Edges

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 its correct digits, or says why there are none: zero (below the format's floor), range (the argument does not fit the word), or the name of the error it reverts with. An empty cell is an op the library does not offer. BL127's own walls are here too: exp past ±88, the word at 1.7 × 10³⁸, erfc past 9.3.

BL127's and BL19's deployed contracts checked live on Sepolia: 276 results at block 12,017,760, 0 mismatches. The other libraries' results reproduce locally with make ladder. 2026-10-29, runner 0xad11E2a6…. 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 BL127 object are warm. The BL127 (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 BL127, 2^63 for 64.64, 2^128 as a bit pattern for binary128; BL19 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 BL127 and BL19 have 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 BL127's, below 2^-128.
versions: Generated 2026-10-28; aave aave-v3-origin@8305565ae342 (MathUtils, transcribed), abdk v3.2, balancer balancer-v3-monorepo@2b9ce4b1b8ad (LogExpMath.sol, vendored), bl127 6c2eeaf, euler evk-periphery@af436991e135 (IRMAdaptiveCurve, ExpLib, transcribed), forge 1.7.1, fraxlend fraxlend@2bed49d4dcc6 (VariableInterestRate V2, transcribed), huff sepolia:0xfae694D0c2c44181791F838c54Ed64C3151FfE30@11857616, maker dss pot.sol rpow via solady v0.1.26, morpho morpho-blue@8e26ca6a8dbc (MathLib, transcribed), openzeppelin openzeppelin-contracts v5.7.0 (Math, SignedMath, vendored), prb-math v4.2.0, snekmate v0.1.2 (math module, vyper 0.4.3), solady v0.1.26, solmate solmate@89365b8 (no tag: v6 predates SignedWadMath), uniswap v4-core v4.0.0 (FullMath, vendored).

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.

		Floating point		Fixed point
probe	truth	BL19	Quad	BL127	64.64	Solady	Solmate	PRB SD	PRB UD	snek	Balancer	OZ	Uniswap	Maker	Aave	Morpho	Frax	Euler
`ln(1+1e-6)`	0.0000009999…	65	28	34	13	12	12	11	11	12	12	—	—	—	—	—	—	—
`ln(1+1e-9)` a per-second rate	0.0000000009…	62	25	30	10	9	9	8	8	9	9	—	—	—	—	—	—	—
`ln(1+1e-12)`	0.0000000000…	59	22	26	7	6	6	5	5	6	5	—	—	—	—	—	—	—
`ln(1+1e-15)`	9.9999999999…	56	19	24	4	15	15	2	2	15	3	—	—	—	—	—	—	—
`ln(1+1e-18)` the last digit of an 18-decimal grid	9.9999999999…	53	16	21	1	18	18	zero	zero	18	zero	—	—	—	—	—	—	—
`ln(1-1e-12)`	-0.000000000…	59	24	26	6	12	12	5	revert	12	12	—	—	—	—	—	—	—
`ln(1e-18)`	-41.44653167…	71	33	22	3	19	19	19	revert	19	20	—	—	—	—	—	—	—
`ln(1e-30)`	-69.07755278…	71	33	10	revert	revert	revert	revert	revert	zero	revert	—	—	—	—	—	—	—
`ln(1e30)`	69.077552789…	71	33	41	range	20	20	18	18	20	20	—	—	—	—	—	—	—
`ln(1e38)`	87.498233533…	71	33	40	range	20	20	18	18	20	20	—	—	—	—	—	—	—
fewest digits		53	16	10	0	0	0	0	0	0	0	—	—	—	—	—	—	—

Revert reasons

PRB UD: PRBMath_UD60x18_Log_InputTooSmall
64.64: bare revert (require without a reason, or out of gas)
Solady: Solady LnWadUndefined()
Solmate: Error(string): a require with a message (Solmate's wadLn and snekmate's wad_ln below zero)
PRB SD: PRBMath_SD59x18_Log_InputTooSmall
Balancer: Balancer OutOfBounds()

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.

