72 digits · 72 vs format · ULP · 267 gas/step
100	10137.9288627234868961823696710416300525454	10137.9288627234868961823696710416300525214 38 digits · 38 vs format · -8175.45 ULP · 104 gas/step	10137.9288627234868961823696710416300525214 38 digits · 38 vs format · -8175.45 ULP · 73 gas/step	10137.9288627234868952345717141305669883877 19 digits · 19 vs format · -17483.8 ULP · 50 gas/step	10137.92886272348689 18 digits · 18 vs format · -6182.37 ULP · 20 gas/step	10137.92886272348688 17 digits · 17 vs format · -16182.4 ULP · 54 gas/step	10137.9288627234868961823696710416295789426 34 digits · 34 vs format · ULP · 111 gas/step	10137.92886272348689 18 digits · 18 vs format · -6182.37 ULP · 34 gas/step	10137.92884801092668615167631 8 digits · 8 vs format · -1.47126e+22 ULP · 16 gas/step	10137.92884799443935 8 digits · 8 vs format · -1.4729e+13 ULP · 14 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	10137.9288627234868961823696710416300525454 71 digits · 71 vs format · ULP · 134 gas/step
365	10512.7109637602403969751763633564522017482	10512.7109637602403969751763633564522017166 38 digits · 39 vs format · -10749.2 ULP · 28 gas/step	10512.7109637602403969751763633564522017166 38 digits · 39 vs format · -10749.2 ULP · 20 gas/step	10512.7109637602403964962843874175746350375 19 digits · 20 vs format · -8834.0 ULP · 14 gas/step	10512.71096376024039 18 digits · 18 vs format · -6975.18 ULP · 5 gas/step	10512.71096376024039 18 digits · 18 vs format · -6975.18 ULP · 15 gas/step	10512.7109637602403969751763633564498826411 33 digits · 33 vs format · ULP · 30 gas/step	10512.71096376024039 18 digits · 18 vs format · -6975.18 ULP · 9 gas/step	10512.70833333333333333333333 6 digits · 6 vs format · -2.63043e+24 ULP · 4 gas/step	10512.70833327093113 6 digits · 6 vs format · -2.63049e+15 ULP · 3 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	10512.7109637602403969751763633564522017482 71 digits · 71 vs format · ULP · 36 gas/step
1000	11468.1243865336300386096785282459267168535	11468.1243865336300386096785282459267167561 38 digits · 38 vs format · -33155.7 ULP · 10 gas/step	11468.1243865336300386096785282459267167561 38 digits · 38 vs format · -33155.7 ULP · 7 gas/step	11468.1243865336300369371994034572992404719 18 digits · 19 vs format · -30851.8 ULP · 5 gas/step	11468.12438653363003 18 digits · 18 vs format · -8609.68 ULP · 2 gas/step	11468.12438653363002 17 digits · 17 vs format · -18609.7 ULP · 5 gas/step	11468.1243865336300386096785282459240321407 33 digits · 33 vs format · ULP · 11 gas/step	11468.12438653363003 18 digits · 18 vs format · -8609.68 ULP · 3 gas/step	11467.97355042753058778065396 4 digits · 4 vs format · -1.50836e+26 ULP · 1 gas/step	11467.97355024109644 4 digits · 4 vs format · -1.50836e+17 ULP · 1 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	11468.1243865336300386096785282459267168535 71 digits · 71 vs format · ULP · 13 gas/step
10000	39348.1164325633456472307592365680379737982	39348.1164325633456472307592365680379712093 37 digits · 38 vs format · -880962.0 ULP · 1 gas/step	39348.1164325633456472307592365680379712093 37 digits · 38 vs format · -880962.0 ULP · 0 gas/step	39348.116432563345597011282561705769467153 17 digits · 19 vs format · -926386.0 ULP · 0 gas/step	39348.1164325633456 17 digits · 17 vs format · -47230.8 ULP · 0 gas/step	39348.11643256334557 17 digits · 17 vs format · -77230.8 ULP · 0 gas/step	39348.1164325633456472307592365680184930427 33 digits · 33 vs format · ULP · 1 gas/step	39348.1164325633456 17 digits · 17 vs format · -47230.8 ULP · 0 gas/step	37365.55643240955193903265578 1 digits · 1 vs format · -1.98256e+30 ULP · 0 gas/step	37365.55642702954313 1 digits · 1 vs format · -1.98256e+21 ULP · 0 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	39348.1164325633456472307592365680379737982 72 digits · 71 vs format · ULP · 1 gas/step

An option walking out of the money

Step: S = S (1 - tick); then C = S N(d1) - K e^(-rT) N(d2). Result: the Black-Scholes call price after N down-ticks of the spot.

