Edges

139 single calls at the end of each library's domain, against 300-digit arithmetic.

The ladder measures a slope. This measures the cliffs: ln within a per-second rate of one, exp(−42), a probability of 10^−18, a year of seconds as an exponent, the normal tail where an option's price lives. A cell has digits, or says why there are none: zero (below the format's floor), revert (with the reason), range (the argument does not fit the word), n/a (the library has no such op). FP127's own walls are here too: exp at ±88, the word at 1.7 × 10^38, erfc past 9.3.

1,511 cells checked live on Sepolia at block 11,928,227, 0 mismatches. 2026-10-16, runner 0xa9f8C514…. Re-run: uv run scripts/ladder/liveedges.py

How this was measured

digits: digits = floor(-log10 |v - t| / |t|), capped at 100 (reported as exact); errUlps is the signed error in units of the representation's last place (none for the floats).
gas: gas is measured inside the adapter around the op, after the adapter and the FP127 object are warm. The FP127 (deployed) form includes the staticcall; the others are internal calls.
live: EdgeRunner.run(op, form, args) on Sepolia evaluates any cell with one eth_call and returns (ok, raw, reason, gas); scripts/ladder/liveedges.py sends every probe's encoded arguments and compares the raw word bit for bit.
range: RANGE means the argument does not fit the representation at all (above 2^127 for FP127, 2^63 for 64.64, 2^128 as a bit pattern for binary128; FP256 holds every probe) and nothing was called.
reproduce: make edges regenerates edges.json and edges.md from test/ladder/Edges.t.sol and this script; make edges-check fails CI when the committed dataset differs.
reverted: A revert is recorded with its selector and name. Unsupported means the library has no way to compute the op (the accrual libraries have no general ops; only FP127 has the secondary Lambert branch; the libraries without erf, erfc or normCdf run the same series and continued fraction through their own ops instead, so those cells are not Unsupported).
truth: Every probe is one op evaluated once. The truth is mpmath at 200 decimal digits from the exact arguments; truthFormat is the same op from the arguments as the representation holds them (floored on a grid, nearest for binary128), so digitsFormat measures the op alone and digits what a caller sees.
zero: A cell is marked zero when the library returned 0 and the truth is not 0: the result fell below the representation's floor. On an 18-decimal grid that is anything below 1e-18; on FP127's, below 2^-128.
versions: Generated 2026-10-15; 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.

ln(x)

The logarithm near one is where a per-second rate lives: ln(1 + r) for r = 1e-9 has eight digits on an 18-decimal grid.

probe	truth	FP127	FP256	ABDK	Solady	PRBMath	Quad	Balancer	Aave	Morpho	Maker
ln(1+1e-6)	0.0000009999995000003333…	34 -0.00841701 ULP	65	13 -1.66367 ULP	12 -0.333333 ULP	11 -8.33333 ULP	28	12 -0.333333 ULP	n/a	n/a	n/a
ln(1+1e-9) a per-second rate	0.0000000009999999995000…	30 -0.323137 ULP	62	10 -1.48618 ULP	9 -0.5 ULP	8 -4.5 ULP	25	9 -0.5 ULP	n/a	n/a	n/a
ln(1+1e-12)	0.0000000000009999999999…	26 -0.627492 ULP	59	7 -1.0737 ULP	6 -1.0 ULP	5 -5.0 ULP	22	5 -2.0 ULP	n/a	n/a	n/a
ln(1+1e-15)	9.9999999999999950000000…	24 -0.146963 ULP	56	4 -1.74407 ULP	15 5.0e-13 ULP	2 -4.0 ULP	19	3 -1.0 ULP	n/a	n/a	n/a
ln(1+1e-18) the last digit of an 18-decimal grid	9.9999999999999999950000…	21 -0.233424 ULP	53	1 -1.44674 ULP	18 5.0e-19 ULP	zero	16	zero	n/a	n/a	n/a
ln(1-1e-12)	-0.000000000001000000000…	26 -0.909044 ULP	59	6 -1.92628 ULP	12 5.0e-7 ULP	5 4.0 ULP	24	12 5.0e-7 ULP	n/a	n/a	n/a
ln(1e-18)	-41.44653167389282231232…	22 -3.74607e+17 ULP	71	3 -4.52243e+17 ULP	19 -0.676154 ULP	19 1.32385 ULP	33	20 0.323846 ULP	n/a	n/a	n/a
ln(1e-30)	-69.07755278982137052053…	10 -9.20938e+29 ULP	71	revert bare revert (require without a reason, or out of gas)	revert unknown selector	revert unknown selector	33	revert Balancer OutOfBounds()	n/a	n/a	n/a
ln(1e30)	69.077552789821370520539…	41 -0.201036 ULP	71	range	20 -0.539744 ULP	18 8.46026 ULP	33	20 -0.539744 ULP	n/a	n/a	n/a
ln(1e38)	87.498233533773735992683…	40 0.478688 ULP	71	range	20 -0.683675 ULP	18 12.3163 ULP	33	20 -0.683675 ULP	n/a	n/a	n/a

exp(x)

exp below -42 is zero on an 18-decimal grid and has one digit at -40: every probability, discount or decay smaller than 1e-18 is gone.

