.\" 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 1.2 1995/10/28 00:56:52 deraadt Exp $ .\" .Dd July 31, 1991 .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 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 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 of the natural logarithm of argument .Fa x. .Pp The .Fn log10 function computes the value of 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 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 and .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 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 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 .