		Floating point		Fixed point
probe	truth	BL19	Quad	BL127	64.64	Solady	Solmate	PRB SD	PRB UD	snek	Balancer	OZ	Uniswap	Maker	Aave	Morpho	Frax	Euler
`exp(-1e-12)`	0.9999999999…	71	34	39	18	24	24	18	revert	24	24	—	—	—	—	—	—	—
`exp(-20)`	0.0000000020…	71	32	31	10	9	9	9	revert	9	9	—	—	—	—	—	—	—
`exp(-30)`	9.3576229688…	71	32	26	6	5	5	5	revert	5	5	—	—	—	—	—	—	—
`exp(-40)` a probability of 4e-18: one digit on an 18-decimal grid	4.2483542552…	71	32	21	2	1	1	1	revert	1	1	—	—	—	—	—	—	—
`exp(-41)`	1.5628821893…	71	32	21	1	0	0	0	revert	0	0	—	—	—	—	—	—	—
`exp(-42)` zero on an 18-decimal grid	5.7495222642…	71	32	21	1	zero	zero	zero	revert	zero	revert	—	—	—	—	—	—	—
`exp(-43)`	2.1151310375…	71	32	21	0	zero	zero	zero	revert	zero	revert	—	—	—	—	—	—	—
`exp(-60)`	8.7565107626…	71	32	13	zero	zero	zero	zero	revert	zero	revert	—	—	—	—	—	—	—
`exp(-88)` two ULPs on BL127's grid	6.0546018954…	71	31	1	zero	zero	zero	zero	revert	zero	revert	—	—	—	—	—	—	—
`exp(-89)` 0.76 ULP on BL127's grid, which rounds to one	2.2273635617…	71	31	0	zero	zero	zero	zero	revert	zero	revert	—	—	—	—	—	—	—
`exp(-100)`	3.7200759760…	71	31	zero	zero	zero	zero	zero	revert	zero	revert	—	—	—	—	—	—	—
`exp(40)`	235385266837…	71	32	57	19	20	20	16	16	20	21	—	—	—	—	—	—	—
`exp(43)`	472783946822…	71	32	58	21	21	21	16	16	21	20	—	—	—	—	—	—	—
`exp(44)` beyond ABDK 64.64's range	128516001143…	71	32	57	revert	20	20	16	16	20	20	—	—	—	—	—	—	—
`exp(60)`	114200738981…	71	32	65	revert	20	20	16	16	20	20	—	—	—	—	—	—	—
`exp(88)` the largest BL127 holds	165163625499…	71	31	77	revert	20	20	16	16	20	20	—	—	—	—	—	—	—
`exp(89)` BL127 reverts; the WAD word still has room	448961281917…	71	31	revert	revert	20	20	16	16	20	20	—	—	—	—	—	—	—
`exp(100)`	2.6881171418…	71	31	revert	revert	21	21	16	16	21	21	—	—	—	—	—	—	—
`exp(130)`	2.8726495508…	71	31	revert	revert	20	20	16	16	20	21	—	—	—	—	—	—	—
`exp(135)`	4.2633899483…	71	31	revert	revert	20	20	revert	revert	20	revert	—	—	—	—	—	—	—
`exp(136)` beyond every fixed-point word	1.1589095424…	71	31	revert	revert	revert	revert	revert	revert	revert	revert	—	—	—	—	—	—	—
fewest digits		71	31	0	0	0	0	0	0	0	0	—	—	—	—	—	—	—

Revert reasons

PRB UD: ParamOutOfRange(uint256): the value does not fit the representation
Balancer: Balancer InvalidExponent()
64.64: bare revert (require without a reason, or out of gas)
BL127: Overflow()
PRB SD: PRBMath_SD59x18_Exp_InputTooBig
PRB UD: PRBMath_UD60x18_Exp_InputTooBig
Solady: Solady ExpOverflow()
Solmate: Error(string): a require with a message (Solmate's wadLn and snekmate's wad_ln below zero)
snek: Error(string): a require with a message (Solmate's wadLn and snekmate's wad_ln below zero)

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.

