libm
- C math library
Synopsis
c99 [flag... ]file...-lm [library... ]
Description
Functions in this library provide common elementary mathematical functions and floating pointenvironment routines defined by System V, ANSI C, POSIX, and so on.Seestandards(5). Additional functions in this library provide extended support for handlingfloating point exceptions.
INTERFACES
The shared objectlibm.so.2 provides the public interfaces defined below. SeeIntro(3)for additional information on shared object interfaces.
acos | acosf | acosh | acoshf | acoshl | acosl | asin | asinf | asinh | asinhf | asinhl | asinl | atan | atan2 | atan2f | atan2l | atanf | atanh | atanhf | atanhl | atanl | cabs | cabsf | cabsl | cacos | cacosf | cacosh | cacoshf | cacoshl | cacosl | carg | cargf | cargl | casin | casinf | casinh | casinhf | casinhl | casinl | catan | catanf | catanh | catanhf | catanhl | catanl | cbrt | cbrtf | cbrtl | ccos | ccosf | ccosh | ccoshf | ccoshl | ccosl | ceil | ceilf | ceill | cexp | cexpf | cexpl | cimag | cimagf | cimagl | clog | clogf | clogl | conj | conjf | conjl | copysign | copysignf | copysignl | cos | cosf | cosh | coshf | coshl | cosl | cpow | cpowf | cpowl | cproj | cprojf | cprojl | creal | crealf | creall | csin | csinf | csinh | csinhf | csinhl | csinl | csqrt | csqrtf | csqrtl | ctan | ctanf | ctanh | ctanhf | ctanhl | ctanl | erf | erfc | erfcf | erfcl | erff | erfl | exp | exp2 | exp2f | exp2l | expf | expl | expm1 | expm1f | expm1l | fabs | fabsf | fabsl | fdim | fdimf | fdiml | feclearexcept | fegetenv | fegetexceptflag | fegetround | feholdexcept | feraiseexcept | fesetenv | fesetexceptflag | fesetround | fetestexcept | feupdateenv | fex_get_handling | fex_get_log | fex_get_log_depth | fex_getexcepthandler | fex_log_entry | fex_merge_flags | fex_set_handling | fex_set_log | fex_set_log_depth | fex_setexcepthandler | floor | floorf | floorl | fma | fmaf | fmal | fmax | fmaxf | fmaxl | fmin | fminf | fminl | fmod | fmodf | fmodl | frexp | frexpf | frexpl | gamma | gamma_r | gammaf | gammaf_r | gammal | gammal_r | hypot | hypotf | hypotl | ilogb | ilogbf | ilogbl | isnan | j0 | j0f | j0l | j1 | j1f | j1l | jn | jnf | jnl | ldexp | ldexpf | ldexpl | lgamma | lgamma_r | lgammaf | lgammaf_r | lgammal | lgammal_r | llrint | llrintf | llrintl | llround | llroundf | llroundl | log | log10 | log10f | log10l | log1p | log1pf | log1pl | log2 | log2f | log2l | logb | logbf | logbl | logf | logl | lrint | lrintf | lrintl | lround | lroundf | lroundl | matherr | modf | modff | modfl | nan | nanf | nanl | nearbyint | nearbyintf | nearbyintl | nextafter | nextafterf | nextafterl | nexttoward | nexttowardf | nexttowardl | pow | powf | powl | remainder | remainderf | remainderl | remquo | remquof | remquol | rint | rintf | rintl | round | roundf | roundl | scalb | scalbf | scalbl | scalbln | scalblnf | scalblnl | scalbn | scalbnf | scalbnl | signgam | signgamf | signgaml | significand | significandf | significandl | sin | sincos | sincosf | sincosl | sinf | sinh | sinhf | sinhl | sinl | sqrt | sqrtf | sqrtl | tan | tanf | tanh | tanhf | tanhl | tanl | tgamma | tgammaf | tgammal | trunc | truncf | truncl | y0 | y0f | y0l | y1 | y1f | y1l | yn | ynf | ynl |
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The following interfaces are unique to the x86 and x64 versions ofthis library:
ACCURACY
ISO/IEC 9899:1999, also known as C99, specifies the functions listed in thefollowing tables and states that the accuracy of these functions is “implementation-defined”.The information below characterizes the accuracy of these functions as implemented inlibm.so.2. For each function, the tables provide an upper bound on the largesterror possible for any argument and the largest error actually observed amonga large sample of arguments. Errors are expressed in “units in thelast place”, or ulps, relative to the exact function value for eachargument (regarding the argument as exact). Ulps depend on the precision of thefloating point format: ify is the exact function value,x andx' are adjacent floating point numbers such thatx <y <x', andx'' is the computed function value, then providedx,x',andx'' all lie in the same binade, the error inx''is |y -x''| / |x -x'| ulps. In particular, when theerror is less than one ulp, the computed value is one ofthe two floating point numbers adjacent to the exact value.
