1 /* @(#)e_exp.c 5.1 93/09/24 */ 2 /* 3 * ==================================================== 4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 5 * 6 * Developed at SunPro, a Sun Microsystems, Inc. business. 7 * Permission to use, copy, modify, and distribute this 8 * software is freely granted, provided that this notice 9 * is preserved. 10 * ==================================================== 11 */ 12 13 #ifndef lint 14 static char rcsid[] = "$Id: e_exp.c,v 1.4 1994/03/03 17:04:10 jtc Exp $"; 15 #endif 16 17 /* __ieee754_exp(x) 18 * Returns the exponential of x. 19 * 20 * Method 21 * 1. Argument reduction: 22 * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658. 23 * Given x, find r and integer k such that 24 * 25 * x = k*ln2 + r, |r| <= 0.5*ln2. 26 * 27 * Here r will be represented as r = hi-lo for better 28 * accuracy. 29 * 30 * 2. Approximation of exp(r) by a special rational function on 31 * the interval [0,0.34658]: 32 * Write 33 * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ... 34 * We use a special Reme algorithm on [0,0.34658] to generate 35 * a polynomial of degree 5 to approximate R. The maximum error 36 * of this polynomial approximation is bounded by 2**-59. In 37 * other words, 38 * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5 39 * (where z=r*r, and the values of P1 to P5 are listed below) 40 * and 41 * | 5 | -59 42 * | 2.0+P1*z+...+P5*z - R(z) | <= 2 43 * | | 44 * The computation of exp(r) thus becomes 45 * 2*r 46 * exp(r) = 1 + ------- 47 * R - r 48 * r*R1(r) 49 * = 1 + r + ----------- (for better accuracy) 50 * 2 - R1(r) 51 * where 52 * 2 4 10 53 * R1(r) = r - (P1*r + P2*r + ... + P5*r ). 54 * 55 * 3. Scale back to obtain exp(x): 56 * From step 1, we have 57 * exp(x) = 2^k * exp(r) 58 * 59 * Special cases: 60 * exp(INF) is INF, exp(NaN) is NaN; 61 * exp(-INF) is 0, and 62 * for finite argument, only exp(0)=1 is exact. 63 * 64 * Accuracy: 65 * according to an error analysis, the error is always less than 66 * 1 ulp (unit in the last place). 67 * 68 * Misc. info. 69 * For IEEE double 70 * if x > 7.09782712893383973096e+02 then exp(x) overflow 71 * if x < -7.45133219101941108420e+02 then exp(x) underflow 72 * 73 * Constants: 74 * The hexadecimal values are the intended ones for the following 75 * constants. The decimal values may be used, provided that the 76 * compiler will convert from decimal to binary accurately enough 77 * to produce the hexadecimal values shown. 78 */ 79 80 #include <math.h> 81 #include <machine/endian.h> 82 83 #if BYTE_ORDER == LITTLE_ENDIAN 84 #define n0 1 85 #else 86 #define n0 0 87 #endif 88 89 #ifdef __STDC__ 90 static const double 91 #else 92 static double 93 #endif 94 one = 1.0, 95 halF[2] = {0.5,-0.5,}, 96 huge = 1.0e+300, 97 twom1000= 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/ 98 o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */ 99 u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */ 100 ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */ 101 -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */ 102 ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */ 103 -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */ 104 invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */ 105 P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ 106 P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ 107 P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ 108 P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ 109 P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */ 110 111 112 #ifdef __STDC__ 113 double __ieee754_exp(double x) /* default IEEE double exp */ 114 #else 115 double __ieee754_exp(x) /* default IEEE double exp */ 116 double x; 117 #endif 118 { 119 double y,hi,lo,c,t; 120 int k,xsb; 121 unsigned hx; 122 123 hx = *(n0+(unsigned*)&x); /* high word of x */ 124 xsb = (hx>>31)&1; /* sign bit of x */ 125 hx &= 0x7fffffff; /* high word of |x| */ 126 127 /* filter out non-finite argument */ 128 if(hx >= 0x40862E42) { /* if |x|>=709.78... */ 129 if(hx>=0x7ff00000) { 130 if(((hx&0xfffff)|*(1-n0+(int*)&x))!=0) 131 return x+x; /* NaN */ 132 else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */ 133 } 134 if(x > o_threshold) return huge*huge; /* overflow */ 135 if(x < u_threshold) return twom1000*twom1000; /* underflow */ 136 } 137 138 /* argument reduction */ 139 if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ 140 if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ 141 hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb; 142 } else { 143 k = invln2*x+halF[xsb]; 144 t = k; 145 hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */ 146 lo = t*ln2LO[0]; 147 } 148 x = hi - lo; 149 } 150 else if(hx < 0x3e300000) { /* when |x|<2**-28 */ 151 if(huge+x>one) return one+x;/* trigger inexact */ 152 } 153 else k = 0; 154 155 /* x is now in primary range */ 156 t = x*x; 157 c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); 158 if(k==0) return one-((x*c)/(c-2.0)-x); 159 else y = one-((lo-(x*c)/(2.0-c))-hi); 160 if(k >= -1021) { 161 *(n0+(int*)&y) += (k<<20); /* add k to y's exponent */ 162 return y; 163 } else { 164 *(n0+(int*)&y) += ((k+1000)<<20);/* add k to y's exponent */ 165 return y*twom1000; 166 } 167 } 168