# CUDA vs OpenCL math builtin precision ## CUDA Guarantees From [Appendix E.1 of the CUDA C Programming Guide][cuda_c_ulp]: > This section specifies the error bounds of each function when executed on the device and also > when executed on the host in the case where the host does not supply the function. > > The error bounds are generated from extensive but not exhaustive tests, so they are not > guaranteed bounds. In [Section 11.1.5 of the CUDA C Best Practices Guide][cuda_best_prac] on Math Libraries and [Section 11.1.6 of the CUDA C Best Practices Guide][cuda_best_prac_precision] on Precision-related Compiler Flags, there are mentions of the precision of math built-ins. [cuda_best_prac]: https://docs.nvidia.com/cuda/cuda-c-best-practices-guide/index.html#math-libraries [cuda_best_prac_precision]: https://docs.nvidia.com/cuda/cuda-c-best-practices-guide/index.html#precision-related-compiler-flags ## Single Precision The following table uses the following sources: - [Section 7.4 of the OpenCL 1.2 Specification][opencl_1.2_ulp] - [Appendix E.1 of the CUDA C Programming Guide][cuda_c_ulp] which is referenced from the [CUDA Math API documentation][cuda_math_single] In addition to the following table, the CUDA documentation also includes: > Addition and multiplication are IEEE-compliant, so have a maximum error of 0.5 ulp. > > The recommended way to round a single-precision floating-point operand to an integer, with > the result being a single-precision floating-point number is rintf(), not roundf(). The reason > is that roundf() maps to an 8-instruction sequence on the device, whereas rintf() maps to a > single instruction. truncf(), ceilf(), and floorf() each map to a single instruction as well. OpenCL defines ULP (units in last place) as: > If x is a real number that lies between two finite consecutive floating-point numbers a and b, > without being equal to one of them, then ulp(x) = |b − a|, otherwise ulp(x) is the distance > between the two non-equal finite floating-point numbers nearest x. Moreover, ulp(NaN) is NaN. Maximum error is defined in the CUDA documentation as: > The maximum error is stated as the absolute value of the difference in ulps between a correctly > rounded single-precision result and the result returned by the CUDA library function. | OpenCL Built-in | OpenCL Min Accuracy (ULP) | CUDA Built-in | CUDA Maximum Error (ULP) | | --------------- | ------------------------- | ------------- | ------------------------ | | `x + y` | Correctly rounded | `x + y` | 0 ulp (IEEE-754 round-to-nearest-even) | | `x - y` | Correctly rounded | N/A | N/A | | `x * y` | Correctly rounded | `x * y` | 0 ulp (IEEE-754 round-to-nearest-even) | | [`1.0 / x`][`recip`] | ≤ 2.5 ulp | `1.0 / x` | 0 ulp (if compute capability ≥ 2 when compiled with `-prec-div=true`), 1 ulp (full range) otherwise | | [`x / y`][`divide`] | ≤ 2.5 ulp | `x / y` | 0 ulp (if compute capability ≥ 2 when compiled with `-prec-div=true`), 2 ulp (full range) otherwise | | [`acos`] | ≤ 4 ulp | [`acosf`] | 3 ulp (full range) | | [`acospi`][`acos`] | ≤ 5 ulp | N/A | N/A | | [`asin`] | ≤ 4 ulp | [`asinf`] | 4 ulp (full range) | | [`asinpi`][`asin`] | ≤ 5 ulp | N/A | N/A | | [`atan`] | ≤ 5 ulp | [`atanf`] | 2 ulp (full range) | | [`atan2`][`atan`] | ≤ 6 ulp | [`atan2f`] | 3 ulp (full range) | | [`atanpi`][`atan`] | ≤ 5 ulp | N/A | N/A | | [`atan2pi`][`atan`] | ≤ 6 ulp | N/A | N/A | | [`acosh`][`acos`] | ≤ 4 ulp | [`acoshf`] | 4 ulp (full range) | | [`asinh`][`asin`] | ≤ 4 ulp | [`asinhf`] | 3 ulp (full range) | | [`atanh`][`atan`] | ≤ 5 ulp | [`atanhf`] | 3 ulp (full range) | | [`cbrt`] | ≤ 2 ulp | [`cbrtf`] | 1 ulp (full range) | | [`ceil`] | Correctly rounded | [`ceilf`] | 0 ulp (full range) | | [`copysign`] | 0 ulp | [`copysignf`] | Undocumented. | | [`cos`] | ≤ 4 ulp | [`cosf`] | 2 ulp (full range) | | [`cosh`][`cos`] | ≤ 4 ulp | [`coshf`] | 2 ulp (full range) | | [`cospi`][`cos`] | ≤ 4 ulp | [`cospi`] | 2 ulp (full range) | | N/A | N/A | [`cyl_bessel_i0f`] | 6 ulp (full range) | | N/A | N/A | [`cyl_bessel_i1f`] | 6 ulp (full range) | | [`erfc`][`erf`] | ≤ 16 ulp | [`erfcf`] | 4 ulp (full range) | | N/A | N/A | [`erfcinvf`] | 2 ulp (full range) | | N/A | N/A | [`erfcxf`] | 4 ulp (full range) | | N/A | N/A | [`erfinvf`] | 2 ulp (full range) | | [`erf`] | ≤ 16 ulp | [`erff`] | 2 ulp (full range) | | [`exp`] | ≤ 3 ulp | [`expf`] | 2 ulp (full range) | | [`exp2`][`exp`] | ≤ 3 ulp | [`exp2f`] | 2 ulp (full range) | | [`exp10`][`exp`] | ≤ 3 ulp | [`exp10f`] | 2 ulp (full range) | | [`expm1`][`exp`] | ≤ 3 ulp | [`expm1f`] | 1 ulp (full range) | | [`fabs`] | 0 ulp | [`fabsf`] | Undocumented. | | [`fdim`] | Correctly rounded | [`fdimf`] | 0 ulp (full range) | | [`floor`] | Correctly rounded | [`floorf`] | 0 ulp (full range) | | [`fma`] | Correctly rounded | [`fmaf`] | 0 ulp (full range) | | [`fmax`] | 0 ulp | [`fmaxf`] | Undocumented. | | [`fmin`] | 0 ulp | [`fminf`] | Undocumented. | | [`fmod`] | 0 ulp | [`fmodf`] | 0 ulp (full range) | | [`fract`] | Correctly rounded | N/A | N/A | | [`frexp`] | 0 ulp | [`frexpf`] | 0 ulp (full range) | | [`hypot`] | ≤ 4 ulp | [`hypotf`] | 3 ulp (full range) | | [`ilogb`] | 0 ulp | [`ilogbf`] | 0 ulp (full range) | | N/A | N/A | [`j0f`] | 9 ulp for `abs(x) < 8`, otherwise `2.2 x 10^(-6)` | | N/A | N/A | [`j1f`] | 9 ulp for `abs(x) < 8`, otherwise `2.2 x 10^(-6)` | | N/A | N/A | [`jnf`] | For `n = 128`, `2.2 x 10^(-6)` | | [`ldexp`] | Correctly rounded | [`ldexpf`] | 0 ulp (full range) | | N/A | N/A | [`lgammaf`] | 6 ulp (outside interval `-10.001 ... -2.264; larger inside`) | | [`log`] | ≤ 3 ulp | [`logf`] | 1 ulp (full range) | | [`log2`][`log`] | ≤ 3 ulp | [`log2f`] | 1 ulp (full range) | | [`log10`][`log`] | ≤ 3 ulp | [`log10f`] | 2 ulp (full range) | | [`log1p`][`log`] | ≤ 2 ulp | [`log1pf`] | 1 ulp (full range) | | [`logb`][`log`] | 0 ulp | [`logbf`] | 0 ulp (full range) | | N/A | N/A | [`lrintf`] | 0 ulp (full range) | | N/A | N/A | [`lroundf`] | 0 ulp (full range) | | N/A | N/A | [`llrintf`] | 0 ulp (full range) | | N/A | N/A | [`llroundf`] | 0 ulp (full range) | | [`mad`] | Any value allowed (infinite ulp) | N/A | N/A | | [`maxmag`][`mag`] | 0 ulp | N/A | N/A | | [`minmag`][`mag`] | 0 ulp | N/A | N/A | | [`modf`] | 0 ulp | [`modff`] | 0 ulp (full range) | | [`nan`] | 0 ulp | [`nanf`] | Undocumented. | | N/A | N/A | [`nearbyintf`] | 0 ulp (full range) | | [`nextafter`] | 0 ulp | [`nextafterf`] | Undocumented. | | N/A | N/A | [`normf`] | 4 ulp (full range) | | N/A | N/A | [`normcdff`] | 5 ulp (full range) | | N/A | N/A | [`normcdfinvf`] | 5 ulp (full range) | | N/A | N/A | [`norm3df`] | 3 ulp (full range) | | N/A | N/A | [`norm4df`] | 3 ulp (full range) | | [`pow(x, y)`][`pow`] | ≤ 16 ulp | [`powf`] | 8 ulp (full range) | | [`pown(x, y)`][`pow`] | ≤ 16 ulp | N/A | N/A | | [`powr(x, y)`][`pow`] | ≤ 16 ulp | N/A | N/A | | N/A | N/A | [`rcbrtf`] | 1 ulp (full range) | | N/A | N/A | [`rhypot`] | 2 ulp (full range) | | N/A | N/A | [`rnormf`] | 3 ulp (full range) | | N/A | N/A | [`rnorm3df`] | 2 ulp (full range) | | N/A | N/A | [`rnorm4df`] | 2 ulp (full range) | | [`remainder`] | 0 ulp | [`remainderf`] | 0 ulp (full range) | | [`remquo`] | 0 ulp | [`remquof`] | 0 ulp (full range) | | [`rint`] | Correctly rounded | [`rintf`] | 0 ulp (full range) | | [`rootn`] | ≤ 16 ulp | N/A | N/A | | [`round`] | Correctly rounded | [`roundf`] | 0 ulp (full range) | | [`rsqrt`][`sqrt`] | ≤ 2 ulp | [`rsqrtf`] | 2 ulp (full range) (applies to `1 / sqrtf(x)` only when converted to `rsqrtf` by compiler) | | N/A | N/A | [`scalbnf`] | 0 ulp (full range) | | N/A | N/A | [`scalblnf`] | 0 ulp (full range) | | [`sin`] | ≤ 4 ulp | [`sinf`] | 2 ulp (full range) | | [`sincos`][`sin`] | ≤ 4 ulp for sine and cosine values | [`sincosf`] | 2 ulp (full range) | | N/A | N/A | [`sincospif`] | 2 ulp (full range) | | [`sinh`][`sin`] | ≤ 4 ulp | [`sinhf`] | 3 ulp (full range) | | [`sinpi`][`sin`] | ≤ 4 ulp | [`sinpif`] | 2 ulp (full range) | | [`sqrt`] | ≤ 3 ulp | [`sqrtf`] | 0 ulp (when compiled with `-prec-sqrt=true`) otherwise 1 ulp if compute capability ≥ 5.2 and 3 ulp otherwise. | | [`tan`] | ≤ 5 ulp | [`tanf`] | 4 ulp (full range) | | [`tanh`][`tan`] | ≤ 5 ulp | [`tanhf`] | 2 ulp (full range) | | [`tanpi`][`tan`] | ≤ 6 ulp | N/A | N/A | | [`tgamma`] | ≤ 16 ulp | [`tgammaf`] | 11 ulp (full range) | | [`trunc`] | Correctly rounded | [`truncf`] | 0 ulp (full range) | | N/A | N/A | [`y0f`] | 9 ulp for `abs(x) < 8`, otherwise `2.2 x 10^(-6)` | | N/A | N/A | [`y1f`] | 9 ulp for `abs(x) < 8`, otherwise `2.2 x 10^(-6)` | | N/A | N/A | [`ynf`] | `ceil(2 + 2.5n)` for `abs(x) < n`, otherwise `2.2 x 10^(-6)` | | N/A | N/A | [`isfinite`] | N/A | | N/A | N/A | [`isinf`] | N/A | | N/A | N/A | [`isnan`] | N/A | | N/A | N/A | [`signbit`] | N/A | OpenCL's `native_` math built-ins map to the same CUDA built-in as the equivalent non-`native_` OpenCL built-in and the precision is implementation-defined: - [`native_cos`][`cos`] - [`native_divide`][`divide`] - [`native_exp`][`exp`] - [`native_exp2`][`exp`] - [`native_exp10`][`exp`] - [`native_log`][`log`] - [`native_log2`][`log`] - [`native_log10`][`log`] - [`native_powr`][`pow`] - [`native_recip`][`recip`] - [`native_rsqrt`][`sqrt`] - [`native_sin`][`sin`] - [`native_sqrt`][`sqrt`] - [`native_tan`][`tan`] In [section 7.4 of the OpenCL 2.1 Specification][opencl_2.1_ulp], `mad` has a different requirement, namely: > Implemented either as a correctly rounded fma or as a multiply followed by an add both of which > are correctly rounded. Precision of SPIR-V math instructions for use in an OpenCL environment, can be [found in this document][opencl_env_ulp]. [cuda_c_ulp]: https://docs.nvidia.com/cuda/cuda-c-programming-guide/index.html#standard-functions [cuda_math_single]: https://docs.nvidia.com/cuda/cuda-math-api/group__CUDA__MATH__SINGLE.html#group__CUDA__MATH__SINGLE [opencl_1.2_ulp]: https://www.khronos.org/registry/OpenCL/specs/opencl-1.2.pdf#page=319 [opencl_2.1_ulp]: https://www.khronos.org/registry/OpenCL/specs/2.2/html/OpenCL_C.html#relative-error-as-ulps [opencl_env_ulp]: https://www.khronos.org/registry/OpenCL/specs/2.2/html/OpenCL_Env.html#relative-error-as-ulps ## Double Precision The following table uses the following sources: - [Section 7.4 of the OpenCL 1.2 Specification][opencl_1.2_dp_ulp] - [Appendix E.1 of the CUDA C Programming Guide][cuda_c_ulp] which is referenced from the [CUDA Math API documentation][cuda_math_double] CUDA defines maximum error in the same way as for single precision, and also includes: > The recommended way to round a double-precision floating-point operand to an integer, with the result being a double-precision > floating-point number is rint(), not round(). The reason is that round() maps to an 8-instruction sequence on the device, > whereas rint() maps to a single instruction. trunc(), ceil(), and floor() each map to a single instruction as well. Only differences from single precision are included. There are only changes to `1.0 / x`, `x / y` and `sqrt` from OpenCL. All built-in names changed for CUDA and many precisions too. | OpenCL Built-in | OpenCL Min Accuracy (ULP) | CUDA Built-in | CUDA Maximum Error (ULP) | | --------------- | ------------------------- | ------------- | ------------------------ | | `x + y` | Correctly rounded | `x + y` | 0 ulp (IEEE-754 round-to-nearest-even) | | `x - y` | Correctly rounded | N/A | N/A | | `x * y` | Correctly rounded | `x * y` | 0 ulp (IEEE-754 round-to-nearest-even) | | [`1.0 / x`][`recip`] | Correctly rounded | `1.0 / x` | 0 ulp (IEEE-754 round-to-nearest-even | | [`x / y`][`divide`] | Correctly rounded | `x / y` | 0 ulp (IEEE-754 round-to-nearest-even) | | [`acos`] | ≤ 4 ulp | [`acos`][`acos`_cuda] | 1 ulp (full range) | | [`acospi`][`acos`] | ≤ 5 ulp | N/A | N/A | | [`asin`] | ≤ 4 ulp | [`asin`][`asin`_cuda] | 2 ulp (full range) | | [`asinpi`][`asin`] | ≤ 5 ulp | N/A | N/A | | [`atan`] | ≤ 5 ulp | [`atan`][`atan`_cuda] | 2 ulp (full range) | | [`atan2`][`atan`] | ≤ 6 ulp | [`atan2`][`atan2`_cuda] | 2 ulp (full range) | | [`atanpi`][`atan`] | ≤ 5 ulp | N/A | N/A | | [`atan2pi`][`atan`] | ≤ 6 ulp | N/A | N/A | | [`acosh`][`acos`] | ≤ 4 ulp | [`acosh`][`acosh`_cuda] | 2 ulp (full range) | | [`asinh`][`asin`] | ≤ 4 ulp | [`asinh`][`asinh`_cuda] | 2 ulp (full range) | | [`atanh`][`atan`] | ≤ 5 ulp | [`atanh`][`atanh`_cuda] | 2 ulp (full range) | | [`cbrt`] | ≤ 2 ulp | [`cbrt`][`cbrt`_cuda] | 1 ulp (full range) | | [`ceil`] | Correctly rounded | [`ceil`][`ceil`_cuda] | 0 ulp (full range) | | [`copysign`] | 0 ulp | [`copysign`][`copysign`_cuda] | Undocumented. | | [`cos`] | ≤ 4 ulp | [`cos`][`cos`_cuda] | 1 ulp (full range) | | [`cosh`][`cos`] | ≤ 4 ulp | [`cosh`][`cosh`_cuda] | 1 ulp (full range) | | [`cospi`][`cos`] | ≤ 4 ulp | [`cospi`][`cospi`_cuda] | 1 ulp (full range) | | N/A | N/A | [`cyl_bessel_i0`][`cyl_bessel_i0`_cuda] | 6 ulp (full range) | | N/A | N/A | [`cyl_bessel_i1`][`cyl_bessel_i1`_cuda] | 6 ulp (full range) | | [`erfc`][`erf`] | ≤ 16 ulp | [`erfc`][`erfc`_cuda] | 4 ulp (full range) | | N/A | N/A | [`erfcinv`][`erfcinv`_cuda] | 6 ulp (full range) | | N/A | N/A | [`erfcx`][`erfcx`_cuda] | 3 ulp (full range) | | N/A | N/A | [`erfinv`][`erfinv`_cuda] | 5 ulp (full range) | | [`erf`] | ≤ 16 ulp | [`erf`][`erf`_cuda] | 2 ulp (full range) | | [`exp`] | ≤ 3 ulp | [`exp`][`exp`_cuda] | 1 ulp (full range) | | [`exp2`][`exp`] | ≤ 3 ulp | [`exp2`][`exp2`_cuda] | 1 ulp (full range) | | [`exp10`][`exp`] | ≤ 3 ulp | [`exp10`][`exp10`_cuda] | 1 ulp (full range) | | [`expm1`][`exp`] | ≤ 3 ulp | [`expm1`][`expm1`_cuda] | 1 ulp (full range) | | [`fabs`] | 0 ulp | [`fabs`][`fabs`_cuda] | Undocumented. | | [`fdim`] | Correctly rounded | [`fdim`][`fdim`_cuda] | 0 ulp (full range) | | [`floor`] | Correctly rounded | [`floor`][`floor`_cuda] | 0 ulp (full range) | | [`fma`] | Correctly rounded | [`fma`][`fma`_cuda] | 0 ulp (IEEE-754 round-to-nearest-even) | | [`fmax`] | 0 ulp | [`fmax`][`fmax`_cuda] | Undocumented. | | [`fmin`] | 0 ulp | [`fmin`][`fmin`_cuda] | Undocumented. | | [`fmod`] | 0 ulp | [`fmod`][`fmod`_cuda] | 0 ulp (full range) | | [`fract`] | Correctly rounded | N/A | N/A | | [`frexp`] | 0 ulp | [`frexp`][`frexp`_cuda] | 0 ulp (full range) | | [`hypot`] | ≤ 4 ulp | [`hypot`][`hypot`_cuda] | 2 ulp (full range) | | [`ilogb`] | 0 ulp | [`ilogb`][`ilogb`_cuda] | 0 ulp (full range) | | N/A | N/A | [`j0`][`j0`_cuda] | 7 ulp for `abs(x) < 8`, otherwise `5 x 10^(-12)` | | N/A | N/A | [`j1`][`j1`_cuda] | 7 ulp for `abs(x) < 8`, otherwise `5 x 10^(-12)` | | N/A | N/A | [`jn`][`jn`_cuda] | For `n = 128`, `5 x 10^(-12)` | | [`ldexp`] | Correctly rounded | [`ldexp`][`ldexp`_cuda] | 0 ulp (full range) | | N/A | N/A | [`lgamma`][`lgamma`_cuda] | 4 ulp (outside interval `-11.0001 ... -2.2637; larger inside`) | | [`log`] | ≤ 3 ulp | [`log`][`log`_cuda] | 1 ulp (full range) | | [`log2`][`log`] | ≤ 3 ulp | [`log2`][`log2`_cuda] | 1 ulp (full range) | | [`log10`][`log`] | ≤ 3 ulp | [`log10`][`log10`_cuda] | 1 ulp (full range) | | [`log1p`][`log`] | ≤ 