1 /* 2 * Copyright (c) 1985, 1993 3 * The Regents of the University of California. All rights reserved. 4 * 5 * %sccs.include.redist.c% 6 */ 7 8 #ifndef lint 9 static char sccsid[] = "@(#)exp.c 8.1 (Berkeley) 06/04/93"; 10 #endif /* not lint */ 11 12 /* EXP(X) 13 * RETURN THE EXPONENTIAL OF X 14 * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS) 15 * CODED IN C BY K.C. NG, 1/19/85; 16 * REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85, 6/14/86. 17 * 18 * Required system supported functions: 19 * scalb(x,n) 20 * copysign(x,y) 21 * finite(x) 22 * 23 * Method: 24 * 1. Argument Reduction: given the input x, find r and integer k such 25 * that 26 * x = k*ln2 + r, |r| <= 0.5*ln2 . 27 * r will be represented as r := z+c for better accuracy. 28 * 29 * 2. Compute exp(r) by 30 * 31 * exp(r) = 1 + r + r*R1/(2-R1), 32 * where 33 * R1 = x - x^2*(p1+x^2*(p2+x^2*(p3+x^2*(p4+p5*x^2)))). 34 * 35 * 3. exp(x) = 2^k * exp(r) . 36 * 37 * Special cases: 38 * exp(INF) is INF, exp(NaN) is NaN; 39 * exp(-INF)= 0; 40 * for finite argument, only exp(0)=1 is exact. 41 * 42 * Accuracy: 43 * exp(x) returns the exponential of x nearly rounded. In a test run 44 * with 1,156,000 random arguments on a VAX, the maximum observed 45 * error was 0.869 ulps (units in the last place). 46 * 47 * Constants: 48 * The hexadecimal values are the intended ones for the following constants. 49 * The decimal values may be used, provided that the compiler will convert 50 * from decimal to binary accurately enough to produce the hexadecimal values 51 * shown. 52 */ 53 54 #include "mathimpl.h" 55 56 vc(ln2hi, 6.9314718055829871446E-1 ,7217,4031,0000,f7d0, 0, .B17217F7D00000) 57 vc(ln2lo, 1.6465949582897081279E-12 ,bcd5,2ce7,d9cc,e4f1, -39, .E7BCD5E4F1D9CC) 58 vc(lnhuge, 9.4961163736712506989E1 ,ec1d,43bd,9010,a73e, 7, .BDEC1DA73E9010) 59 vc(lntiny,-9.5654310917272452386E1 ,4f01,c3bf,33af,d72e, 7,-.BF4F01D72E33AF) 60 vc(invln2, 1.4426950408889634148E0 ,aa3b,40b8,17f1,295c, 1, .B8AA3B295C17F1) 61 vc(p1, 1.6666666666666602251E-1 ,aaaa,3f2a,a9f1,aaaa, -2, .AAAAAAAAAAA9F1) 62 vc(p2, -2.7777777777015591216E-3 ,0b60,bc36,ec94,b5f5, -8,-.B60B60B5F5EC94) 63 vc(p3, 6.6137563214379341918E-5 ,b355,398a,f15f,792e, -13, .8AB355792EF15F) 64 vc(p4, -1.6533902205465250480E-6 ,ea0e,b6dd,5f84,2e93, -19,-.DDEA0E2E935F84) 65 vc(p5, 4.1381367970572387085E-8 ,bb4b,3431,2683,95f5, -24, .B1BB4B95F52683) 66 67 #ifdef vccast 68 #define ln2hi vccast(ln2hi) 69 #define ln2lo vccast(ln2lo) 70 #define lnhuge vccast(lnhuge) 71 #define lntiny vccast(lntiny) 72 #define invln2 vccast(invln2) 73 #define p1 vccast(p1) 74 #define p2 vccast(p2) 75 #define p3 vccast(p3) 76 #define p4 vccast(p4) 77 #define