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[] = "$FreeBSD$"; 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 "math_private.h" 82 83 #ifdef __STDC__ 84 static const double 85 #else 86 static double 87 #endif 88 one = 1.0, 89 halF[2] = {0.5,-0.5,}, 90 huge = 1.0e+300, 91 twom1000= 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/ 92 o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */ 93 u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */ 94 ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */ 95 -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */ 96 ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */ 97 -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */ 98 invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */ 99 P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ 100 P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ 101 P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ 102 P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ 103 P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */ 104 105 106 #ifdef __STDC__ 107 double __generic___ieee754_exp(double x) /* default IEEE double exp */ 108 #else 109 double __generic___ieee754_exp(x) /* default IEEE double exp */ 110 double x; 111 #endif 112 { 113 double y,hi=0.0,lo=0.0,c,t; 114 int32_t k=0,xsb; 115 u_int32_t hx; 116 117 GET_HIGH_WORD(hx,x); 118 xsb = (hx>>31)&1; /* sign bit of x */ 119 hx &= 0x7fffffff; /* high word of |x| */ 120 121 /* filter out non-finite argument */ 122 if(hx >= 0x40862E42) { /* if |x|>=709.78... */ 123 if(hx>=0x7ff00000) { 124 u_int32_t lx; 125 GET_LOW_WORD(lx,x); 126 if(((hx&0xfffff)|lx)!=0) 127 return x+x; /* NaN */ 128 else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */ 129 } 130 if(x > o_threshold) return huge*huge; /* overflow */ 131 if(x < u_threshold) return twom1000*twom1000; /* underflow */ 132 } 133 134 /* argument reduction */ 135 if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ 136 if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ 137 hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb; 138 } else { 139 k = invln2*x+halF[xsb]; 140 t = k; 141 hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */ 142 lo = t*ln2LO[0]; 143 } 144 x = hi - lo; 145 } 146 else if(hx < 0x3e300000) { /* when |x|<2**-28 */ 147 if(huge+x>one) return one+x;/* trigger inexact */ 148 } 149 else k = 0; 150 151 /* x is now in primary range */ 152 t = x*x; 153 c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); 154 if(k==0) return one-((x*c)/(c-2.0)-x); 155 else y = one-((lo-(x*c)/(2.0-c))-hi); 156 if(k >= -1021) { 157 u_int32_t hy; 158 GET_HIGH_WORD(hy,y); 159 SET_HIGH_WORD(y,hy+(k<<20)); /* add k to y's exponent */ 160 return y; 161 } else { 162 u_int32_t hy; 163 GET_HIGH_WORD(hy,y); 164 SET_HIGH_WORD(y,hy+((k+1000)<<20)); /* add k to y's exponent */ 165 return y*twom1000; 166 } 167 } 168