2 ulp | [`log1p`][`log1p`_cuda] | 1 ulp (full range) | | [`logb`][`log`] | 0 ulp | [`logb`][`logb`_cuda] | 0 ulp (full range) | | N/A | N/A | [`lrint`][`lrint`_cuda] | 0 ulp (full range) | | N/A | N/A | [`lround`][`lround`_cuda] | 0 ulp (full range) | | N/A | N/A | [`llrint`][`llrint`_cuda] | 0 ulp (full range) | | N/A | N/A | [`llround`][`llround`_cuda] | 0 ulp (full range) | | [`mad`] | Any value allowed (infinite ulp) | N/A | N/A | | [`maxmag`][`mag`] | 0 ulp | N/A | N/A | | [`minmag`][`mag`] | 0 ulp | N/A | N/A | | [`modf`] | 0 ulp | [`mod`][`mod`_cuda] (might be called `modf`, the documentation is inconsistent) | 0 ulp (full range) | | [`nan`] | 0 ulp | [`nan`][`nan`_cuda] | Undocumented. | | N/A | N/A | [`nearbyint`][`nearbyint`_cuda] | 0 ulp (full range) | | [`nextafter`] | 0 ulp | [`nextafter`][`nextafter`_cuda] | Undocumented. | | N/A | N/A | [`norm`][`norm`_cuda] | 3 ulp (full range) | | N/A | N/A | [`normcdf`][`normcdf`_cuda] | 5 ulp (full range) | | N/A | N/A | [`normcdfinv`][`normcdfinv`_cuda] | 7 ulp (full range) | | N/A | N/A | [`norm3d`][`norm3d`_cuda] | 2 ulp (full range) | | N/A | N/A | [`norm4d`][`norm4d`_cuda] | 2 ulp (full range) | | [`pow(x, y)`][`pow`] | ≤ 16 ulp | [`pow`][`pow`_cuda] | 2 ulp (full range) | | [`pown(x, y)`][`pow`] | ≤ 16 ulp | N/A | N/A | | [`powr(x, y)`][`pow`] | ≤ 16 ulp | N/A | N/A | | N/A | N/A | [`rcbrt`][`rcbrt`_cuda] | 1 ulp (full range) | | N/A | N/A | [`rhypot`][`rhypot`_cuda] | 1 ulp (full range) | | N/A | N/A | [`rnorm`][`rnorm`_cuda] | 2 ulp (full range) | | N/A | N/A | [`rnorm3d`][`rnorm3d`_cuda] | 1 ulp (full range) | | N/A | N/A | [`rnorm4d`][`rnorm4d`_cuda] | 1 ulp (full range) | | [`remainder`] | 0 ulp | [`remainder`][`remainder`_cuda] | 0 ulp (full range) | | [`remquo`] | 0 ulp | [`remquo`][`remquo`_cuda] | 0 ulp (full range) | | [`rint`] | Correctly rounded | [`rint`][`rint`_cuda] | 0 ulp (full range) | | [`rootn`] | ≤ 16 ulp | N/A | N/A | | [`round`] | Correctly rounded | [`round`][`round`_cuda] | 0 ulp (full range) | | [`rsqrt`][`sqrt`] | ≤ 2 ulp | [`rsqrt`][`rsqrt`_cuda] | 1 ulp (full range) | | N/A | N/A | [`scalbn`][`scalbn`_cuda] | 0 ulp (full range) | | N/A | N/A | [`scalbln`][`scalbln`_cuda] | 0 ulp (full range) | | [`sin`] | ≤ 4 ulp | [`sin`][`sin`_cuda] | 1 ulp (full range) | | [`sincos`][`sin`] | ≤ 4 ulp for sine and cosine values | [`sincos`][`sincos`_cuda] | 1 ulp (full range) | | N/A | N/A | [`sincospi`][`sincospi`_cuda] | 1 ulp (full range) | | [`sinh`][`sin`] | ≤ 4 ulp | [`sinh`][`sinh`_cuda] | 1 ulp (full range) | | [`sinpi`][`sin`] | ≤ 4 ulp | [`sinpi`][`sinpi`_cuda] | 1 ulp (full range) | | [`sqrt`] | Correctly rounded | [`sqrt`][`sqrt`_cuda] | 0 ulp (IEEE-754 round-to-nearest-even) | | [`tan`] | ≤ 5 ulp | [`tan`][`tan`_cuda] | 2 ulp (full range) | | [`tanh`][`tan`] | ≤ 5 ulp | [`tanh`][`tanh`_cuda] | 1 ulp (full range) | | [`tanpi`][`tan`] | ≤ 6 ulp | N/A | N/A | | [`tgamma`] | ≤ 16 ulp | [`tgamma`][`tgamma`_cuda] | 8 ulp (full range) | | [`trunc`] | Correctly rounded | [`trunc`][`trunc`_cuda] | 0 ulp (full range) | | N/A | N/A | [`y0`][`y0`_cuda] | 7 ulp for `abs(x) < 8`, otherwise `5 x 10^(-12)` | | N/A | N/A | [`y1`][`y1`_cuda] | 7 ulp for `abs(x) < 8`, otherwise `5 x 10^(-12)` | | N/A | N/A | [`yn`][`yn`_cuda] | For `abs(x) > 1.5n`, otherwise `5 x 10^(-12)` | | N/A | N/A | [`isfinite`][`isfinite`_cuda] | N/A | | N/A | N/A | [`isinf`][`isinf`_cuda] | N/A | | N/A | N/A | [`isnan`][`isnan`_cuda] | N/A | | N/A | N/A | [`signbit`][`signbit`_cuda] | N/A | [cuda_math_double]: https://docs.nvidia.com/cuda/cuda-math-api/group__CUDA__MATH__DOUBLE.html#group__CUDA__MATH__DOUBLE [opencl_1.2_dp_ulp]: https://www.khronos.org/registry/OpenCL/specs/opencl-1.2.pdf#page=322 ## Half Precision The following tables