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### 19.4 Exponentiation and Logarithms

Function: double exp (double x)
Function: float expf (float x)
Function: long double expl (long double x)
Function: _FloatN expfN (_FloatN x)
Function: _FloatNx expfNx (_FloatNx x)

Preliminary: | MT-Safe | AS-Safe | AC-Safe | See POSIX Safety Concepts.

These functions compute `e` (the base of natural logarithms) raised to the power x.

If the magnitude of the result is too large to be representable, `exp` signals overflow.

Function: double exp2 (double x)
Function: float exp2f (float x)
Function: long double exp2l (long double x)
Function: _FloatN exp2fN (_FloatN x)
Function: _FloatNx exp2fNx (_FloatNx x)

Preliminary: | MT-Safe | AS-Safe | AC-Safe | See POSIX Safety Concepts.

These functions compute `2` raised to the power x. Mathematically, `exp2 (x)` is the same as `exp (x * log (2))`.

Function: double exp10 (double x)
Function: float exp10f (float x)
Function: long double exp10l (long double x)
Function: _FloatN exp10fN (_FloatN x)
Function: _FloatNx exp10fNx (_FloatNx x)

Preliminary: | MT-Safe | AS-Safe | AC-Safe | See POSIX Safety Concepts.

These functions compute `10` raised to the power x. Mathematically, `exp10 (x)` is the same as `exp (x * log (10))`.

The `exp10` functions are from TS 18661-4:2015.

Function: double log (double x)
Function: float logf (float x)
Function: long double logl (long double x)
Function: _FloatN logfN (_FloatN x)
Function: _FloatNx logfNx (_FloatNx x)

Preliminary: | MT-Safe | AS-Safe | AC-Safe | See POSIX Safety Concepts.

These functions compute the natural logarithm of x. ```exp (log (x))``` equals x, exactly in mathematics and approximately in C.

If x is negative, `log` signals a domain error. If x is zero, it returns negative infinity; if x is too close to zero, it may signal overflow.

Function: double log10 (double x)
Function: float log10f (float x)
Function: long double log10l (long double x)
Function: _FloatN log10fN (_FloatN x)
Function: _FloatNx log10fNx (_FloatNx x)

Preliminary: | MT-Safe | AS-Safe | AC-Safe | See POSIX Safety Concepts.

These functions return the base-10 logarithm of x. `log10 (x)` equals `log (x) / log (10)`.

Function: double log2 (double x)
Function: float log2f (float x)
Function: long double log2l (long double x)
Function: _FloatN log2fN (_FloatN x)
Function: _FloatNx log2fNx (_FloatNx x)

Preliminary: | MT-Safe | AS-Safe | AC-Safe | See POSIX Safety Concepts.

These functions return the base-2 logarithm of x. `log2 (x)` equals `log (x) / log (2)`.

Function: double logb (double x)
Function: float logbf (float x)
Function: long double logbl (long double x)
Function: _FloatN logbfN (_FloatN x)
Function: _FloatNx logbfNx (_FloatNx x)

Preliminary: | MT-Safe | AS-Safe | AC-Safe | See POSIX Safety Concepts.

These functions extract the exponent of x and return it as a floating-point value. If `FLT_RADIX` is two, `logb` is equal to `floor (log2 (x))`, except it’s probably faster.

If x is de-normalized, `logb` returns the exponent x would have if it were normalized. If x is infinity (positive or negative), `logb` returns &infin;. If x is zero, `logb` returns &infin;. It does not signal.

Function: int ilogb (double x)
Function: int ilogbf (float x)
Function: int ilogbl (long double x)
Function: int ilogbfN (_FloatN x)
Function: int ilogbfNx (_FloatNx x)
Function: long int llogb (double x)
Function: long int llogbf (float x)
Function: long int llogbl (long double x)
Function: long int llogbfN (_FloatN x)
Function: long int llogbfNx (_FloatNx x)

Preliminary: | MT-Safe | AS-Safe | AC-Safe | See POSIX Safety Concepts.

