## NAME

`exp`

, `expf`

,
`expl`

, `exp2`

,
`exp2f`

, `exp2l`

,
`expm1`

, `expm1f`

,
`expm1l`

, `log`

,
`logf`

, `logl`

,
`log2`

, `log2f`

,
`log2l`

, `log10`

,
`log10f`

, `log10l`

,
`log1p`

, `log1pf`

,
`log1pl`

, `pow`

,
`powf`

, `powl`

—
exponential, logarithm, power
functions

## SYNOPSIS

```
#include
<math.h>
```

`double`

`exp`

(`double
x`);

`float`

`expf`

(`float
x`);

`long double`

`expl`

(`long
double x`);

`double`

`exp2`

(`double
x`);

`float`

`exp2f`

(`float
x`);

`long double`

`exp2l`

(`long
double x`);

`double`

`expm1`

(`double
x`);

`float`

`expm1f`

(`float
x`);

`long double`

`expm1l`

(`long
double x`);

`double`

`log`

(`double
x`);

`float`

`logf`

(`float
x`);

`long double`

`logl`

(`long
double x`);

`double`

`log2`

(`double
x`);

`float`

`log2f`

(`float
x`);

`long double`

`log2l`

(`long
double x`);

`double`

`log10`

(`double
x`);

`float`

`log10f`

(`float
x`);

`long double`

`log10l`

(`long
double x`);

`double`

`log1p`

(`double
x`);

`float`

`log1pf`

(`float
x`);

`long double`

`log1pl`

(`long
double x`);

`double`

`pow`

(`double
x`, `double y`);

`float`

`powf`

(`float
x`, `float y`);

`long double`

`powl`

(`long
double x`, `long double
y`);

## DESCRIPTION

The
`exp`

()
function computes the base e exponential value of
the given argument `x`. The
`expf`

()
function is a single precision version of `exp`

(). The
`expl`

()
function is an extended precision version of
`exp`

().

The
`exp2`

()
function computes the base 2 exponential of the given argument
`x`. The
`exp2f`

()
function is a single precision version of `exp2`

().
The
`exp2l`

()
function is an extended precision version of
`exp2`

().

The
`expm1`

()
function computes the value exp(x) − 1 accurately even for tiny
argument `x`. The
`expm1f`

()
function is a single precision version of `expm1`

().
The
`expm1l`

()
function is an extended precision version of
`expm1`

().

The
`log`

() function
computes the value of the natural logarithm of argument
`x`. The
`logf`

()
function is a single precision version of `log`

(). The
`logl`

()
function is an extended precision version of
`log`

().

The
`log2`

()
function computes the value of the logarithm of argument
`x` to base 2. The
`log2f`

()
function is a single precision version of `log2`

().
The
`log2l`

()
function is an extended precision version of
`log2`

().

The
`log10`

()
function computes the value of the logarithm of argument
`x` to base 10. The
`log10f`

()
function is a single precision version of `log10`

().
The
`log10l`

()
function is an extended precision version of
`log10`

().

The
`log1p`

()
function computes the value of log(1 + x) accurately even for tiny argument
`x`. The
`log1pf`

()
function is a single precision version of `log1p`

().
The
`log1pl`

()
function is an extended precision version of
`log1p`

().

The
`pow`

() function
computes the value of `x` to the exponent
`y`. The
`powf`

()
function is a single precision version of `pow`

(). The
`powl`

()
function is an extended precision version of
`pow`

().

## RETURN VALUES

These functions will return the appropriate computation unless an
error occurs or an argument is out of range. The functions
`exp`

(), `expm1`

() and
`pow`

() detect if the computed value will overflow and
set the global variable `errno` to
`ERANGE`

. The function
`pow`

(`x`,
`y`) checks to see if `x` < 0 and
`y` is not an integer, in the event this is true, the
global variable `errno` is set to
`EDOM`

.

## ERRORS (due to Roundoff etc.)

exp(x), log(x), expm1(x) and log1p(x) are accurate to within an
*ulp*, and log10(x) to within about 2
*ulps*; an *ulp* is one
*Unit* in
the *Last*
*Place*.
The error in `pow`

(`x`,
`y`) is below about 2 *ulps* when its
magnitude is moderate, but increases as
`pow`

(`x`,
`y`) approaches the over/underflow thresholds until
almost as many bits could be lost as are occupied by the floating-point
format's exponent field; that is 11 bits for IEEE 754 Double. No such
drastic loss has been exposed by testing; the worst errors observed have
been below 300 *ulps* for IEEE 754 Double. Moderate values
of `pow`

() are accurate enough that
`pow`

(`integer`,
`integer`) is exact until it is bigger than 2**53 for
IEEE 754.

## NOTES

The functions exp(x) − 1 and log(1 + x) are called expm1 and logp1 in BASIC on the Hewlett-Packard HP-71B and APPLE Macintosh, EXP1 and LN1 in Pascal, exp1 and log1 in C on APPLE Macintoshes, where they have been provided to make sure financial calculations of ((1 + x)**n − 1) / x, namely expm1(n * log1p(x)) / x, will be accurate when x is tiny. They also provide accurate inverse hyperbolic functions.

The function
`pow`

(`x`,
`0`) returns x**0 = 1 for all x including x = 0 and
infinity. Previous implementations of `pow`

() may have
defined x**0 to be undefined in some or all of these cases. Here are reasons
for returning x**0 = 1 always:

- Any program that already tests whether x is zero (or infinite or NaN) before computing x**0 cannot care whether 0**0 = 1 or not. Any program that depends upon 0**0 to be invalid is dubious anyway since that expression's meaning and, if invalid, its consequences vary from one computer system to another.
- Some Algebra texts (e.g., Sigler's) define x**0 = 1 for all x, including x
= 0. This is compatible with the convention that accepts a[0] as the value
of polynomial
p(x) = a[0]*x**0 + a[1]*x**1 + a[2]*x**2 +...+ a[n]*x**n

at x = 0 rather than reject a[0]*0**0 as invalid.

- Analysts will accept 0**0 = 1 despite that x**y can approach anything or
nothing as x and y approach 0 independently. The reason for setting 0**0 =
1 anyway is this:
If x(z) and y(z) are
*any*functions analytic (expandable in power series) in z around z = 0, and if there x(0) = y(0) = 0, then x(z)**y(z) → 1 as z → 0. - If 0**0 = 1, then infinity**0 = 1/0**0 = 1 too; and then NaN**0 = 1 too because x**0 = 1 for all finite and infinite x, i.e., independently of x.

## SEE ALSO

## HISTORY

The `exp`

() and
`log`

() functions first appeared in
Version 1 AT&T UNIX;
`pow`

() in Version 3 AT&T
UNIX; `log10`

() in
Version 7 AT&T UNIX;
`log1p`

() and `expm1`

() in
4.3BSD.