Options protocols price with N(d), the normal distribution function, and the spot walks. Near the money every library agrees; past d = -8 an 18-decimal N(d) is zero and the price with it, and before that e^(-d^2/2) has lost most of its digits. FP127 has normCdf as an op; the other libraries run the same series and continued fraction through their own exp, sqrt, mul and div, with pi by Ramanujan through the same ops, so each is judged on its primitives.

Over all 6 inputs

vs exact inputs (what a protocol sees) vs inputs as each format rounds them (the arithmetic alone)

N	FP127	FP127Lib	ABDK	Solady	PRBMath	Quad	Balancer	Aave	Morpho	Maker	FP256
1	37 (36–38)	37 (36–38)	37 (36–38)	37 (36–38)	18 (17–18)	18 (17–18)	16 (16–16)	16 (16–16)	16 (16–17)	16 (16–17)	32 (31–32)	32 (31–32)	16 (15–16)	16 (15–16)	— 6 rev	— 6 rev	— 6 rev	— 6 rev	— 6 rev	— 6 rev	70 (69–71)	70 (69–71)
2	37 (36–38)	38 (36–38)	37 (36–38)	38 (36–38)	17.5 (17–18)	17.5 (17–18)	16 (16–17)	16 (16–17)	16 (16–16)	16 (16–16)	32 (31–32)	32 (31–32)	16 (16–17)	16 (16–17)	— 6 rev	— 6 rev	— 6 rev	— 6 rev	— 6 rev	— 6 rev	70 (69–71)	70 (69–71)
5	37 (36–38)	37.5 (37–39)	37 (36–38)	37.5 (37–39)	17.5 (17–18)	17.5 (17–18)	16 (16–17)	16 (16–17)	16 (16–16)	16 (16–16)	32 (31–32)	32 (31–32)	16 (15–17)	16 (15–17)	— 6 rev	— 6 rev	— 6 rev	— 6 rev	— 6 rev	— 6 rev	70 (68–70)	70 (68–71)
10	36 (36–38)	37.5 (37–38)	36 (36–38)	37.5 (37–38)	17 (17–18)	18 (16–18)	16 (15–17)	16 (15–17)	16 (15–16)	16 (15–16)	31 (31–32)	31 (31–32)	16 (15–17)	16 (15–17)	— 6 rev	— 6 rev	— 6 rev	— 6 rev	— 6 rev	— 6 rev	70 (68–71)	70 (68–71)
20	36 (35–38)	37.5 (37–38)	36 (35–38)	37.5 (37–38)	17 (16–17)	17.5 (16–18)	16 (15–16)	16 (15–16)	16 (16–16)	16 (16–16)	31 (30–32)	31 (30–32)	16 (15–16)	16 (15–16)	— 6 rev	— 6 rev	— 6 rev	— 6 rev	— 6 rev	— 6 rev	69 (68–71)	69 (68–71)
50	36 (35–38)	37 (37–39)	36 (35–38)	37 (37–39)	16.5 (16–18)	17.5 (17–18)	16 (16–16)	16 (16–16)	16 (16–17)	16 (16–17)	31 (30–31)	31 (30–31)	16 (15–16)	16 (15–16)	— 6 rev	— 6 rev	— 6 rev	— 6 rev	— 6 rev	— 6 rev	69 (68–70)	69 (68–70)
100	35 (35–38)	37 (36–38)	35 (35–38)	37 (36–38)	16 (15–19)	17 (17–18)	16 (16–16)	16 (16–16)	16 (16–19)	16 (16–19)	30 (30–31)	30 (30–31)	16 (15–16)	16 (15–16)	— 6 rev	— 6 rev	— 6 rev	— 6 rev	— 6 rev	— 6 rev	68 (67–69)	68 (67–69)
365	34.5 (34–37)	37 (36–37)	34.5 (34–37)	37 (36–37)	15.5 (15–17)	17 (16–17)	16 (15–16)	16 (15–16)	16 (15–16)	16 (15–16)	30 (29–30)	30 (29–30)	15.5 (15–16)	15.5 (15–16)	— 6 rev	— 6 rev	— 6 rev	— 6 rev	— 6 rev	— 6 rev	68 (67–68)	68 (67–68)
1000	34 (33–36)	35.5 (34–37)	34 (33–36)	35.5 (34–37)	14.5 (12–17)	16 (12–17)	15 (11–15)	15 (11–15)	15 (10–15)	15 (10–15)	29 (27–30)	29 (27–30)	15 (10–16)	15 (10–16)	— 6 rev	— 6 rev	— 6 rev	— 6 rev	— 6 rev	— 6 rev	67 (66–68)	67 (66–68)
10000	0.5 (0–25)	0.5 (0–25)	0.5 (0–25)	0.5 (0–25)	0 (0–6)	0 (0–6)	0 (0–4)	0 (0–4)	0 (0–4)	0 (0–4)	27 (26–28)	27 (26–28)	4 (4–4) 5 rev	4 (4–4) 5 rev	— 6 rev	— 6 rev	— 6 rev	— 6 rev	— 6 rev	— 6 rev	65 (53–66)	65 (53–66)

