Math Functions¶
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:
To check math functions enabled for Windows:
windows-aarch64 - to be added
To check math functions enabled for macOS:
To check math functions enabled for GPU:
To check math functions enabled for embedded system:
baremetal-riscv32 - to be added
Basic Operations¶
<Func> |
<Func_f> (float) |
<Func> (double) |
<Func_l> (long double) |
<Func_f16> (float16) |
<Func_f128> (float128) |
C23 Definition Section |
C23 Error Handling Section |
---|---|---|---|---|---|---|---|
ceil |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.9.1 |
F.10.6.1 |
canonicalize |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.11.7 |
F.10.8.7 |
copysign |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.11.1 |
F.10.8.1 |
dadd |
N/A |
N/A |
✅ |
N/A |
✅* |
7.12.14.1 |
F.10.11 |
ddiv |
N/A |
N/A |
✅ |
N/A |
✅* |
7.12.14.4 |
F.10.11 |
dfma |
N/A |
N/A |
✅ |
N/A |
✅* |
7.12.14.5 |
F.10.11 |
dmul |
N/A |
N/A |
✅ |
N/A |
✅* |
7.12.14.3 |
F.10.11 |
dsub |
N/A |
N/A |
✅ |
N/A |
✅* |
7.12.14.2 |
F.10.11 |
f16add |
✅* |
✅* |
✅* |
N/A |
✅ |
7.12.14.1 |
F.10.11 |
f16div |
✅* |
✅* |
✅* |
N/A |
✅ |
7.12.14.4 |
F.10.11 |
f16fma |
✅* |
✅* |
✅* |
N/A |
✅ |
7.12.14.5 |
F.10.11 |
f16mul |
✅* |
✅* |
✅* |
N/A |
✅ |
7.12.14.5 |
F.10.11 |
f16sub |
✅* |
✅* |
✅* |
N/A |
✅ |
7.12.14.2 |
F.10.11 |
fabs |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.7.3 |
F.10.4.3 |
fadd |
N/A |
✅ |
✅ |
N/A |
✅ |
7.12.14.1 |
F.10.11 |
fdim |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.12.1 |
F.10.9.1 |
fdiv |
N/A |
✅ |
✅ |
N/A |
✅* |
7.12.14.4 |
F.10.11 |
ffma |
N/A |
✅ |
✅ |
N/A |
✅* |
7.12.14.5 |
F.10.11 |
floor |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.9.2 |
F.10.6.2 |
fmax |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.12.2 |
F.10.9.2 |
fmaximum |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.12.4 |
F.10.9.4 |
fmaximum_mag |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.12.6 |
F.10.9.4 |
fmaximum_mag_num |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.12.10 |
F.10.9.5 |
fmaximum_num |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.12.8 |
F.10.9.5 |
fmin |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.12.3 |
F.10.9.3 |
fminimum |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.12.5 |
F.10.9.4 |
fminimum_mag |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.12.7 |
F.10.9.4 |
fminimum_mag_num |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.12.11 |
F.10.9.5 |
fminimum_num |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.12.9 |
F.10.9.5 |
fmod |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.10.1 |
F.10.7.1 |
fmul |
N/A |
✅ |
✅ |
N/A |
✅* |
7.12.14.3 |
F.10.11 |
frexp |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.6.7 |
F.10.3.7 |
fromfp |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.9.10 |
F.10.6.10 |
fromfpx |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.9.11 |
F.10.6.11 |
fsub |
N/A |
✅ |
✅ |
N/A |
✅* |
7.12.14.2 |
F.10.11 |
getpayload |
✅ |
✅ |
✅ |
✅ |
✅ |
F.10.13.1 |
N/A |
ilogb |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.6.8 |
F.10.3.8 |
iscanonical |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.3.2 |
N/A |
issignaling |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.3.8 |
N/A |
ldexp |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.6.9 |
F.10.3.9 |
llogb |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.6.10 |
F.10.3.10 |
llrint |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.9.5 |
F.10.6.5 |
llround |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.9.7 |
F.10.6.7 |
logb |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.6.17 |
F.10.3.17 |
lrint |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.9.5 |
