# lgamma, lgammaf, lgammal

< c‎ | numeric‎ | math

Common mathematical functions
Functions
Basic operations
 abslabsllabsimaxabs(C99)(C99) fabs divldivlldivimaxdiv(C99)(C99)
 fmod remainder(C99) remquo(C99) fma(C99) fdim(C99) nannanfnanlnandN(C99)(C99)(C99)(C23)
Maximum/minimum operations
 fmax(C99) fmaximum(C23) fmaximum_mag(C23) fmaximum_num(C23) fmaximum_mag_num(C23)
 fmin(C99) fminimum(C23) fminimum_mag(C23) fminimum_num(C23) fminimum_mag_num(C23)
Exponential functions
 exp exp10(C23) exp2(C99) expm1(C99) exp10m1(C23) exp2m1(C23)
 log log10 log2(C99) log1plogp1(C99)(C23) log10p1(C23) log2p1(C23)
Power functions
 sqrt cbrt(C99) rootn(C23) rsqrt(C23)
 hypot(C99) compound(C23) pow pown(C23) powr(C23)
Trigonometric and hyperbolic functions
 sin cos tan asin acos atan atan2 sinpi(C23) cospi(C23) tanpi(C23)
 asinpi(C23) acospi(C23) atanpi(C23) atan2pi(C23) sinh cosh tanh asinh(C99) acosh(C99) atanh(C99)
Error and gamma functions
 erf(C99) erfc(C99)
 lgamma(C99) tgamma(C99)
Nearest integer floating-point operations
 ceil floor roundlroundllround(C99)(C99)(C99) roundeven(C23) trunc(C99)
 nearbyint(C99) rintlrintllrint(C99)(C99)(C99) fromfpfromfpxufromfpufromfpx(C23)(C23)(C23)(C23)
Floating-point manipulation functions
 ldexp frexp scalbnscalbln(C99)(C99) ilogbllogb(C99)(C23) logb(C99)
 modf nextafternexttoward(C99)(C99) nextupnextdown(C23)(C23) copysign(C99) canonicalize(C23)
Narrowing operations
 fdiv(C23) ffma(C23) fsqrt(C23)
Quantum and quantum exponent functions
 quantizedN(C23) samequantumdN(C23)
 quantumdN(C23) llquantexpdN(C23)
Decimal re-encoding functions
 encodedecdN(C23) decodedecdN(C23)
 encodebindN(C23) decodebindN(C23)
Classification
 fpclassify(C99) iscanonical(C23) isfinite(C99) isinf(C99) isnan(C99) isnormal(C99) signbit(C99) issubnormal(C23) iszero(C23)
 isgreater(C99) isgreaterequal(C99) isless(C99) islessequal(C99) islessgreater(C99) isunordered(C99) issignaling(C23) iseqsig(C23)
Types
 div_tldiv_tlldiv_timaxdiv_t(C99)(C99)
 float_tdouble_t(C99)(C99) _Decimal32_t_Decimal64_t(C23)(C23)
Macro constants
Special floating-point values
 HUGE_VALFHUGE_VALHUGE_VALLHUGE_VALDN(C99)(C99)(C23)
 INFINITYDEC_INFINITY(C99)(C23) NANDEC_NAN(C99)(C23)
Arguments and return values
 FP_ILOGB0FP_ILOGBNAN(C99)(C99) FP_INT_UPWARDFP_INT_DOWNWARDFP_INT_TOWARDZEROFP_INT_TONEARESTFROMZEROFP_INT_TONEAREST(C23)(C23)(C23)(C23)(C23)
 FP_LLOGB0FP_LLOGBNAN(C23)(C23) FP_NORMALFP_SUBNORMALFP_ZEROFP_INFINITEFP_NAN(C99)(C99)(C99)(C99)(C99)
Error handling
Fast operation indicators

 Defined in header  float       lgammaf( float arg ); (1) (since C99) double      lgamma( double arg ); (2) (since C99) long double lgammal( long double arg ); (3) (since C99) Defined in header  #define lgamma( arg ) (4) (since C99)
1-3) Computes the natural logarithm of the absolute value of the gamma function of arg.
4) Type-generic macro: If arg has type long double, lgammal is called. Otherwise, if arg has integer type or the type double, lgamma is called. Otherwise, lgammaf is called.

