libm

- C math library

Synopsis

c99 [ flag... ] file... -lm [ library... ] 

Description

Functions in this library provide common elementary mathematical functions and floating point environment routines defined by System V, ANSI C, POSIX, and so on. See standards(5). Additional functions in this library provide extended support for handling floating point exceptions.

INTERFACES

The shared object libm.so.2 provides the public interfaces defined below. See Intro(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

The following interfaces are unique to the x86 and x64 versions of this library:

fegetprec fesetprec

ACCURACY

ISO/IEC 9899:1999, also known as C99, specifies the functions listed in the following tables and states that the accuracy of these functions is “implementation-defined”. The information below characterizes the accuracy of these functions as implemented in libm.so.2. For each function, the tables provide an upper bound on the largest error possible for any argument and the largest error actually observed among a large sample of arguments. Errors are expressed in “units in the last place”, or ulps, relative to the exact function value for each argument (regarding the argument as exact). Ulps depend on the precision of the floating point format: if y is the exact function value, x and x' are adjacent floating point numbers such that x < y < x', and x'' is the computed function value, then provided x, x', and x'' all lie in the same binade, the error in x'' is |y - x''| / |x - x'| ulps. In particular, when the error is less than one ulp, the computed value is one of the two floating point numbers adjacent to the exact value.

The bounds and observed errors listed below apply only in the default floating point modes. Specifically, on SPARC, these bounds assume the rounding direction is round-to-nearest and non-standard mode is disabled. On x86, the bounds assume the rounding direction is round-to-nearest and the rounding precision is round-to-64-bits. Moreover, on x86, floating point function values are returned in a floating point register in extended double precision format, but the bounds below assume that the result value is then stored to memory in the format corresponding to the function's type. On x64, the bounds assume the rounding direction in both the x87 floating point control word and the MXCSR is round-to-nearest, the rounding precision in the x87 control word is round-to-64-bits, and the FTZ and DAZ modes are disabled.

The error bounds listed below are believed to be correct, but smaller bounds might be proved later. The observed errors are the largest ones currently known, but larger errors might be discovered later. Numbers in the notes 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

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

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

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

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

Notes:

[1]

On SPARC and x64, sqrtf, sqrt, and sqrtl 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, but sqrt 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 functions cabsf, cabs, cabsl, cargf, carg, and cargl are equivalent to the real functions hypotf, hypot, hypotl, atan2f, atan2, and atan2l, respectively. The error bounds and observed errors given above for the latter functions also apply to the former.

The complex functions listed below are complex-valued. For each function, the error bound shown applies separately to both the real and imaginary parts of the result. (For example, both the real and imaginary parts of cacosf(z) are accurate to within 1 ulp regardless of their magnitudes.) Similarly, the largest observed error shown is the largest error found in either the real or the imaginary part of the result.

Single precision complex functions (SPARC and x64)

error bound largest error function (ulps) observed (ulps) notes 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

Single precision complex functions (x86)

error bound largest error 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

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

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

Quadruple precision complex functions (SPARC)

error bound largest error 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

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

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 by cpowf, cpow, and cpowl 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(|Re z|, |Im z|, |Re w|, |Im w|) <= 1}.

Files

/lib/libm.so.2

shared object

/lib/64/libm.so.2

64-bit shared object

Attributes

See attributes(5) for descriptions of the following attributes:

ATTRIBUTE TYPE ATTRIBUTE VALUE Availability system/library/math MT-Level Safe with exceptions

As described on the lgamma(3M) manual page, gamma() and lgamma() and their float and long double counterparts are Unsafe. All other functions in libm.so.2 are MT-Safe.

See Also

Intro(3), lgamma(3M), math.h(3HEAD), attributes(5), standards(5)