(math)=

math.h

<style> .green {color:green} </style>

.. role:: green

:hidden: true

log

Source Locations

• The main source is located at: libc/src/math.
• The tests are located at: libc/test/src/math.
• The floating point utilities are located at: libc/src/__support/FPUtil.

Implementation Requirements / Goals

• The highest priority is to be as accurate as possible, according to the C and IEEE 754 standards. By default, we will aim to be correctly rounded for all rounding modes. The current rounding mode of the floating point environment is used to perform computations and produce the final results.

  • To test for correctness, we compare the outputs with other correctly rounded multiple-precision math libraries such as the GNU MPFR library or the CORE-MATH library.

• Our next requirement is that the outputs are consistent across all platforms. Notice that the consistency requirement will be satisfied automatically if the implementation is correctly rounded.

• Our last requirement for the implementations is to have good and predicable performance:

  • The average performance should be comparable to other libc implementations.
  • The worst case performance should be within 10X-20X of the average.
  • Platform-specific implementations or instructions could be added whenever it makes sense and provides significant performance boost.

• For other use cases that have strict requirements on the code size, memory footprint, or latency, such as embedded systems, we will aim to be as accurate as possible within the memory or latency budgets, and consistent across all platforms.

Add a new math function to LLVM libc

• To add a new math function, follow the steps at: libc/src/math/docs/add_math_function.md.

Implementation Status

• To check math functions enabled for Linux:

  • linux-x86_64
  • linux-aarch64
  • linux-aarch32
  • linux-riscv and linux-riscv32

• To check math functions enabled for Windows:

  • windows-x86_64
  • windows-aarch64 - to be added

• To check math functions enabled for macOS:

  • darwin-x86_64
  • darwin-aarch64

• To check math functions enabled for GPU:

  • amdgpu-entrypoints
  • nvptx-entrypoints

• To check math functions enabled for embedded system:

  • baremetal-aarch64
  • baremetal-aarch32
  • baremetal-riscv32

Basic Operations

<Func>	<Func_f> (float)	<Func> (double)	<Func_l> (long double)	<Func_f16> (float16)	<Func_f128> (float128)	<Func_bf16> (bfloat16)	C23 Definition Section	C23 Error Handling Section
ceil	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.9.1	F.10.6.1
canonicalize	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.11.7	F.10.8.7
copysign	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.11.1	F.10.8.1
dadd	N/A	N/A	{{ check }}	N/A	{{ check }}*	N/A	7.12.14.1	F.10.11
ddiv	N/A	N/A	{{ check }}	N/A	{{ check }}*	N/A	7.12.14.4	F.10.11
dfma	N/A	N/A	{{ check }}	N/A	{{ check }}*	N/A	7.12.14.5	F.10.11
dmul	N/A	N/A	{{ check }}	N/A	{{ check }}*	N/A	7.12.14.3	F.10.11
dsub	N/A	N/A	{{ check }}	N/A	{{ check }}*	N/A	7.12.14.2	F.10.11
f16add	{{ check }}*	{{ check }}*	{{ check }}*	N/A	{{ check }}	N/A	7.12.14.1	F.10.11
f16div	{{ check }}*	{{ check }}*	{{ check }}*	N/A	{{ check }}	N/A	7.12.14.4	F.10.11
f16fma	{{ check }}*	{{ check }}*	{{ check }}*	N/A	{{ check }}	N/A	7.12.14.5	F.10.11
f16mul	{{ check }}*	{{ check }}*	{{ check }}*	N/A	{{ check }}	N/A	7.12.14.3	F.10.11
f16sub	{{ check }}*	{{ check }}*	{{ check }}*	N/A	{{ check }}	N/A	7.12.14.2	F.10.11
bf16add	{{ check }}*	{{ check }}*	{{ check }}*	N/A	{{ check }}	N/A	7.12.14.1	F.10.11
bf16div	{{ check }}*	{{ check }}*	{{ check }}*	N/A	{{ check }}	N/A	7.12.14.4	F.10.11
bf16fma	{{ check }}*	{{ check }}*	{{ check }}*	N/A	{{ check }}	N/A	7.12.14.5	F.10.11
bf16mul	{{ check }}*	{{ check }}*	{{ check }}*	N/A	{{ check }}	N/A	7.12.14.3	F.10.11
bf16sub	{{ check }}*	{{ check }}*	{{ check }}*	N/A	{{ check }}	N/A	7.12.14.2	F.10.11
fabs	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.7.3	F.10.4.3
fadd	N/A	{{ check }}	{{ check }}	N/A	{{ check }}	N/A	7.12.14.1	F.10.11
fdim	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.12.1	F.10.9.1
fdiv	N/A	{{ check }}	{{ check }}	N/A	{{ check }}*	N/A	7.12.14.4	F.10.11
ffma	N/A	{{ check }}	{{ check }}	N/A	{{ check }}*	N/A	7.12.14.5	F.10.11
floor	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.9.2	F.10.6.2
fmax	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.12.2	F.10.9.2
fmaximum	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.12.4	F.10.9.4
fmaximum_mag	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.12.6	F.10.9.4
fmaximum_mag_num	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.12.10	F.10.9.5
fmaximum_num	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.12.8	F.10.9.5
fmin	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.12.3	F.10.9.3
fminimum	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.12.5	F.10.9.4
fminimum_mag	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.12.7	F.10.9.4
fminimum_mag_num	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.12.11	F.10.9.5
fminimum_num	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.12.9	F.10.9.5
fmod	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.10.1	F.10.7.1
fmul	N/A	{{ check }}	{{ check }}	N/A	{{ check }}*	N/A	7.12.14.3	F.10.11
frexp	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.6.7	F.10.3.7
fromfp	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.9.10	F.10.6.10
fromfpx	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.9.11	F.10.6.11
fsub	N/A	{{ check }}	{{ check }}	N/A	{{ check }}*	N/A	7.12.14.2	F.10.11
getpayload	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	F.10.13.1	N/A
ilogb	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.6.8	F.10.3.8
iscanonical	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.3.2	N/A
issignaling	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.3.8	N/A
ldexp	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.6.9	F.10.3.9
llogb	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.6.10	F.10.3.10
llrint	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.9.5	F.10.6.5
llround	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.9.7	F.10.6.7
logb	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.6.17	F.10.3.17
lrint	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.9.5	F.10.6.5
lround	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.9.7	F.10.6.7
modf	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.6.18	F.10.3.18
nan	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.11.2	F.10.8.2
nearbyint	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.9.3	F.10.6.3
nextafter	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.11.3	F.10.8.3
nextdown	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.11.6	F.10.8.6
nexttoward	{{ check }}	{{ check }}	{{ check }}	{{ check }}	N/A	{{ check }}	7.12.11.4	F.10.8.4
nextup	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.11.5	F.10.8.5
remainder	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.10.2	F.10.7.2
remquo	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.10.3	F.10.7.3
rint	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.9.4	F.10.6.4
round	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.9.6	F.10.6.6
roundeven	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.9.8	F.10.6.8
scalbln	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.6.19	F.10.3.19
scalbn	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.6.19	F.10.3.19
setpayload	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	F.10.13.2	N/A
setpayloadsig	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	F.10.13.3	N/A
totalorder	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	F.10.12.1	N/A
totalordermag	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	F.10.12.2	N/A
trunc	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.9.9	F.10.6.9
ufromfp	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.9.10	F.10.6.10
ufromfpx	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.9.11	F.10.6.11