## Contents

### Parameters

 arg - floating point value

### Return value

If no errors occur, the value of the logarithm of the gamma function of arg, that is log
e
|
0
targ-1
e-t dt|
, is returned.

If a pole error occurs, +HUGE_VAL, +HUGE_VALF, or +HUGE_VALL is returned.

If a range error due to overflow occurs, ±HUGE_VAL, ±HUGE_VALF, or ±HUGE_VALL is returned.

### Error handling

Errors are reported as specified in math_errhandling.

If arg is zero or is an integer less than zero, a pole error may occur.

If the implementation supports IEEE floating-point arithmetic (IEC 60559),

• If the argument is 1, +0 is returned.
• If the argument is 2, +0 is returned.
• If the argument is ±0, +∞ is returned and FE_DIVBYZERO is raised.
• If the argument is a negative integer, +∞ is returned and FE_DIVBYZERO is raised.
• If the argument is ±∞, +∞ is returned.
• If the argument is NaN, NaN is returned.

### Notes

If arg is a natural number, lgamma(arg) is the logarithm of the factorial of arg - 1.

The POSIX version of lgamma is not thread-safe: each execution of the function stores the sign of the gamma function of arg in the static external variable signgam. Some implementations provide lgamma_r, which takes a pointer to user-provided storage for singgam as the second parameter, and is thread-safe.

There is a non-standard function named gamma in various implementations, but its definition is inconsistent. For example, glibc and 4.2BSD version of gamma executes lgamma, but 4.4BSD version of gamma executes tgamma.

### Example

#include <stdio.h>
#include <math.h>
#include <float.h>
#include <errno.h>
#include <fenv.h>
// #pragma STDC FENV_ACCESS ON

int main(void)
{
printf("lgamma(10) = %f, log(9!) = %f\n", lgamma(10), log(2*3*4*5*6*7*8*9));
const double pi = acos(-1);
printf("lgamma(0.5) = %f, log(sqrt(pi)) = %f\n", log(sqrt(pi)), lgamma(0.5));
// special values
printf("lgamma(1) = %f\n", lgamma(1));
printf("lgamma(+Inf) = %f\n", lgamma(INFINITY));

// error handling
errno = 0; feclearexcept(FE_ALL_EXCEPT);
printf("lgamma(0) = %f\n", lgamma(0));
if (errno == ERANGE)
perror("    errno == ERANGE");
if (fetestexcept(FE_DIVBYZERO))
puts("    FE_DIVBYZERO raised");
}

Possible output:

lgamma(10) = 12.801827, log(9!) = 12.801827
lgamma(0.5) = 0.572365, log(sqrt(pi)) = 0.572365
lgamma(1) = 0.000000
lgamma(+Inf) = inf
lgamma(0) = inf
errno == ERANGE: Numerical result out of range
FE_DIVBYZERO raised

### References

• C17 standard (ISO/IEC 9899:2018):
• 7.12.8.3 The lgamma functions (p: 182)
• 7.25 Type-generic math <tgmath.h> (p: 272-273)
• F.10.5.3 The lgamma functions (p: 383)
• C11 standard (ISO/IEC 9899:2011):
• 7.12.8.3 The lgamma functions (p: 250)
• 7.25 Type-generic math <tgmath.h> (p: 373-375)
• F.10.5.3 The lgamma functions (p: 525)
• C99 standard (ISO/IEC 9899:1999):
• 7.12.8.3 The lgamma functions (p: 231)
• 7.22 Type-generic math <tgmath.h> (p: 335-337)
• F.9.5.3 The lgamma functions (p: 462)