uses the following sources: - [Section 7.4 of the OpenCL 1.2 Specification][opencl_1.2_dp_ulp] - [CUDA Math API documentation][cuda_math_half] CUDA doesn't specify the ULP values for any of its half precision math builtins: | OpenCL Built-in | OpenCL Min Accuracy (ULP) | CUDA Built-in | CUDA Maximum Error (ULP) | | --------------- | ------------------------- | ------------- | ------------------------ | | N/A | N/A | [`_hadd`] | Undocumented (only specifies "round-to-nearest-even mode") | | N/A | N/A | [`_hadd_sat`] | Undocumented (only specifies "round-to-nearest-even mode") | | N/A | N/A | [`hceil`] | Undocumented | | [`half_cos`][`cos`] | ≤ 8192 ulp | [`hcos`] | Undocumented (only specifies "round-to-nearest-even mode") | | [`half_divide`][`divide`] | ≤ 8192 ulp | [`_hdiv`] | Undocumented (only specifies "round-to-nearest mode") | | N/A | N/A | [`_heq`] | Undocumented | | N/A | N/A | [`_hequ`] | Undocumented | | [`half_exp`][`exp`] | ≤ 8192 ulp | [`hexp`] | Undocumented (only specifies "round-to-nearest-even mode") | | [`half_exp2`][`exp`] | ≤ 8192 ulp | [`hexp2`] | Undocumented (only specifies "round-to-nearest-even mode") | | [`half_exp10`][`exp`] | ≤ 8192 ulp | [`hexp10`] | Undocumented (only specifies "round-to-nearest-even mode") | | N/A | N/A | [`hfloor`] | Undocumented | | N/A | N/A | [`_hfma`] | Undocumented (only specifies "round-to-nearest-even mode") | | N/A | N/A | [`_hfma_sat`] | Undocumented (only specifies "round-to-nearest-even mode") | | N/A | N/A | [`_hge`] | Undocumented | | N/A | N/A | [`_hgeu`] | Undocumented | | N/A | N/A | [`_hgt`] | Undocumented | | N/A | N/A | [`_hgtu`] | Undocumented | | N/A | N/A | [`_hisinf`] | Undocumented | | N/A | N/A | [`_hisnan`] | Undocumented | | N/A | N/A | [`_hle`] | Undocumented | | N/A | N/A | [`_hleu`] | Undocumented | | [`half_log`][`log`] | ≤ 8192 ulp | [`hlog`] | Undocumented (only specifies "round-to-nearest-even mode") | | [`half_log2`][`log`] | ≤ 8192 ulp | [`hlog2`] | Undocumented (only specifies "round-to-nearest-even mode") | | [`half_log10`][`log`] | ≤ 8192 ulp | [`hlog10`] | Undocumented (only specifies "round-to-nearest-even mode") | | N/A | N/A | [`_hlt`] | Undocumented | | N/A | N/A | [`_hltu`] | Undocumented | | N/A | N/A | [`_hmul`] | Undocumented (only specifies "round-to-nearest mode") | | N/A | N/A | [`_hmul_sat`] | Undocumented (only specifies "round-to-nearest mode") | | N/A | N/A | [`_hneg`] | Undocumented | | N/A | N/A | [`_hne`] | Undocumented | | N/A | N/A | [`_hneu`] | Undocumented | | [`half_powr`][`pow`] | ≤ 8192 ulp | N/A | N/A | | [`half_recip`][`recip`] | ≤ 8192 ulp | [`hrcp`] | Undocumented (only specifies "round-to-nearest-even mode") | | N/A | N/A | [`hrint`] | Undocumented (only specifies "halfway cases rounded to nearest even integer value") | | [`half_rsqrt`][`sqrt`] | ≤ 8192 ulp | [`hrqsrt`] | Undocumented (only specifies "round-to-nearest mode") | | [`half_sin`][`sin`] | ≤ 8192 ulp | [`hsin`] | Undocumented (only specifies "round-to-nearest-even mode") | | [`half_sqrt`][`sqrt`] | ≤ 8192 ulp | [`hsqrt`] | Undocumented (only specifies "round-to-nearest-even mode") | | N/A | N/A | [`_hsub`] | Undocumented (only specifies "round-to-nearest mode") | | N/A | N/A | [`_hsub_sat`] | Undocumented (only specifies "round-to-nearest mode") | | [`half_tan`][`tan`] | ≤ 8192 ulp | N/A | N/A | | N/A | N/A | [`htrunc`] | Undocumented | CUDA also defines math builtins that operate on a `half2` type to which there is no OpenCL parallel: | CUDA Built-in | CUDA Maximum Error (ULP) | | ------------- | ------------------------ | | [`_h2div`] | Undocumented (only specifies "round-to-nearest mode") | | [`_hadd2_sat`] | Undocumented (only specifies "round-to-nearest-even mode") | | [`_hadd2`] | Undocumented (only