probe	truth	FP127	FP256	ABDK	Solady	PRBMath	Quad	Balancer	Aave	Morpho	Maker
exp(-1e-12)	0.9999999999990000000000…	39 -0.0862357 ULP	71	18 -1.9263 ULP	24 -5.0e-7 ULP	18 1.0 ULP	34	24 -5.0e-7 ULP	n/a	n/a	n/a
exp(-20)	0.0000000020611536224385…	31 0.0325112 ULP	72	10 -0.723441 ULP	9 -0.438558 ULP	9 -0.438558 ULP	32	9 -0.438558 ULP	n/a	n/a	n/a
exp(-30)	9.3576229688401746049158…	26 0.17186 ULP	71	6 -0.760445 ULP	5 -0.229688 ULP	5 -0.229688 ULP	32	5 -0.229688 ULP	n/a	n/a	n/a
exp(-40) a probability of 4e-18: one digit on an 18-decimal grid	4.2483542552915889953292…	21 0.479968 ULP	71	2 -0.368304 ULP	1 -0.248354 ULP	1 -0.248354 ULP	32	1 -0.248354 ULP	n/a	n/a	n/a
exp(-41)	1.5628821893349887680908…	21 0.169562 ULP	71	1 -0.830088 ULP	0 -0.562882 ULP	0 -0.562882 ULP	32	0 -0.562882 ULP	n/a	n/a	n/a
exp(-42) zero on an 18-decimal grid	5.7495222642935598066643…	21 -0.123147 ULP	71	1 -0.605997 ULP	zero	zero	32	revert Balancer InvalidExponent()	n/a	n/a	n/a
exp(-43)	2.1151310375910804866314…	21 -0.0187405 ULP	71	0 -0.901728 ULP	zero	zero	32	revert Balancer InvalidExponent()	n/a	n/a	n/a
exp(-60)	8.7565107626965203384887…	13 -0.0440478 ULP	71	zero	zero	zero	32	revert Balancer InvalidExponent()	n/a	n/a	n/a
exp(-88) the last value above FP127's floor	6.0546018954011858845318…	1 -0.0602743 ULP	71	zero	zero	zero	31	revert Balancer InvalidExponent()	n/a	n/a	n/a
exp(-89) zero on FP127's grid	2.2273635617957437392229…	zero	71	zero	zero	zero	31	revert Balancer InvalidExponent()	n/a	n/a	n/a
exp(-100)	3.7200759760208359629596…	zero	71	zero	zero	zero	31	revert Balancer InvalidExponent()	n/a	n/a	n/a
exp(40)	235385266837019985.40789…	43 8.0567e+11 ULP	72	19 -8.82694e+16 ULP	20 1.47502e+15 ULP	16 -2.35725e+18 ULP	32	21 -2.18129e+14 ULP	n/a	n/a	n/a
exp(43)	4727839468229346561.4744…	43 -2.62103e+13 ULP	71	21 -2.1582e+16 ULP	21 1.81375e+15 ULP	16 -5.1073e+19 ULP	32	20 -5.15474e+15 ULP	n/a	n/a	n/a
exp(44) beyond ABDK 64.64's range	12851600114359308275.809…	43 4.68139e+13 ULP	72	revert bare revert (require without a reason, or out of gas)	20 1.22003e+17 ULP	16 -1.40317e+20 ULP	32	20 -1.43248e+16 ULP	n/a	n/a	n/a
exp(60)	114200738981568428366295…	45 -2.17756e+19 ULP	71	revert bare revert (require without a reason, or out of gas)	20 -1.19245e+23 ULP	16 -1.72561e+27 ULP	32	20 -1.66152e+23 ULP	n/a	n/a	n/a
exp(88) the largest FP127 holds	165163625499400185552832…	44 2.69603e+32 ULP	71	revert bare revert (require without a reason, or out of gas)	20 8.43627e+35 ULP	16 -3.6327e+39 ULP	31	20 -2.3506e+35 ULP	n/a	n/a	n/a
exp(89) FP127 reverts; the WAD word still has room	448961281917434524628424…	revert Overflow()	72	revert bare revert (require without a reason, or out of gas)	20 2.66688e+36 ULP	16 -9.97864e+39 ULP	31	20 -6.8644e+35 ULP	n/a	n/a	n/a
exp(100)	2.6881171418161354484126…	revert Overflow()	71	revert bare revert (require without a reason, or out of gas)	21 8.41059e+39 ULP	16 -6.74115e+44 ULP	31	21 5.71174e+39 ULP	n/a	n/a	n/a
exp(130)	2.8726495508178319332673…	revert Overflow()	72	revert bare revert (require without a reason, or out of gas)	20 6.70025e+53 ULP	16 -9.32383e+57 ULP	31	21 -2.14179e+53 ULP	n/a	n/a	n/a
exp(135)	4.2633899483147210448936…	revert Overflow()	72	revert bare revert (require without a reason, or out of gas)	20 -4.90946e+55 ULP	revert unknown selector	31	revert Balancer InvalidExponent()	n/a	n/a	n/a
exp(136) beyond every fixed-point word	1.1589095424138854283480…	revert Overflow()	71	revert bare revert (require without a reason, or out of gas)	revert unknown selector	revert unknown selector	31	revert Balancer InvalidExponent()	n/a	n/a	n/a

sqrt(x)

