1 /* 2 * Copyright (c) 1985 Regents of the University of California. 3 * 4 * Use and reproduction of this software are granted in accordance with 5 * the terms and conditions specified in the Berkeley Software License 6 * Agreement (in particular, this entails acknowledgement of the programs' 7 * source, and inclusion of this notice) with the additional understanding 8 * that all recipients should regard themselves as participants in an 9 * ongoing research project and hence should feel obligated to report 10 * their experiences (good or bad) with these elementary function codes, 11 * using "sendbug 4bsd-bugs@BERKELEY", to the authors. 12 */ 13 14 #ifndef lint 15 static char sccsid[] = 16 "@(#)exp.c 4.3 (Berkeley) 8/21/85; 1.5 (ucb.elefunt) 10/18/86"; 17 #endif not lint 18 19 /* EXP(X) 20 * RETURN THE EXPONENTIAL OF X 21 * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS) 22 * CODED IN C BY K.C. NG, 1/19/85; 23 * REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85, 6/14/86. 24 * 25 * Required system supported functions: 26 * scalb(x,n) 27 * copysign(x,y) 28 * finite(x) 29 * 30 * Method: 31 * 1. Argument Reduction: given the input x, find r and integer k such 32 * that 33 * x = k*ln2 + r, |r| <= 0.5*ln2 . 34 * r will be represented as r := z+c for better accuracy. 35 * 36 * 2. Compute exp(r) by 37 * 38 * exp(r) = 1 + r + r*R1/(2-R1), 39 * where 40 * R1 = x - x^2*(p1+x^2*(p2+x^2*(p3+x^2*(p4+p5*x^2)))). 41 * 42 * 3. exp(x) = 2^k * exp(r) . 43 * 44 * Special cases: 45 * exp(INF) is INF, exp(NaN) is NaN; 46 * exp(-INF)= 0; 47 * for finite argument, only exp(0)=1 is exact. 48 * 49 * Accuracy: 50 * exp(x) returns the exponential of x nearly rounded. In a test run 51 * with 1,156,000 random arguments on a VAX, the maximum observed 52 * error was 0.869 ulps (units in the last place). 53 * 54 * Constants: 55 * The hexadecimal values are the intended ones for the following constants. 56 * The decimal values may be used, provided that the compiler will convert 57 * from decimal to binary accurately enough to produce the hexadecimal values 58 * shown. 59 */ 60 61 #ifdef VAX /* VAX D format */ 62 /* static double */ 63 /* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */ 64 /* ln2lo = 1.6465949582897081279E-12 , Hex 2^-39 * .E7BCD5E4F1D9CC */ 65 /* lnhuge = 9.4961163736712506989E1 , Hex 2^ 7 * .BDEC1DA73E9010 */ 66 /* lntiny = -9.5654310917272452386E1 , Hex 2^ 7 * -.BF4F01D72E33AF */ 67 /* invln2 = 1.4426950408889634148E0 ; Hex 2^ 1 * .B8AA3B295C17F1 */ 68 /* p1 = 1.6666666666666602251E-1 , Hex 2^-2 * .AAAAAAAAAAA9F1 */ 69 /* p2 = -2.7777777777015591216E-3 , Hex 2^-8 * -.B60B60B5F5EC94 */ 70 /* p3 = 6.6137563214379341918E-5 , Hex 2^-13 * .8AB355792EF15F */ 71 /* p4 = -1.6533902205465250480E-6 , Hex 2^-19 * -.DDEA0E2E935F84 */ 72 /* p5 = 4.1381367970572387085E-8 , Hex 2^-24 * .B1BB4B95F52683 */ 73 static long ln2hix[] = { 0x72174031, 0x0000f7d0}; 74 static long ln2lox[] = { 0xbcd52ce7, 0xd9cce4f1}; 75 static long lnhugex[] = { 0xec1d43bd, 0x9010a73e}; 76 static long lntinyx[] = { 0x4f01c3bf, 0x33afd72e}; 77 static long invln2x[] = { 0xaa3b40b8, 0x17f1295c}; 78 static long p1x[] = { 0xaaaa3f2a, 0xa9f1aaaa}; 79 static long p2x[] = { 0x0b60bc36, 0xec94b5f5}; 80 static long p3x[] = { 0xb355398a, 0xf15f792e}; 81 static long p4x[] = { 0xea0eb6dd, 0x5f842e93}; 82 static long p5x[] = { 0xbb4b3431, 0x268395f5}; 83 #define ln2hi (*(double*)ln2hix) 84 #define ln2lo (*(double*)ln2lox) 85 #define lnhuge (*(double*)lnhugex) 86 #define lntiny (*(double*)lntinyx) 87 #define invln2 (*(double*)invln2x) 88 #define p1 (*(double*)p1x) 89 #define p2 (*(double*)p2x) 90 #define p3 (*(double*)p3x) 91 #define p4 (*(double*)p4x) 92 #define p5 (*(double*)p5x) 93 94 #else /* IEEE double */ 95 static double 96 p1 = 1.6666666666666601904E-1 , /*Hex 2^-3 * 1.555555555553E */ 97 p2 = -2.7777777777015593384E-3 , /*Hex 2^-9 * -1.6C16C16BEBD93 */ 98 p3 = 6.6137563214379343612E-5 , /*Hex 2^-14 * 1.1566AAF25DE2C */ 99 p4 = -1.6533902205465251539E-6 , /*Hex 2^-20 * -1.BBD41C5D26BF1 */ 100 p5 = 4.1381367970572384604E-8 , /*Hex 2^-25 * 1.6376972BEA4D0 */ 101 ln2hi = 6.9314718036912381649E-1 , /*Hex 2^ -1 * 1.62E42FEE00000 */ 102 ln2lo = 1.9082149292705877000E-10 , /*Hex 2^-33 * 1.A39EF35793C76 */ 103 lnhuge = 7.1602103751842355450E2 , /*Hex 2^ 9 * 1.6602B15B7ECF2 */ 104 lntiny = -7.5137154372698068983E2 , /*Hex 2^ 9 * -1.77AF8EBEAE354 */ 105 invln2 = 1.4426950408889633870E0 ; /*Hex 2^ 0 * 1.71547652B82FE */ 106 #endif 107 108 double exp(x) 109 double x; 110 { 111 double scalb(), copysign(), z,hi,lo,c; 112 int k,finite(); 113 114 #ifndef VAX 115 if(x!=x) return(x); /* x is NaN */ 116 #endif 117 if( x <= lnhuge ) { 118 if( x >= lntiny ) { 119 120 /* argument reduction : x --> x - k*ln2 */ 121 122 k=invln2*x+copysign(0.5,x); /* k=NINT(x/ln2) */ 123 124 /* express x-k*ln2 as hi-lo and let x=hi-lo rounded */ 125 126 hi=x-k*ln2hi; 127 x=hi-(lo=k*ln2lo); 128 129 /* return 2^k*[1+x+x*c/(2+c)] */ 130 z=x*x; 131 c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5)))); 132 return scalb(1.0+(hi-(lo-(x*c)/(2.0-c))),k); 133 134 } 135 /* end of x > lntiny */ 136 137 else 138 /* exp(-big#) underflows to zero */ 139 if(finite(x)) return(scalb(1.0,-5000)); 140 141 /* exp(-INF) is zero */ 142 else return(0.0); 143 } 144 /* end of x < lnhuge */ 145 146 else 147 /* exp(INF) is INF, exp(+big#) overflows to INF */ 148 return( finite(x) ? scalb(1.0,5000) : x); 149 } 150