324 lines
7.0 KiB
Groff
324 lines
7.0 KiB
Groff
.\" 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.
|
|
.\" 3. All advertising materials mentioning features or use of this software
|
|
.\" must display the following acknowledgement:
|
|
.\" This product includes software developed by the University of
|
|
.\" California, Berkeley and its contributors.
|
|
.\" 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
|
|
.\" $Id: exp.3,v 1.1.1.1 1994/08/19 09:39:42 jkh Exp $
|
|
.\"
|
|
.Dd July 31, 1991
|
|
.Dt EXP 3
|
|
.Os BSD 4
|
|
.Sh NAME
|
|
.Nm exp ,
|
|
.Nm expf ,
|
|
.Nm exp2 ,
|
|
.Nm exp2f ,
|
|
.Nm exp10 ,
|
|
.Nm exp10f ,
|
|
.Nm expm1 ,
|
|
.Nm expm1f ,
|
|
.Nm log ,
|
|
.Nm logf ,
|
|
.Nm log2 ,
|
|
.Nm log2f ,
|
|
.Nm log10 ,
|
|
.Nm log10f ,
|
|
.Nm log1p ,
|
|
.Nm log1pf ,
|
|
.Nm pow ,
|
|
.Nm powf
|
|
.Nd exponential, logarithm, power functions
|
|
.Sh SYNOPSIS
|
|
.Fd #include <math.h>
|
|
.Ft double
|
|
.Fn exp "double x"
|
|
.Ft float
|
|
.Fn expf "float x"
|
|
.Ft double
|
|
.Fn expm1 "double x"
|
|
.Ft float
|
|
.Fn expm1f "float x"
|
|
.Ft double
|
|
.Fn log "double x"
|
|
.Ft float
|
|
.Fn logf "float x"
|
|
.Ft double
|
|
.Fn log10 "double x"
|
|
.Ft float
|
|
.Fn log10f "float x"
|
|
.Ft double
|
|
.Fn log1p "double x"
|
|
.Ft float
|
|
.Fn log1pf "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 exponential value 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 log
|
|
and the
|
|
.Fn logf
|
|
functions compute the value of the natural logarithm of argument
|
|
.Fa x.
|
|
.Pp
|
|
The
|
|
.Fn log10
|
|
and the
|
|
.Fn log10f
|
|
functions compute the value of the logarithm of argument
|
|
.Fa x
|
|
to base 10.
|
|
.Pp
|
|
The
|
|
.Fn log1p
|
|
and the
|
|
.Fn log1pf
|
|
functions compute
|
|
the value of log(1+x) 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.)
|
|
.Fn exp(x) ,
|
|
.Fn log(x) ,
|
|
.Fn expm1(x) and
|
|
.Fn log1p(x)
|
|
are accurate to within
|
|
an
|
|
.Em ulp ,
|
|
and log10(x) to within about 2
|
|
.Em ulps ;
|
|
an
|
|
.Em ulp
|
|
is one
|
|
.Em Unit
|
|
in the
|
|
.Em Last
|
|
.Em Place .
|
|
The error in
|
|
.Fn pow x y
|
|
is below about 2
|
|
.Em ulps
|
|
when its
|
|
magnitude is moderate, but increases as
|
|
.Fn 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 8 bits for
|
|
.Tn "VAX D"
|
|
and 11 bits for IEEE 754 Double.
|
|
No such drastic loss has been exposed by testing; the worst
|
|
errors observed have been below 20
|
|
.Em ulps
|
|
for
|
|
.Tn "VAX D" ,
|
|
300
|
|
.Em ulps
|
|
for
|
|
.Tn IEEE
|
|
754 Double.
|
|
Moderate values of
|
|
.Fn pow
|
|
are accurate enough that
|
|
.Fn pow integer integer
|
|
is exact until it is bigger than 2**56 on a
|
|
.Tn VAX ,
|
|
2**53 for
|
|
.Tn IEEE
|
|
754.
|
|
.Sh RETURN VALUES
|
|
These functions will return the appropriate computation unless an error
|
|
occurs or an argument is out of range.
|
|
The functions
|
|
.Fn exp ,
|
|
.Fn expm1 ,
|
|
.Fn pow
|
|
detect if the computed value will overflow,
|
|
set the global variable
|
|
.Va errno to
|
|
.Er ERANGE
|
|
and cause a reserved operand fault on a
|
|
.Tn VAX
|
|
or
|
|
.Tn Tahoe .
|
|
The functions
|
|
.Fn pow x y
|
|
checks to see if
|
|
.Fa x
|
|
< 0 and
|
|
.Fa y
|
|
is not an integer, in the event this is true,
|
|
the global variable
|
|
.Va errno
|
|
is set to
|
|
.Er EDOM
|
|
and on the
|
|
.Tn VAX
|
|
and
|
|
.Tn Tahoe
|
|
generate a reserved operand fault.
|
|
On a
|
|
.Tn VAX
|
|
and
|
|
.Tn Tahoe ,
|
|
.Va errno
|
|
is set to
|
|
.Er EDOM
|
|
and the reserved operand is returned
|
|
by log unless
|
|
.Fa x
|
|
> 0, by
|
|
.Fn log1p
|
|
unless
|
|
.Fa x
|
|
> \-1.
|
|
.Sh NOTES
|
|
The functions exp(x)\-1 and log(1+x) are called
|
|
expm1 and logp1 in
|
|
.Tn BASIC
|
|
on the Hewlett\-Packard
|
|
.Tn HP Ns \-71B
|
|
and
|
|
.Tn APPLE
|
|
Macintosh,
|
|
.Tn EXP1
|
|
and
|
|
.Tn LN1
|
|
in Pascal, exp1 and log1 in C
|
|
on
|
|
.Tn 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.
|
|
.Pp
|
|
The function
|
|
.Fn pow x 0
|
|
returns x**0 = 1 for all x including x = 0,
|
|
.if n \
|
|
Infinity
|
|
.if t \
|
|
\(if
|
|
(not found on a
|
|
.Tn VAX ) ,
|
|
and
|
|
.Em NaN
|
|
(the reserved
|
|
operand on a
|
|
.Tn VAX ) . 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 -filled -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 n \
|
|
infinity**0 = 1/0**0 = 1 too; and
|
|
.if t \
|
|
\(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 math 3 ,
|
|
.Xr infnan 3
|
|
.Sh HISTORY
|
|
A
|
|
.Fn exp ,
|
|
.Fn log
|
|
and
|
|
.Fn pow
|
|
functions
|
|
appeared in
|
|
.At v6 .
|
|
A
|
|
.Fn log10
|
|
function
|
|
appeared in
|
|
.At v7 .
|
|
The
|
|
.Fn log1p
|
|
and
|
|
.Fn expm1
|
|
functions appeared in
|
|
.Bx 4.3 .
|