Median (min–max) correct digits over the inputs, capped at 80; "rev" counts inputs that reverted at that N and are excluded.

The inputs and the ops

input	parameters	why
reference reference	T = 0.5, r = 0.05, sigma = 0.2, spot = 100, strike = 100, tick = 0.000173	at the money, 20% vol, six months; d1 reaches -12 at N = 10,000
slow-walk	T = 0.5, r = 0.05, sigma = 0.2, spot = 100, strike = 100, tick = 0.0001	a slower walk: d1 reaches -7 at N = 10,000, where an 18-decimal N(d) has a few digits left
fast-walk	T = 0.5, r = 0.05, sigma = 0.2, spot = 100, strike = 100, tick = 0.0005	a faster walk: d1 passes -12 near N = 3,500; beyond it the price is below every grid
low-vol	T = 0.5, r = 0.05, sigma = 0.05, spot = 100, strike = 100, tick = 0.00005	5% vol: d moves four times faster per tick
high-vol	T = 0.5, r = 0.05, sigma = 0.8, spot = 100, strike = 100, tick = 0.0007	80% vol: a long way to the tail
short-dated	T = 0.01, r = 0.05, sigma = 0.2, spot = 100, strike = 100, tick = 0.00003	ten days to expiry: sigma sqrt(T) is 0.02 and the tail comes fast

Library calls per step: FP127 mul, sqrt, ln, div, add, exp, sub, normCdf; FP127Lib mul, sqrt, ln, div, add, exp, sub, normCdf; ABDK mul, sqrt, ln, div, add, exp, sub, normCdf by the series/fraction through these; Solady sMulWad, sqrtWad, lnWad, sDivWad, expWad, normCdf by the series/fraction through these; PRBMath mul, sqrt, ln, div, exp, normCdf by the series/fraction through these; Quad mul, sqrt, ln, div, add, exp, sub, normCdf by the series/fraction through these; Balancer mulDown, pow(x, 1/2), ln, divDown, exp, normCdf by the series/fraction through these; Aave unsupported; Morpho unsupported; Maker unsupported; FP256 mul, sqrt, ln, div, add, exp, sub, normCdf by the series/fraction through these. Parameters reach each adapter floored into its representation (ladder-inputs.json).

The reference input in full: T = 0.5, r = 0.05, sigma = 0.2, spot = 100, strike = 100, tick = 0.000173

