1 /* 2 * Copyright (c) 1985 Regents of the University of California. 3 * All rights reserved. 4 * 5 * Redistribution and use in source and binary forms are permitted 6 * provided that this notice is preserved and that due credit is given 7 * to the University of California at Berkeley. The name of the University 8 * may not be used to endorse or promote products derived from this 9 * software without specific prior written permission. This software 10 * is provided ``as is'' without express or implied warranty. 11 * 12 * All recipients should regard themselves as participants in an ongoing 13 * research project and hence should feel obligated to report their 14 * experiences (good or bad) with these elementary function codes, using 15 * the sendbug(8) program, to the authors. 16 */ 17 18 #ifndef lint 19 static char sccsid[] = "@(#)exp.c 5.2 (Berkeley) 04/29/88"; 20 #endif /* not lint */ 21 22 /* EXP(X) 23 * RETURN THE EXPONENTIAL OF X 24 * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS) 25 * CODED IN C BY K.C. NG, 1/19/85; 26 * REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85, 6/14/86. 27 * 28 * Required system supported functions: 29 * scalb(x,n) 30 * copysign(x,y) 31 * finite(x) 32 * 33 * Method: 34 * 1. Argument Reduction: given the input x, find r and integer k such 35 * that 36 * x = k*ln2 + r, |r| <= 0.5*ln2 . 37 * r will be represented as r := z+c for better accuracy. 38 * 39 * 2. Compute exp(r) by 40 * 41 * exp(r) = 1 + r + r*R1/(2-R1), 42 * where 43 * R1 = x - x^2*(p1+x^2*(p2+x^2*(p3+x^2*(p4+p5*x^2)))). 44 * 45 * 3. exp(x) = 2^k * exp(r) . 46 * 47 * Special cases: 48 * exp(INF) is INF, exp(NaN) is NaN; 49 * exp(-INF)= 0; 50 * for finite argument, only exp(0)=1 is exact. 51 * 52 * Accuracy: 53 * exp(x) returns the exponential of x nearly rounded. In a test run 54 * with 1,156,000 random arguments on a VAX, the maximum observed 55 * error was 0.869 ulps (units in the last place). 56 * 57 * Constants: 58 * The hexadecimal values are the intended ones for the following constants. 59 * The decimal values may be used, provided that the compiler will convert 60 * from decimal to binary accurately enough to produce the hexadecimal values 61 * shown. 62 */ 63 64 #if defined(vax)||defined(tahoe) /* VAX D format */ 65 #ifdef vax 66 #define _0x(A,B) 0x/**/A/**/B 67 #else /* vax */ 68 #define _0x(A,B) 0x/**/B/**/A 69 #endif /* vax */ 70 /* static double */ 71 /* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */ 72 /* ln2lo = 1.6465949582897081279E-12 , Hex 2^-39 * .E7BCD5E4F1D9CC */ 73 /* lnhuge = 9.4961163736712506989E1 , Hex 2^ 7 * .BDEC1DA73E9010 */ 74 /* lntiny = -9.5654310917272452386E1 , Hex 2^ 7 * -.BF4F01D72E33AF */ 75 /* invln2 = 1.4426950408889634148E0 ; Hex 2^ 1 * .B8AA3B295C17F1 */ 76 /* p1 = 1.6666666666666602251E-1 , Hex 2^-2 * .AAAAAAAAAAA9F1 */ 77 /* p2 = -2.7777777777015591216E-3 , Hex 2^-8 * -.B60B60B5F5EC94 */ 78 /* p3 = 6.6137563214379341918E-5 , Hex 2^-13 * .8AB355792EF15F */ 79 /* p4 = -1.6533902205465250480E-6 , Hex 2^-19 * -.DDEA0E2E935F84 */ 80 /* p5 = 4.1381367970572387085E-8 , Hex 2^-24 * .B1BB4B95F52683 */ 81 static long ln2hix[] = { _0x(7217,4031), _0x(0000,f7d0)}; 82 static long ln2lox[] = { _0x(bcd5,2ce7), _0x(d9cc,e4f1)}; 83 static long lnhugex[] = { _0x(ec1d,43bd), _0x(9010,a73e)}; 84 static long lntinyx[] = { _0x(4f01,c3bf), _0x(33af,d72e)}; 85 static long invln2x[] = { _0x(aa3b,40b8), _0x(17f1,295c)}; 86 static long p1x[] = { _0x(aaaa,3f2a), _0x(a9f1,aaaa)}; 87 static long p2x[] = { _0x(0b60,bc36), _0x(ec94,b5f5)}; 88 static long p3x[] = { _0x(b355,398a), _0x(f15f,792e)}; 89 static long p4x[] = { _0x(ea0e,b6dd), _0x(5f84,2e93)}; 90 static long p5x[] = { _0x(bb4b,3431), _0x(2683,95f5)}; 91 #define ln2hi (*(double*)ln2hix) 92 #define ln2lo (*(double*)ln2lox) 93 #define lnhuge (*(double*)lnhugex) 94 #define lntiny (*(double*)lntinyx) 95 #define invln2 (*(double*)invln2x) 96 #define p1 (*(double*)p1x) 97 #define p2 (*(double*)p2x) 98 #define p3 (*(double*)p3x) 99 #define p4 (*(double*)p4x) 100 #define p5 (*(double*)p5x) 101 102 #else /* defined(vax)||defined(tahoe) */ 103 static double 104 p1 = 1.6666666666666601904E-1 , /*Hex 2^-3 * 1.555555555553E */ 105 p2 = -2.7777777777015593384E-3 , /*Hex 2^-9 * -1.6C16C16BEBD93 */ 106 p3 = 6.6137563214379343612E-5 , /*Hex 2^-14 * 1.1566AAF25DE2C */ 107 p4 = -1.6533902205465251539E-6 , /*Hex 2^-20 * -1.BBD41C5D26BF1 */ 108 p5 = 4.1381367970572384604E-8 , /*Hex 2^-25 * 1.6376972BEA4D0 */ 109 ln2hi = 6.9314718036912381649E-1 , /*Hex 2^ -1 * 1.62E42FEE00000 */ 110 ln2lo = 1.9082149292705877000E-10 , /*Hex 2^-33 * 1.A39EF35793C76 */ 111 lnhuge = 7.1602103751842355450E2 , /*Hex 2^ 9 * 1.6602B15B7ECF2 */ 112 lntiny = -7.5137154372698068983E2 , /*Hex 2^ 9 * -1.77AF8EBEAE354 */ 113 invln2 = 1.4426950408889633870E0 ; /*Hex 2^ 0 * 1.71547652B82FE */ 114 #endif /* defined(vax)||defined(tahoe) */ 115 116 double exp(x) 117 double x; 118 { 119 double scalb(), copysign(), z,hi,lo,c; 120 int k,finite(); 121 122 #if !defined(vax)&&!defined(tahoe) 123 if(x!=x) return(x); /* x is NaN */ 124 #endif /* !defined(vax)&&!defined(tahoe) */ 125 if( x <= lnhuge ) { 126 if( x >= lntiny ) { 127 128 /* argument reduction : x --> x - k*ln2 */ 129 130 k=invln2*x+copysign(0.5,x); /* k=NINT(x/ln2) */ 131 132 /* express x-k*ln2 as hi-lo and let x=hi-lo rounded */ 133 134 hi=x-k*ln2hi; 135 x=hi-(lo=k*ln2lo); 136 137 /* return 2^k*[1+x+x*c/(2+c)] */ 138 z=x*x; 139 c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5)))); 140 return scalb(1.0+(hi-(lo-(x*c)/(2.0-c))),k); 141 142 } 143 /* end of x > lntiny */ 144 145 else 146 /* exp(-big#) underflows to zero */ 147 if(finite(x)) return(scalb(1.0,-5000)); 148 149 /* exp(-INF) is zero */ 150 else return(0.0); 151 } 152 /* end of x < lnhuge */ 153 154 else 155 /* exp(INF) is INF, exp(+big#) overflows to INF */ 156 return( finite(x) ? scalb(1.0,5000) : x); 157 } 158