The root of a tiny number: sqrt(1e-20) is 1e-10, which the grid holds, but 1e-20 itself is zero on it.

probe	truth	FP127	FP256	ABDK	Solady	PRBMath	Quad	Balancer	Aave	Morpho	Maker
sqrt(1e-6)	0.001	33 -106.456 ULP	71	13 -276.616 ULP	exact 0 ULP	exact 0 ULP	34	exact 0 ULP	n/a	n/a	n/a
sqrt(1e-12)	0.000001	27 -215884.0 ULP	71	8 -36855.6 ULP	exact 0 ULP	exact 0 ULP	34	exact 0 ULP	n/a	n/a	n/a
sqrt(1e-18)	0.000000001	21 -1.87304e+8 ULP	71	1 -2.24741e+8 ULP	exact 0 ULP	exact 0 ULP	34	exact 0 ULP	n/a	n/a	n/a
sqrt(1e-20) the input is zero on an 18-decimal grid	0.0000000001	19 -3.16873e+9 ULP	71	zero	zero	zero	34	zero	n/a	n/a	n/a
sqrt(1e-30)	1.0e-15	8 -4.60469e+14 ULP	71	zero	zero	zero	34	zero	n/a	n/a	n/a
sqrt(1e30)	1000000000000000.0	exact 0 ULP	exact	range	exact 0 ULP	exact 0 ULP	exact	18 -3.08076e+14 ULP	n/a	n/a	n/a
sqrt(1e36)	1000000000000000000.0	exact 0 ULP	exact	range	exact 0 ULP	exact 0 ULP	exact	18 -4.53569e+17 ULP	n/a	n/a	n/a
sqrt(1e38)	10000000000000000000.0	exact 0 ULP	exact	range	exact 0 ULP	exact 0 ULP	exact	17 -2.1889e+19 ULP	n/a	n/a	n/a
sqrt(1e40) beyond FP127's word	100000000000000000000.0	range	exact	range	exact 0 ULP	exact 0 ULP	exact	18 -3.96527e+19 ULP	n/a	n/a	n/a
sqrt(1e50)	100000000000000000000000…	range	exact	range	exact 0 ULP	revert unknown selector	exact	18 -4.82281e+24 ULP	range	n/a	range

pow(x, y)

A base near one to a large power: per-second rates over years, and Uniswap's tick range 1.0001^(+-887272).

probe	truth	FP127	FP256	ABDK	Solady	PRBMath	Quad	Balancer	Aave	Morpho	Maker
pow(1+0.05/31536000, 31536000) 5% APR per second, one year	1.0512710963343545550116…	31 -1.67642e+7 ULP	63	11 -5.1987e+7 ULP	10 -3.08766e+7 ULP	9 -3.06013e+8 ULP	27	11 -6.24035e+6 ULP	n/a	n/a	n/a
pow(1+0.001/31536000, 31536000) 0.1% APR per second, one year	1.0010005001666924709092…	31 -3.12555e+7 ULP	63	11 -4.24835e+7 ULP	10 -3.10392e+7 ULP	9 -2.10973e+8 ULP	26	10 -3.1055e+7 ULP	n/a	n/a	n/a
pow(1+0.05/31536000, 315360000) ten years	1.6487212700466205410050…	30 -2.62915e+8 ULP	62	10 -8.15318e+8 ULP	9 -4.84242e+8 ULP	8 -4.79924e+9 ULP	26	10 -9.78681e+7 ULP	n/a	n/a	n/a
pow(1+0.05/31536000/12, 31536000*12) twelve times a second	1.0512710963725515826369…	30 -3.1514e+8 ULP	62	10 -4.58222e+8 ULP	10 -1.02227e+8 ULP	8 -2.15962e+9 ULP	25	9 -1.05699e+8 ULP	n/a	n/a	n/a
pow(1.0001, 887272) Uniswap's top tick, 3.4e38: one bit past FP127's word	340256786836388094050805…	revert Overflow()	65	revert bare revert (require without a reason, or out of gas)	12 -1.01237e+44 ULP	11 -1.71142e+45 ULP	29	17 -4.15871e+38 ULP	n/a	n/a	n/a
pow(1.0001, -887272) Uniswap's bottom tick, 2.9e-39: one ULP of FP127	2.9389568075855848388747…	4 -7.51788e-5 ULP	65	zero	zero	zero	29	revert Balancer: negative base or exponent	n/a	n/a	n/a
pow(0.999999, 1000000)	0.3678792572316450942857…	32 -290089.0 ULP	65	13 -464489.0 ULP	12 -245253.0 ULP	11 2.28596e+6 ULP	28	18 -0.285798 ULP	n/a	n/a	n/a
pow(2, 0.5)	1.4142135623730950488016…	39 -0.229961 ULP	71	18 -2.69914 ULP	17 -1.80169 ULP	18 -0.801689 ULP	33	17 -1.80169 ULP	n/a	n/a	n/a
pow(10, -18)	1.0e-18	20 -0.374607 ULP	71	1 -0.446744 ULP	exact 0 ULP	exact 0 ULP	31	revert Balancer: negative base or exponent	n/a	n/a	n/a
pow(1e-9, 0.5)	0.0000316227766016837933…	29 -12147.3 ULP	71	10 -11219.6 ULP	9 -15811.8 ULP	9 -15811.8 ULP	32	9 -15811.8 ULP	n/a	n/a	n/a
pow(1e-18, 0.5)	0.000000001	21 -1.87304e+8 ULP	71	1 -2.24741e+8 ULP	exact 0 ULP	exact 0 ULP	32	exact 0 ULP	n/a	n/a	n/a

