290 lines
6.5 KiB
Groff
290 lines
6.5 KiB
Groff
.\" Copyright (c) 1985, 1991, 1993
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.\" The Regents of the University of California. 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. All advertising materials mentioning features or use of this software
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.\" must display the following acknowledgement:
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.\" This product includes software developed by the University of
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.\" California, Berkeley and its contributors.
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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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.\" @(#)exp.3 8.2 (Berkeley) 4/19/94
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.\" $FreeBSD$
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.\"
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.Dd April 19, 1994
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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 expm1 ,
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.Nm log ,
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.Nm log10 ,
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.Nm log1p ,
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.Nm pow
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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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.Fd #include <math.h>
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.Ft double
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.Fn exp "double x"
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.Ft double
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.Fn expm1 "double x"
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.Ft double
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.Fn log "double x"
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.Ft double
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.Fn log10 "double x"
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.Ft double
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.Fn log1p "double x"
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.Ft double
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.Fn pow "double x" "double y"
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.Sh DESCRIPTION
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The
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.Fn exp
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function computes the 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 expm1
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function 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 for the natural logarithm of
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the argument 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 for the logarithm of
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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 pow
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computes 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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exp(x), log(x), expm1(x) and log1p(x) are accurate to within
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an
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.Em up ,
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and log10(x) to within about 2
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.Em ups ;
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an
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.Em up
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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 ups
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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 ups
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for
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.Tn "VAX D" ,
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300
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.Em ups
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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
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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 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 to
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.Er RANGE
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and cause a reserved operand fault on a
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.Tn VAX
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or
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.Tn Tahoe .
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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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< 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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and
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.Tn Tahoe
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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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and
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.Tn Tahoe ,
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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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> 0, by
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.Fn log1p
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unless
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.Fa x
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> \-1.
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.Sh NOTES
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The functions exp(x)\-1 and log(1+x) are called
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expm1 and logp1 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, exp1 and log1 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. 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 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 infnan 3 ,
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.Xr math 3
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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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function
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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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