DoubleFloats Accuracy Report

Generated by docs/reports/accuracies.jl on Julia 1.13.0-DEV.1123.

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	0.16	0.60
`exp`	(-660, -600)	250	0.02	0.12
`expm1`	(-1, 1)	250	0.11	0.70
`log`	(0, 100)	250	0.06	0.32
`log`	1 ± 2⁻³¹	250	0.05	1.04
`log1p`	(-1, 1)	250	0.05	0.90
`log2`	(0, 100)	250	0.08	0.49
`log10`	(0, 100)	250	0.67	1.02

Trigonometric Functions

function	domain	samples	median (ulps)	worst (ulps)
`sin`	(-100, 100)	250	0.02	0.23
`cos`	(-100, 100)	250	0.02	0.22
`tan`	(-1, 1)	250	0.14	0.85
`sin`	near π/2	250	0.03	0.23
`cos`	near π/2	250	0.07	0.48
`cos`	near π	250	0.02	0.19

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	0.03	0.42

Hyperbolic and Inverse Hyperbolic Functions

function	domain	samples	median (ulps)	worst (ulps)
`sinh`	(-300, 300)	250	0.17	0.96
`cosh`	(-100, 100)	250	0.13	0.72
`tanh`	(-1, 1)	250	0.13	1.16
`asinh`	(-100, 100)	250	0.07	0.37
`asinh`	±2⁻⁴⁰	250	0.08	0.91
`acosh`	(1, 101)	250	0.07	0.32
`acosh`	1 + ε	250	0.12	1.09
`atanh`	(-1, 1)	250	0.06	0.90
`atanh`	±2⁻⁴⁰	250	0.00	0.99

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	0.02	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. 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	0.38	0.88
`exp∘log roundtrip`	8×8	25	1.90	3.22
`sqrt squared`	8×8	25	2.03	3.57
`sin²+cos²−I`	8×8	25	6.13	13.86

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.