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.\" Copyright (c) 1985, 1991 Regents of the University of California.
.\" All rights reserved.
.\"
.\" Redistribution and use in source and binary forms, with or without
.\" modification, are permitted provided that the following conditions
.\" are met:
.\" 1. Redistributions of source code must retain the above copyright
.\" notice, this list of conditions and the following disclaimer.
.\" 2. Redistributions in binary form must reproduce the above copyright
.\" notice, this list of conditions and the following disclaimer in the
.\" documentation and/or other materials provided with the distribution.
.\" 4. Neither the name of the University nor the names of its contributors
.\" may be used to endorse or promote products derived from this software
.\" without specific prior written permission.
.\"
.\" THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
.\" ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
.\" IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
.\" ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
.\" FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
.\" DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
.\" OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
.\" HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
.\" LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
.\" OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
.\" SUCH DAMAGE.
.\"
.\" from: @(#)exp.3 6.12 (Berkeley) 7/31/91
.\" $FreeBSD$
.\"
.Dd January 17, 2008
.Dt EXP 3
.Os
.Sh NAME
.Nm exp ,
.Nm expf ,
.\" The sorting error is intentional. exp and expf should be adjacent.
.Nm exp2 ,
.Nm exp2f ,
.Nm exp2l ,
.Nm expm1 ,
.Nm expm1f ,
.Nm pow ,
.Nm powf
.Nd exponential and power functions
.Sh LIBRARY
.Lb libm
.Sh SYNOPSIS
.In math.h
.Ft double
.Fn exp "double x"
.Ft float
.Fn expf "float x"
.Ft double
.Fn exp2 "double x"
.Ft float
.Fn exp2f "float x"
.Ft long double
.Fn exp2l "long double x"
.Ft double
.Fn expm1 "double x"
.Ft float
.Fn expm1f "float x"
.Ft double
.Fn pow "double x" "double y"
.Ft float
.Fn powf "float x" "float y"
.Sh DESCRIPTION
The
.Fn exp
and the
.Fn expf
functions compute the base
.Ms e
exponential value of the given argument
.Fa x .
.Pp
The
.Fn exp2 ,
.Fn exp2f ,
and
.Fn exp2l
functions compute the base 2 exponential of the given argument
.Fa x .
.Pp
The
.Fn expm1
and the
.Fn expm1f
functions compute the value exp(x)\-1 accurately even for tiny argument
.Fa x .
.Pp
The
.Fn pow
and the
.Fn powf
functions compute the value
of
.Ar x
to the exponent
.Ar y .
.Sh ERROR (due to Roundoff etc.)
The values of
.Fn exp 0 ,
.Fn expm1 0 ,
.Fn exp2 integer ,
and
.Fn pow integer integer
are exact provided that they are representable.
.\" XXX Is this really true for pow()?
Otherwise the error in these functions is generally below one
.Em ulp .
.Sh RETURN VALUES
These functions will return the appropriate computation unless an error
occurs or an argument is out of range.
The functions
.Fn pow x y
and
.Fn powf x y
raise an invalid exception and return an \*(Na if
.Fa x
< 0 and
.Fa y
is not an integer.
.Sh NOTES
The function
.Fn pow x 0
returns x**0 = 1 for all x including x = 0, \*(If, and \*(Na .
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:
.Bl -enum -width indent
.It
Any program that already tests whether x is zero (or
infinite or \*(Na) 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.
.It
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
.Bd -literal -offset indent
p(x) = a[0]\(**x**0 + a[1]\(**x**1 + a[2]\(**x**2 +...+ a[n]\(**x**n
.Ed
.Pp
at x = 0 rather than reject a[0]\(**0**0 as invalid.
.It
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:
.Bd -ragged -offset indent
If x(z) and y(z) are
.Em 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.
.Ed
.It
If 0**0 = 1, then
\*(If**0 = 1/0**0 = 1 too; and
then \*(Na**0 = 1 too because x**0 = 1 for all finite
and infinite x, i.e., independently of x.
.El
.Sh SEE ALSO
.Xr fenv 3 ,
.Xr ldexp 3 ,
.Xr log 3 ,
.Xr math 3
.Sh STANDARDS
These functions conform to
.St -isoC-99 .
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