N	truth	FP127 (deployed, staticcall)	FP127Lib (inline)	ABDKMath64x64	Solady FixedPointMathLib	PRBMath SD59x18	ABDKMathQuad (binary128)	Balancer LogExpMath	Aave v3 MathUtils (RAY)	Morpho Blue MathLib	Maker rpow (RAY)	FP256 (float, inline)
1	6.87839186610381111236482092683425007076306	6.87839186610381111236482092683425007066222 37 digits · 37 vs format · -34.313 ULP · 53688 gas/step	6.87839186610381111236482092683425007066222 37 digits · 37 vs format · -34.313 ULP · 44487 gas/step	6.8783918661038111084723989385736331314547 18 digits · 18 vs format · -71.8025 ULP · 103328 gas/step	6.878391866103810997 16 digits · 16 vs format · -115.365 ULP · 77511 gas/step	6.878391866103810933 16 digits · 16 vs format · -179.365 ULP · 142047 gas/step	6.87839186610381111236482092683423446296251 32 digits · 32 vs format · ULP · 345298 gas/step	6.878391866103810948 16 digits · 16 vs format · -164.365 ULP · 126930 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	6.87839186610381111236482092683425007076306 70 digits · 70 vs format · ULP · 556086 gas/step
2	6.86806513271493210766538832843009014451141	6.86806513271493210766538832843009014466145 37 digits · 38 vs format · 51.0579 ULP · 27655 gas/step	6.86806513271493210766538832843009014466145 37 digits · 38 vs format · 51.0579 ULP · 22422 gas/step	6.86806513271493210157298339724185609611595 18 digits · 17 vs format · -112.385 ULP · 51807 gas/step	6.868065132714932044 17 digits · 17 vs format · -63.6654 ULP · 38872 gas/step	6.868065132714931973 16 digits · 16 vs format · -134.665 ULP · 71318 gas/step	6.86806513271493210766538832843007004985001 32 digits · 32 vs format · ULP · 173114 gas/step	6.868065132714932095 17 digits · 17 vs format · -12.6654 ULP · 63717 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	6.86806513271493210766538832843009014451141 70 digits · 70 vs format · ULP · 278662 gas/step
5	6.83714479479769759938000331844859304685863	6.83714479479769759938000331844859304728414 37 digits · 39 vs format · 144.794 ULP · 11988 gas/step	6.83714479479769759938000331844859304728414 37 digits · 39 vs format · 144.794 ULP · 9132 gas/step	6.83714479479769760228652711320229684588412 18 digits · 18 vs format · 53.6159 ULP · 20874 gas/step	6.837144794797697437 16 digits · 16 vs format · -162.38 ULP · 15688 gas/step	6.83714479479769745 16 digits · 16 vs format · -149.38 ULP · 28891 gas/step	6.83714479479769759938000331844854958785622 32 digits · 32 vs format · ULP · 69826 gas/step	6.837144794797697487 16 digits · 16 vs format · -112.38 ULP · 25789 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	6.83714479479769759938000331844859304685863 70 digits · 70 vs format · ULP · 112206 gas/step
10	6.78581039602925175035404532612573043320465	6.78581039602925175035404532612573043420648 36 digits · 38 vs format · 340.904 ULP · 6805 gas/step	6.78581039602925175035404532612573043420648 36 digits · 38 vs format · 340.904 ULP · 4745 gas/step	6.78581039602925175654264330904474888939149 18 digits · 18 vs format · 114.159 ULP · 10586 gas/step	6.785810396029251611 16 digits · 16 vs format · -139.354 ULP · 7960 gas/step	6.785810396029251528 16 digits · 16 vs format · -222.354 ULP · 14740 gas/step	6.78581039602925175035404532612564075836968 31 digits · 31 vs format · ULP · 35008 gas/step	6.785810396029251659 16 digits · 16 vs format · -91.354 ULP · 13146 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	6.78581039602925175035404532612573043320465 69 digits · 69 vs format · ULP · 56718 gas/step
20	6.68388936636074529143221680202703623201243	6.68388936636074529143221680202703623396278 36 digits · 38 vs format · 663.67 ULP · 4213 gas/step	6.68388936636074529143221680202703623396278 36 digits · 38 vs format · 663.67 ULP · 2551 gas/step	6.6838893663607453099907013127634058946569 17 digits · 17 vs format · 342.344 ULP · 5433 gas/step	6.683889366360745126 16 digits · 16 vs format · -165.432 ULP · 4052 gas/step	6.68388936636074514 16 digits · 16 vs format · -151.432 ULP · 7586 gas/step	6.68388936636074529143221680202685090471822 31 digits · 31 vs format · ULP · 17772 gas/step	6.683889366360745176 16 digits · 16 vs format · -115.432 ULP · 6752 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	6.68388936636074529143221680202703623201243 71 digits · 71 vs format · ULP · 28452 gas/step