mul(a, b)

Tiny times tiny: a fee on a fee, a unit price times a unit size, below 1e-18 is zero.

probe	truth	FP127	FP256	ABDK	Solady	PRBMath	Quad	Balancer	Aave	Morpho	Maker
mul(1e-9, 1e-9) the last digit of an 18-decimal grid	1.0e-18	20 -0.374607 ULP	71	1 -0.446744 ULP	zero	zero	33	zero	n/a	n/a	n/a
mul(1e-10, 1e-10) zero on it	1.0e-20	18 -0.633746 ULP	71	zero	zero	zero	34	zero	n/a	n/a	n/a
mul(1e-12, 1e-8)	1.0e-20	18 -0.633746 ULP	72	zero	zero	zero	34	zero	n/a	n/a	n/a
mul(1e19, 1e19)	100000000000000000000000…	exact 0 ULP	exact	range	exact 0 ULP	exact 0 ULP	exact	exact 0 ULP	n/a	n/a	n/a
mul(1e20, 1e20) beyond FP127's word	100000000000000000000000…	revert Overflow()	exact	range	exact 0 ULP	exact 0 ULP	exact	exact 0 ULP	n/a	n/a	n/a

div(a, b)

Small over large, and one over tiny.

probe	truth	FP127	FP256	ABDK	Solady	PRBMath	Quad	Balancer	Aave	Morpho	Maker
div(1, 3)	0.3333333333333333333333…	38 -0.333333 ULP	71	19 -0.333333 ULP	17 -0.333333 ULP	17 -0.333333 ULP	34	17 -0.333333 ULP	n/a	n/a	n/a
div(1e-12, 1e9)	1.0e-21	17 -0.463375 ULP	71	zero	zero	zero	34	zero	n/a	n/a	n/a
div(1e-9, 1e12)	1.0e-21	17 -0.463375 ULP	71	zero	zero	zero	33	zero	n/a	n/a	n/a
div(1e30, 1e-9)	100000000000000000000000…	revert Overflow()	exact	range	8 1.0e+48 ULP	8 1.0e+48 ULP	exact	8 1.0e+48 ULP	n/a	n/a	n/a
div(1, 1e-30)	100000000000000000000000…	8 9.20938e+59 ULP	exact	revert bare revert (require without a reason, or out of gas)	revert unknown selector	revert Panic (checked arithmetic overflow or division by zero)	exact	revert Panic (checked arithmetic overflow or division by zero)	n/a	n/a	n/a

W0(x)

The principal Lambert branch near its branch point -1/e, where the slope is infinite; only Solady has it among the others.