		Floating point		Fixed point
probe	truth	BL19	Quad	BL127	64.64	Solady	Solmate	PRB SD	PRB UD	snek	Balancer	OZ	Uniswap	Maker	Aave	Morpho	Frax	Euler
`sqrt(1e-6)`	0.001	71	34	33	13	exact	exact	exact	exact	exact	exact	exact	—	—	—	—	—	—
`sqrt(1e-12)`	0.000001	71	34	27	8	exact	exact	exact	exact	exact	exact	exact	—	—	—	—	—	—
`sqrt(1e-18)`	0.000000001	71	34	21	1	exact	exact	exact	exact	exact	exact	exact	—	—	—	—	—	—
`sqrt(1e-20)` the input is zero on an 18-decimal grid	0.0000000001	71	34	19	zero	zero	zero	zero	zero	zero	zero	zero	—	—	—	—	—	—
`sqrt(1e-30)`	1.0e-15	71	34	8	zero	zero	zero	zero	zero	zero	zero	zero	—	—	—	—	—	—
`sqrt(1e30)`	100000000000…	exact	exact	exact	range	exact	exact	exact	exact	exact	18	exact	—	—	—	—	—	—
`sqrt(1e36)`	100000000000…	exact	exact	exact	range	exact	exact	exact	exact	exact	18	exact	—	—	—	—	—	—
`sqrt(1e38)`	100000000000…	exact	exact	exact	range	exact	exact	exact	exact	exact	17	exact	—	—	—	—	—	—
`sqrt(1e40)` beyond BL127's word	100000000000…	exact	exact	range	range	exact	exact	exact	exact	exact	18	exact	—	—	—	—	—	—
`sqrt(1e50)`	100000000000…	exact	exact	range	range	exact	revert	revert	revert	revert	18	revert	—	range	range	—	—	—
fewest digits		71	34	0	0	0	0	0	0	0	0	0	—	0	0	—	—	—

Revert reasons

Solmate: Panic (checked arithmetic overflow or division by zero)
PRB SD: PRBMath_SD59x18_Sqrt_Overflow
PRB UD: PRBMath_UD60x18_Sqrt_Overflow
snek: bare revert (require without a reason, or out of gas)
OZ: Panic (checked arithmetic overflow or division by zero)

pow(x, y)

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

		Floating point		Fixed point
probe	truth	BL19	Quad	BL127	64.64	Solady	Solmate	PRB SD	PRB UD	snek	Balancer	OZ	Uniswap	Maker	Aave	Morpho	Frax	Euler
`pow(1+0.05/31536000, 31536000)` 5% APR per second, one year	1.0512710963…	63	27	31	11	10	10	9	9	10	11	—	—	—	—	—	—	—
`pow(1+0.001/31536000, 31536000)` 0.1% APR per second, one year	1.0010005001…	63	26	31	11	10	10	9	9	10	10	—	—	—	—	—	—	—
`pow(1+0.05/31536000, 315360000)` ten years	1.6487212700…	62	26	30	10	9	9	8	8	9	10	—	—	—	—	—	—	—
`pow(1+0.05/31536000/12, 31536000*12)` twelve times a second	1.0512710963…	62	25	30	10	10	10	8	8	10	9	—	—	—	—	—	—	—
`pow(1.0001, 887272)` Uniswap's top tick, 3.4e38: one bit past BL127's word	340256786836…	65	29	revert	revert	12	12	11	11	12	17	—	—	—	—	—	—	—
`pow(1.0001, -887272)` Uniswap's bottom tick, 2.9e-39: one ULP of BL127	2.9389568075…	65	29	4	zero	zero	zero	zero	revert	zero	revert	—	—	—	—	—	—	—
`pow(0.999999, 1000000)`	0.3678792572…	65	28	32	13	12	12	11	11	12	18	—	—	—	—	—	—	—
`pow(2, 0.5)`	1.4142135623…	71	33	39	18	17	17	18	18	17	17	—	—	—	—	—	—	—
`pow(10, -18)`	1.0e-18	71	31	20	1	exact	exact	exact	revert	exact	revert	—	—	—	—	—	—	—
`pow(1e-9, 0.5)`	0.0000316227…	71	32	29	10	9	9	9	9	9	9	—	—	—	—	—	—	—
`pow(1e-18, 0.5)`	0.000000001	71	32	21	1	exact	exact	exact	exact	exact	exact	—	—	—	—	—	—	—
fewest digits		62	25	0	0	0	0	0	0	0	0	—	—	—	—	—	—	—