50	6.38410014515129075799369654827204817146431	6.38410014515129075799369654827204817621643 36 digits · 38 vs format · 1617.06 ULP · 2641 gas/step	6.38410014515129075799369654827204817621643 36 digits · 38 vs format · 1617.06 ULP · 1216 gas/step	6.38410014515129082038856489500489033162012 17 digits · 18 vs format · 1150.98 ULP · 2323 gas/step	6.384100145151290678 16 digits · 16 vs format · -79.9937 ULP · 1743 gas/step	6.384100145151290676 16 digits · 16 vs format · -81.9937 ULP · 3352 gas/step	6.38410014515129075799369654827160797782368 31 digits · 31 vs format · ULP · 7628 gas/step	6.38410014515129063 16 digits · 16 vs format · -127.994 ULP · 2974 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	6.38410014515129075799369654827204817146431 70 digits · 70 vs format · ULP · 11785 gas/step
100	5.90429198156954564974290965066758972962968	5.90429198156954564974290965066758973868736 35 digits · 37 vs format · 3082.17 ULP · 2116 gas/step	5.90429198156954564974290965066758973868736 35 digits · 37 vs format · 3082.17 ULP · 769 gas/step	5.90429198156954576133135723092770774655946 16 digits · 17 vs format · 2058.44 ULP · 1283 gas/step	5.904291981569545436 16 digits · 16 vs format · -213.743 ULP · 971 gas/step	5.904291981569545509 16 digits · 16 vs format · -140.743 ULP · 1941 gas/step	5.90429198156954564974290965066675510723534 30 digits · 30 vs format · ULP · 4131 gas/step	5.904291981569545387 16 digits · 16 vs format · -262.743 ULP · 1711 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	5.90429198156954564974290965066758972962968 69 digits · 69 vs format · ULP · 6188 gas/step
365	3.76450175108813956006578640941955904105981	3.76450175108813956006578640941955906555828 35 digits · 37 vs format · 8336.4 ULP · 1767 gas/step	3.76450175108813956006578640941955906555828 35 digits · 37 vs format · 8336.4 ULP · 481 gas/step	3.76450175108813987908361331502504754098481 16 digits · 17 vs format · 5884.84 ULP · 577 gas/step	3.764501751088139439 16 digits · 16 vs format · -121.066 ULP · 450 gas/step	3.764501751088139409 16 digits · 16 vs format · -151.066 ULP · 990 gas/step	3.76450175108813956006578640941740283703066 30 digits · 30 vs format · ULP · 1938 gas/step	3.764501751088139392 16 digits · 16 vs format · -168.066 ULP · 859 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	3.76450175108813956006578640941955904105981 68 digits · 68 vs format · ULP · 2853 gas/step
1000	0.974187219763713465710122175526671930776915	0.974187219763713465710122175526671954175089 34 digits · 36 vs format · 7961.99 ULP · 1683 gas/step	0.974187219763713465710122175526671954175089 34 digits · 36 vs format · 7961.99 ULP · 410 gas/step	0.974187219763713765485063644788965575571638 15 digits · 16 vs format · 5529.87 ULP · 409 gas/step	0.974187219763713273 15 digits · 15 vs format · -192.71 ULP · 325 gas/step	0.974187219763713293 15 digits · 15 vs format · -172.71 ULP · 763 gas/step	0.974187219763713465710122175524596531068713 29 digits · 29 vs format · ULP · 1410 gas/step	0.974187219763713135 15 digits · 15 vs format · -330.71 ULP · 655 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	0.974187219763713465710122175526671930776915 68 digits · 68 vs format · ULP · 2018 gas/step
10000	4.26395835385621571476498764907019748044021e-34	4.26325352420350177671331445478421006817079e-34 3 digits · 3 vs format · -23.9841 ULP · 1648 gas/step	4.26325352420350177671331445478421006817079e-34 3 digits · 3 vs format · -23.9841 ULP · 364 gas/step	0.0 0 digits · 0 vs format · -7.86561e-15 ULP · 317 gas/step	0.0 0 digits · 0 vs format · -4.26396e-16 ULP · 256 gas/step	0.0 0 digits · 0 vs format · -4.26396e-16 ULP · 640 gas/step	4.26395835385621571476498764850805535968142e-34 27 digits · 27 vs format · ULP · 1087 gas/step	reverts Balancer InvalidExponent() 0xd4794efd	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	4.26395835385621571476498764907019748044021e-34 66 digits · 66 vs format · ULP · 1407 gas/step