Higher Math Functions

<Func>	<Func_f> (float)	<Func> (double)	<Func_l> (long double)	<Func_f16> (float16)	<Func_f128> (float128)	<Func_bf16> (bfloat16)	C23 Definition Section	C23 Error Handling Section
acos	{{ check }}	{{ check }}		{{ check }}		{{ check }}	7.12.4.1	F.10.1.1
acosh	{{ check }}			{{ check }}			7.12.5.1	F.10.2.1
acospi				{{ check }}			7.12.4.8	F.10.1.8
asin	{{ check }}	{{ check }}		{{ check }}		{{ check }}	7.12.4.2	F.10.1.2
asinh	{{ check }}			{{ check }}			7.12.5.2	F.10.2.2
asinpi	{{ check }}	{{ check }}		{{ check }}			7.12.4.9	F.10.1.9
atan	{{ check }}	1 ULP		{{ check }}		{{ check }}	7.12.4.3	F.10.1.3
atan2	{{ check }}	1 ULP		{{ check }}	1 ULP		7.12.4.4	F.10.1.4
atan2pi							7.12.4.11	F.10.1.11
atanh	{{ check }}			{{ check }}			7.12.5.3	F.10.2.3
atanpi				{{ check }}			7.12.4.10	F.10.1.10
cbrt	{{ check }}	{{ check }}		{{ check }}		{{ check }}	7.12.7.1	F.10.4.1
compoundn							7.12.7.2	F.10.4.2
cos	{{ check }}	{{ check }}		{{ check }}			7.12.4.5	F.10.1.5
cosh	{{ check }}			{{ check }}			7.12.5.4	F.10.2.4
cospi	{{ check }}			{{ check }}			7.12.4.12	F.10.1.12
dsqrt	N/A	N/A	{{ check }}	N/A	{{ check }}*		7.12.14.6	F.10.11
erf	{{ check }}						7.12.8.1	F.10.5.1
erfc							7.12.8.2	F.10.5.2
exp	{{ check }}	{{ check }}		{{ check }}		{{ check }}	7.12.6.1	F.10.3.1
exp10	{{ check }}	{{ check }}		{{ check }}			7.12.6.2	F.10.3.2
exp10m1	{{ check }}			{{ check }}			7.12.6.3	F.10.3.3
exp2	{{ check }}	{{ check }}		{{ check }}		{{ check }}	7.12.6.4	F.10.3.4
exp2m1	{{ check }}			{{ check }}			7.12.6.5	F.10.3.5
expm1	{{ check }}	{{ check }}		{{ check }}			7.12.6.6	F.10.3.6
fma	{{ check }}	{{ check }}		{{ check }}		{{ check }}	7.12.13.1	F.10.10.1
f16sqrt	{{ check }}*	{{ check }}*	{{ check }}*	N/A	{{ check }}		7.12.14.6	F.10.11
fsqrt	N/A	{{ check }}	{{ check }}	N/A	{{ check }}*		7.12.14.6	F.10.11
hypot	{{ check }}	{{ check }}		{{ check }}		{{ check }}	7.12.7.4	F.10.4.4
lgamma				{{ check }}			7.12.8.3	F.10.5.3
log	{{ check }}	{{ check }}		{{ check }}		{{ check }} ?	7.12.6.11	F.10.3.11
log10	{{ check }}	{{ check }}		{{ check }}			7.12.6.12	F.10.3.12
log10p1				{{ check }}			7.12.6.13	F.10.3.13
log1p	{{ check }}	{{ check }}					7.12.6.14	F.10.3.14
log2	{{ check }}	{{ check }}		{{ check }}			7.12.6.15	F.10.3.15
log2p1				{{ check }}			7.12.6.16	F.10.3.16
logp1							7.12.6.14	F.10.3.14
pow	{{ check }}	{{ check }}					7.12.7.5	F.10.4.5
powi*
pown							7.12.7.6	F.10.4.6
powr							7.12.7.7	F.10.4.7
rootn							7.12.7.8	F.10.4.8
rsqrt	{{ check }}			{{ check }}			7.12.7.9	F.10.4.9
sin	{{ check }}	{{ check }}		{{ check }}			7.12.4.6	F.10.1.6
sincos	{{ check }}	{{ check }}
sinh	{{ check }}			{{ check }}			7.12.5.5	F.10.2.5
sinpi	{{ check }}			{{ check }}			7.12.4.13	F.10.1.13
sqrt	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	{{ check }}	7.12.7.10	F.10.4.10
tan	{{ check }}	{{ check }}		{{ check }}		{{ check }}	7.12.4.7	F.10.1.7
tanh	{{ check }}			{{ check }}			7.12.5.6	F.10.2.6
tanpi	{{ check }}			{{ check }}			7.12.4.14	F.10.1.14
tgamma							7.12.8.4	F.10.5.4

Legends:

• {{ check }} : correctly rounded for all 4 rounding modes.
• CR: correctly rounded for the default rounding mode (round-to-the-nearest, tie-to-even).
• x ULPs: largest errors recorded.
• N/A: Not defined in the standard or will not be added.
• *: LLVM libc extension.
• ? Because of a conflict between float16 logb function and bfloat16 log function, the latter is implemented as log_bf16.

% TODO(lntue): Add a new page to discuss about the algorithms used in the % implementations and include the link here.

