/* * Copyright (c) 1985 Regents of the University of California. * * Use and reproduction of this software are granted in accordance with * the terms and conditions specified in the Berkeley Software License * Agreement (in particular, this entails acknowledgement of the programs' * source, and inclusion of this notice) with the additional understanding * that all recipients should regard themselves as participants in an * ongoing research project and hence should feel obligated to report * their experiences (good or bad) with these elementary function codes, * using "sendbug 4bsd-bugs@BERKELEY", to the authors. */ #ifndef lint static char sccsid[] = "@(#)exp.c 4.3 (Berkeley) 8/21/85; 1.6 (ucb.elefunt) 07/07/87"; #endif not lint /* EXP(X) * RETURN THE EXPONENTIAL OF X * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS) * CODED IN C BY K.C. NG, 1/19/85; * REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85, 6/14/86. * * Required system supported functions: * scalb(x,n) * copysign(x,y) * finite(x) * * Method: * 1. Argument Reduction: given the input x, find r and integer k such * that * x = k*ln2 + r, |r| <= 0.5*ln2 . * r will be represented as r := z+c for better accuracy. * * 2. Compute exp(r) by * * exp(r) = 1 + r + r*R1/(2-R1), * where * R1 = x - x^2*(p1+x^2*(p2+x^2*(p3+x^2*(p4+p5*x^2)))). * * 3. exp(x) = 2^k * exp(r) . * * Special cases: * exp(INF) is INF, exp(NaN) is NaN; * exp(-INF)= 0; * for finite argument, only exp(0)=1 is exact. * * Accuracy: * exp(x) returns the exponential of x nearly rounded. In a test run * with 1,156,000 random arguments on a VAX, the maximum observed * error was 0.869 ulps (units in the last place). * * Constants: * The hexadecimal values are the intended ones for the following constants. * The decimal values may be used, provided that the compiler will convert * from decimal to binary accurately enough to produce the hexadecimal values * shown. */ #if (defined(VAX)||defined(TAHOE)) /* VAX D format */ /* static double */ /* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */ /* ln2lo = 1.6465949582897081279E-12 , Hex 2^-39 * .E7BCD5E4F1D9CC */ /* lnhuge = 9.4961163736712506989E1 , Hex 2^ 7 * .BDEC1DA73E9010 */ /* lntiny = -9.5654310917272452386E1 , Hex 2^ 7 * -.BF4F01D72E33AF */ /* invln2 = 1.4426950408889634148E0 ; Hex 2^ 1 * .B8AA3B295C17F1 */ /* p1 = 1.6666666666666602251E-1 , Hex 2^-2 * .AAAAAAAAAAA9F1 */ /* p2 = -2.7777777777015591216E-3 , Hex 2^-8 * -.B60B60B5F5EC94 */ /* p3 = 6.6137563214379341918E-5 , Hex 2^-13 * .8AB355792EF15F */ /* p4 = -1.6533902205465250480E-6 , Hex 2^-19 * -.DDEA0E2E935F84 */ /* p5 = 4.1381367970572387085E-8 , Hex 2^-24 * .B1BB4B95F52683 */ static long ln2hix[] = { 0x72174031, 0x0000f7d0}; static long ln2lox[] = { 0xbcd52ce7, 0xd9cce4f1}; static long lnhugex[] = { 0xec1d43bd, 0x9010a73e}; static long lntinyx[] = { 0x4f01c3bf, 0x33afd72e}; static long invln2x[] = { 0xaa3b40b8, 0x17f1295c}; static long p1x[] = { 0xaaaa3f2a, 0xa9f1aaaa}; static long p2x[] = { 0x0b60bc36, 0xec94b5f5}; static long p3x[] = { 0xb355398a, 0xf15f792e}; static long p4x[] = { 0xea0eb6dd, 0x5f842e93}; static long p5x[] = { 0xbb4b3431, 0x268395f5}; #define ln2hi (*(double*)ln2hix) #define ln2lo (*(double*)ln2lox) #define lnhuge (*(double*)lnhugex) #define lntiny (*(double*)lntinyx) #define invln2 (*(double*)invln2x) #define p1 (*(double*)p1x) #define p2 (*(double*)p2x) #define p3 (*(double*)p3x) #define p4 (*(double*)p4x) #define p5 (*(double*)p5x) #else /* IEEE double */ static double p1 = 1.6666666666666601904E-1 , /*Hex 2^-3 * 1.555555555553E */ p2 = -2.7777777777015593384E-3 , /*Hex 2^-9 * -1.6C16C16BEBD93 */ p3 = 6.6137563214379343612E-5 , /*Hex 2^-14 * 1.1566AAF25DE2C */ p4 = -1.6533902205465251539E-6 , /*Hex 2^-20 * -1.BBD41C5D26BF1 */ p5 = 4.1381367970572384604E-8 , /*Hex 2^-25 * 1.6376972BEA4D0 */ ln2hi = 6.9314718036912381649E-1 , /*Hex 2^ -1 * 1.62E42FEE00000 */ ln2lo = 1.9082149292705877000E-10 , /*Hex 2^-33 * 1.A39EF35793C76 */ lnhuge = 7.1602103751842355450E2 , /*Hex 2^ 9 * 1.6602B15B7ECF2 */ lntiny = -7.5137154372698068983E2 , /*Hex 2^ 9 * -1.77AF8EBEAE354 */ invln2 = 1.4426950408889633870E0 ; /*Hex 2^ 0 * 1.71547652B82FE */ #endif double exp(x) double x; { double scalb(), copysign(), z,hi,lo,c; int k,finite(); #if (!defined(VAX)&&!defined(TAHOE)) if(x!=x) return(x); /* x is NaN */ #endif if( x <= lnhuge ) { if( x >= lntiny ) { /* argument reduction : x --> x - k*ln2 */ k=invln2*x+copysign(0.5,x); /* k=NINT(x/ln2) */ /* express x-k*ln2 as hi-lo and let x=hi-lo rounded */ hi=x-k*ln2hi; x=hi-(lo=k*ln2lo); /* return 2^k*[1+x+x*c/(2+c)] */ z=x*x; c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5)))); return scalb(1.0+(hi-(lo-(x*c)/(2.0-c))),k); } /* end of x > lntiny */ else /* exp(-big#) underflows to zero */ if(finite(x)) return(scalb(1.0,-5000)); /* exp(-INF) is zero */ else return(0.0); } /* end of x < lnhuge */ else /* exp(INF) is INF, exp(+big#) overflows to INF */ return( finite(x) ? scalb(1.0,5000) : x); }