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 5.4 (Berkeley) 09/22/88"; 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 #include "mathimpl.h" 70 71 vc(ln2hi, 6.9314718055829871446E-1 ,7217,4031,0000,f7d0, 0, .B17217F7D00000) 72 vc(ln2lo, 1.6465949582897081279E-12 ,bcd5,2ce7,d9cc,e4f1, -39, .E7BCD5E4F1D9CC) 73 vc(lnhuge, 9.4961163736712506989E1 ,ec1d,43bd,9010,a73e, 7, .BDEC1DA73E9010) 74 vc(lntiny,-9.5654310917272452386E1 ,4f01,c3bf,33af,d72e, 7,-.BF4F01D72E33AF) 75 vc(invln2, 1.4426950408889634148E0 ,aa3b,40b8,17f1,295c, 1, .B8AA3B295C17F1) 76 vc(p1, 1.6666666666666602251E-1 ,aaaa,3f2a,a9f1,aaaa, -2, .AAAAAAAAAAA9F1) 77 vc(p2, -2.7777777777015591216E-3 ,0b60,bc36,ec94,b5f5, -8,-.B60B60B5F5EC94) 78 vc(p3, 6.6137563214379341918E-5 ,b355,398a,f15f,792e, -13, .8AB355792EF15F) 79 vc(p4, -1.6533902205465250480E-6 ,ea0e,b6dd,5f84,2e93, -19,-.DDEA0E2E935F84) 80 vc(p5, 4.1381367970572387085E-8 ,bb4b,3431,2683,95f5, -24, .B1BB4B95F52683) 81 82 #ifdef vccast 83 #define ln2hi vccast(ln2hi) 84 #define ln2lo vccast(ln2lo) 85 #define lnhuge vccast(lnhuge) 86 #define lntiny vccast(lntiny) 87 #define invln2 vccast(invln2) 88 #define p1 vccast(p1) 89 #define p2 vccast(p2) 90 #define p3 vccast(p3) 91 #define p4 vccast(p4) 92 #define p5 vccast(p5) 93 #endif 94 95 ic(p1, 1.6666666666666601904E-1, -3, 1.555555555553E) 96 ic(p2, -2.7777777777015593384E-3, -9, -1.6C16C16BEBD93) 97 ic(p3, 6.6137563214379343612E-5, -14, 1.1566AAF25DE2C) 98 ic(p4, -1.6533902205465251539E-6, -20, -1.BBD41C5D26BF1) 99 ic(p5, 4.1381367970572384604E-8, -25, 1.6376972BEA4D0) 100 ic(ln2hi, 6.9314718036912381649E-1, -1, 1.62E42FEE00000) 101 ic(ln2lo, 1.9082149292705877000E-10,-33, 1.A39EF35793C76) 102 ic(lnhuge, 7.1602103751842355450E2, 9, 1.6602B15B7ECF2) 103 ic(lntiny,-7.5137154372698068983E2, 9, -1.77AF8EBEAE354) 104 ic(invln2, 1.4426950408889633870E0, 0, 1.71547652B82FE) 105 106 double exp(x) 107 double x; 108 { 109 double z,hi,lo,c; 110 int k; 111 112 #if !defined(vax)&&!defined(tahoe) 113 if(x!=x) return(x); /* x is NaN */ 114 #endif /* !defined(vax)&&!defined(tahoe) */ 115 if( x <= lnhuge ) { 116 if( x >= lntiny ) { 117 118 /* argument reduction : x --> x - k*ln2 */ 119 120 k=invln2*x+copysign(0.5,x); /* k=NINT(x/ln2) */ 121 122 /* express x-k*ln2 as hi-lo and let x=hi-lo rounded */ 123 124 hi=x-k*ln2hi; 125 x=hi-(lo=k*ln2lo); 126 127 /* return 2^k*[1+x+x*c/(2+c)] */ 128 z=x*x; 129 c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5)))); 130 return scalb(1.0+(hi-(lo-(x*c)/(2.0-c))),k); 131 132 } 133 /* end of x > lntiny */ 134 135 else 136 /* exp(-big#) underflows to zero */ 137 if(finite(x)) return(scalb(1.0,-5000)); 138 139 /* exp(-INF) is zero */ 140 else return(0.0); 141 } 142 /* end of x < lnhuge */ 143 144 else 145 /* exp(INF) is INF, exp(+big#) overflows to INF */ 146 return( finite(x) ? scalb(1.0,5000) : x); 147 } 148