3ffff4bad5
as a fairly faithful implementation of the algorithm found in PTP Tang, "Table-driven implementation of the Expm1 function in IEEE floating-point arithmetic," ACM Trans. Math. Soft., 18, 211-222 (1992). Over the last 18-24 months, the code has under gone significant optimization and testing. Reviewed by: bde Obtained from: bde (most of the optimizations)
186 lines
4.8 KiB
Groff
186 lines
4.8 KiB
Groff
.\" Copyright (c) 1985, 1991 Regents of the University of California.
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.\" All rights reserved.
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.\"
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.\" Redistribution and use in source and binary forms, with or without
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.\" modification, are permitted provided that the following conditions
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.\" are met:
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.\" 1. Redistributions of source code must retain the above copyright
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.\" notice, this list of conditions and the following disclaimer.
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.\" 2. Redistributions in binary form must reproduce the above copyright
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.\" notice, this list of conditions and the following disclaimer in the
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.\" documentation and/or other materials provided with the distribution.
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.\" 4. Neither the name of the University nor the names of its contributors
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.\" may be used to endorse or promote products derived from this software
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.\" without specific prior written permission.
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.\"
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.\" THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
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.\" ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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.\" IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
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.\" ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
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.\" FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
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.\" DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
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.\" OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
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.\" HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
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.\" LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
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.\" OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
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.\" SUCH DAMAGE.
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.\"
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.\" from: @(#)exp.3 6.12 (Berkeley) 7/31/91
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.\" $FreeBSD$
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.\"
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.Dd June 3, 2013
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.Dt EXP 3
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.Os
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.Sh NAME
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.Nm exp ,
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.Nm expf ,
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.Nm expl ,
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.\" The sorting error is intentional. exp, expf, and expl should be adjacent.
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.Nm exp2 ,
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.Nm exp2f ,
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.Nm exp2l ,
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.Nm expm1 ,
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.Nm expm1f ,
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.Nm expm1l ,
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.Nm pow ,
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.Nm powf
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.Nd exponential and power functions
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.Sh LIBRARY
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.Lb libm
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.Sh SYNOPSIS
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.In math.h
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.Ft double
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.Fn exp "double x"
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.Ft float
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.Fn expf "float x"
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.Ft long double
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.Fn expl "long double x"
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.Ft double
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.Fn exp2 "double x"
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.Ft float
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.Fn exp2f "float x"
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.Ft long double
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.Fn exp2l "long double x"
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.Ft double
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.Fn expm1 "double x"
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.Ft float
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.Fn expm1f "float x"
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.Ft long double
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.Fn expm1l "long double x"
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.Ft double
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.Fn pow "double x" "double y"
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.Ft float
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.Fn powf "float x" "float y"
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.Sh DESCRIPTION
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The
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.Fn exp ,
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.Fn expf ,
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and
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.Fn expl
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functions compute the base
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.Ms e
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exponential value of the given argument
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.Fa x .
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.Pp
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The
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.Fn exp2 ,
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.Fn exp2f ,
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and
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.Fn exp2l
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functions compute the base 2 exponential of the given argument
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.Fa x .
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.Pp
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The
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.Fn expm1 ,
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.Fn expm1f ,
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and the
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.Fn expm1l
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functions compute the value exp(x)\-1 accurately even for tiny argument
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.Fa x .
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.Pp
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The
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.Fn pow
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and the
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.Fn powf
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functions compute the value
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of
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.Ar x
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to the exponent
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.Ar y .
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.Sh ERROR (due to Roundoff etc.)
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The values of
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.Fn exp 0 ,
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.Fn expm1 0 ,
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.Fn exp2 integer ,
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and
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.Fn pow integer integer
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are exact provided that they are representable.
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.\" XXX Is this really true for pow()?
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Otherwise the error in these functions is generally below one
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.Em ulp .
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.Sh RETURN VALUES
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These functions will return the appropriate computation unless an error
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occurs or an argument is out of range.
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The functions
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.Fn pow x y
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and
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.Fn powf x y
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raise an invalid exception and return an \*(Na if
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.Fa x
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< 0 and
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.Fa y
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is not an integer.
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.Sh NOTES
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The function
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.Fn pow x 0
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returns x**0 = 1 for all x including x = 0, \*(If, and \*(Na .
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Previous implementations of pow may
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have defined x**0 to be undefined in some or all of these
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cases.
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Here are reasons for returning x**0 = 1 always:
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.Bl -enum -width indent
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.It
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Any program that already tests whether x is zero (or
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infinite or \*(Na) before computing x**0 cannot care
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whether 0**0 = 1 or not.
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Any program that depends
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upon 0**0 to be invalid is dubious anyway since that
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expression's meaning and, if invalid, its consequences
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vary from one computer system to another.
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.It
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Some Algebra texts (e.g.\& Sigler's) define x**0 = 1 for
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all x, including x = 0.
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This is compatible with the convention that accepts a[0]
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as the value of polynomial
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.Bd -literal -offset indent
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p(x) = a[0]\(**x**0 + a[1]\(**x**1 + a[2]\(**x**2 +...+ a[n]\(**x**n
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.Ed
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.Pp
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at x = 0 rather than reject a[0]\(**0**0 as invalid.
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.It
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Analysts will accept 0**0 = 1 despite that x**y can
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approach anything or nothing as x and y approach 0
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independently.
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The reason for setting 0**0 = 1 anyway is this:
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.Bd -ragged -offset indent
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If x(z) and y(z) are
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.Em any
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functions analytic (expandable
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in power series) in z around z = 0, and if there
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x(0) = y(0) = 0, then x(z)**y(z) \(-> 1 as z \(-> 0.
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.Ed
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.It
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If 0**0 = 1, then
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\*(If**0 = 1/0**0 = 1 too; and
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then \*(Na**0 = 1 too because x**0 = 1 for all finite
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and infinite x, i.e., independently of x.
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.El
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.Sh SEE ALSO
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.Xr fenv 3 ,
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.Xr ldexp 3 ,
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.Xr log 3 ,
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.Xr math 3
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.Sh STANDARDS
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These functions conform to
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.St -isoC-99 .
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