probe	truth	FP127	FP256	ABDK	Solady	PRBMath	Quad	Balancer	Aave	Morpho	Maker
lambertW0(-1/e+1e-6)	-0.997670166272007889180…	35 486.696 ULP	n/a	n/a	15 -751.82 ULP	n/a	n/a	n/a	n/a	n/a	n/a
lambertW0(-1/e+1e-9)	-0.999926268755381939963…	34 -31840.1 ULP	n/a	n/a	13 -40286.0 ULP	n/a	n/a	n/a	n/a	n/a	n/a
lambertW0(-1/e+1e-12)	-0.999997668357830588824…	32 -683938.0 ULP	n/a	n/a	12 -631979.0 ULP	n/a	n/a	n/a	n/a	n/a	n/a
lambertW0(-1/e+1e-15)	-0.999999926266945067481…	30 -3.74715e+7 ULP	n/a	n/a	10 -3.27666e+7 ULP	n/a	n/a	n/a	n/a	n/a	n/a
lambertW0(-1/e+1e-18) one grid step from the branch point	-0.999999997668356020215…	29 -5.9868e+8 ULP	n/a	n/a	8 -2.07323e+9 ULP	n/a	n/a	n/a	n/a	n/a	n/a
lambertW0(-1e-18)	-1.000000000000000001000…	20 -0.343026 ULP	n/a	n/a	17 1.0e-18 ULP	n/a	n/a	n/a	n/a	n/a	n/a
lambertW0(1e-18)	9.9999999999999999900000…	20 -2.09224 ULP	n/a	n/a	18 1.0e-18 ULP	n/a	n/a	n/a	n/a	n/a	n/a
lambertW0(1e30)	64.904633770046124079605…	41 -0.128278 ULP	n/a	range	20 -0.605604 ULP	n/a	n/a	n/a	n/a	n/a	n/a
lambertW0(1e38)	83.078448213164095927204…	40 -0.55261 ULP	n/a	range	20 -0.204104 ULP	n/a	n/a	n/a	n/a	n/a	n/a

W-1(x)

The secondary branch, toward zero from below; FP127 only.

probe	truth	FP127	FP256	ABDK	Solady	PRBMath	Quad	Balancer	Aave	Morpho	Maker
lambertWm1(-1/e+1e-9)	-1.000073734868993836022…	34 32964.0 ULP	n/a	n/a	n/a	n/a	n/a	n/a	n/a	n/a	n/a
lambertWm1(-1e-10)	-26.29523881924692569411…	30 8.55722e+9 ULP	n/a	n/a	n/a	n/a	n/a	n/a	n/a	n/a	n/a
lambertWm1(-1e-30)	-73.37311031382297679706…	11 8.0154e+28 ULP	n/a	n/a	n/a	n/a	n/a	n/a	n/a	n/a	n/a

log1p(x)

ln(1 + x) as an op: the libraries without it compute ln(1 + x), and 1 + x is where x loses its digits.

probe	truth	FP127	FP256	ABDK	Solady	PRBMath	Quad	Balancer	Aave	Morpho	Maker
log1p(1e-6)	0.0000009999995000003333…	34 -0.00841701 ULP	65	13 -1.66367 ULP	12 -0.333333 ULP	11 -8.33333 ULP	28	12 -0.333333 ULP	n/a	n/a	n/a
log1p(1e-9) a per-second rate	0.0000000009999999995000…	30 -0.323137 ULP	62	10 -1.48618 ULP	8 -1.5 ULP	8 -6.5 ULP	25	8 -1.5 ULP	n/a	n/a	n/a
log1p(1e-12)	0.0000000000009999999999…	26 -0.627492 ULP	59	7 -1.0737 ULP	12 5.0e-7 ULP	5 -4.0 ULP	22	6 -1.0 ULP	n/a	n/a	n/a
log1p(1e-15)	9.9999999999999950000000…	24 -0.146963 ULP	56	4 -1.74407 ULP	15 5.0e-13 ULP	2 -4.0 ULP	18	3 -1.0 ULP	n/a	n/a	n/a
log1p(1e-18)	9.9999999999999999950000…	21 -0.233424 ULP	53	1 -1.44674 ULP	18 5.0e-19 ULP	zero	15	zero	n/a	n/a	n/a
log1p(1e-20) below an 18-decimal grid	9.9999999999999999999500…	18 -0.616732 ULP	51	zero	zero	zero	14	zero	n/a	n/a	n/a
log1p(1e-30)	9.9999999999999999999999…	8 -0.920938 ULP	41	zero	zero	zero	4	zero	n/a	n/a	n/a
log1p(-1e-9)	-0.000000001000000000500…	30 -0.161072 ULP	63	10 -1.06708 ULP	9 -0.5 ULP	8 5.5 ULP	25	9 0.5 ULP	n/a	n/a	n/a
log1p(1e6)	13.815511557963774104441…	40 -0.248624 ULP	71	20 -0.632972 ULP	19 -0.441282 ULP	18 -6.44128 ULP	33	19 -0.441282 ULP	n/a	n/a	n/a

expm1(x)

e^x - 1 as an op: the libraries without it compute exp(x) - 1 and the one cancels the digits.

