# log1p, log1pf, log1pl

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< c‎ | numeric‎ | math

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.

### Parameters

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

### Notes

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.

### Example

Run this code

    #include <stdio.h>
    #include <math.h>
    #include <float.h>
    #include <errno.h>
    #include <fenv.h>
    #pragma STDC FENV_ACCESS ON
    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",
               100*expm1(2*log1p(0.01/360)));
        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

### References

- C11 standard (ISO/IEC 9899:2011):

  - 7.12.6.9 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):

  - 7.12.6.9 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

log logf logl (C99)(C99) | computes natural (base-e) logarithm (ln(x)) (function)
log10 log10f log10l (C99)(C99) | computes common (base-10) logarithm (log10(x)) (function)
log2 log2f log2l (C99)(C99)(C99) | computes base-2 logarithm (log2(x)) (function)
expm1 expm1f expm1l (C99)(C99)(C99) | computes e raised to the given power, minus one (ex-1) (function)

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