Revert reasons

BL127: Overflow()
64.64: bare revert (require without a reason, or out of gas)
PRB UD: ParamOutOfRange(uint256): the value does not fit the representation
Balancer: Negative(): a negative value where the library is unsigned (Balancer pow, PRBMath UD60x18, the WAD roots)

mul(a, b)

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

		Floating point		Fixed point
probe	truth	BL19	Quad	BL127	64.64	Solady	Solmate	PRB SD	PRB UD	snek	Balancer	OZ	Uniswap	Maker	Aave	Morpho	Frax	Euler
`mul(1e-9, 1e-9)` the last digit of an 18-decimal grid	1.0e-18	71	33	20	1	zero	zero	zero	zero	zero	zero	zero	zero	—	—	—	—	—
`mul(1e-10, 1e-10)` zero on it	1.0e-20	71	34	18	zero	zero	zero	zero	zero	zero	zero	zero	zero	—	—	—	—	—
`mul(1e-12, 1e-8)`	1.0e-20	71	34	18	zero	zero	zero	zero	zero	zero	zero	zero	zero	—	—	—	—	—
`mul(1e19, 1e19)`	100000000000…	exact	exact	exact	range	exact	exact	exact	exact	exact	exact	exact	exact	—	—	—	—	—
`mul(1e20, 1e20)` beyond BL127's word	100000000000…	exact	exact	revert	range	exact	exact	exact	exact	exact	exact	exact	exact	—	—	—	—	—
fewest digits		71	33	0	0	0	0	0	0	0	0	0	0	—	—	—	—	—

Revert reasons

BL127: Overflow()

div(a, b)

Small over large, and one over tiny.

		Floating point		Fixed point
probe	truth	BL19	Quad	BL127	64.64	Solady	Solmate	PRB SD	PRB UD	snek	Balancer	OZ	Uniswap	Maker	Aave	Morpho	Frax	Euler
`div(1, 3)`	0.3333333333…	71	34	38	19	17	17	17	17	17	17	17	17	—	—	—	—	—
`div(1e-12, 1e9)`	1.0e-21	71	34	17	zero	zero	zero	zero	zero	zero	zero	zero	zero	—	—	—	—	—
`div(1e-9, 1e12)`	1.0e-21	71	33	17	zero	zero	zero	zero	zero	zero	zero	zero	zero	—	—	—	—	—
`div(1e30, 1e-9)`	100000000000…	exact	exact	revert	range	8	8	8	8	8	8	8	8	—	—	—	—	—
`div(1, 1e-30)`	100000000000…	exact	exact	8	revert	revert	revert	revert	revert	revert	revert	revert	revert	—	—	—	—	—
fewest digits		71	33	0	0	0	0	0	0	0	0	0	0	—	—	—	—	—

Revert reasons

BL127: Overflow()
64.64: bare revert (require without a reason, or out of gas)
Solady: Solady SDivWadFailed()
Solmate: bare revert (require without a reason, or out of gas)
PRB SD: Panic (checked arithmetic overflow or division by zero)
PRB UD: Panic (checked arithmetic overflow or division by zero)
snek: Error(string): a require with a message (Solmate's wadLn and snekmate's wad_ln below zero)
Balancer: Panic (checked arithmetic overflow or division by zero)
OZ: Panic (checked arithmetic overflow or division by zero)
Uniswap: bare revert (require without a reason, or out of gas)

W0(x)