These functions are equivalent to the corresponding `logb` functions except that they return signed integer values. The `ilogb`, `ilogbf`, and `ilogbl` functions are from ISO C99; the `llogb`, `llogbf`, `llogbl` functions are from TS 18661-1:2014; the `ilogbfN`, `ilogbfNx`, `llogbfN`, and `llogbfNx` functions are from TS 18661-3:2015.

Since integers cannot represent infinity and NaN, `ilogb` instead returns an integer that can’t be the exponent of a normal floating-point number. math.h defines constants so you can check for this.

Macro: int FP_ILOGB0

`ilogb` returns this value if its argument is `0`. The numeric value is either `INT_MIN` or `-INT_MAX`.

This macro is defined in ISO C99.

Macro: long int FP_LLOGB0

`llogb` returns this value if its argument is `0`. The numeric value is either `LONG_MIN` or `-LONG_MAX`.

This macro is defined in TS 18661-1:2014.

Macro: int FP_ILOGBNAN

`ilogb` returns this value if its argument is `NaN`. The numeric value is either `INT_MIN` or `INT_MAX`.

This macro is defined in ISO C99.

Macro: long int FP_LLOGBNAN

`llogb` returns this value if its argument is `NaN`. The numeric value is either `LONG_MIN` or `LONG_MAX`.

This macro is defined in TS 18661-1:2014.

These values are system specific. They might even be the same. The proper way to test the result of `ilogb` is as follows:

```i = ilogb (f);
if (i == FP_ILOGB0 || i == FP_ILOGBNAN)
{
if (isnan (f))
{
/* Handle NaN.  */
}
else if (f  == 0.0)
{
/* Handle 0.0.  */
}
else
{
/* Some other value with large exponent,
perhaps +Inf.  */
}
}
```
Function: double pow (double base, double power)
Function: float powf (float base, float power)
Function: long double powl (long double base, long double power)
Function: _FloatN powfN (_FloatN base, _FloatN power)
Function: _FloatNx powfNx (_FloatNx base, _FloatNx power)

Preliminary: | MT-Safe | AS-Safe | AC-Safe | See POSIX Safety Concepts.

These are general exponentiation functions, returning base raised to power.

Mathematically, `pow` would return a complex number when base is negative and power is not an integral value. `pow` can’t do that, so instead it signals a domain error. `pow` may also underflow or overflow the destination type.

Function: double sqrt (double x)
Function: float sqrtf (float x)
Function: long double sqrtl (long double x)
Function: _FloatN sqrtfN (_FloatN x)
Function: _FloatNx sqrtfNx (_FloatNx x)

Preliminary: | MT-Safe | AS-Safe | AC-Safe | See POSIX Safety Concepts.

These functions return the nonnegative square root of x.

If x is negative, `sqrt` signals a domain error. Mathematically, it should return a complex number.

Function: double cbrt (double x)
Function: float cbrtf (float x)
Function: long double cbrtl (long double x)
Function: _FloatN cbrtfN (_FloatN x)
Function: _FloatNx cbrtfNx (_FloatNx x)

Preliminary: | MT-Safe | AS-Safe | AC-Safe | See POSIX Safety Concepts.

These functions return the cube root of x. They cannot fail; every representable real value has a representable real cube root.

Function: double hypot (double x, double y)
Function: float hypotf (float x, float y)
Function: long double hypotl (long double x, long double y)
Function: _FloatN hypotfN (_FloatN x, _FloatN y)
Function: _FloatNx hypotfNx (_FloatNx x, _FloatNx y)

Preliminary: | MT-Safe | AS-Safe | AC-Safe | See POSIX Safety Concepts.

These functions return ```sqrt (x*x + y*y)```. This is the length of the hypotenuse of a right triangle with sides of length x and y, or the distance of the point (x, y) from the origin. Using this function instead of the direct formula is wise, since the error is much smaller. See also the function `cabs` in Absolute Value.