The bounds and observed errors listed below apply only in the defaultfloating point modes. Specifically, on SPARC, these bounds assume the rounding directionis round-to-nearest and non-standard mode is disabled. On x86, the bounds assumethe rounding direction is round-to-nearest and the rounding precision is round-to-64-bits. Moreover, onx86, floating point function values are returned in a floating point registerin extended double precision format, but the bounds below assume that theresult value is then stored to memory in the format corresponding tothe function's type. On x64, the bounds assume the rounding direction in boththe x87 floating point control word and the MXCSR is round-to-nearest, therounding precision in the x87 control word is round-to-64-bits, and the FTZand DAZ modes are disabled.
The error bounds listed below are believed to be correct, but smallerbounds might be proved later. The observed errors are the largest onescurrently known, but larger errors might be discovered later. Numbers in thenotes column refer to the notes following the tables.
Real Functions
Single precision real functions (SPARC, x86, and x64)
| error bound | largest error | |
|---|
function | (ulps) | observed(ulps) | notes |
|---|
acosf | 1.0 | < 1 | | acoshf | 1.0 | < 1 | | asinf | 1.0 | < 1 | | asinhf | 1.0 | < 1 | | atanf | 1.0 | < 1 | | atan2f | 1.0 | < 1 | | atanhf | 1.0 | < 1 | | cbrtf | 1.0 | < 1 | | cosf | 1.0 | < 1 | | coshf | 1.0 | < 1 | | erff | 1.0 | < 1 | | erfcf | 1.0 | <1 | | expf | 1.0 | < 1 | | exp2f | 1.0 | < 1 | | expm1f | 1.0 | < 1 | | hypotf | 1.0 | < 1 | | lgammaf | 1.0 | < 1 | | logf | 1.0 | < 1 | | log10f | 1.0 | < 1 | | log1pf | 1.0 | < 1 | | log2f | 1.0 | < 1 | | powf | 1.0 | < 1 | | sinf | 1.0 | < 1 | | sinhf | 1.0 | <1 | | sqrtf | 0.5 | 0.500 | [1] | tanf | 1.0 | < 1 | | tanhf | 1.0 | < 1 | | tgammaf | 1.0 | < 1 | |
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Double precision real functions (SPARC and x64)
| error bound | largest error | |
|---|
function | (ulps) | observed (ulps) | notes |
|---|
acos | 1.0 | < 1 | | acosh | 4.0 | 1.878 | | asin | 1.0 | < 1 | | asinh | 7.0 | 1.653 | | atan | 1.0 | <1 | | atan2 | 2.5 | 1.475 | | atanh | 4.0 | 1.960 | | cbrt | 1.0 | < 1 | | cos | 1.0 | < 1 | | cosh | 3.0 | 1.168 | | erf | 4.0 | 0.959 | | erfc | 6.0 | 2.816 | | exp | 1.0 | <1 | | exp2 | 2.0 | 1.050 | | expm1 | 1.0 | < 1 | | hypot | 1.0 | < 1 | | lgamma | 61.5 | 5.629 | [2] | log | 1.0 | < 1 | | log10 | 3.5 | 1.592 | | log1p | 1.0 | < 1 | | log2 | 1.0 | < 1 | | pow | 1.0 | < 1 | | sin | 1.0 | < 1 | | sinh | 4.0 | 2.078 | | sqrt | 0.5 | 0.500 | [1] | tan | 1.0 | < 1 | | tanh | 3.5 | 2.136 | | tgamma | 1.0 | < 1 | |
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Double precision real functions (x86)
| error bound | largest error | |
|---|
function | (ulps) | observed (ulps) | notes |
|---|
acos | 1.0 | < 1 | | acosh | 4.0 | 1.694 | | asin | 1.0 | < 1 | | asinh | 7.0 | 1.493 | | atan | 1.0 | < 1 | | atan2 | 1.0 | < 1 | | atanh | 4.0 | 1.445 | | cbrt | 1.0 | < 1 | | cos | 1.0 | < 1 | | cosh | 3.0 | 1.001 | | erf | 4.0 | 0.932 | | erfc | 6.0 | 2.728 | | exp | 1.0 | < 1 | | exp2 | 1.0 | < 1 | | expm1 | 1.0 | < 1 | | hypot | 1.0 | < 1 | | lgamma | 61.5 | 2.654 | [2] | log | 1.0 | <1 | | log10 | 1.0 | < 1 | | log1p | 1.0 | < 1 | | log2 | 1.0 | < 1 | | pow | 1.0 | < 1 | | sin | 1.0 | < 1 | | sinh | 4.0 | 1.458 | | sqrt | 0.5003 | 0.500 | [1] | tan | 1.0 | < 1 | | tanh | 3.5 | 1.592 | | tgamma | 1.0 | < 1 | |