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

		Floating point		Fixed point
probe	truth	BL19	Quad	BL127	64.64	Solady	Solmate	PRB SD	PRB UD	snek	Balancer	OZ	Uniswap	Maker	Aave	Morpho	Frax	Euler
`lambertW0(-1/e+1e-6)`	-0.997670166…	69	—	35	—	15	—	—	revert	—	—	—	—	—	—	—	—	—
`lambertW0(-1/e+1e-9)`	-0.999926268…	67	—	34	—	13	—	—	revert	—	—	—	—	—	—	—	—	—
`lambertW0(-1/e+1e-12)`	-0.999997668…	65	—	32	—	12	—	—	revert	—	—	—	—	—	—	—	—	—
`lambertW0(-1/e+1e-15)`	-0.999999926…	64	—	31	—	10	—	—	revert	—	—	—	—	—	—	—	—	—
`lambertW0(-1/e+1e-18)` one grid step from the branch point	-0.999999997…	62	—	29	—	8	—	—	revert	—	—	—	—	—	—	—	—	—
`lambertW0(-1e-18)`	-1.000000000…	71	—	20	—	17	—	—	revert	—	—	—	—	—	—	—	—	—
`lambertW0(1e-18)`	9.9999999999…	71	—	21	—	18	—	—	—	—	—	—	—	—	—	—	—	—
`lambertW0(1e30)`	64.904633770…	71	—	41	range	20	—	—	—	—	—	—	—	—	—	—	—	—
`lambertW0(1e38)`	83.078448213…	71	—	40	range	20	—	—	—	—	—	—	—	—	—	—	—	—
fewest digits		62	—	20	0	8	—	—	0	—	—	—	—	—	—	—	—	—

Revert reasons

PRB UD: ParamOutOfRange(uint256): the value does not fit the representation

W-1(x)

The secondary branch, toward zero from below; BL127 only.

		Floating point		Fixed point
probe	truth	BL19	Quad	BL127	64.64	Solady	Solmate	PRB SD	PRB UD	snek	Balancer	OZ	Uniswap	Maker	Aave	Morpho	Frax	Euler
`lambertWm1(-1/e+1e-9)`	-1.000073734…	67	—	34	—	—	—	—	revert	—	—	—	—	—	—	—	—	—
`lambertWm1(-1e-10)`	-26.29523881…	71	—	30	—	—	—	—	revert	—	—	—	—	—	—	—	—	—
`lambertWm1(-1e-30)`	-73.37311031…	71	—	11	—	—	—	—	revert	—	—	—	—	—	—	—	—	—
fewest digits		67	—	11	—	—	—	—	0	—	—	—	—	—	—	—	—	—

Revert reasons

PRB UD: ParamOutOfRange(uint256): the value does not fit the representation

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.

		Floating point		Fixed point
probe	truth	BL19	Quad	BL127	64.64	Solady	Solmate	PRB SD	PRB UD	snek	Balancer	OZ	Uniswap	Maker	Aave	Morpho	Frax	Euler
`log1p(1e-6)`	0.0000009999…	71	28	34	13	12	12	11	11	12	12	—	—	—	—	—	—	—
`log1p(1e-9)` a per-second rate	0.0000000009…	71	25	30	10	8	8	8	8	8	8	—	—	—	—	—	—	—
`log1p(1e-12)`	0.0000000000…	71	22	26	7	12	12	5	5	12	6	—	—	—	—	—	—	—
`log1p(1e-15)`	9.9999999999…	71	18	24	4	15	15	2	2	15	3	—	—	—	—	—	—	—
`log1p(1e-18)`	9.9999999999…	71	15	21	1	18	18	zero	zero	18	zero	—	—	—	—	—	—	—
`log1p(1e-20)` below an 18-decimal grid	9.9999999999…	71	14	18	zero	zero	zero	zero	zero	zero	zero	—	—	—	—	—	—	—
`log1p(1e-30)`	9.9999999999…	71	4	8	zero	zero	zero	zero	zero	zero	zero	—	—	—	—	—	—	—
`log1p(-1e-9)`	-0.000000001…	71	25	30	10	9	9	8	revert	9	9	—	—	—	—	—	—	—
`log1p(1e6)`	13.815511557…	71	33	40	20	19	19	18	18	19	19	—	—	—	—	—	—	—
fewest digits		71	4	8	0	0	0	0	0	0	0	—	—	—	—	—	—	—

Revert reasons

PRB UD: ParamOutOfRange(uint256): the value does not fit the representation

expm1(x)