Function: double expm1 (double x)
Function: float expm1f (float x)
Function: long double expm1l (long double x)
Function: _FloatN expm1fN (_FloatN x)
Function: _FloatNx expm1fNx (_FloatNx x)

Preliminary: | MT-Safe | AS-Safe | AC-Safe | See POSIX Safety Concepts.

These functions return a value equivalent to `exp (x) - 1`. They are computed in a way that is accurate even if x is near zero—a case where `exp (x) - 1` would be inaccurate owing to subtraction of two numbers that are nearly equal.

Function: double log1p (double x)
Function: float log1pf (float x)
Function: long double log1pl (long double x)
Function: _FloatN log1pfN (_FloatN x)
Function: _FloatNx log1pfNx (_FloatNx x)

Preliminary: | MT-Safe | AS-Safe | AC-Safe | See POSIX Safety Concepts.

These functions return a value equivalent to `log (1 + x)`. They are computed in a way that is accurate even if x is near zero.

ISO C99 defines complex variants of some of the exponentiation and logarithm functions.

Function: complex double cexp (complex double z)
Function: complex float cexpf (complex float z)
Function: complex long double cexpl (complex long double z)
Function: complex _FloatN cexpfN (complex _FloatN z)
Function: complex _FloatNx cexpfNx (complex _FloatNx z)

Preliminary: | MT-Safe | AS-Safe | AC-Safe | See POSIX Safety Concepts.

These functions return `e` (the base of natural logarithms) raised to the power of z. Mathematically, this corresponds to the value

exp (z) = exp (creal (z)) * (cos (cimag (z)) + I * sin (cimag (z)))

Function: complex double clog (complex double z)
Function: complex float clogf (complex float z)
Function: complex long double clogl (complex long double z)
Function: complex _FloatN clogfN (complex _FloatN z)
Function: complex _FloatNx clogfNx (complex _FloatNx z)

Preliminary: | MT-Safe | AS-Safe | AC-Safe | See POSIX Safety Concepts.

These functions return the natural logarithm of z. Mathematically, this corresponds to the value

log (z) = log (cabs (z)) + I * carg (z)

`clog` has a pole at 0, and will signal overflow if z equals or is very close to 0. It is well-defined for all other values of z.

Function: complex double clog10 (complex double z)
Function: complex float clog10f (complex float z)
Function: complex long double clog10l (complex long double z)
Function: complex _FloatN clog10fN (complex _FloatN z)
Function: complex _FloatNx clog10fNx (complex _FloatNx z)

Preliminary: | MT-Safe | AS-Safe | AC-Safe | See POSIX Safety Concepts.

These functions return the base 10 logarithm of the complex value z. Mathematically, this corresponds to the value

log10 (z) = log10 (cabs (z)) + I * carg (z) / log (10)

All these functions, including the `_FloatN` and `_FloatNx` variants, are GNU extensions.

Function: complex double csqrt (complex double z)
Function: complex float csqrtf (complex float z)
Function: complex long double csqrtl (complex long double z)
Function: complex _FloatN csqrtfN (_FloatN z)
Function: complex _FloatNx csqrtfNx (complex _FloatNx z)

Preliminary: | MT-Safe | AS-Safe | AC-Safe | See POSIX Safety Concepts.

These functions return the complex square root of the argument z. Unlike the real-valued functions, they are defined for all values of z.

Function: complex double cpow (complex double base, complex double power)
Function: complex float cpowf (complex float base, complex float power)
Function: complex long double cpowl (complex long double base, complex long double power)
Function: complex _FloatN cpowfN (complex _FloatN base, complex _FloatN power)
Function: complex _FloatNx cpowfNx (complex _FloatNx base, complex _FloatNx power)

Preliminary: | MT-Safe | AS-Safe | AC-Safe | See POSIX Safety Concepts.

These functions return base raised to the power of power. This is equivalent to `cexp (y * clog (x))`

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