log1p, log1pf, log1pl

Defined in header <math.h>
float       log1pf( float arg );
(1) (since C99)
double      log1p( double arg );
(2) (since C99)
long double log1pl( long double arg );
(3) (since C99)
Defined in header <tgmath.h>
#define log1p( arg )
(4) (since C99)
1-3) Computes the natural (base e) logarithm of 1+arg. This function is more precise than the expression log(1+arg) if arg is close to zero.
4) Type-generic macro: If arg has type long double, log1pl is called. Otherwise, if arg has integer type or the type double, log1p is called. Otherwise, log1pf is called.


arg - floating point value

Return value

If no errors occur ln(1+arg) is returned.

If a domain error occurs, an implementation-defined value is returned (NaN where supported).

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

If a range error occurs due to underflow, the correct result (after rounding) is returned.

Error handling

Errors are reported as specified in math_errhandling.

Domain error occurs if arg is less than -1.

Pole error may occur if arg is -1.

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

  • If the argument is ±0, it is returned unmodified
  • If the argument is -1, -∞ is returned and FE_DIVBYZERO is raised.
  • If the argument is less than -1, NaN is returned and FE_INVALID is raised.
  • If the argument is +∞, +∞ is returned
  • If the argument is NaN, NaN is returned


The functions expm1 and log1p are useful for financial calculations, for example, when calculating small daily interest rates: (1+x)n
-1 can be expressed as expm1(n * log1p(x)). These functions also simplify writing accurate inverse hyperbolic functions.


#include <stdio.h>
#include <math.h>
#include <float.h>
#include <errno.h>
#include <fenv.h>
int main(void)
    printf("log1p(0) = %f\n", log1p(0));
    printf("Interest earned in 2 days on $100, compounded daily at 1%%\n"
           " on a 30/360 calendar = %f\n",
    printf("log(1+1e-16) = %g, but log1p(1e-16) = %g\n",
           log(1+1e-16), log1p(1e-16));
    // special values
    printf("log1p(-0) = %f\n", log1p(-0.0));
    printf("log1p(+Inf) = %f\n", log1p(INFINITY));
    //error handling
    errno = 0; feclearexcept(FE_ALL_EXCEPT);
    printf("log1p(-1) = %f\n", log1p(-1));
    if(errno == ERANGE) perror("    errno == ERANGE");
    if(fetestexcept(FE_DIVBYZERO)) puts("    FE_DIVBYZERO raised");

Possible output:

log1p(0) = 0.000000
Interest earned in 2 days on $100, compounded daily at 1%
 on a 30/360 calendar = 0.005556
log(1+1e-16) = 0, but log1p(1e-16) = 1e-16
log1p(-0) = -0.000000
log1p(+Inf) = Inf
log1p(-1) = -Inf
    errno == ERANGE: Result too large
    FE_DIVBYZERO raised


  • C11 standard (ISO/IEC 9899:2011):
    • The log1p functions (p: 245)
    • 7.25 Type-generic math <tgmath.h> (p: 373-375)
    • F.10.3.9 The log1p functions (p: 522)
  • C99 standard (ISO/IEC 9899:1999):
    • The log1p functions (p: 226)
    • 7.22 Type-generic math <tgmath.h> (p: 335-337)
    • F.9.3.9 The log1p functions (p: 459)

See also

computes natural (base-e) logarithm (\({\small \ln{x} }\)ln(x))
computes common (base-10) logarithm (\({\small \log_{10}{x} }\)log10(x))
computes base-2 logarithm (\({\small \log_{2}{x} }\)log2(x))
computes e raised to the given power, minus one (\({\small e^x-1}\)ex-1)
C++ documentation for log1p

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