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

		Floating point		Fixed point
probe	truth	BL19	Quad	BL127	64.64	Solady	Solmate	PRB SD	PRB UD	snek	Balancer	OZ	Uniswap	Maker	Aave	Morpho	Frax	Euler
`expm1(1e-6)`	0.0000010000…	71	27	33	13	12	12	11	11	12	12	—	—	—	—	—	—	—
`expm1(1e-9)`	0.0000000010…	71	24	29	9	8	8	8	8	8	8	—	—	—	—	—	—	—
`expm1(1e-12)`	0.0000000000…	71	21	26	7	12	12	5	5	12	12	—	—	—	—	—	—	—
`expm1(1e-15)`	1.0000000000…	71	18	23	4	15	15	2	2	15	15	—	—	—	—	—	—	—
`expm1(1e-18)`	1.0000000000…	71	15	20	1	18	18	zero	zero	18	18	—	—	—	—	—	—	—
`expm1(1e-20)`	1.0000000000…	71	14	18	zero	zero	zero	zero	zero	zero	zero	—	—	—	—	—	—	—
`expm1(-1e-9)`	-0.000000000…	71	25	30	10	9	9	9	revert	9	9	—	—	—	—	—	—	—
`expm1(-40)`	-0.999999999…	71	34	38	19	18	18	18	revert	18	18	—	—	—	—	—	—	—
`expm1(-88)`	-0.999999999…	71	34	39	38	38	38	38	revert	38	revert	—	—	—	—	—	—	—
`expm1(20)`	485165194.40…	71	32	47	19	19	19	17	17	19	21	—	—	—	—	—	—	—
fewest digits		71	14	18	0	0	0	0	0	0	0	—	—	—	—	—	—	—

Revert reasons

PRB UD: ParamOutOfRange(uint256): the value does not fit the representation
Balancer: Balancer InvalidExponent()

compound(r, n)

(1 + r)^n for an integer n: per-second rates over years. BL127 and Maker take an exact integer power; the rest exp(n ln(1 + r)).

		Floating point		Fixed point
probe	truth	BL19	Quad	BL127	64.64	Solady	Solmate	PRB SD	PRB UD	snek	Balancer	OZ	Uniswap	Maker	Aave	Morpho	Frax	Euler
`compound(0.05/31536000, 31536000)` 5% APR per second, one year	1.0512710963…	71	26	31	11	12	12	10	10	10	11	—	—	19	—	—	—	—
`compound(0.05/31536000, 315360000)` ten years	1.6487212700…	71	25	30	10	11	11	9	9	9	10	—	—	18	—	—	—	—
`compound(0.001/31536000, 31536000)` 0.1% APR per second	1.0010005001…	71	26	31	12	10	10	10	10	10	10	—	—	19	—	—	—	—
`compound(0.1/365, 365)` the ladder's headline cell	1.1051557816…	71	31	36	17	17	17	15	15	15	16	—	—	24	—	—	—	—
`compound(1e-12, 1000000000)` a rate of 1e-12 over a billion steps	1.0010005001…	71	25	29	11	12	12	12	12	15	18	—	—	22	—	—	—	—
`compound(-0.5, 10)`	0.0009765625	exact	32	exact	exact	exact	exact	exact	revert	14	exact	—	—	revert	—	—	—	—
`compound(0.000000001, 10000000)`	1.0100501670…	71	27	31	12	10	10	10	10	10	10	—	—	19	—	—	—	—
fewest digits		71	25	29	10	10	10	9	0	9	10	—	—	0	—	—	—	—

Revert reasons

PRB UD: ParamOutOfRange(uint256): the value does not fit the representation
Maker: ParamOutOfRange(uint256): the value does not fit the representation

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; BL127 on a 256-bit mantissa.