GPU Conformance

• Conformance tests are located at: offload/unittests/Conformance.
• The math functions for GPUs are compiled with the following optimization options: LIBC_MATH_SKIP_ACCURATE_PASS, LIBC_MATH_INTERMEDIATE_COMP_IN_FLOAT, LIBC_MATH_SMALL_TABLES, LIBC_MATH_NO_ERRNO, and LIBC_MATH_NO_EXCEPT.
• The conformance test results for higher math functions on GPUs are reported in the table below. The results show the maximum observed ULP distance when comparing a given GPU implementation against the corresponding correctly rounded implementation from LLVM libc, which is computed on the host CPU and serves as the reference. For comparison purposes, results for CUDA Math and HIP Math against the same reference are also included.

+------------------------+-------------+---------------+-----------------------------------------------------------------------------------+
| Function               | Test Method | ULP Tolerance | Max ULP Distance                                                                  |
|                        |             |               +--------------------+--------------------+--------------------+--------------------+
|                        |             |               | LLVM libc          | LLVM libc          | CUDA Math          | HIP Math           |
|                        |             |               | (AMDGPU)           | (CUDA)             | (CUDA)             | (AMDGPU)           |
+========================+=============+===============+====================+====================+====================+====================+
| acos                   | Randomized  | 4             | 6 (FAILED)         | 6 (FAILED)         | 1                  | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| acosf                  | Exhaustive  | 4             | 1                  | 1                  | 1                  | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| acosf16                | Exhaustive  | 2             | 1                  | 1                  |                    | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| acoshf                 | Exhaustive  | 4             | 1                  | 1                  | 2                  | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| acoshf16               | Exhaustive  | 2             | 0                  | 0                  |                    | 0                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| acospif16              | Exhaustive  | 2             | 0                  | 0                  |                    |                    |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| asin                   | Randomized  | 4             | 6 (FAILED)         | 6 (FAILED)         | 2                  | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| asinf                  | Exhaustive  | 4             | 1                  | 1                  | 1                  | 3                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| asinf16                | Exhaustive  | 2             | 0                  | 0                  |                    | 2                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| asinhf                 | Exhaustive  | 4             | 1                  | 1                  | 2                  | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| asinhf16               | Exhaustive  | 2             | 1                  | 1                  |                    | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| atanf                  | Exhaustive  | 5             | 0                  | 0                  | 1                  | 2                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| atanf16                | Exhaustive  | 2             | 1                  | 1                  |                    | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| atan2f                 | Randomized  | 6             | 1                  | 1                  | 2                  | 3                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| atanhf                 | Exhaustive  | 5             | 0                  | 0                  | 3                  | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| atanhf16               | Exhaustive  | 2             | 0                  | 0                  |                    | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| cbrt                   | Randomized  | 2             | 1                  | 1                  | 1                  | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| cbrtf                  | Exhaustive  | 2             | 0                  | 0                  | 1                  | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| cos                    | Randomized  | 4             | 1                  | 1                  | 2                  | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| cosf                   | Exhaustive  | 4             | 1                  | 1                  | 2                  | 2                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| cosf16                 | Exhaustive  | 2             | 1                  | 1                  |                    | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| coshf                  | Exhaustive  | 4             | 0                  | 0                  | 2                  | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| coshf16                | Exhaustive  | 2             | 1                  | 0                  |                    | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| cospif                 | Exhaustive  | 4             | 0                  | 0                  | 1                  | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| cospif16               | Exhaustive  | 2             | 0                  | 0                  |                    |                    |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| erff                   | Exhaustive  | 16            | 0                  | 0                  | 1                  | 2                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| exp                    | Randomized  | 3             | 1                  | 1                  | 1                  | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| expf                   | Exhaustive  | 3             | 0                  | 0                  | 2                  | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| expf16                 | Exhaustive  | 2             | 1                  | 1                  |                    | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| exp10                  | Randomized  | 3             | 1                  | 1                  | 1                  | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| exp10f                 | Exhaustive  | 3             | 0                  | 0                  | 2                  | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| exp10f16               | Exhaustive  | 2             | 1                  | 1                  |                    | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| exp2                   | Randomized  | 3             | 1                  | 1                  | 1                  | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| exp2f                  | Exhaustive  | 3             | 1                  | 1                  | 2                  | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| exp2f16                | Exhaustive  | 2             | 1                  | 1                  |                    | 0                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| expm1                  | Randomized  | 3             | 0                  | 0                  | 1                  | 2                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| expm1f                 | Exhaustive  | 3             | 1                  | 1                  | 1                  | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| expm1f16               | Exhaustive  | 2             | 1                  | 1                  |                    | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| hypot                  | Randomized  | 4             | 0                  | 0                  | 2                  | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| hypotf                 | Randomized  | 4             | 0                  | 0                  | 1                  | 2                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| hypotf16               | Exhaustive  | 2             | 0                  | 0                  |                    |                    |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| log                    | Randomized  | 3             | 1                  | 1                  | 1                  | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| logf                   | Exhaustive  | 3             | 1                  | 1                  | 1                  | 2                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| logf16                 | Exhaustive  | 2             | 1                  | 1                  |                    | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| log10                  | Randomized  | 3             | 1                  | 1                  | 1                  | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| log10f                 | Exhaustive  | 3             | 1                  | 1                  | 2                  | 2                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| log10f16               | Exhaustive  | 2             | 1                  | 1                  |                    | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| log1p                  | Randomized  | 2             | 1                  | 1                  | 1                  | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| log1pf                 | Exhaustive  | 2             | 1                  | 1                  | 1                  | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| log2                   | Randomized  | 3             | 1                  | 1                  | 1                  | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| log2f                  | Exhaustive  | 3             | 0                  | 0                  | 1                  | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| log2f16                | Exhaustive  | 2             | 1                  | 1                  |                    | 0                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| powf (integer exp.)    | Randomized  | 16            | 0                  | 0                  | 2                  | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| powf (real exp.)       | Randomized  | 16            | 0                  | 0                  | 2                  | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| sin                    | Randomized  | 4             | 1                  | 1                  | 1                  | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| sinf                   | Exhaustive  | 4             | 1                  | 1                  | 1                  | 2                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| sinf16                 | Exhaustive  | 2             | 1                  | 1                  |                    | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| sincos (cos part)      | Randomized  | 4             | 1                  | 1                  | 2                  | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| sincos (sin part)      | Randomized  | 4             | 1                  | 1                  | 1                  | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| sincosf (cos part)     | Exhaustive  | 4             | 1                  | 1                  | 2                  | 2                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| sincosf (sin part)     | Exhaustive  | 4             | 1                  | 1                  | 1                  | 2                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| sinhf                  | Exhaustive  | 4             | 1                  | 1                  | 3                  | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| sinhf16                | Exhaustive  | 2             | 1                  | 1                  |                    | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| sinpif                 | Exhaustive  | 4             | 0                  | 0                  | 1                  | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| sinpif16               | Exhaustive  | 2             | 0                  | 0                  |                    |                    |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| tan                    | Randomized  | 5             | 2                  | 2                  | 2                  | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| tanf                   | Exhaustive  | 5             | 0                  | 0                  | 3                  | 2                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| tanf16                 | Exhaustive  | 2             | 1                  | 1                  |                    | 2                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| tanhf                  | Exhaustive  | 5             | 0                  | 0                  | 2                  | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| tanhf16                | Exhaustive  | 2             | 0                  | 0                  |                    | 1                  |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| tanpif                 | Exhaustive  | 6             | 0                  | 0                  |                    |                    |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| tanpif16               | Exhaustive  | 2             | 1                  | 1                  |                    |                    |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+

