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 the above copyright notice and this paragraph are 7 * duplicated in all such forms and that any documentation, 8 * advertising materials, and other materials related to such 9 * distribution and use acknowledge that the software was developed 10 * by the University of California, Berkeley. The name of the 11 * University may not be used to endorse or promote products derived 12 * from this software without specific prior written permission. 13 * THIS SOFTWARE IS PROVIDED ``AS IS'' AND WITHOUT ANY EXPRESS OR 14 * IMPLIED WARRANTIES, INCLUDING, WITHOUT LIMITATION, THE IMPLIED 15 * WARRANTIES OF MERCHANTIBILITY AND FITNESS FOR A PARTICULAR PURPOSE. 16 * 17 * All recipients should regard themselves as participants in an ongoing 18 * research project and hence should feel obligated to report their 19 * experiences (good or bad) with these elementary function codes, using 20 * the sendbug(8) program, to the authors. 21 */ 22 23 #ifndef lint 24 static char sccsid[] = "@(#)exp.c 5.3 (Berkeley) 06/30/88"; 25 #endif /* not lint */ 26 27 /* EXP(X) 28 * RETURN THE EXPONENTIAL OF X 29 * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS) 30 * CODED IN C BY K.C. NG, 1/19/85; 31 * REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85, 6/14/86. 32 * 33 * Required system supported functions: 34 * scalb(x,n) 35 * copysign(x,y) 36 * finite(x) 37 * 38 * Method: 39 * 1. Argument Reduction: given the input x, find r and integer k such 40 * that 41 * x = k*ln2 + r, |r| <= 0.5*ln2 . 42 * r will be represented as r := z+c for better accuracy. 43 * 44 * 2. Compute exp(r) by 45 * 46 * exp(r) = 1 + r + r*R1/(2-R1), 47 * where 48 * R1 = x - x^2*(p1+x^2*(p2+x^2*(p3+x^2*(p4+p5*x^2)))). 49 * 50 * 3. exp(x) = 2^k * exp(r) . 51 * 52 * Special cases: 53 * exp(INF) is INF, exp(NaN) is NaN; 54 * exp(-INF)= 0; 55 * for finite argument, only exp(0)=1 is exact. 56 * 57 * Accuracy: 58 * exp(x) returns the exponential of x nearly rounded. In a test run 59 * with 1,156,000 random arguments on a VAX, the maximum observed 60 * error was 0.869 ulps (units in the last place). 61 * 62 * Constants: 63 * The hexadecimal values are the intended ones for the following constants. 64 * The decimal values may be used, provided that the compiler will convert 65 * from decimal to binary accurately enough to produce the hexadecimal values 66 * shown. 67 */ 68 69 #if defined(vax)||defined(tahoe) /* VAX D format */ 70 #ifdef vax 71 #define _0x(A,B) 0x/**/A/**/B 72 #else /* vax */ 73 #define _0x(A,B) 0x/**/B/**/A 74 #endif /* vax */ 75 /* static double */ 76 /* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */ 77 /* ln2lo = 1.6465949582897081279E-12 , Hex 2^-39 * .E7BCD5E4F1D9CC */ 78 /* lnhuge = 9.4961163736712506989E1 , Hex 2^ 7 * .BDEC1DA73E9010 */ 79 /* lntiny = -9.5654310917272452386E1 , Hex 2^ 7 * -.BF4F01D72E33AF */ 80 /* invln2 = 1.4426950408889634148E0 ; Hex 2^ 1 * .B8AA3B295C17F1 */ 81 /* p1 = 1.6666666666666602251E-1 , Hex 2^-2 * .AAAAAAAAAAA9F1 */ 82 /* p2 = -2.7777777777015591216E-3 , Hex 2^-8 * -.B60B60B5F5EC94 */ 83 /* p3 = 6.6137563214379341918E-5 , Hex 2^-13 * .8AB355792EF15F */ 84 /* p4 = -1.6533902205465250480E-6 , Hex 2^-19 * -.DDEA0E2E935F84 */ 85 /* p5 = 4.1381367970572387085E-8 , Hex 2^-24 * .B1BB4B95F52683 */ 86 static long ln2hix[] = { _0x(7217,4031), _0x(0000,f7d0)}; 87 static long ln2lox[] = { _0x(bcd5,2ce7), _0x(d9cc,e4f1)}; 88 static long lnhugex[] = { _0x(ec1d,43bd), _0x(9010,a73e)}; 89 static long lntinyx[] = { _0x(4f01,c3bf), _0x(33af,d72e)}; 90 static long invln2x[] = { _0x(aa3b,40b8), _0x(17f1,295c)}; 91 static long p1x[] = { _0x(aaaa,3f2a), _0x(a9f1,aaaa)}; 92 static long p2x[] = { _0x(0b60,bc36), _0x(ec94,b5f5)}; 93 static long p3x[] = { _0x(b355,398a), _0x(f15f,792e)}; 94 static long p4x[] = { _0x(ea0e,b6dd), _0x(5f84,2e93)}; 95 static long p5x[] = { _0x(bb4b,3431), _0x(2683,95f5)}; 96 #define ln2hi (*(double*)ln2hix) 97 #define ln2lo (*(double*)ln2lox) 98 #define lnhuge (*(double*)lnhugex) 99 #define lntiny (*(double*)lntinyx) 100 #define invln2 (*(double*)invln2x) 101 #define p1 (*(double*)p1x) 102 #define p2 (*(double*)p2x) 103 #define p3 (*(double*)p3x) 104 #define p4 (*(double*)p4x) 105 #define p5 (*(double*)p5x) 106 107 #else /* defined(vax)||defined(tahoe) */ 108 static double 109 p1 = 1.6666666666666601904E-1 , /*Hex 2^-3 * 1.555555555553E */ 110 p2 = -2.7777777777015593384E-3 , /*Hex 2^-9 * -1.6C16C16BEBD93 */ 111 p3 = 6.6137563214379343612E-5 , /*Hex 2^-14 * 1.1566AAF25DE2C */ 112 p4 = -1.6533902205465251539E-6 , /*Hex 2^-20 * -1.BBD41C5D26BF1 */ 113 p5 = 4.1381367970572384604E-8 , /*Hex 2^-25 * 1.6376972BEA4D0 */ 114 ln2hi = 6.9314718036912381649E-1 , /*Hex 2^ -1 * 1.62E42FEE00000 */ 115 ln2lo = 1.9082149292705877000E-10 , /*Hex 2^-33 * 1.A39EF35793C76 */ 116 lnhuge = 7.1602103751842355450E2 , /*Hex 2^ 9 * 1.6602B15B7ECF2 */ 117 lntiny = -7.5137154372698068983E2 , /*Hex 2^ 9 * -1.77AF8EBEAE354 */ 118 invln2 = 1.4426950408889633870E0 ; /*Hex 2^ 0 * 1.71547652B82FE */ 119 #endif /* defined(vax)||defined(tahoe) */ 120 121 double exp(x) 122 double x; 123 { 124 double scalb(), copysign(), z,hi,lo,c; 125 int k,finite(); 126 127 #if !defined(vax)&&!defined(tahoe) 128 if(x!=x) return(x); /* x is NaN */ 129 #endif /* !defined(vax)&&!defined(tahoe) */ 130 if( x <= lnhuge ) { 131 if( x >= lntiny ) { 132 133 /* argument reduction : x --> x - k*ln2 */ 134 135 k=invln2*x+copysign(0.5,x); /* k=NINT(x/ln2) */ 136 137 /* express x-k*ln2 as hi-lo and let x=hi-lo rounded */ 138 139 hi=x-k*ln2hi; 140 x=hi-(lo=k*ln2lo); 141 142 /* return 2^k*[1+x+x*c/(2+c)] */ 143 z=x*x; 144 c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5)))); 145 return scalb(1.0+(hi-(lo-(x*c)/(2.0-c))),k); 146 147 } 148 /* end of x > lntiny */ 149 150 else 151 /* exp(-big#) underflows to zero */ 152 if(finite(x)) return(scalb(1.0,-5000)); 153 154 /* exp(-INF) is zero */ 155 else return(0.0); 156 } 157 /* end of x < lnhuge */ 158 159 else 160 /* exp(INF) is INF, exp(+big#) overflows to INF */ 161 return( finite(x) ? scalb(1.0,5000) : x); 162 } 163