| 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; 1.6 (ucb.elefunt) %G%"; |
| 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 | #if (defined(VAX)||defined(TAHOE)) /* 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 | #if (!defined(VAX)&&!defined(TAHOE)) |
| 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 | } |