specifies "round-to-nearest-even mode") | | [`_hbeg2`] | Undocumented | | [`_hbegu2`] | Undocumented | | [`_hbge2`] | Undocumented | | [`_hbgeu2`] | Undocumented | | [`_hbgt2`] | Undocumented | | [`_hbgtu2`] | Undocumented | | [`_hble2`] | Undocumented | | [`_hbleu2`] | Undocumented | | [`_hblt2`] | Undocumented | | [`_hbltu2`] | Undocumented | | [`_hbne2`] | Undocumented | | [`_hbneu2`] | Undocumented | | [`_heq2`] | Undocumented | | [`_hequ2`] | Undocumented | | [`_hfma2_sat`] | Undocumented (only specifies "round-to-nearest-even mode") | | [`_hfma2`] | Undocumented (only specifies "round-to-nearest-even mode") | | [`_hge2`] | Undocumented | | [`_hgeu2`] | Undocumented | | [`_hgt2`] | Undocumented | | [`_hgtu2`] | Undocumented | | [`_hisnan2`] | Undocumented | | [`_hle2`] | Undocumented | | [`_hleu2`] | Undocumented | | [`_hlt2`] | Undocumented | | [`_hltu2`] | Undocumented | | [`_hmul2_sat`] | Undocumented (only specifies "round-to-nearest-even mode") | | [`_hmul2`] | Undocumented (only specifies "round-to-nearest-even mode") | | [`_hne2`] | Undocumented | | [`_hneg2`] | Undocumented | | [`_hneu2`] | Undocumented | | [`_hsub2_sat`] | Undocumented (only specifies "round-to-nearest-even mode") | | [`_hsub2`] | Undocumented (only specifies "round-to-nearest-even mode") | | [`h2ceil`] | Undocumented | | [`h2cos`] | Undocumented (only specifies "round-to-nearest-even mode") | | [`h2exp10`] | Undocumented (only specifies "round-to-nearest-even mode") | | [`h2exp2`] | Undocumented (only specifies "round-to-nearest-even mode") | | [`h2exp`] | Undocumented (only specifies "round-to-nearest mode") | | [`h2floor`] | Undocumented | | [`h2log10`] | Undocumented (only specifies "round-to-nearest-even mode") | | [`h2log2`] | Undocumented (only specifies "round-to-nearest-even mode") | | [`h2log`] | Undocumented (only specifies "round-to-nearest-even mode") | | [`h2rcp`] | Undocumented (only specifies "round-to-nearest-even mode") | | [`h2rint`] | Undocumented (only specifies "halfway cases rounded to nearest even integer value") | | [`h2rsqrt`] | Undocumented (only specifies "round-to-nearest-even mode") | | [`h2trunc`] | Undocumented | Further, CUDA defines conversion and data movement functions: | CUDA Built-in | CUDA Maximum Error (ULP) | | ------------- | ------------------------ | | [`__float22half2_rn`] | Undocumented (only specifies "round-to-nearest-even mode") | | [`__float2half2_rn`] | Undocumented (only specifies "round-to-nearest-even mode") | | [`__float2half_rd`] | Undocumented (only specifies "round-down mode") | | [`__float2half_rn`] | Undocumented (only specifies "round-to-nearest-even mode") | | [`__float2half_ru`] | Undocumented (only specifies "round-up mode") | | [`__float2half_rz`] | Undocumented (only specifies "round-towards-zero mode") | | [`__float2half`] | Undocumented (only specifies "round-to-nearest-even mode") | | [`__floats2half2_rn`] | Undocumented (only specifies "round-to-nearest-even mode") | | [`__half22float2`] | Undocumented | | [`__half2float`] | Undocumented | | [`__half2half2`] | Undocumented | | [`__half2int_rd`] | Undocumented (only specifies "round-down mode") | | [`__half2int_rn`] | Undocumented (only specifies "round-to-nearest-even mode") | | [`__half2int_ru`] | Undocumented (only specifies "round-up mode") | | [`__half2int_rz`] | Undocumented (only specifies "round-towards-zero mode") | | [`__half2ll_rd`] | Undocumented (only specifies "round-down mode") | | [`__half2ll_rn`] | Undocumented (only specifies "round-to-nearest-even mode") | | [`__half2ll_ru`] | Undocumented (only specifies "round-up mode") | | [`__half2ll_rz`] | Undocumented (only specifies "round-towards-zero mode") | | [`__half2short_rd`] | Undocumented (only specifies "round-down mode") | | [`__half2short_rn`] | Undocumented (only specifies "round-to-nearest-even mode") | | [`__half2short_ru`] | Undocumented (only specifies "round-up mode") | | [`__half2short_rz`] | Undocumented (only specifies "round-towards-zero mode") | | [`__half2uint_rd`] | Undocumented (only specifies "round-down mode") | | [`__half2uint_rn`] | Undocumented (only specifies "round-to-nearest-even mode") | | [`__half2uint_ru`] | Undocumented (only specifies "round-up mode") | | [`__half2uint_rz`] | Undocumented (only specifies "round-towards-zero mode") | | [`__half2ull_rd`] | Undocumented (only specifies "round-down mode") | | [`__half2ull_rn`] | Undocumented (only specifies "round-to-nearest-even mode") | | [`__half2ull_ru`] | Undocumented (only specifies "round-up mode") | | [`__half2ull_rz`] | Undocumented (only specifies "round-towards-zero mode") | | [`__half2ushort_rd`] | Undocumented (only specifies "round-down mode") | | [`__half2ushort_rn`] | Undocumented (only specifies "round-to-nearest-even mode") | | [`__half2ushort_ru`] | Undocumented (only specifies "round-up mode") | | [`__half2ushort_rz`] | Undocumented (only specifies "round-towards-zero mode") | | [`__half_as_short`] | Undocumented | | [`__half_as_ushort`] | Undocumented | | [`__halves2half2`] | Undocumented | | [`__high2float`] | Undocumented | | [`__high2half2`] | Undocumented | | [`__high2half`] | Undocumented | | [`__highs2half2`] | Undocumented | | [`__int2half_rd`] | Undocumented (only specifies "round-down mode") | | [`__int2half_rn`] | Undocumented (only specifies "round-to-nearest-even mode") | | [`__int2half_ru`] | Undocumented (only specifies "round-up mode") | | [`__int2half_rz`] | Undocumented (only specifies "round-towards-zero mode") | | [`__ll2half_rd`] | Undocumented (only specifies "round-down mode") | | [`__ll2half_rn`] | Undocumented (only specifies "round-to-nearest-even mode") | | [`__ll2half_ru`] | Undocumented (only specifies "round-up mode") | | [`__ll2half_rz`] | Undocumented (only specifies "round-towards-zero mode") | | [`__low2float`] | Undocumented | | [`__low2half2`] | Undocumented | | [`__low2half`] | Undocumented | | [`__lowhigh2highlow`] | Undocumented | | [`__lows2half2`] | Undocumented | | [`__shfl_down_sync`] | Undocumented | | [`__shfl_sync`] | Undocumented | | [`__shfl_up_sync`] | Undocumented | | [`__shfl_xor_sync`] | Undocumented | | [`__short2half_rd`] | Undocumented (only specifies "round-down mode") | | [`__short2half_rn`] | Undocumented (only specifies "round-to-nearest-even mode") | | [`__short2half_ru`] | Undocumented (only specifies "round-up mode") | | [`__short2half_rz`] | Undocumented (only specifies "round-towards-zero mode") | | [`__short_as_half`] | Undocumented | | [`__uint2half_rd`] | Undocumented (only specifies "round-down mode") | | [`__uint2half_rn`] | Undocumented (only specifies "round-to-nearest-even mode") | | [`__uint2half_ru`] | Undocumented (only specifies "round-up mode") | | [`__uint2half_rz`] | Undocumented (only specifies "round-towards-zero mode") | | [`__ull2half_rd`] | Undocumented (only specifies "round-down mode") | | [`__ull2half_rn`] | Undocumented (only specifies "round-to-nearest-even mode") | | [`__ull2half_ru`] | Undocumented (only specifies "round-up mode") | | [`__ull2half_rz`] | Undocumented (only specifies "round-towards-zero mode") | | [`__ushort2half_rd`] | Undocumented (only specifies "round-down mode") | | [`__ushort2half_rn`] | Undocumented (only specifies "round-to-nearest-even mode") | | [`__ushort2half_ru`] | Undocumented (only specifies "round-up mode") | | [`__ushort2half_rz`] | Undocumented (only specifies "round-towards-zero mode") | | [`__ushort_as_half`] | 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