0dc9e0996a
As the current libc includes a libm implementation, with the new libc this is needed. Unneeded (for the moment) archs have been removed.
386 lines
7.6 KiB
Groff
386 lines
7.6 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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.\" 3. 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: src/lib/msun/man/exp.3,v 1.24 2008/01/18 21:43:00 das Exp $
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.\" $NetBSD: exp.3,v 1.26 2010/05/03 05:35:58 jruoho Exp $
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.\"
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.Dd May 3, 2010
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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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.\" The sorting error is intentional. exp and expf 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 log ,
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.Nm logf ,
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.Nm log2 ,
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.Nm log2f ,
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.Nm log10 ,
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.Nm log10f ,
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.Nm log1p ,
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.Nm log1pf ,
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.Nm pow ,
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.Nm powf
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.Nd exponential, logarithm, 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 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 double
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.Fn log "double x"
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.Ft float
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.Fn logf "float x"
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.Ft double
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.Fn log2 "double x"
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.Ft float
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.Fn log2f "float x"
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.Ft double
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.Fn log10 "double x"
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.Ft float
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.Fn log10f "float x"
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.Ft double
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.Fn log1p "double x"
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.Ft float
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.Fn log1pf "float 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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and the
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.Fn expf
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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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and
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.Fn exp2f
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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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and the
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.Fn expm1f
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functions computes 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 log
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function computes the value of the natural logarithm of argument
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.Fa x .
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.Pp
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The
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.Fn log10
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function computes the value of the logarithm of argument
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.Fa x
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to base 10.
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.Pp
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The
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.Fn log1p
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function computes
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the value of log(1+x) 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 log2
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and the
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.Fn log2f
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functions compute the value of the logarithm of argument
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.Fa x
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to base 2.
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.Pp
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The
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.Fn pow
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and
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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 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 exp ,
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.Fn expm1
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and
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.Fn pow
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detect if the computed value will overflow,
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set the global variable
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.Va errno
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to
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.Er ERANGE
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and cause a reserved operand fault on a
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.Tn VAX .
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The function
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.Fn pow x y
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checks to see if
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.Fa x
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\*[Lt] 0 and
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.Fa y
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is not an integer, in the event this is true,
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the global variable
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.Va errno
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is set to
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.Er EDOM
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and on the
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.Tn VAX
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generate a reserved operand fault.
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On a
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.Tn VAX ,
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.Va errno
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is set to
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.Er EDOM
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and the reserved operand is returned
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by log unless
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.Fa x
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\*[Gt] 0, by
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.Fn log1p
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unless
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.Fa x
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\*[Gt] \-1.
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.Sh ERRORS
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The values of
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.Fn exp x ,
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.Fn expm1 x ,
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.Fn exp2 x ,
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.Fn log x ,
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and
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.Fn log1p x ,
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are exact provided that they are representable.
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Otherwise the error in these functions is generally below one
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.Em ulp .
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The values of
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.Fn log10 x
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are within about 2
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.Em ulps ;
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an
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.Em ulp
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is one
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.Em Unit
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in the
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.Em Last
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.Em Place .
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The error in
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.Fn pow x y
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is below about 2
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.Em ulps
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when its
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magnitude is moderate, but increases as
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.Fn pow x y
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approaches
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the over/underflow thresholds until almost as many bits could be
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lost as are occupied by the floating\-point format's exponent
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field; that is 8 bits for
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.Tn "VAX D"
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and 11 bits for IEEE 754 Double.
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No such drastic loss has been exposed by testing; the worst
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errors observed have been below 20
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.Em ulps
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for
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.Tn "VAX D" ,
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300
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.Em ulps
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for
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.Tn IEEE
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754 Double.
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Moderate values of
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.Fn pow x y
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are accurate enough that
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.Fn pow integer integer
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is exact until it is bigger than 2**56 on a
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.Tn VAX ,
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2**53 for
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.Tn IEEE
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754.
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.Sh NOTES
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The functions
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.Fn exp x\ \-\ 1
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and
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.Fn log 1\ \+\ x
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are called
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.Fn expm1 x
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and
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.Fn logp1 x
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in
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.Tn BASIC
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on the Hewlett\-Packard
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.Tn HP Ns \-71B
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and
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.Tn APPLE
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Macintosh,
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.Tn EXP1
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and
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.Tn LN1
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in Pascal,
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.Fn exp1 x
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and
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.Fn log1 x
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in C
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on
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.Tn APPLE
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Macintoshes, where they have been provided to make
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sure financial calculations of ((1+x)**n\-1)/x, namely
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expm1(n*log1p(x))/x, will be accurate when x is tiny.
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They also provide accurate inverse hyperbolic functions.
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.Pp
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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,
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.if n \
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Infinity
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.if t \
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\(if
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(not found on a
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.Tn VAX ) ,
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and
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.Em NaN
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(the reserved
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operand on a
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.Tn VAX ) .
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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 -filled -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) \(-\*[Gt] 1 as z \(-\*[Gt] 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 n \
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infinity**0 = 1/0**0 = 1 too; and
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.if t \
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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 math 3
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.Sh STANDARDS
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The
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.Fn exp ,
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.Fn log ,
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.Fn log10
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and
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.Fn pow
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functions conform to
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.St -ansiC .
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The
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.Fn exp2 ,
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.Fn exp2f ,
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.Fn expf ,
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.Fn expm1 ,
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.Fn expm1f ,
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.Fn log1p ,
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.Fn log1pf ,
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.Fn log2 ,
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.Fn log2f ,
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.Fn log10f ,
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.Fn logf ,
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and
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.Fn powf
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functions conform to
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.St -isoC-99 .
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.Sh HISTORY
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A
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.Fn exp ,
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.Fn log
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and
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.Fn pow
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functions
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appeared in
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.At v6 .
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A
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.Fn log10
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function
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appeared in
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.At v7 .
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The
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.Fn log1p
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and
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.Fn expm1
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functions appeared in
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.Bx 4.3 .
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