p5 vccast(p5) 78 #endif 79 80 ic(p1, 1.6666666666666601904E-1, -3, 1.555555555553E) 81 ic(p2, -2.7777777777015593384E-3, -9, -1.6C16C16BEBD93) 82 ic(p3, 6.6137563214379343612E-5, -14, 1.1566AAF25DE2C) 83 ic(p4, -1.6533902205465251539E-6, -20, -1.BBD41C5D26BF1) 84 ic(p5, 4.1381367970572384604E-8, -25, 1.6376972BEA4D0) 85 ic(ln2hi, 6.9314718036912381649E-1, -1, 1.62E42FEE00000) 86 ic(ln2lo, 1.9082149292705877000E-10,-33, 1.A39EF35793C76) 87 ic(lnhuge, 7.1602103751842355450E2, 9, 1.6602B15B7ECF2) 88 ic(lntiny,-7.5137154372698068983E2, 9, -1.77AF8EBEAE354) 89 ic(invln2, 1.4426950408889633870E0, 0, 1.71547652B82FE) 90 91 double exp(x) 92 double x; 93 { 94 double z,hi,lo,c; 95 int k; 96 97 #if !defined(vax)&&!defined(tahoe) 98 if(x!=x) return(x); /* x is NaN */ 99 #endif /* !defined(vax)&&!defined(tahoe) */ 100 if( x <= lnhuge ) { 101 if( x >= lntiny ) { 102 103 /* argument reduction : x --> x - k*ln2 */ 104 105 k=invln2*x+copysign(0.5,x); /* k=NINT(x/ln2) */ 106 107 /* express x-k*ln2 as hi-lo and let x=hi-lo rounded */ 108 109 hi=x-k*ln2hi; 110 x=hi-(lo=k*ln2lo); 111 112 /* return 2^k*[1+x+x*c/(2+c)] */ 113 z=x*x; 114 c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5)))); 115 return scalb(1.0+(hi-(lo-(x*c)/(2.0-c))),k); 116 117 } 118 /* end of x > lntiny */ 119 120 else 121 /* exp(-big#) underflows to zero */ 122 if(finite(x)) return(scalb(1.0,-5000)); 123 124 /* exp(-INF) is zero */ 125 else return(0.0); 126 } 127 /* end of x < lnhuge */ 128 129 else 130 /* exp(INF) is INF, exp(+big#) overflows to INF */ 131 return( finite(x) ? scalb(1.0,5000) : x); 132 } 133 134 /* returns exp(r = x + c) for |c| < |x| with no overlap. */ 135 136 double __exp__D(x, c) 137 double x, c; 138 { 139 double z,hi,lo, t; 140 int k; 141 142 #if !defined(vax)&&!defined(tahoe) 143 if (x!=x) return(x); /* x is NaN */ 144 #endif /* !defined(vax)&&!defined(tahoe) */ 145 if ( x <= lnhuge ) { 146 if ( x >= lntiny ) { 147 148 /* argument reduction : x --> x - k*ln2 */ 149 z = invln2*x; 150 k = z + copysign(.5, x); 151 152 /* express (x+c)-k*ln2 as hi-lo and let x=hi-lo rounded */ 153 154 hi=(x-k*ln2hi); /* Exact. */ 155 x= hi - (lo = k*ln2lo-c); 156 /* return 2^k*[1+x+x*c/(2+c)] */ 157 z=x*x; 158 c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5)))); 159 c = (x*c)/(2.0-c); 160 161 return scalb(1.+(hi-(lo - c)), k); 162 } 163 /* end of x > lntiny */ 164 165 else 166 /* exp(-big#) underflows to zero */ 167 if(finite(x)) return(scalb(1.0,-5000)); 168 169 /* exp(-INF) is zero */ 170 else return(0.0); 171 } 172 /* end of x < lnhuge */ 173 174 else 175 /* exp(INF) is INF, exp(+big#) overflows to INF */ 176 return( finite(x) ? scalb(1.0,5000) : x); 177 } 178