DoubleFloats Accuracy Report

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

This report samples a representative subset; it does not cover all available functions.

Double64

Errors are measured against a 512-bit BigFloat reference in units of the Double64 working precision, 1 ulp = 2⁻¹⁰⁴ (about 4.9e-32) relative. A worst case of about 1 ulp means the function is as accurate as the format allows.

Exponentials and Logarithms

function	domain	samples	median (ulps)	worst (ulps)
`exp`	(-100, 100)	250	0.34	3.33
`expm1`	(-1, 1)	250	0.34	4.46
`exp2`	(-100, 100)	250	0.32	2.40
`log`	(0, 100)	250	0.08	0.49
`log`	1 ± 2⁻³¹	250	0.14	1.01
`log1p`	(-1, 1)	250	0.09	0.58
`log2`	(0, 100)	250	0.04	0.36

Trigonometric Functions

function	domain	samples	median (ulps)	worst (ulps)
`sin`	(-100, 100)	250	0.02	0.30
`cos`	(-100, 100)	250	0.02	0.24
`tan`	(-1, 1)	250	0.13	1.44
`sin`	near π/2	250	0.02	0.21
`cos`	near π/2	250	0.07	0.41
`cos`	near π	250	0.02	0.23

Inverse Trigonometric Functions

function	domain	samples	median (ulps)	worst (ulps)
`asin`	(-1, 1)	250	0.02	0.12
`acos`	(-1, 1)	250	0.03	0.12
`atan`	(-100, 100)	250	0.03	0.34

Hyperbolic and Inverse Hyperbolic Functions

function	domain	samples	median (ulps)	worst (ulps)
`sinh`	(-2, 2)	250	0.41	3.51
`sinh`	(-9, 9)	250	0.39	3.13
`cosh`	(-2, 2)	250	0.22	3.13
`cosh`	(-9, 9)	250	0.28	2.81
`tanh`	(-2, 2)	250	0.16	2.97
`tanh`	(-9, 9)	250	0.10	0.95
`asinh`	(-100, 100)	250	0.08	0.43
`asinh`	±2⁻⁴⁰	250	0.12	0.79
`acosh`	(1, 101)	250	0.07	0.66
`acosh`	1 + ε	250	0.17	0.99
`atanh`	(-1, 1)	250	0.10	2.12
`atanh`	±2⁻⁴⁰	250	0.09	1.11

Special Functions

erf, erfc, and gamma are computed via Float128; its 113-bit significand caps their accuracy at roughly half an ulp of Double64.

function	domain	samples	median (ulps)	worst (ulps)
`erf`	(-1, 1)	250	0.03	0.11
`erfc`	(-1, 1)	250	0.02	0.11
`gamma`	(0.1, 10)	250	0.02	0.11
`ellipk`	(0, 1)	250	0.12	1.06
`ellipk`	1 - 2⁻²⁰⁻⁵⁰	250	0.17	1.02

Matrix Functions (identity residuals)

Matrix routines are checked through their defining identities on random matrices, normalized per matrix dimension. An unavailable row identifies a routine that cannot currently execute for that precision.

identity	size	samples	median (ulps)	worst (ulps)
`eigen residual ‖AV−VΛ‖`	8×8	25	0.39	0.66
`exp∘log roundtrip`	8×8	25	2.26	2.78
`sqrt squared`	8×8	25	2.01	4.13
`sin²+cos²−I`	8×8	25	5.30	12.20

Double32

Errors are measured against a 512-bit BigFloat reference in units of the Double32 working precision, 1 ulp = 2⁻⁴⁶ (about 1.4e-14) relative. A worst case of about 1 ulp means the function is as accurate as the format allows.