1.\" Copyright (c) 1985, 1991 Regents of the University of California. 2.\" All rights reserved. 3.\" 4.\" Redistribution and use in source and binary forms, with or without 5.\" modification, are permitted provided that the following conditions 6.\" are met: 7.\" 1. Redistributions of source code must retain the above copyright 8.\" notice, this list of conditions and the following disclaimer. 9.\" 2. Redistributions in binary form must reproduce the above copyright 10.\" notice, this list of conditions and the following disclaimer in the 11.\" documentation and/or other materials provided with the distribution. 12.\" 3. All advertising materials mentioning features or use of this software 13.\" must display the following acknowledgement: 14.\" This product includes software developed by the University of 15.\" California, Berkeley and its contributors. 16.\" 4. Neither the name of the University nor the names of its contributors 17.\" may be used to endorse or promote products derived from this software 18.\" without specific prior written permission. 19.\" 20.\" THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND 21.\" ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 22.\" IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 23.\" ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE 24.\" FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 25.\" DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 26.\" OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 27.\" HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 28.\" LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 29.\" OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 30.\" SUCH DAMAGE. 31.\" 32.\" from: @(#)exp.3 6.12 (Berkeley) 7/31/91 33.\" $Id: exp.3,v 1.2 1993/08/01 07:41:21 mycroft Exp $ 34.\" 35.Dd July 31, 1991 36.Dt EXP 3 37.Os BSD 4 38.Sh NAME 39.Nm exp , 40.Nm expm1 , 41.Nm log , 42.Nm log10 , 43.Nm log1p , 44.Nm pow 45.Nd exponential, logarithm, power functions 46.Sh SYNOPSIS 47.Fd #include <math.h> 48.Ft double 49.Fn exp "double x" 50.Ft double 51.Fn expm1 "double x" 52.Ft double 53.Fn log "double x" 54.Ft double 55.Fn log10 "double x" 56.Ft double 57.Fn log1p "double x" 58.Ft double 59.Fn pow "double x" "double y" 60.Sh DESCRIPTION 61The 62.Fn exp 63function computes the exponential value of the given argument 64.Fa x . 65.Pp 66The 67.Fn expm1 68function computes the value exp(x)\-1 accurately even for tiny argument 69.Fa x . 70.Pp 71The 72.Fn log 73function computes the value for the natural logarithm of 74the argument x. 75.Pp 76The 77.Fn log10 78function computes the value for the logarithm of 79argument 80.Fa x 81to base 10. 82.Pp 83The 84.Fn log1p 85function computes 86the value of log(1+x) accurately even for tiny argument 87.Fa x . 88.Pp 89The 90.Fn pow 91computes the value 92of 93.Ar x 94to the exponent 95.Ar y . 96.Sh ERROR (due to Roundoff etc.) 97exp(x), log(x), expm1(x) and log1p(x) are accurate to within 98an 99.Em up , 100and log10(x) to within about 2 101.Em ups ; 102an 103.Em up 104is one 105.Em Unit 106in the 107.Em Last 108.Em Place . 109The error in 110.Fn pow x y 111is below about 2 112.Em ups 113when its 114magnitude is moderate, but increases as 115.Fn pow x y 116approaches 117the over/underflow thresholds until almost as many bits could be 118lost as are occupied by the floating\-point format's exponent 119field; that is 8 bits for 120.Tn "VAX D" 121and 11 bits for IEEE 754 Double. 122No such drastic loss has been exposed by testing; the worst 123errors observed have been below 20 124.Em ups 125for 126.Tn "VAX D" , 127300 128.Em ups 129for 130.Tn IEEE 131754 Double. 