libm/common_source/exp.3

.\" Copyright (c) 1985, 1991, 1993
.\"	The Regents of the University of California.  All rights reserved.
.\"
.\" %sccs.include.redist.roff%
.\"
.\"     @(#)exp.3	8.2 (Berkeley) 04/19/94
.\"
.Dd
.Dt EXP 3
.Os BSD 4
.Sh NAME
.Nm exp ,
.Nm expm1 ,
.Nm log ,
.Nm log10 ,
.Nm log1p ,
.Nm pow
.Nd exponential, logarithm, power functions
.Sh SYNOPSIS
.Fd #include <math.h>
.Ft double
.Fn exp "double x"
.Ft double
.Fn expm1 "double x"
.Ft double
.Fn log "double x"
.Ft double
.Fn log10 "double x"
.Ft double
.Fn log1p "double x"
.Ft double
.Fn pow "double x" "double y"
.Sh DESCRIPTION
The
.Fn exp
function computes the exponential value of the given argument
.Fa x .
.Pp
The
.Fn expm1
function computes the value exp(x)\-1 accurately even for tiny argument
.Fa x .
.Pp
The
.Fn log
function computes the value for the natural logarithm of
the argument x.
.Pp
The
.Fn log10
function computes the value for the logarithm of
argument
.Fa x
to base 10.
.Pp
The
.Fn log1p
function computes
the value of log(1+x) accurately even for tiny argument
.Fa x .
.Pp
The
.Fn pow
computes the value
of
.Ar x
to the exponent
.Ar y .
.Sh ERROR (due to Roundoff etc.)
exp(x), log(x), expm1(x) and log1p(x) are accurate to within
an
.Em up ,
and log10(x) to within about 2
.Em ups ;
an
.Em up
is one
.Em Unit
in the
.Em Last
.Em Place .
The error in
.Fn pow x y
is below about 2
.Em ups
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 ups
for
.Tn "VAX D" ,
300
.Em ups
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
and
.Fn pow
detect if the computed value will overflow,
set the global variable
.Va errno to
.Er RANGE
and cause a reserved operand fault on a
.Tn VAX
or
.Tn Tahoe .
The function
.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
function
appeared in
.At v6 .
A
.Fn log10
function
appeared in
.At v7 .
The
.Fn log1p
and
.Fn expm1
functions appeared in
.Bx 4.3 .