Decay, the share released

Step: D = D e^(-eps); result 1 - D. Result: the share released after N steps of decay.

Vesting, TWAP weights and exponential moving averages multiply by e^(-dt/tau) every step. The factor is a hair below one, and on an 18-decimal grid e^(-eps) rounds to a grid point whose distance from one is wrong by up to 1e-18 relative, every step: at eps = 1e-12 that is six digits of the released share, at 1e-15 three, at 1e-18 nothing is released at all.

Over all 5 inputs

vs exact inputs (what a protocol sees) vs inputs as each format rounds them (the arithmetic alone)

N	FP127	FP127Lib	ABDK	Solady	PRBMath	Quad	Balancer	Aave	Morpho	Maker	FP256
1	26 (21–32)	26 (22–33)	26 (21–32)	26 (22–33)	7 (2–13)	7 (2–13)	12 (9–17)	12 (9–17)	6 (1–12)	6 (1–12)	22 (17–28)	22 (17–28)	12 (9–17)	12 (9–17)	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	59 (54–66)	59 (54–66)
2	26 (21–32)	26 (22–33)	26 (21–32)	26 (22–33)	7 (2–13)	7 (2–13)	12 (9–16)	12 (9–16)	6 (1–18)	6 (1–18)	22 (17–28)	22 (17–28)	12 (9–16)	12 (9–16)	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	59 (54–66)	59 (54–66)
5	26 (22–33)	27 (21–34)	26 (22–33)	27 (21–34)	7 (2–13)	7 (2–13)	12 (8–16)	12 (8–16)	6 (1–12)	6 (1–12)	22 (17–28)	22 (17–28)	12 (8–16)	12 (8–16)	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	59 (54–66)	59 (54–66)
10	26 (22–33)	27 (21–34)	26 (22–33)	27 (21–34)	7 (2–13)	7 (2–13)	12 (8–16)	12 (8–16)	6 (1–12)	6 (1–12)	22 (16–28)	22 (16–28)	12 (8–16)	12 (8–16)	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	59 (54–66)	59 (54–66)
20	26 (22–33)	27 (21–33)	26 (22–33)	27 (21–33)	7 (2–13)	7 (2–13)	12 (8–15)	12 (8–15)	6 (1–12)	6 (1–12)	22 (16–28)	22 (16–28)	12 (8–15)	12 (8–15)	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	59 (54–66)	59 (54–66)
50	26 (22–33)	28 (21–33)	26 (22–33)	28 (21–33)	7 (2–13)	7 (2–13)	12 (8–15)	12 (8–15)	6 (1–12)	6 (1–12)	22 (16–28)	22 (16–28)	12 (8–15)	12 (8–15)	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	59 (54–66)	59 (54–66)
100	26 (22–33)	28 (21–33)	26 (22–33)	28 (21–33)	7 (2–13)	7 (2–13)	12 (8–15)	12 (8–15)	6 (1–12)	6 (1–12)	22 (16–28)	22 (16–28)	12 (8–15)	12 (8–15)	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	59 (54–66)	59 (54–66)
365	26 (22–33)	29 (21–33)	26 (22–33)	29 (21–33)	7 (2–13)	7 (2–13)	11 (8–14)	11 (8–14)	6 (1–12)	6 (1–12)	22 (16–28)	22 (16–28)	11 (8–14)	11 (8–14)	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	59 (54–66)	59 (54–66)
1000	26 (22–33)	28 (21–33)	26 (22–33)	28 (21–33)	7 (2–13)	7 (2–13)	11 (8–14)	11 (8–14)	6 (1–12)	6 (1–12)	22 (16–28)	22 (16–28)	11 (8–14)	11 (8–14)	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	59 (54–66)	59 (54–66)
10000	26 (22–33)	28 (21–33)	26 (22–33)	28 (21–33)	7 (2–13)	7 (2–13)	11 (8–13)	11 (8–13)	6 (1–12)	6 (1–12)	21 (16–28)	21 (16–28)	11 (8–13)	11 (8–13)	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	— 5 rev	59 (54–66)	59 (54–66)

Median (min–max) correct digits over the inputs, capped at 80; "rev" counts inputs that reverted at that N and are excluded.

The inputs and the ops

input	parameters	why
reference reference	eps = 0.000001	one millionth a step: a one-second step against a twelve-day time constant
nano	eps = 0.000000001	1e-9 a step: a one-second step against a thirty-year time constant
pico	eps = 0.000000000001	1e-12 a step: the factor is within WAD's last six digits of one
femto	eps = 0.000000000000001	1e-15 a step: three digits of the factor's distance from one survive on WAD
atto	eps = 0.00000000000000001	1e-17 a step: below WAD's grid, e^(-eps) is one and nothing decays

Library calls per step: FP127 exp, mul, sub; FP127Lib exp, mul, sub; ABDK exp, mul, sub; Solady expWad, sMulWad; PRBMath exp, mul; Quad exp, mul, sub; Balancer exp, mulDown; Aave unsupported; Morpho unsupported; Maker unsupported; FP256 exp, mul, sub. Parameters reach each adapter floored into its representation (ladder-inputs.json).

