xtd

C++ functions for CPU and GPUs with a consistent interface.

Accuracy of mathematical functions

For a detailed comparison of the accuracy of different math library implementations, see

“Accuracy of Mathematical Functions in Single, Double, Double Extended, and Quadruple Precision”, Brian Gladman, Vincenzo Innocente, John Mather, Paul Zimmermann, hal-03141101v8, https://inria.hal.science/hal-03141101 .

GNU libc

On a Linux system the math functions are usually provided by GNU libc. The accuracy of the GNU libc math functions implementation is described in https://www.gnu.org/software/libc/manual/htmlnode/Errors-in-Math-Functions.html and summarised below for an x8664 host system:

function | float | double -----------------|------------|------------ acosf / acos | 1 | 1 acoshf / acosh | 2 | 2 asinf / asin | 1 | 1 asinhf / asinh | 2 | 2 atan2f / atan2 | 2 | 1 atanf / atan | 1 | 1 atanhf / atanh | 2 | 2 cbrtf / cbrt | 1 | 4 cosf / cos | 1 | 1 coshf / cosh | 2 | 2 erfcf / erfc | 3 | 5 erff / erf | 1 | 1 exp10f / exp10 | 1 | 2 exp2f / exp2 | 1 | 1 expf / exp | 1 | 1 expm1f / expm1 | 1 | 1 fabsf / fabs | 0 | 0 fmaf / fma | 0 | 0 fmodf / fmod | 0 | 0 hypotf / hypot | 1 | 1 j0f / j0 | 9 | 3 j1f / j1 | 9 | 4 jnf / jn | 4 | 4 lgammaf / lgamma | 7 | 4 log10f / log10 | 2 | 2 log1pf / log1p | 1 | 1 log2f / log2 | 1 | 2 logf / log | 1 | 1 pow10f / pow10 | n/a | n/a powf / pow | 1 | 1 sincosf / sincos | 1 | 1 sinf / sin | 1 | 1 sinhf / sinh | 2 | 2 sqrtf / sqrt | 0 | 0 tanf / tan | 1 | 1 tanhf / tanh | 2 | 2 tgammaf / tgamma | 8 | 9 y0f / y0 | 9 | 3 y1f / y1 | 9 | 6 ynf / yn | 3 | 3

NVIDIA CUDA

The accuracy of the NVIDIA CUDA math functions implementation is described in https://docs.nvidia.com/cuda/cuda-c-programming-guide/#standard-functions and summarised below for GPUs with compute capability 5.2 or higher:

function | float | double -----------------|------------|------------ acosf / acos | 2 | 2 acoshf / acosh | 4 | 3 asinf / asin | 2 | 2 asinhf / asinh | 3 | 3 atan2f / atan2 | 3 | 2 atanf / atan | 2 | 2 atanhf / atanh | 3 | 2 cbrtf / cbrt | 1 | 1 cosf / cos | 2† | 2 coshf / cosh | 2 | 1 erfcf / erfc | 4 | 5 erff / erf | 2 | 2 exp10f / exp10 | 2† | 1 exp2f / exp2 | 2 | 1 expf / exp | 2† | 1 expm1f / expm1 | 1 | 1 fabsf / fabs | 0 | 0 fmaf / fma | 0 | 0 fmodf / fmod | 0 | 0 hypotf / hypot | 3 | 2 j0f / j0 | 9* | 7ᕯ j1f / j1 | 9* | 7ᕯ jnf / jn | 2 + 2.5×n* | n/aᕯ lgammaf / lgamma | 6' | 4" log10f / log10 | 2† | 1 log1pf / log1p | 1 | 1 log2f / log2 | 1† | 1 logf / log | 1† | 1 pow10f / pow10 | n/a | n/a powf / pow | 4† | 2 sincosf / sincos | 2† | 2 sinf / sin | 2† | 2 sinhf / sinh | 3 | 2 sqrtf / sqrt | 0‡ | 0 tanf / tan | 4† | 2 tanhf / tanh | 2† | 1 tgammaf / tgamma | 5 | 10 y0f / y0 | 9* | 7ᕯ y1f / y1 | 9* | 7ᕯ ynf / yn | 2 + 2.5×n* | n/a

