libm/man/exp.3 - android_bionic - Gitiles

 .\" 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
 .\" $FreeBSD: src/lib/msun/man/exp.3,v 1.22 2005/04/05 02:57:28 das Exp $
 .\"
 .Dd April 5, 2005
 .Dt EXP 3
 .Os
 .Sh NAME
 .Nm exp ,
 .Nm expf ,
 .\" The sorting error is intentional.  exp and expf should be adjacent.
 .Nm exp2 ,
 .Nm exp2f ,
 .Nm expm1 ,
 .Nm expm1f ,
 .Nm log ,
 .Nm logf ,
 .Nm log10 ,
 .Nm log10f ,
 .Nm log1p ,
 .Nm log1pf ,
 .Nm pow ,
 .Nm powf
 .Nd exponential, logarithm, power functions
 .Sh LIBRARY
 .Lb libm
 .Sh SYNOPSIS
 .In math.h
 .Ft double
 .Fn exp "double x"
 .Ft float
 .Fn expf "float x"
 .Ft double
 .Fn exp2 "double x"
 .Ft float
 .Fn exp2f "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 base
 .Ms e
 exponential value of the given argument
 .Fa x .
 .Pp
 The
 .Fn exp2
 and the
 .Fn exp2f
 functions compute the base 2 exponential 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.)
 The values of
 .Fn exp 0 ,
 .Fn expm1 0 ,
 .Fn exp2 integer ,
 and
 .Fn pow integer integer
 are exact provided that they are representable.
 .\" XXX Is this really true for pow()?
 Otherwise the error in these functions is generally below one
 .Em ulp .
 .Sh RETURN VALUES
 These functions will return the appropriate computation unless an error
 occurs or an argument is out of range.
 The functions
 .Fn pow x y
 and
 .Fn powf x y
 raise an invalid exception and return an \*(Na if
 .Fa x
 < 0 and
 .Fa y
 is not an integer.
 An attempt to take the logarithm of \*(Pm0 will result in
 a divide-by-zero exception, and an infinity will be returned.
 An attempt to take the logarithm of a negative number will
 result in an invalid exception, and an \*(Na will be generated.
 .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, and \*(Na .
 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 -ragged -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**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 fenv 3 ,
 .Xr math 3
	.\" 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
	.\" $FreeBSD: src/lib/msun/man/exp.3,v 1.22 2005/04/05 02:57:28 das Exp $
	.\"
	.Dd April 5, 2005
	.Dt EXP 3
	.Os
	.Sh NAME
	.Nm exp ,
	.Nm expf ,
	.\" The sorting error is intentional. exp and expf should be adjacent.
	.Nm exp2 ,
	.Nm exp2f ,
	.Nm expm1 ,
	.Nm expm1f ,
	.Nm log ,
	.Nm logf ,
	.Nm log10 ,
	.Nm log10f ,
	.Nm log1p ,
	.Nm log1pf ,
	.Nm pow ,
	.Nm powf
	.Nd exponential, logarithm, power functions
	.Sh LIBRARY
	.Lb libm
	.Sh SYNOPSIS
	.In math.h
	.Ft double
	.Fn exp "double x"
	.Ft float
	.Fn expf "float x"
	.Ft double
	.Fn exp2 "double x"
	.Ft float
	.Fn exp2f "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 base
	.Ms e
	exponential value of the given argument
	.Fa x .
	.Pp
	The
	.Fn exp2
	and the
	.Fn exp2f
	functions compute the base 2 exponential 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.)
	The values of
	.Fn exp 0 ,
	.Fn expm1 0 ,
	.Fn exp2 integer ,
	and
	.Fn pow integer integer
	are exact provided that they are representable.
	.\" XXX Is this really true for pow()?
	Otherwise the error in these functions is generally below one
	.Em ulp .
	.Sh RETURN VALUES
	These functions will return the appropriate computation unless an error
	occurs or an argument is out of range.
	The functions
	.Fn pow x y
	and
	.Fn powf x y
	raise an invalid exception and return an \*(Na if
	.Fa x
	< 0 and
	.Fa y
	is not an integer.
	An attempt to take the logarithm of \*(Pm0 will result in
	a divide-by-zero exception, and an infinity will be returned.
	An attempt to take the logarithm of a negative number will
	result in an invalid exception, and an \*(Na will be generated.
	.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, and \*(Na .
	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]\(x0 + a[1]\(x1 + a[2]\(x2 +...+ a[n]\(xn
	.Ed
	.Pp
	at x = 0 rather than reject a[0]\(00 as invalid.
	.It
	Analysts will accept 00 = 1 despite that xy can
	approach anything or nothing as x and y approach 0
	independently.
	The reason for setting 0**0 = 1 anyway is this:
	.Bd -ragged -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
	\(If0 = 1/0*0 = 1 too; and
	then \(Na0 = 1 too because x*0 = 1 for all finite
	and infinite x, i.e., independently of x.
	.El
	.Sh SEE ALSO
	.Xr fenv 3 ,
	.Xr math 3