probe	truth	FP127	FP256	ABDK	Solady	PRBMath	Quad	Balancer	Aave	Morpho	Maker
expm1(1e-6)	0.0000010000005000001666…	33 -0.122708 ULP	65	13 -1.66293 ULP	12 -0.166667 ULP	11 -1.16667 ULP	27	12 -0.166667 ULP	n/a	n/a	n/a
expm1(1e-9)	0.0000000010000000005000…	29 -0.790358 ULP	62	9 -1.93292 ULP	8 -1.5 ULP	8 -1.5 ULP	24	8 -1.5 ULP	n/a	n/a	n/a
expm1(1e-12)	0.0000000000010000000000…	26 -0.377228 ULP	59	7 -1.07372 ULP	12 -5.0e-7 ULP	5 -1.0 ULP	21	12 -5.0e-7 ULP	n/a	n/a	n/a
expm1(1e-15)	1.0000000000000005000000…	23 -1.0679 ULP	57	4 -1.74407 ULP	15 -5.0e-13 ULP	2 -1.0 ULP	18	15 -5.0e-13 ULP	n/a	n/a	n/a
expm1(1e-18)	1.0000000000000000005000…	20 -0.515791 ULP	53	1 -1.44674 ULP	18 -5.0e-19 ULP	zero	15	18 -5.0e-19 ULP	n/a	n/a	n/a
expm1(1e-20)	1.0000000000000000000050…	18 -0.65076 ULP	51	zero	zero	zero	14	zero	n/a	n/a	n/a
expm1(-1e-9)	-0.000000000999999999500…	30 0.0588864 ULP	63	10 -1.51382 ULP	9 -0.5 ULP	9 0.5 ULP	25	9 -0.5 ULP	n/a	n/a	n/a
expm1(-40)	-0.999999999999999995751…	38 0.479968 ULP	71	19 -0.368304 ULP	18 -0.248354 ULP	18 -0.248354 ULP	34	18 -0.248354 ULP	n/a	n/a	n/a
expm1(-88)	-0.999999999999999999999…	39 -0.0602743 ULP	72	38 -1.11688e-19 ULP	38 -6.0546e-21 ULP	38 -6.0546e-21 ULP	34	revert Balancer InvalidExponent()	n/a	n/a	n/a
expm1(20)	485165194.40979027796910…	44 -1135.17 ULP	71	19 -2.59114e+8 ULP	19 5.67027e+6 ULP	17 -2.42983e+9 ULP	32	21 -256274.0 ULP	n/a	n/a	n/a

compound(r, n)

(1 + r)^n for an integer n: per-second rates over years. FP127 and Maker take an exact integer power; the rest exp(n ln(1 + r)).

probe	truth	FP127	FP256	ABDK	Solady	PRBMath	Quad	Balancer	Aave	Morpho	Maker
compound(0.05/31536000, 31536000) 5% APR per second, one year	1.0512710963343545550116…	31 -1.67642e+7 ULP	63	11 -2.49841e+7 ULP	12 1.04809e+6 ULP	10 -1.88494e+7 ULP	26	11 -6.24035e+6 ULP	n/a	n/a	19 -1.53971e+7 ULP
compound(0.05/31536000, 315360000) ten years	1.6487212700466205410050…	30 -2.62915e+8 ULP	62	10 -3.91828e+8 ULP	11 1.64374e+7 ULP	9 -2.95618e+8 ULP	25	10 -9.78681e+7 ULP	n/a	n/a	18 -2.41475e+8 ULP
compound(0.001/31536000, 31536000) 0.1% APR per second	1.0010005001666924709092…	31 -3.12555e+7 ULP	63	12 -1.12088e+7 ULP	10 -3.16372e+7 ULP	10 -3.17366e+7 ULP	26	10 -3.1055e+7 ULP	n/a	n/a	19 -1.05987e+7 ULP
compound(0.1/365, 365) the ladder's headline cell	1.1051557816162643739380…	36 -194.363 ULP	68	17 -146.12 ULP	17 2.06199 ULP	15 -278.938 ULP	31	16 -11.938 ULP	n/a	n/a	24 -166.681 ULP
compound(1e-12, 1000000000) a rate of 1e-12 over a billion steps	1.0010005001667078411678…	29 -4.322e+8 ULP	62	11 -7.37833e+7 ULP	12 -572106.0 ULP	12 -721907.0 ULP	25	18 -0.167806 ULP	n/a	n/a	22 -47615.0 ULP
compound(-0.5, 10)	0.0009765625	exact 0 ULP	exact	exact 0 ULP	exact 0 ULP	exact 0 ULP	32	exact 0 ULP	n/a	n/a	revert ParamOutOfRange(uint256): the value does not fit the representation
compound(0.000000001, 10000000)	1.0100501670791178067101…	31 -7.75932e+6 ULP	64	12 -7.16683e+6 ULP	10 -1.01009e+7 ULP	10 -1.89434e+7 ULP	27	10 -1.01005e+7 ULP	n/a	n/a	19 -1.01154e+7 ULP

powu(x, n)

An integer power. Solady's rpow, PRBMath's powu, ABDK's pow and Maker's rpow square at the format's precision; FP127 on a 256-bit mantissa.