• † unless compiled with --use_fast_math
• ‡ unless compiled with --use_fast_math or --prec-sqrt=false
• * for |x| < 8, otherwise, the maximum absolute error is 2.2×10⁻⁶
• ᕯ for |x| < 8, otherwise, the maximum absolute error is 5×10⁻¹²
• ' outside interval -10.001 … -2.264; larger inside
• " outside interval -23.0001 … -2.2637; larger inside

AMD HIP/ROCm

The accuracy of the AMD ROCm math functions implementation is described in https://rocm.docs.amd.com/projects/HIP/en/latest/reference/math_api.html and summarised below:

function | float | double -----------------|------------|------------ acosf / acos | 1 | 1 acoshf / acosh | 1 | 1 asinf / asin | 2 | 1 asinhf / asinh | 1 | 1 atan2f / atan2 | 3† | 2† atanf / atan | 2 | 1 atanhf / atanh | 1 | 1 cbrtf / cbrt | 2 | 1 cosf / cos | 1 | 1 coshf / cosh | 1 | 1 erfcf / erfc | 2 | 2 erff / erf | 4 | 4 exp10f / exp10 | 1 | 1 exp2f / exp2 | 1 | 1 expf / exp | 1 | 1 expm1f / expm1 | 1 | 2† fabsf / fabs | 0 | 0 fmaf / fma | 0 | 0 fmodf / fmod | 0 | 0 hypotf / hypot | 1 | 1 j0f / j0 | n/a | n/a j1f / j1 | n/a | n/a jnf / jn | n/a | n/a lgammaf / lgamma | 4 | 2 log10f / log10 | 2 | 1 log1pf / log1p | 1 | 1 log2f / log2 | 1 | 1 logf / log | 2 | 1 pow10f / pow10 | n/a | n/a powf / pow | 1 | 1 sincosf / sincos | 1 | 1 sinf / sin | 1 | 1 sinhf / sinh | 1 | 1 sqrtf / sqrt | 1 | 1 tanf / tan | 2† | 2† tanhf / tanh | 2 | 1 tgammaf / tgamma | 6 | 6 y0f / y0 | n/a | n/a y1f / y1 | n/a | n/a ynf / yn | n/a | n/a

• † 1 ULP for a reduced input range, according to the documentation

Note: in some cases the accuracy is documented only for a small range of values.

Intel oneAPI

The SYCL specification does not describe the accuracy of the mathematical functions. Since the SYCL programming model was originally designed as a high-level model for the OpenCL API, the accuracy of the SYCL built-in functions is the same as the corresponding OpenCL functions.

The accuracy of the OpenCL built-in functions is described in https://registry.khronos.org/OpenCL/specs/3.0-unified/html/OpenCL_C.html#relative-error-as-ulps and summarised below:

function | float | double -----------------|------------|------------ acosf / acos | 4 | 4 acoshf / acosh | 4 | 4 asinf / asin | 4 | 4 asinhf / asinh | 4 | 4 atan2f / atan2 | 6 | 6 atanf / atan | 5 | 5* atanhf / atanh | 5 | 5 cbrtf / cbrt | 2 | 2 cosf / cos | 4 | 4 coshf / cosh | 4 | 4 erfcf / erfc | 16 | 16 erff / erf | 16 | 16 exp10f / exp10 | 3 | 3 exp2f / exp2 | 3 | 3 expf / exp | 3 | 3 expm1f / expm1 | 3 | 3 fabsf / fabs | 0 | 0 fmaf / fma | 0 | 0 fmaxf / fmax | 0 | 0 fminf / fmin | 0 | 0 fmodf / fmod | 0 | 0 hypotf / hypot | 4 | 4 j0f / j0 | n/a | n/a j1f / j1 | n/a | n/a jnf / jn | n/a | n/a lgammaf / lgamma | n/a | n/a log10f / log10 | 3 | 3 log1pf / log1p | 2 | 2 log2f / log2 | 3 | 3 logf / log | 3 | 3 pow10f / pow10 | n/a | n/a powf / pow | 16 | 16 sincosf / sincos | 4 | 4 sinf / sin | 4 | 4 sinhf / sinh | 4 | 4 sqrtf / sqrt | 3† | 0† tanf / tan | 5 | 5 tanhf / tanh | 5 | 5 tgammaf / tgamma | 16 | 16 y0f / y0 | n/a | n/a y1f / y1 | n/a | n/a ynf / yn | n/a | n/a

• † may be affected by the -ffast-math compiler option.