		Floating point		Fixed point
probe	truth	BL19	Quad	BL127	64.64	Solady	Solmate	PRB SD	PRB UD	snek	Balancer	OZ	Uniswap	Maker	Aave	Morpho	Frax	Euler
`powu(2, 10)`	1024.0	exact	32	exact	exact	exact	exact	exact	exact	17	17	—	—	exact	—	—	—	—
`powu(1.0001, 887272)` Uniswap's top tick	340256786836…	65	29	revert	revert	13	13	12	12	12	17	—	—	revert	—	—	—	—
`powu(1.0001, -887272)` Uniswap's bottom tick	2.9389568075…	65	29	4	revert	zero	zero	zero	zero	zero	zero	—	—	—	—	—	—	—
`powu(0.999999, 1000000)`	0.3678792572…	65	28	32	13	14	14	13	13	12	18	—	—	23	—	—	—	—
`powu(1+0.05/31536000, 31536000)`	1.0512710963…	63	27	31	11	12	12	10	10	10	11	—	—	19	—	—	—	—
`powu(10, 38)`	100000000000…	exact	31	exact	revert	exact	exact	exact	exact	16	16	—	—	revert	—	—	—	—
`powu(3, -5)`	0.0041152263…	71	32	36	17	15	15	15	15	15	15	—	—	—	—	—	—	—
`powu(1.1, 100)`	13780.612339…	69	32	36	17	19	19	19	19	17	15	—	—	29	—	—	—	—
fewest digits		63	27	0	0	0	0	0	0	0	0	—	—	0	—	—	—	—

Revert reasons

BL127: Overflow()
64.64: bare revert (require without a reason, or out of gas)
Maker: Solady RPowOverflow() (Maker's rpow)

erf(x)

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

		Floating point		Fixed point
probe	truth	BL19	Quad	BL127	64.64	Solady	Solmate	PRB SD	PRB UD	snek	Balancer	OZ	Uniswap	Maker	Aave	Morpho	Frax	Euler
`erf(1e-10)`	0.0000000001…	71	33	28	9	8	8	8	8	8	8	—	—	—	—	—	—	—
`erf(0.5)`	0.5204998778…	71	32	39	17	16	16	16	revert	16	16	—	—	—	—	—	—	—
`erf(1)`	0.8427007929…	71	32	39	17	16	16	16	revert	16	16	—	—	—	—	—	—	—
`erf(3)`	0.9999779095…	71	32	39	15	14	14	14	revert	14	14	—	—	—	—	—	—	—
`erf(4)`	0.9999999845…	71	32	39	12	11	11	11	revert	11	11	—	—	—	—	—	—	—
`erf(5)`	0.9999999999…	71	34	38	19	18	18	18	revert	18	18	—	—	—	—	—	—	—
`erf(6)`	0.9999999999…	71	34	39	19	18	18	18	revert	18	18	—	—	—	—	—	—	—
`erf(8)`	0.9999999999…	71	34	39	28	28	28	28	revert	28	revert	—	—	—	—	—	—	—
`erf(9)`	0.9999999999…	71	34	39	36	36	36	36	revert	36	revert	—	—	—	—	—	—	—
fewest digits		71	32	28	9	8	8	8	0	8	0	—	—	—	—	—	—	—

Revert reasons

PRB UD: Negative(): a negative value where the library is unsigned (Balancer pow, PRBMath UD60x18, the WAD roots)
Balancer: Balancer InvalidExponent()

erfc(x)

The tail of the normal distribution: erfc(6) is 2e-17, erfc(9.3) one ULP of BL127, zero on an 18-decimal grid from about 6.4.