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Quadruple precision real functions (SPARC)
| error bound | largest error | |
|---|
function | (ulps) | observed (ulps) | notes |
|---|
acosl | 3.5 | 1.771 | | acoshl | 8.0 | 1.275 | | asinl | 4.0 | 2.007 | | asinhl | 9.0 | 1.823 | | atanl | 1.0 | <1 | | atan2l | 2.5 | 1.102 | | atanhl | 4.0 | 1.970 | | cbrtl | 1.0 | < 1 | | cosl | 1.0 | < 1 | | coshl | 3.5 | 0.985 | | erfl | 2.0 | 0.779 | | erfcl | 68.5 | 13.923 | | expl | 1.0 | < 1 | | exp2l | 2.0 | 0.714 | | expm1l | 2.0 | 1.020 | | hypotl | 1.0 | < 1 | | lgammal | 18.5 | 2.916 | [2] | logl | 1.0 | < 1 | | log10l | 3.5 | 1.156 | | log1pl | 2.0 | 1.216 | | log2l | 3.5 | 1.675 | | powl | 1.0 | < 1 | | sinl | 1.0 | < 1 | | sinhl | 4.5 | 1.589 | | sqrtl | 0.5 | 0.500 | [1] | tanl | 4.5 | 2.380 | | tanhl | 4.5 | 1.692 | | tgammal | 1.0 | < 1 | |
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Extended precision real functions (x86 and x64)
| error bound | largest error | |
|---|
function | (ulps) | observed(ulps) | notes |
|---|
acosl | 3.0 | 1.868 | | acoshl | 8.0 | 2.352 | | asinl | 3.0 | 1.716 | | asinhl | 9.0 | 2.346 | | atanl | 1.0 | < 1 | | atan2l | 1.0 | < 1 | | atanhl | 4.0 | 2.438 | | cbrtl | 1.0 | < 1 | | cosl | 1.0 | < 1 | | coshl | 3.5 | 1.288 | | erfl | 1.0 | < 1 | | erfcl | 78.5 | 13.407 | | expl | 3.5 | 1.291 | | exp2l | 1.5 | 0.807 | | expm1l | 4.0 | 1.936 | | hypotl | 3.5 | 2.087 | | lgammal | 22.5 | 4.197 | [2] | logl | 2.0 | 0.881 | | log10l | 2.0 | 1.284 | | log1pl | 5.0 | 2.370 | | log2l | 1.0 | < 1 | | powl | 32770.0 | 4478.132 | | sinl | 1.0 | < 1 | | sinhl | 4.5 | 2.356 | | sqrtl | 0.5 | 0.500 | [1] | tanl | 4.5 | 2.366 | | tanhl | 4.5 | 2.417 | | tgammal | 1.0 | < 1 | |
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Notes:
- [1]
- On SPARC and x64,sqrtf,sqrt, andsqrtl are correctly rounded in accordance with IEEE 754. On x86,sqrtl is correctly rounded,sqrtf is correctly rounded provided the result is narrowed to single precision as discussed above, butsqrt might not be correctly rounded due to “double rounding”: when the intermediate value computed to extended precision lies exactly halfway between two representable numbers in double precision, the result of rounding the intermediate value to double precision is determined by the round-ties-to-even rule. If this rule causes the second rounding to round in the same direction as the first, the net rounding error can exceed 0.5 ulps. (The error is bounded instead by 0.5*(1 + 2^-11) ulps.)
- [2]
Error bounds for lgamma and lgammal apply only for positive arguments.
Complex functions
The real-valued complex functionscabsf,cabs,cabsl,cargf,carg, andcargl areequivalent to the real functionshypotf,hypot,hypotl,atan2f,atan2, andatan2l,respectively. The error bounds and observed errors given above for the latterfunctions also apply to the former.
The complex functions listed below are complex-valued. For each function, the errorbound shown applies separately to both the real and imaginary parts ofthe result. (For example, both the real and imaginary parts ofcacosf(z)are accurate to within 1 ulp regardless of their magnitudes.) Similarly, the largestobserved error shown is the largest error found in either the realor the imaginary part of the result.
