DoubleFloats Accuracy Report

Generated by docs/reports/accuracies.jl on Julia 1.12.6.

Each function is evaluated at random in-domain points and compared against a 512-bit BigFloat reference. Errors are reported in units of the Double64 working precision, 1 ulp = 2⁻¹⁰⁴ relative (about 4.9e-32); a worst case of ~1 ulp means the function is as accurate as the format allows. This report samples a representative subset; it does not cover all available functions.

Exponentials and Logarithms

function	domain	samples	median (ulps)	worst (ulps)
`exp`	(-100, 100)	250	0.13	0.81
`exp`	(-660, -600)	250	0.02	0.12
`expm1`	(-1, 1)	250	0.02	0.12
`log`	(0, 100)	250	0.05	0.51
`log`	1 ± 2⁻³¹	250	0.05	1.04
`log1p`	(-1, 1)	250	0.10	1.00
`log2`	(0, 100)	250	0.06	0.49
`log10`	(0, 100)	250	0.66	0.98

Note: exp(x) for x below about −670 is representation-limited — the result's low word falls below the subnormal floor, so no algorithm returning a Double64 can do better; errors there are excluded from the table because they measure the format, not the function.

Trigonometric Functions

function	domain	samples	median (ulps)	worst (ulps)
`sin`	(-100, 100)	250	0.02	0.67
`cos`	(-100, 100)	250	0.02	10.26
`tan`	(-1, 1)	250	0.04	0.66
`sin`	near π/2	250	0.08	0.51
`cos`	near π/2	250	0.66	209.14
`cos`	near π	250	0.07	0.50

Note: near the zeros of a function (e.g. cos at π/2) the relative measure used here inflates, because the mathematical condition number is unbounded there. The large worst-ulp entries for cos come from samples landing very close to a root; absolute error stays at the 2⁻¹⁰⁴ level throughout.

Inverse Trigonometric Functions

function	domain	samples	median (ulps)	worst (ulps)
`asin`	(-1, 1)	250	0.02	0.12
`acos`	(-1, 1)	250	0.02	0.12
`atan`	(-100, 100)	250	0.02	0.09

Hyperbolic and Inverse Hyperbolic Functions

function	domain	samples	median (ulps)	worst (ulps)
`sinh`	(-300, 300)	250	0.15	1.02
`cosh`	(-100, 100)	250	0.12	1.03
`tanh`	(-1, 1)	250	0.23	16.32
`asinh`	(-100, 100)	250	0.04	0.31
`asinh`	±2⁻⁴⁰	250	0.08	0.91
`acosh`	(1, 101)	250	0.05	0.25
`acosh`	1 + ε	250	0.12	1.09
`atanh`	(-1, 1)	250	0.09	0.98
`atanh`	±2⁻⁴⁰	250	0.00	0.99

Note: sinh and tanh lose a few bits for very small arguments to cancellation in exp(x) - exp(-x); the worst cases above (~16 ulps) come from that region.

Special Functions

erf, erfc, and gamma are computed via Float128; its 113-bit significand caps their accuracy at roughly half an ulp of Double64, and the measured values below reflect that route.

function	domain	samples	median (ulps)	worst (ulps)
`erf`	(-1, 1)	250	0.02	0.11
`erfc`	(-1, 1)	250	0.02	0.11
`gamma`	(0.1, 10)	250	0.02	0.11
`ellipk`	(0, 1)	250	0.11	0.70
`ellipk`	1 - 2⁻²⁰⁻⁵⁰	250	0.16	0.93

Matrix Functions (identity residuals)

Matrix routines are checked through their defining identities on random matrices, normalized per matrix dimension.

identity	size	samples	median (ulps)	worst (ulps)
`eigen residual ‖AV−VΛ‖`	8×8	25	0.38	0.88
`exp∘log roundtrip`	8×8	25	1.90	3.22
`sqrt squared`	8×8	25	1.98	3.57
`sin²+cos²−I`	8×8	25	0.38	0.78