probe	truth	FP127	FP256	ABDK	Solady	PRBMath	Quad	Balancer	Aave	Morpho	Maker
powu(2, 10)	1024.0	exact 0 ULP	exact	exact 0 ULP	exact 0 ULP	exact 0 ULP	32	17 -4352.0 ULP	n/a	n/a	exact 0 ULP
powu(1.0001, 887272) Uniswap's top tick	340256786836388094050805…	revert Overflow()	65	revert bare revert (require without a reason, or out of gas)	13 2.31812e+43 ULP	12 -4.02888e+43 ULP	29	17 -4.15871e+38 ULP	n/a	n/a	revert Solady RPowOverflow() (Maker's rpow)
powu(1.0001, -887272) Uniswap's bottom tick	2.9389568075855848388747…	4 -7.51788e-5 ULP	65	revert bare revert (require without a reason, or out of gas)	zero	zero	29	zero	n/a	n/a	n/a
powu(0.999999, 1000000)	0.3678792572316450942857…	32 -290089.0 ULP	65	13 -164952.0 ULP	14 432.714 ULP	13 -5731.29 ULP	28	18 -0.285798 ULP	n/a	n/a	23 2266.73 ULP
powu(1+0.05/31536000, 31536000)	1.0512710963343545550116…	31 -1.67642e+7 ULP	63	11 -2.49841e+7 ULP	12 1.04809e+6 ULP	10 -1.88494e+7 ULP	27	11 -6.24035e+6 ULP	n/a	n/a	19 -1.53971e+7 ULP
powu(10, 38)	100000000000000000000000…	exact 0 ULP	exact	revert bare revert (require without a reason, or out of gas)	exact 0 ULP	exact 0 ULP	31	16 -4.04798e+39 ULP	n/a	n/a	revert Solady RPowOverflow() (Maker's rpow)
powu(3, -5)	0.0041152263374485596707…	36 0.205761 ULP	73	17 -0.547325 ULP	15 -0.670782 ULP	15 -0.670782 ULP	32	15 -0.670782 ULP	n/a	n/a	n/a
powu(1.1, 100)	13780.612339822270184118…	36 -751670.0 ULP	69	17 -751670.0 ULP	19 -886.337 ULP	19 -917.337 ULP	32	15 -1.4391e+6 ULP	n/a	n/a	29 -72.6368 ULP

erf(x)

The error function; the libraries without it run the same series and continued fraction through their own ops.

probe	truth	FP127	FP256	ABDK	Solady	PRBMath	Quad	Balancer	Aave	Morpho	Maker
erf(1e-10)	0.0000000001128379167095…	28 -0.512361 ULP	71	9 -1.35165 ULP	8 -0.709551 ULP	8 -0.709551 ULP	33	8 -0.709551 ULP	n/a	n/a	n/a
erf(0.5)	0.5204998778130465376827…	39 -0.0325755 ULP	70	17 -16.5024 ULP	16 -8.68275 ULP	16 -8.68275 ULP	32	16 -9.68275 ULP	n/a	n/a	n/a
erf(1)	0.8427007929497148693412…	39 -0.131046 ULP	70	17 -28.5021 ULP	16 -15.3412 ULP	16 -15.3412 ULP	32	16 -17.3412 ULP	n/a	n/a	n/a
erf(3)	0.9999779095030014145586…	39 0.276513 ULP	71	15 -9432.37 ULP	14 -6132.56 ULP	14 -6132.56 ULP	32	14 -6132.56 ULP	n/a	n/a	n/a
erf(4)	0.9999999845827420997199…	39 -0.0912579 ULP	71	12 -1.00476e+7 ULP	11 -5.24154e+6 ULP	11 -5.24154e+6 ULP	32	11 -5.24154e+6 ULP	n/a	n/a	n/a
erf(5)	0.9999999999984625402055…	38 -0.444951 ULP	68	19 0.351432 ULP	18 0.794428 ULP	18 0.794428 ULP	34	18 0.794428 ULP	n/a	n/a	n/a
erf(6)	0.9999999999999999784802…	39 -0.0593803 ULP	71	19 0.969076 ULP	18 0.519737 ULP	18 0.519737 ULP	34	18 0.519737 ULP	n/a	n/a	n/a
erf(8)	0.9999999999999999999999…	39 0.0466287 ULP	72	28 2.07052e-10 ULP	28 1.12243e-11 ULP	28 1.12243e-11 ULP	34	revert Balancer InvalidExponent()	n/a	n/a	n/a
erf(9)	0.9999999999999999999999…	39 -0.224105 ULP	72	36 7.63148e-18 ULP	36 4.13703e-19 ULP	36 4.13703e-19 ULP	34	revert Balancer InvalidExponent()	n/a	n/a	n/a

erfc(x)

The tail of the normal distribution: erfc(6) is 2e-17, erfc(9.3) one ULP of FP127, zero on an 18-decimal grid from about 6.4.

