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 1.8 1995/04/14 18:40:07 jtc 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 log2 , 43.Nm log10 , 44.Nm log1p , 45.Nm pow 46.Nd exponential, logarithm, power functions 47.Sh SYNOPSIS 48.Fd #include <math.h> 49.Ft double 50.Fn exp "double x" 51.Ft float 52.Fn expf "float x" 53.Ft double 54.Fn expm1 "double x" 55.Ft float 56.Fn expm1f "float x" 57.Ft double 58.Fn log "double x" 59.Ft float 60.Fn logf "float x" 61.Ft double 62.Fn log10 "double x" 63.Ft float 64.Fn log10f "float x" 65.Ft double 66.Fn log1p "double x" 67.Ft float 68.Fn log1pf "float x" 69.Ft double 70.Fn pow "double x" "double y" 71.Ft float 72.Fn powf "float x" float y" 73.Sh DESCRIPTION 74The 75.Fn exp 76function computes the exponential value of the given argument 77.Fa x . 78.Pp 79The 80.Fn expm1 81function computes the value exp(x)\-1 accurately even for tiny argument 82.Fa x . 83.Pp 84The 85.Fn log 86function computes the value of the natural logarithm of argument 87.Fa x. 88.Pp 89The 90.Fn log10 91function computes the value of the logarithm of argument 92.Fa x 93to base 10. 94.Pp 95The 96.Fn log1p 97function computes 98the value of log(1+x) accurately even for tiny argument 99.Fa x . 100.Pp 101The 102.Fn pow 103computes the value 104of 105.Ar x 106to the exponent 107.Ar y . 108.Sh ERROR (due to Roundoff etc.) 109exp(x), log(x), expm1(x) and log1p(x) are accurate to within 110an 111.Em ulp , 112and log10(x) to within about 2 113.Em ulps ; 114an 115.Em ulp 116is one 117.Em Unit 118in the 119.Em Last 120.Em Place . 121The error in 122.Fn pow x y 123is below about 2 124.Em ulps 125when its 126magnitude is moderate, but increases as 127.Fn pow x y 128approaches 129the over/underflow thresholds until almost as many bits could be 130lost as are occupied by the floating\-point format's exponent 131field; that is 8 bits for 132.Tn "VAX D" 133and 11 bits for IEEE 754 Double. 134No such drastic loss has been exposed by testing; the worst 135errors observed have been below 20 136.Em ulps 137for 138.Tn "VAX D" , 139300 140.Em ulps 141for 142.Tn IEEE 143754 Double. 144Moderate values of 145.Fn pow 146are accurate enough that 147.Fn pow integer integer 148is exact until it is bigger than 2**56 on a 149.Tn VAX , 1502**53 for 151.Tn IEEE 152754. 153.Sh RETURN VALUES 154These functions will return the appropriate computation unless an error 155occurs or an argument is out of range. 156The functions 157.Fn exp , 158.Fn expm1 159and 160.Fn pow 161detect if the computed value will overflow, 162set the global variable 163.Va errno to 164.Er ERANGE 165and cause a reserved operand fault on a 166.Tn VAX 167or 168.Tn Tahoe . 169The function 170.Fn pow x y 171checks to see if 172.Fa x 173< 0 and 174.Fa y 175is not an integer, in the event this is true, 176the global variable 177.Va errno 178is set to 179.Er EDOM 180and on the 181.Tn VAX 182and 183.Tn Tahoe 184generate a reserved operand fault. 185On a 186.Tn VAX 187and 188.Tn Tahoe , 189.Va errno 190is set to 191.Er EDOM 192and the reserved operand is returned 193by log unless 194.Fa x 195> 0, by 196.Fn log1p 197unless 198.Fa x 199> \-1. 200.Sh NOTES 201The functions exp(x)\-1 and log(1+x) are called 202expm1 and logp1 in 203.Tn BASIC 204on the Hewlett\-Packard 205.Tn HP Ns \-71B 206and 207.Tn APPLE 208Macintosh, 209.Tn EXP1 210and 211.Tn LN1 212in Pascal, exp1 and log1 in C 213on 214.Tn APPLE 215Macintoshes, where they have been provided to make 216sure financial calculations of ((1+x)**n\-1)/x, namely 217expm1(n\(**log1p(x))/x, will be accurate when x is tiny. 218They also provide accurate inverse hyperbolic functions. 219.Pp 220The function 221.Fn pow x 0 222returns x**0 = 1 for all x including x = 0, 223.if n \ 224Infinity 225.if t \ 226\(if 227(not found on a 228.Tn VAX ) , 229and 230.Em NaN 231(the reserved 232operand on a 233.Tn VAX ) . Previous implementations of pow may 234have defined x**0 to be undefined in some or all of these 235cases. Here are reasons for returning x**0 = 1 always: 236.Bl -enum -width indent 237.It 238Any program that already tests whether x is zero (or 239infinite or \*(Na) before computing x**0 cannot care 240whether 0**0 = 1 or not. Any program that depends 241upon 0**0 to be invalid is dubious anyway since that 242expression's meaning and, if invalid, its consequences 243vary from one computer system to another. 244.It 245Some Algebra texts (e.g. Sigler's) define x**0 = 1 for 246all x, including x = 0. 247This is compatible with the convention that accepts a[0] 248as the value of polynomial 249.Bd -literal -offset indent 250p(x) = a[0]\(**x**0 + a[1]\(**x**1 + a[2]\(**x**2 +...+ a[n]\(**x**n 251.Ed 252.Pp 253at x = 0 rather than reject a[0]\(**0**0 as invalid. 254.It 255Analysts will accept 0**0 = 1 despite that x**y can 256approach anything or nothing as x and y approach 0 257independently. 258The reason for setting 0**0 = 1 anyway is this: 259.Bd -filled -offset indent 260If x(z) and y(z) are 261.Em any 262functions analytic (expandable 263in power series) in z around z = 0, and if there 264x(0) = y(0) = 0, then x(z)**y(z) \(-> 1 as z \(-> 0. 265.Ed 266.It 267If 0**0 = 1, then 268.if n \ 269infinity**0 = 1/0**0 = 1 too; and 270.if t \ 271\(if**0 = 1/0**0 = 1 too; and 272then \*(Na**0 = 1 too because x**0 = 1 for all finite 273and infinite x, i.e., independently of x. 274.El 275.Sh SEE ALSO 276.Xr math 3 , 277.Xr infnan 3 278.Sh HISTORY 279A 280.Fn exp , 281.Fn log 282and 283.Fn pow 284functions 285appeared in 286.At v6 . 287A 288.Fn log10 289function 290appeared in 291.At v7 . 292The 293.Fn log1p 294and 295.Fn expm1 296functions appeared in 297.Bx 4.3 . 298