log1p, log1pf, log1pl

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< c‎ | numeric‎ | math
Common mathematical functions
Basic operations
Exponential functions
Power functions
Trigonometric and hyperbolic functions
Error and gamma functions
Nearest integer floating point operations
Floating point manipulation functions
Macro constants
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
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 (ln(x))
computes common (base-10) logarithm (log10(x))
computes base-2 logarithm (log2(x))
computes e raised to the given power, minus one (ex-1)