132Moderate values of 133.Fn pow 134are accurate enough that 135.Fn pow integer integer 136is exact until it is bigger than 2**56 on a 137.Tn VAX , 1382**53 for 139.Tn IEEE 140754. 141.Sh RETURN VALUES 142These functions will return the approprate computation unless an error 143occurs or an argument is out of range. 144The functions 145.Fn exp , 146.Fn expm1 147and 148.Fn pow 149detect if the computed value will overflow, 150set the global variable 151.Va errno to 152.Er RANGE 153and cause a reserved operand fault on a 154.Tn VAX 155or 156.Tn Tahoe . 157The function 158.Fn pow x y 159checks to see if 160.Fa x 161< 0 and 162.Fa y 163is not an integer, in the event this is true, 164the global variable 165.Va errno 166is set to 167.Er EDOM 168and on the 169.Tn VAX 170and 171.Tn Tahoe 172generate a reserved operand fault. 173On a 174.Tn VAX 175and 176.Tn Tahoe , 177.Va errno 178is set to 179.Er EDOM 180and the reserved operand is returned 181by log unless 182.Fa x 183> 0, by 184.Fn log1p 185unless 186.Fa x 187> \-1. 188.Sh NOTES 189The functions exp(x)\-1 and log(1+x) are called 190expm1 and logp1 in 191.Tn BASIC 192on the Hewlett\-Packard 193.Tn HP Ns \-71B 194and 195.Tn APPLE 196Macintosh, 197.Tn EXP1 198and 199.Tn LN1 200in Pascal, exp1 and log1 in C 201on 202.Tn APPLE 203Macintoshes, where they have been provided to make 204sure financial calculations of ((1+x)**n\-1)/x, namely 205expm1(n\(**log1p(x))/x, will be accurate when x is tiny. 206They also provide accurate inverse hyperbolic functions. 207.Pp 208The function 209.Fn pow x 0 210returns x**0 = 1 for all x including x = 0, 211.if n \ 212Infinity 213.if t \ 214\(if 215(not found on a 216.Tn VAX ) , 217and 218.Em NaN 219(the reserved 220operand on a 221.Tn VAX ) . Previous implementations of pow may 222have defined x**0 to be undefined in some or all of these 223cases. Here are reasons for returning x**0 = 1 always: 224.Bl -enum -width indent 225.It 226Any program that already tests whether x is zero (or 227infinite or \*(Na) before computing x**0 cannot care 228whether 0**0 = 1 or not. Any program that depends 229upon 0**0 to be invalid is dubious anyway since that 230expression's meaning and, if invalid, its consequences 231vary from one computer system to another. 232.It 233Some Algebra texts (e.g. Sigler's) define x**0 = 1 for 234all x, including x = 0. 235This is compatible with the convention that accepts a[0] 236as the value of polynomial 237.Bd -literal -offset indent 238p(x) = a[0]\(**x**0 + a[1]\(**x**1 + a[2]\(**x**2 +...+ a[n]\(**x**n 239.Ed 240.Pp 241at x = 0 rather than reject a[0]\(**0**0 as invalid. 242.It 243Analysts will accept 0**0 = 1 despite that x**y can 244approach anything or nothing as x and y approach 0 245independently. 246The reason for setting 0**0 = 1 anyway is this: 247.Bd -filled -offset indent 248If x(z) and y(z) are 249.Em any 250functions analytic (expandable 251in power series) in z around z = 0, and if there 252x(0) = y(0) = 0, then x(z)**y(z) \(-> 1 as z \(-> 0. 253.Ed 254.It 255If 0**0 = 1, then 256.if n \ 257infinity**0 = 1/0**0 = 1 too; and 258.if t \ 259\(if**0 = 1/0**0 = 1 too; and 260then \*(Na**0 = 1 too because x**0 = 1 for all finite 261and infinite x, i.e., independently of x. 262.El 263.Sh SEE ALSO 264.Xr math 3 , 265.Xr infnan 3 266.Sh HISTORY 267A 268.Fn exp , 269.Fn log 270and 271.Fn pow 272function 273appeared in 274.At v6 . 275A 276.Fn log10 277function 278appeared in 279.At v7 . 280The 281.Fn log1p 282and 283.Fn expm1 284functions appeared in 285.Bx 4.3 . 286