F.10.6.5 |
lround |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.9.7 |
F.10.6.7 |
modf |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.6.18 |
F.10.3.18 |
nan |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.11.2 |
F.10.8.2 |
nearbyint |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.9.3 |
F.10.6.3 |
nextafter |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.11.3 |
F.10.8.3 |
nextdown |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.11.6 |
F.10.8.6 |
nexttoward |
✅ |
✅ |
✅ |
✅ |
N/A |
7.12.11.4 |
F.10.8.4 |
nextup |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.11.5 |
F.10.8.5 |
remainder |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.10.2 |
F.10.7.2 |
remquo |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.10.3 |
F.10.7.3 |
rint |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.9.4 |
F.10.6.4 |
round |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.9.6 |
F.10.6.6 |
roundeven |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.9.8 |
F.10.6.8 |
scalbln |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.6.19 |
F.10.3.19 |
scalbn |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.6.19 |
F.10.3.19 |
setpayload |
✅ |
✅ |
✅ |
✅ |
✅ |
F.10.13.2 |
N/A |
setpayloadsig |
✅ |
✅ |
✅ |
✅ |
✅ |
F.10.13.3 |
N/A |
totalorder |
✅ |
✅ |
✅ |
✅ |
✅ |
F.10.12.1 |
N/A |
totalordermag |
✅ |
✅ |
✅ |
✅ |
✅ |
F.10.12.2 |
N/A |
trunc |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.9.9 |
F.10.6.9 |
ufromfp |
✅ |
✅ |
✅ |
✅ |
✅ |
7.12.9.10 |
F.10.6.10 |
ufromfpx |
✅ |
✅ |
✅ |
✅ |
✅ |
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) |
C23 Definition Section |
C23 Error Handling Section |
---|---|---|---|---|---|---|---|
acos |
✅ |
7.12.4.1 |
F.10.1.1 |
||||
acosh |
✅ |
7.12.5.1 |
F.10.2.1 |
||||
acospi |
7.12.4.8 |
F.10.1.8 |
|||||
asin |
✅ |
7.12.4.2 |
F.10.1.2 |
||||
asinh |
✅ |
7.12.5.2 |
F.10.2.2 |
||||
asinpi |
7.12.4.9 |
F.10.1.9 |
|||||
atan |
✅ |
7.12.4.3 |
F.10.1.3 |
||||
atan2 |
✅ |
1 ULP |
7.12.4.4 |
F.10.1.4 |
|||
atan2pi |
7.12.4.11 |
F.10.1.11 |
|||||
atanh |
✅ |
7.12.5.3 |
F.10.2.3 |
||||
atanpi |
7.12.4.10 |
F.10.1.10 |
|||||
cbrt |
✅ |
✅ |
7.12.7.1 |
F.10.4.1 |
|||
compoundn |
7.12.7.2 |
F.10.4.2 |
|||||
cos |
✅ |
✅ |
7.12.4.5 |
F.10.1.5 |
|||
cosh |
✅ |
7.12.5.4 |
F.10.2.4 |
||||
cospi |
✅ |
7.12.4.12 |
F.10.1.12 |
||||
dsqrt |
N/A |
N/A |
✅ |
N/A |
✅* |
7.12.14.6 |
F.10.11 |
erf |
✅ |
7.12.8.1 |
F.10.5.1 |
||||
erfc |
7.12.8.2 |
F.10.5.2 |
|||||
exp |
✅ |
✅ |
✅ |
7.12.6.1 |
F.10.3.1 |
||
exp10 |
✅ |
✅ |
✅ |
7.12.6.2 |
F.10.3.2 |
||
exp10m1 |
7.12.6.3 |
F.10.3.3 |
|||||
exp2 |
✅ |
✅ |
✅ |
7.12.6.4 |
F.10.3.4 |
||
exp2m1 |
✅ |
7.12.6.5 |
F.10.3.5 |
||||
expm1 |
✅ |
✅ |
✅ |
7.12.6.6 |
F.10.3.6 |
||
fma |
✅ |
✅ |
7.12.13.1 |
F.10.10.1 |
|||
f16sqrt |
✅* |
✅* |
✅* |
N/A |
✅ |
7.12.14.6 |
F.10.11 |
fsqrt |
N/A |
✅ |
✅ |
N/A |
✅* |
7.12.14.6 |
F.10.11 |
hypot |
✅ |
✅ |
7.12.7.4 |
F.10.4.4 |
|||
lgamma |
7.12.8.3 |
F.10.5.3 |
|||||
log |
✅ |
✅ |
7.12.6.11 |
F.10.3.11 |
|||
log10 |
✅ |
✅ |
7.12.6.12 |
F.10.3.12 |
|||
log10p1 |
7.12.6.13 |
F.10.3.13 |
|||||
log1p |
✅ |
✅ |
7.12.6.14 |
F.10.3.14 |
|||
log2 |
✅ |
✅ |
7.12.6.15 |
F.10.3.15 |
|||
log2p1 |
7.12.6.16 |
F.10.3.16 |
|||||
logp1 |
7.12.6.14 |
F.10.3.14 |
|||||
pow |
✅ |
1 ULP |
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 |
7.12.7.9 |
F.10.4.9 |
|||||
sin |
✅ |
✅ |
7.12.4.6 |
F.10.1.6 |
|||
sincos |
✅ |
✅ |
|||||
sinh |
✅ |
7.12.5.5 |
F.10.2.5 |
||||
sinpi |
✅ |
7.12.4.13 |
F.10.1.13 |
||||
sqrt |
✅ |
✅ |
✅ |
✅ |
7.12.7.10 |
F.10.4.10 |
|
tan |
✅ |
✅ |
7.12.4.7 |
F.10.1.7 |
|||
tanh |
✅ |
7.12.5.6 |
F.10.2.6 |
||||
tanpi |
7.12.4.14 |
F.10.1.14 |
|||||
tgamma |
7.12.8.4 |
F.10.5.4 |
Legends:
✅ : 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.