		Floating point		Fixed point
probe	truth	BL19	Quad	BL127	64.64	Solady	Solmate	PRB SD	PRB UD	snek	Balancer	OZ	Uniswap	Maker	Aave	Morpho	Frax	Euler
`erfc(1)`	0.1572992070…	71	31	38	17	16	16	16	revert	16	15	—	—	—	—	—	—	—
`erfc(3)`	0.0000220904…	71	27	34	10	9	9	9	revert	9	9	—	—	—	—	—	—	—
`erfc(4)`	0.0000000154…	71	24	31	4	3	3	3	revert	3	3	—	—	—	—	—	—	—
`erfc(5)`	0.0000000000…	71	32	27	7	6	6	6	revert	6	6	—	—	—	—	—	—	—
`erfc(6)` 2e-17: the last digit of an 18-decimal grid	2.1519736712…	71	32	23	2	1	1	1	revert	1	1	—	—	—	—	—	—	—
`erfc(7)`	4.1838256077…	71	32	16	zero	zero	zero	zero	revert	zero	revert	—	—	—	—	—	—	—
`erfc(8)`	1.1224297172…	71	32	10	zero	zero	zero	zero	revert	zero	revert	—	—	—	—	—	—	—
`erfc(9)`	4.1370317465…	71	32	2	zero	zero	zero	zero	revert	zero	revert	—	—	—	—	—	—	—
`erfc(9.3)` one ULP of BL127	1.6532441840…	69	34	0	zero	zero	zero	zero	revert	zero	revert	—	—	—	—	—	—	—
`erfc(10)`	2.0884875837…	71	31	zero	zero	zero	zero	zero	revert	zero	revert	—	—	—	—	—	—	—
fewest digits		69	24	0	0	0	0	0	0	0	0	—	—	—	—	—	—	—

Revert reasons

PRB UD: Negative(): a negative value where the library is unsigned (Balancer pow, PRBMath UD60x18, the WAD roots)
Balancer: Balancer InvalidExponent()

normCdf(x)

N(d) of Black-Scholes: zero on an 18-decimal grid from d = -8.3, two ULPs of BL127 at -13.

		Floating point		Fixed point
probe	truth	BL19	Quad	BL127	64.64	Solady	Solmate	PRB SD	PRB UD	snek	Balancer	OZ	Uniswap	Maker	Aave	Morpho	Frax	Euler
`normCdf(1e-12)`	0.5000000000…	71	34	38	19	18	18	18	18	18	18	—	—	—	—	—	—	—
`normCdf(-1)`	0.1586552539…	71	32	38	17	16	16	16	revert	16	16	—	—	—	—	—	—	—
`normCdf(-3)`	0.0013498980…	71	29	35	14	13	13	13	revert	13	13	—	—	—	—	—	—	—
`normCdf(-5)`	0.0000002866…	71	26	32	8	6	6	6	revert	6	6	—	—	—	—	—	—	—
`normCdf(-6)` a one-in-a-billion event	0.0000000009…	71	32	30	10	8	8	8	revert	8	8	—	—	—	—	—	—	—
`normCdf(-8)`	6.2209605742…	71	31	24	4	3	3	3	revert	3	3	—	—	—	—	—	—	—
`normCdf(-8.3)` the last value above an 18-decimal grid	5.2055697448…	69	32	23	3	2	2	2	revert	2	2	—	—	—	—	—	—	—
`normCdf(-9)`	1.1285884059…	71	31	21	1	zero	zero	zero	revert	zero	zero	—	—	—	—	—	—	—
`normCdf(-10)`	7.6198530241…	71	32	15	zero	zero	zero	zero	revert	zero	revert	—	—	—	—	—	—	—
`normCdf(-13)` two ULPs of BL127	6.1171643995…	71	31	1	zero	zero	zero	zero	revert	zero	revert	—	—	—	—	—	—	—
`normCdf(-14)`	7.7935368191…	71	31	zero	zero	zero	zero	zero	revert	zero	revert	—	—	—	—	—	—	—
`normCdf(5)`	0.9999997133…	71	32	39	14	12	12	12	revert	12	12	—	—	—	—	—	—	—
fewest digits		69	26	0	0	0	0	0	0	0	0	—	—	—	—	—	—	—

Revert reasons

PRB UD: ParamOutOfRange(uint256): the value does not fit the representation
Balancer: Balancer InvalidExponent()
PRB UD: Negative(): a negative value where the library is unsigned (Balancer pow, PRBMath UD60x18, the WAD roots)

A cell is the correct significant digits against the exact value; hover it for the value and its error in ULP. zero: below the format's floor; range: the argument does not fit the word; any other red word is a revert, named by its error (hover for the full reason). A — is an op the library does not offer. BL19's digits are shown to 71, its precision: a 72 or 73 is where the truth happens to sit near the grid. The last row is each library's fewest digits on the op, with a zero, revert or range counted as 0. Full dataset: edges.json. Regenerate with make edges.