The reference input in full: eps = 0.000001

N	truth	FP127 (deployed, staticcall)	FP127Lib (inline)	ABDKMath64x64	Solady FixedPointMathLib	PRBMath SD59x18	ABDKMathQuad (binary128)	Balancer LogExpMath	Aave v3 MathUtils (RAY)	Morpho Blue MathLib	Maker rpow (RAY)	FP256 (float, inline)
1	0.000000999999500000166666625000008333331944444643	0.000000999999500000166666625000008333330356286197 32 digits · 33 vs format · -0.540422 ULP · 7978 gas/step	0.000000999999500000166666625000008333330356286197 32 digits · 33 vs format · -0.540422 ULP · 6642 gas/step	0.000000999999500000188935172618087676710274536163 13 digits · 13 vs format · 0.410782 ULP · 4981 gas/step	0.000000999999500001 12 digits · 12 vs format · 0.833333 ULP · 1824 gas/step	0.0000009999995 12 digits · 12 vs format · -0.166667 ULP · 4038 gas/step	0.000000999999500000166666625000008351268400079745 28 digits · 28 vs format · ULP · 11056 gas/step	0.000000999999500001 12 digits · 12 vs format · 0.833333 ULP · 2685 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	0.000000999999500000166666625000008333331944444643 66 digits · 66 vs format · ULP · 11151 gas/step
2	0.00000199999800000133333266666693333324444446984	0.00000199999800000133333266666693333324146219963 32 digits · 33 vs format · -1.01481 ULP · 7570 gas/step	0.00000199999800000133333266666693333324146219963 32 digits · 33 vs format · -1.01481 ULP · 6225 gas/step	0.00000199999800000141185821189715454693214269355 13 digits · 13 vs format · 1.44854 ULP · 4529 gas/step	0.000001999998000003 12 digits · 12 vs format · 1.66667 ULP · 1424 gas/step	0.000001999998000001 12 digits · 12 vs format · -0.333333 ULP · 3638 gas/step	0.0000019999980000013333326666669334537325718948 28 digits · 28 vs format · ULP · 9651 gas/step	0.000001999998000003 12 digits · 12 vs format · 1.66667 ULP · 2285 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	0.00000199999800000133333266666693333324444446984 66 digits · 66 vs format · ULP · 10163 gas/step
5	0.00000499998750002083330729169270831163195994543	0.00000499998750002083330729169270831162916850658 33 digits · 34 vs format · -0.949877 ULP · 7326 gas/step	0.00000499998750002083330729169270831162916850658 33 digits · 34 vs format · -0.949877 ULP · 5975 gas/step	0.00000499998750002109658419069404544643475674093 13 digits · 13 vs format · 4.8566 ULP · 4258 gas/step	0.000004999987500025 12 digits · 12 vs format · 4.16669 ULP · 1184 gas/step	0.00000499998750002 12 digits · 12 vs format · -0.833307 ULP · 3398 gas/step	0.00000499998750002083330729169270858171956675914 28 digits · 28 vs format · ULP · 8801 gas/step	0.000004999987500025 12 digits · 12 vs format · 4.16669 ULP · 2045 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	0.00000499998750002083330729169270831163195994543 66 digits · 66 vs format · ULP · 9565 gas/step
10	0.00000999995000016666625000083333194444642856895	0.00000999995000016666625000083333194443983689778 33 digits · 34 vs format · -2.24303 ULP · 7244 gas/step	0.00000999995000016666625000083333194443983689778 33 digits · 34 vs format · -2.24303 ULP · 5892 gas/step	0.00000999995000016713894361308767599894054001197 13 digits · 13 vs format · 8.71966 ULP · 4168 gas/step	0.000009999950000175 12 digits · 12 vs format · 8.33375 ULP · 1104 gas/step	0.000009999950000165 12 digits · 12 vs format · -1.66625 ULP · 3318 gas/step	0.00000999995000016666625000083333252890150559837 28 digits · 28 vs format · ULP · 8511 gas/step	0.000009999950000175 12 digits · 12 vs format · 8.33375 ULP · 1965 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	0.00000999995000016666625000083333194444642856895 66 digits · 66 vs format · ULP · 9367 gas/step
20	0.0000199998000013333266666933332444446984120635	0.000019999800001333326666693333244444693890718 33 digits · 33 vs format · -1.53853 ULP · 7203 gas/step	0.000019999800001333326666693333244444693890718 33 digits · 33 vs format · -1.53853 ULP · 5850 gas/step	0.0000199998000013342576604291611630515035358258 13 digits · 13 vs format · 17.1738 ULP · 4123 gas/step	0.00001999980000135 12 digits · 12 vs format · 16.6733 ULP · 1064 gas/step	0.00001999980000133 12 digits · 12 vs format · -3.32667 ULP · 3278 gas/step	0.0000199998000013333266666933332455969252083375 28 digits · 28 vs format · ULP · 8372 gas/step	0.00001999980000135 12 digits · 12 vs format · 16.6733 ULP · 1925 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	0.0000199998000013333266666933332444446984120635 66 digits · 66 vs format · ULP · 9269 gas/step