probe	truth	FP127	FP256	ABDK	Solady	PRBMath	Quad	Balancer	Aave	Morpho	Maker
erfc(1)	0.1572992070502851306587…	38 0.131046 ULP	70	17 28.5021 ULP	16 15.3412 ULP	16 15.3412 ULP	31	15 17.3412 ULP	n/a	n/a	n/a
erfc(3)	0.0000220904969985854413…	34 -0.276513 ULP	66	10 9432.37 ULP	9 6132.56 ULP	9 6132.56 ULP	27	9 6132.56 ULP	n/a	n/a	n/a
erfc(4)	0.0000000154172579002800…	31 0.0912579 ULP	63	4 1.00476e+7 ULP	3 5.24154e+6 ULP	3 5.24154e+6 ULP	24	3 5.24154e+6 ULP	n/a	n/a	n/a
erfc(5)	0.0000000000015374597944…	27 0.444951 ULP	56	7 -0.351432 ULP	6 -0.794428 ULP	6 -0.794428 ULP	32	6 -0.794428 ULP	n/a	n/a	n/a
erfc(6) 2e-17: the last digit of an 18-decimal grid	2.1519736712498913116593…	23 0.0593803 ULP	66	2 -0.969076 ULP	1 -0.519737 ULP	1 -0.519737 ULP	32	1 -0.519737 ULP	n/a	n/a	n/a
erfc(7)	4.1838256077794143986140…	16 0.357181 ULP	70	zero	zero	zero	32	revert Balancer InvalidExponent()	n/a	n/a	n/a
erfc(8)	1.1224297172982927079967…	10 -0.0466287 ULP	71	zero	zero	zero	32	revert Balancer InvalidExponent()	n/a	n/a	n/a
erfc(9)	4.1370317465138102380539…	2 0.224105 ULP	71	zero	zero	zero	32	revert Balancer InvalidExponent()	n/a	n/a	n/a
erfc(9.3) one ULP of FP127	1.6532441840301569593991…	0 0.43743 ULP	69	zero	zero	zero	34	revert Balancer InvalidExponent()	n/a	n/a	n/a
erfc(10)	2.0884875837625447570007…	zero	70	zero	zero	zero	31	revert Balancer InvalidExponent()	n/a	n/a	n/a

normCdf(x)

N(d) of Black-Scholes: zero on an 18-decimal grid from d = -8.3, two ULPs of FP127 at -13.

probe	truth	FP127	FP256	ABDK	Solady	PRBMath	Quad	Balancer	Aave	Morpho	Maker
normCdf(1e-12)	0.5000000000003989422804…	38 -0.550282 ULP	72	19 -0.146747 ULP	18 -0.280401 ULP	18 -0.280401 ULP	34	18 -0.280401 ULP	n/a	n/a	n/a
normCdf(-1)	0.1586552539314570514147…	38 -0.324212 ULP	71	17 10.7466 ULP	16 5.58523 ULP	16 5.58523 ULP	32	16 6.58523 ULP	n/a	n/a	n/a
normCdf(-3)	0.0013498980316300945266…	35 -0.498911 ULP	68	14 55.9323 ULP	13 56.4733 ULP	13 56.4733 ULP	29	13 54.4733 ULP	n/a	n/a	n/a
normCdf(-5)	0.0000002866515718791939…	32 0.0998993 ULP	64	8 49315.0 ULP	6 133747.0 ULP	6 133747.0 ULP	26	6 133740.0 ULP	n/a	n/a	n/a
normCdf(-6) a one-in-a-billion event	0.0000000009865876450376…	30 -0.330608 ULP	49	10 -0.294221 ULP	8 -1.0377 ULP	8 -1.0377 ULP	32	8 -1.0377 ULP	n/a	n/a	n/a
normCdf(-8)	6.2209605742717841235159…	24 0.15449 ULP	63	4 -0.646761 ULP	3 -0.0960574 ULP	3 -0.0960574 ULP	31	3 -0.0960574 ULP	n/a	n/a	n/a
normCdf(-8.3) the last value above an 18-decimal grid	5.2055697448902851579958…	23 -0.126918 ULP	65	3 -0.258128 ULP	2 -0.0556974 ULP	2 -0.0556974 ULP	32	2 -0.0556974 ULP	n/a	n/a	n/a
normCdf(-9)	1.1285884059538406477355…	21 -0.0118453 ULP	70	1 -0.0818781 ULP	zero	zero	31	zero	n/a	n/a	n/a
normCdf(-10)	7.6198530241605260659733…	15 0.290579 ULP	69	zero	zero	zero	32	revert Balancer InvalidExponent()	n/a	n/a	n/a
normCdf(-13) two ULPs of FP127	6.1171643995498796822752…	1 -0.0815632 ULP	68	zero	zero	zero	31	revert Balancer InvalidExponent()	n/a	n/a	n/a
normCdf(-14)	7.7935368191928002543596…	zero	69	zero	zero	zero	31	revert Balancer InvalidExponent()	n/a	n/a	n/a
normCdf(5)	0.9999997133484281208060…	39 -0.0998993 ULP	70	14 -49315.0 ULP	12 -133747.0 ULP	12 -133747.0 ULP	32	12 -133740.0 ULP	n/a	n/a	n/a

A cell is correct significant digits against the exact value (hover for it), or zero, revert, range or n/a. Full dataset: edges.json. Regenerate with make edges.