Notes:

• Exhaustive tests check every representable point in the input space. This method is used for half-precision functions and single-precision univariate functions.
• Randomized tests check a large, deterministic subset of the input space, typically using 2{sup}32 samples. This method is used for functions with larger input spaces, such as single-precision bivariate and double-precision functions.
• ULP tolerances are based on The Khronos Group, The OpenCL C Specification v3.0.19, Sec. 7.4, Khronos Registry [July 10, 2025].
• The AMD GPU used for testing is AMD Radeon RX 6950 XT.
• The NVIDIA GPU used for testing is NVIDIA RTX 4000 SFF Ada Generation.

Performance

• Simple performance testings are located at: libc/test/src/math/performance_testing.
• We also use the perf tool from the CORE-MATH project: link. The performance results from the CORE-MATH's perf tool are reported in the table below, using the system library as reference (such as the GNU C library on Linux). Fmod performance results obtained with “performance_testing”.

+--------------+-------------------------------+-------------------------------+-------------------------------------+----------------------------------------------------------------------+
| <Func>       | Reciprocal throughput (clk)   | Latency (clk)                 | Testing ranges                      | Testing configuration                                                |
|              +-----------+-------------------+-----------+-------------------+                                     +-------------+-------------------------+--------------+---------------+
|              | LLVM libc | Reference (glibc) | LLVM libc | Reference (glibc) |                                     | CPU         | OS                      | Compiler     | Special flags |
+==============+===========+===================+===========+===================+=====================================+=============+=========================+==============+===============+
| acosf        |        24 |                29 |        62 |                77 | :math:`[-1, 1]`                     | Ryzen 1700  | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA           |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| acoshf       |        18 |                26 |        73 |                74 | :math:`[1, 21]`                     | Ryzen 1700  | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA           |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| asinf        |        23 |                27 |        62 |                62 | :math:`[-1, 1]`                     | Ryzen 1700  | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA           |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| asinhf       |        21 |                39 |        77 |                91 | :math:`[-10, 10]`                   | Ryzen 1700  | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA           |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| atanf        |        27 |                29 |        79 |                68 | :math:`[-10, 10]`                   | Ryzen 1700  | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA           |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| atanhf       |        18 |                66 |        68 |               133 | :math:`[-1, 1]`                     | Ryzen 1700  | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA           |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| cosf         |        13 |                32 |        53 |                59 | :math:`[0, 2\pi]`                   | Ryzen 1700  | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA           |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| coshf        |        14 |                20 |        50 |                48 | :math:`[-10, 10]`                   | Ryzen 1700  | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA           |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| expf         |         9 |                 7 |        44 |                38 | :math:`[-10, 10]`                   | Ryzen 1700  | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA           |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| exp10f       |        10 |                 8 |        40 |                38 | :math:`[-10, 10]`                   | Ryzen 1700  | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA           |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| exp2f        |         9 |                 6 |        35 |                31 | :math:`[-10, 10]`                   | Ryzen 1700  | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA           |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| expm1f       |         9 |                44 |        42 |               121 | :math:`[-10, 10]`                   | Ryzen 1700  | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA           |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| fmodf        |        73 |               263 |        -  |                 - | [MIN_NORMAL, MAX_NORMAL]            | i5 mobile   | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 |               |
|              +-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
|              |         9 |                11 |        -  |                 - | [0, MAX_SUBNORMAL]                  | i5 mobile   | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 |               |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| fmod         |       595 |              3297 |        -  |                 - | [MIN_NORMAL, MAX_NORMAL]            | i5 mobile   | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 |               |
|              +-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
|              |        14 |                13 |        -  |                 - | [0, MAX_SUBNORMAL]                  | i5 mobile   | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 |               |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| hypotf       |        25 |                15 |        64 |                49 | :math:`[-10, 10] \times [-10, 10]`  | Ryzen 1700  | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 |               |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| logf         |        12 |                10 |        56 |                46 | :math:`[e^{-1}, e]`                 | Ryzen 1700  | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA           |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| log10f       |         9 |                17 |        35 |                48 | :math:`[e^{-1}, e]`                 | Ryzen 5900X | Ubuntu 22.04 LTS x86_64 | Clang 15.0.6 | FMA           |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| log1pf       |        16 |                33 |        61 |                97 | :math:`[e^{-0.5} - 1, e^{0.5} - 1]` | Ryzen 1700  | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA           |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| log2f        |        13 |                10 |        57 |                46 | :math:`[e^{-1}, e]`                 | Ryzen 1700  | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA           |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| sinf         |        12 |                25 |        51 |                57 | :math:`[-\pi, \pi]`                 | Ryzen 1700  | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA           |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| sincosf      |        19 |                30 |        57 |                68 | :math:`[-\pi, \pi]`                 | Ryzen 1700  | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA           |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| sinhf        |        13 |                63 |        48 |               137 | :math:`[-10, 10]`                   | Ryzen 1700  | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA           |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| tanf         |        16 |                50 |        61 |               107 | :math:`[-\pi, \pi]`                 | Ryzen 1700  | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA           |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| tanhf        |        13 |                55 |        57 |               123 | :math:`[-10, 10]`                   | Ryzen 1700  | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA           |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+

Algorithms + Implementation Details

• {doc}log
• Trigonometric range reduction

Fixed-point Arithmetics

• {doc}../stdfix

References

• CRLIBM.
• RLIBM.
• Sollya.
• The CORE-MATH Project.
• The GNU C Library (glibc).
• The GNU MPFR Library.
• C++23 Standard.
• BFloat16.