50	0.0000499987500208330729192708116320994533962727	0.0000499987500208330729192708116320994423985862 33 digits · 33 vs format · -3.74232 ULP · 7179 gas/step	0.0000499987500208330729192708116320994423985862 33 digits · 33 vs format · -3.74232 ULP · 5825 gas/step	0.000049998750020835685398123970912820368539542 13 digits · 13 vs format · 48.1917 ULP · 4096 gas/step	0.000049998750020875 12 digits · 12 vs format · 41.9271 ULP · 1040 gas/step	0.000049998750020825 12 digits · 12 vs format · -8.07292 ULP · 3254 gas/step	0.0000499987500208330729192708116354756796406027 28 digits · 28 vs format · ULP · 8289 gas/step	0.000049998750020875 12 digits · 12 vs format · 41.9271 ULP · 1901 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	0.0000499987500208330729192708116320994533962727 66 digits · 66 vs format · ULP · 9209 gas/step
100	0.000099995000166662500083331944464285466272597	0.0000999950001666625000833319444642854496080643 33 digits · 33 vs format · -5.67065 ULP · 7171 gas/step	0.0000999950001666625000833319444642854496080643 33 digits · 33 vs format · -5.67065 ULP · 5817 gas/step	0.0000999950001666675362260727766283707751426846 13 digits · 13 vs format · 92.9004 ULP · 4087 gas/step	0.00009999500016675 12 digits · 12 vs format · 87.4999 ULP · 1032 gas/step	0.00009999500016665 12 digits · 12 vs format · -12.5001 ULP · 3246 gas/step	0.0000999950001666625000833319444707690150656674 28 digits · 28 vs format · ULP · 8261 gas/step	0.00009999500016675 12 digits · 12 vs format · 87.4999 ULP · 1893 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	0.000099995000166662500083331944464285466272597 66 digits · 66 vs format · ULP · 9188 gas/step
365	0.000364933395603781349790247076049260633375857	0.00036493339560378134979024707604926057849132 33 digits · 33 vs format · -18.6762 ULP · 7168 gas/step	0.00036493339560378134979024707604926057849132 33 digits · 33 vs format · -18.6762 ULP · 5811 gas/step	0.000364933395603799567031427786290009862568695 13 digits · 13 vs format · 336.049 ULP · 4080 gas/step	0.000364933395604218 11 digits · 11 vs format · 436.65 ULP · 1026 gas/step	0.000364933395603853 12 digits · 12 vs format · 71.6502 ULP · 3240 gas/step	0.000364933395603781349790247076073267291133876 28 digits · 28 vs format · ULP · 8240 gas/step	0.000364933395604218 11 digits · 11 vs format · 436.65 ULP · 1887 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	0.000364933395603781349790247076049260633375857 66 digits · 66 vs format · ULP · 9174 gas/step
1000	0.000999500166625008331944642832344025297644098	0.000999500166625008331944642832344025157086449 33 digits · 33 vs format · -47.8293 ULP · 7171 gas/step	0.000999500166625008331944642832344025157086449 33 digits · 33 vs format · -47.8293 ULP · 5809 gas/step	0.00099950016662505854030118523922965323436074 13 digits · 13 vs format · 926.181 ULP · 4078 gas/step	0.000999500166626284 11 digits · 11 vs format · 1275.67 ULP · 1024 gas/step	0.000999500166625284 12 digits · 12 vs format · 275.668 ULP · 3238 gas/step	0.000999500166625008331944642832410424715795436 28 digits · 28 vs format · ULP · 8235 gas/step	0.000999500166626284 11 digits · 11 vs format · 1275.67 ULP · 1885 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	0.000999500166625008331944642832344025297644098 66 digits · 66 vs format · ULP · 9170 gas/step
10000	0.00995016625083194642609402281996344222792092	0.0099501662508319464260940228199634411929768 33 digits · 33 vs format · -352.173 ULP · 7241 gas/step	0.0099501662508319464260940228199634411929768 33 digits · 33 vs format · -352.173 ULP · 5809 gas/step	0.00995016625083243862149792735571907087432919 13 digits · 13 vs format · 9079.4 ULP · 4078 gas/step	0.009950166250845127 11 digits · 11 vs format · 13180.6 ULP · 1024 gas/step	0.009950166250835246 12 digits · 12 vs format · 3299.57 ULP · 3238 gas/step	0.00995016625083194642609402282061759264376801 28 digits · 28 vs format · ULP · 8233 gas/step	0.009950166250845127 11 digits · 11 vs format · 13180.6 ULP · 1885 gas/step	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	reverts Unsupported(uint8): the library has no model for this scenario 0x7bb9ae15	0.00995016625083194642609402281996344222792092 66 digits · 66 vs format · ULP · 9169 gas/step

Full dataset: ladder.json. Regenerate with make ladder.