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 |
\([-1, 1]\) |
Ryzen 1700 |
Ubuntu 22.04 LTS x86_64 |
Clang 14.0.0 |
FMA |
acoshf |
18 |
26 |
73 |
74 |
\([1, 21]\) |
Ryzen 1700 |
Ubuntu 22.04 LTS x86_64 |
Clang 14.0.0 |
FMA |
asinf |
23 |
27 |
62 |
62 |
\([-1, 1]\) |
Ryzen 1700 |
Ubuntu 22.04 LTS x86_64 |
Clang 14.0.0 |
FMA |
asinhf |
21 |
39 |
77 |
91 |
\([-10, 10]\) |
Ryzen 1700 |
Ubuntu 22.04 LTS x86_64 |
Clang 14.0.0 |
FMA |
atanf |
27 |
29 |
79 |
68 |
\([-10, 10]\) |
Ryzen 1700 |
Ubuntu 22.04 LTS x86_64 |
Clang 14.0.0 |
FMA |
atanhf |
18 |
66 |
68 |
133 |
\([-1, 1]\) |
Ryzen 1700 |
Ubuntu 22.04 LTS x86_64 |
Clang 14.0.0 |
FMA |
cosf |
13 |
32 |
53 |
59 |
\([0, 2\pi]\) |
Ryzen 1700 |
Ubuntu 20.04 LTS x86_64 |
Clang 12.0.0 |
FMA |
coshf |
14 |
20 |
50 |
48 |
\([-10, 10]\) |
Ryzen 1700 |
Ubuntu 22.04 LTS x86_64 |
Clang 14.0.0 |
FMA |
expf |
9 |
7 |
44 |
38 |
\([-10, 10]\) |
Ryzen 1700 |
Ubuntu 20.04 LTS x86_64 |
Clang 12.0.0 |
FMA |
exp10f |
10 |
8 |
40 |
38 |
\([-10, 10]\) |
Ryzen 1700 |
Ubuntu 22.04 LTS x86_64 |
Clang 14.0.0 |
FMA |
exp2f |
9 |
6 |
35 |
31 |
\([-10, 10]\) |
Ryzen 1700 |
Ubuntu 22.04 LTS x86_64 |
Clang 14.0.0 |
FMA |
expm1f |
9 |
44 |
42 |
121 |
\([-10, 10]\) |
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 |
\([-10, 10] \times [-10, 10]\) |
Ryzen 1700 |
Ubuntu 20.04 LTS x86_64 |
Clang 12.0.0 |
|
logf |
12 |
10 |
56 |
46 |
\([e^{-1}, e]\) |
Ryzen 1700 |
Ubuntu 20.04 LTS x86_64 |
Clang 12.0.0 |
FMA |
log10f |
9 |
17 |
35 |
48 |
\([e^{-1}, e]\) |
Ryzen 5900X |
Ubuntu 22.04 LTS x86_64 |
Clang 15.0.6 |
FMA |
log1pf |
16 |
33 |
61 |
97 |
\([e^{-0.5} - 1, e^{0.5} - 1]\) |
Ryzen 1700 |
Ubuntu 20.04 LTS x86_64 |
Clang 12.0.0 |
FMA |
log2f |
13 |
10 |
57 |
46 |
\([e^{-1}, e]\) |
Ryzen 1700 |
Ubuntu 20.04 LTS x86_64 |
Clang 12.0.0 |
FMA |
sinf |
12 |
25 |
51 |
57 |
\([-\pi, \pi]\) |
Ryzen 1700 |
Ubuntu 20.04 LTS x86_64 |
Clang 12.0.0 |
FMA |
sincosf |
19 |
30 |
57 |
68 |
\([-\pi, \pi]\) |
Ryzen 1700 |
Ubuntu 20.04 LTS x86_64 |
Clang 12.0.0 |
FMA |
sinhf |
13 |
63 |
48 |
137 |
\([-10, 10]\) |
Ryzen 1700 |
Ubuntu 22.04 LTS x86_64 |
Clang 14.0.0 |
FMA |
tanf |
16 |
50 |
61 |
107 |
\([-\pi, \pi]\) |
Ryzen 1700 |
Ubuntu 22.04 LTS x86_64 |
Clang 14.0.0 |
FMA |
tanhf |
13 |
55 |
57 |
123 |
\([-10, 10]\) |
Ryzen 1700 |
Ubuntu 22.04 LTS x86_64 |
Clang 14.0.0 |
FMA |