Single precision complex functions (SPARC and x64)
| error bound | largest error | |
|---|
function | (ulps) | observed (ulps) | notes |
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cacosf,cacoshf | 1 | <1 | [1] | casinf,casinhf | 1 | < 1 | | catanf,catanhf | 6 | < 1 | | ccosf,ccoshf | 10 | 2.012 | | cexpf | 3 | 2.239 | | clogf | 3 | < 1 | | cpowf | — | < 1 | [2] | csinf,csinhf | 10 | 2.009 | | csqrtf | 4 | < 1 | | ctanf,ctanhf | 13 | 6.987 | |
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Single precision complex functions (x86)
| error bound | largesterror | |
|---|
function | (ulps) | observed (ulps) | notes |
|---|
cacosf,cacoshf | 1 | < 1 | [1] | casinf,casinhf | 1 | < 1 | | catanf,catanhf | 6 | < 1 | | ccosf,ccoshf | 10 | 1.984 | | cexpf | 3 | 1.984 | | clogf | 3 | < 1 | | cpowf | — | < 1 | [2] | csinf,csinhf | 10 | 1.973 | | csqrtf | 4 | < 1 | | ctanf,ctanhf | 13 | 4.657 | |
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Double precision complex functions (SPARC and x64)
| error bound | largest error | |
|---|
function | (ulps) | observed (ulps) | notes |
|---|
cacos,cacosh | 9 | 3.831 | [1] | casin,casinh | 9 | 3.732 | | catan,catanh | 6 | 4.179 | | ccos,ccosh | 10 | 3.832 | | cexp | 3 | 2.255 | | clog | 3 | 2.870 | | cpow | - | - | [2] | csin,csinh | 10 | 3.722 | | csqrt | 4 | 3.204 | | ctan,ctanh | 13 | 7.143 | |
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Double precision complex functions (x86)
| error bound | largest error | |
|---|
function | (ulps) | observed (ulps) | notes |
|---|
cacos,cacosh | 9 | 3.624 | [1] | casin,casinh | 9 | 3.624 | | catan,catanh | 6 | 2.500 | | ccos,ccosh | 10 | 2.929 | | cexp | 3 | 2.147 | | clog | 3 | 1.927 | | cpow | - | - | [2] | csin,csinh | 10 | 2.918 | | csqrt | 4 | 1.914 | | ctan,ctanh | 13 | 4.630 | |
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Quadruple precision complex functions (SPARC)
| error bound | largesterror | |
|---|
function | (ulps) | observed (ulps) | notes |
|---|
cacosl,cacoshl | 9 | 3 | [1] | casinl,casinhl | 9 | 3 | | catanl,catanhl | 6 | 3 | | ccosl,ccoshl | 10 | 3 | | cexpl | 3 | 2 | | clogl | 3 | 2 | | cpowl | - | - | [2] | csinl,csinhl | 10 | 3 | | csqrtl | 4 | 3 | | ctanl,ctanhl | 13 | 5 | |
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Extended precision complex functions (x86 and x64)
| error bound | largest error | |
|---|
function | (ulps) | observed(ulps) | notes |
|---|
cacosl,cacoshl | 9 | 2 | [1] | casinl,casinhl | 9 | 2 | | catanl,catanhl | 6 | 2 | | ccosl,ccoshl | 10 | 3 | | cexpl | 3 | 2.699 | | clogl | 3 | 1 | | cpowl | - | - | [2] | csinl,csinhl | 10 | 3 | | csqrtl | 4 | 1.452 | | ctanl,ctanhl | 13 | 5 | |
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Notes:
- [1]
- The complex hyperbolic trigonometric functions are equivalent by symmetries to their circular trigonometric counterparts. Because the implementations of these functions exploit these symmetries, corresponding functions have the same error bounds and observed errors.
- [2]
For large arguments, the results computed bycpowf,cpow, andcpowl can have unbounded relative error. It might be possible to give error bounds for specific domains, but no such bounds are currently available. The observed errors shown are for the domain {(z,w) :max(|Rez|, |Imz|, |Rew|, |Imw|) <= 1}.
Files
- /lib/libm.so.2
shared object
- /lib/64/libm.so.2
64-bit shared object
Attributes
Seeattributes(5) for descriptions of the following attributes:
ATTRIBUTE TYPE | ATTRIBUTE VALUE |
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Availability | system/library/math | MT-Level | Safe with exceptions |
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As described on thelgamma(3M) manual page,gamma() andlgamma() and theirfloat andlong double counterparts are Unsafe. All other functions inlibm.so.2 areMT-Safe.
See Also
Intro(3),lgamma(3M),math.h(3HEAD),attributes(5),standards(5)
