1 /* derived from /netlib/fdlibm */ 2 3 /* @(#)e_exp.c 1.3 95/01/18 */ 4 /* 5 * ==================================================== 6 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 7 * 8 * Developed at SunSoft, a Sun Microsystems, Inc. business. 9 * Permission to use, copy, modify, and distribute this 10 * software is freely granted, provided that this notice 11 * is preserved. 12 * ==================================================== 13 */ 14 15 /* __ieee754_exp(x) 16 * Returns the exponential of x. 17 * 18 * Method 19 * 1. Argument reduction: 20 * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658. 21 * Given x, find r and integer k such that 22 * 23 * x = k*ln2 + r, |r| <= 0.5*ln2. 24 * 25 * Here r will be represented as r = hi-lo for better 26 * accuracy. 27 * 28 * 2. Approximation of exp(r) by a special rational function on 29 * the interval [0,0.34658]: 30 * Write 31 * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ... 32 * We use a special Reme algorithm on [0,0.34658] to generate 33 * a polynomial of degree 5 to approximate R. The maximum error 34 * of this polynomial approximation is bounded by 2**-59. In 35 * other words, 36 * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5 37 * (where z=r*r, and the values of P1 to P5 are listed below) 38 * and 39 * | 5 | -59 40 * | 2.0+P1*z+...+P5*z - R(z) | <= 2 41 * | | 42 * The computation of exp(r) thus becomes 43 * 2*r 44 * exp(r) = 1 + ------- 45 * R - r 46 * r*R1(r) 47 * = 1 + r + ----------- (for better accuracy) 48 * 2 - R1(r) 49 * where 50 * 2 4 10 51 * R1(r) = r - (P1*r + P2*r + ... + P5*r ). 52 * 53 * 3. Scale back to obtain exp(x): 54 * From step 1, we have 55 * exp(x) = 2^k * exp(r) 56 * 57 * Special cases: 58 * exp(INF) is INF, exp(NaN) is NaN; 59 * exp(-INF) is 0, and 60 * for finite argument, only exp(0)=1 is exact. 61 * 62 * Accuracy: 63 * according to an error analysis, the error is always less than 64 * 1 ulp (unit in the last place). 65 * 66 * Misc. info. 67 * For IEEE double 68 * if x > 7.09782712893383973096e+02 then exp(x) overflow 69 * if x < -7.45133219101941108420e+02 then exp(x) underflow 70 * 71 * Constants: 72 * The hexadecimal values are the intended ones for the following 73 * constants. The decimal values may be used, provided that the 74 * compiler will convert from decimal to binary accurately enough 75 * to produce the hexadecimal values shown. 76 */ 77 78 #include "fdlibm.h" 79 80 static const double 81 one = 1.0, 82 halF[2] = {0.5,-0.5,}, 83 Huge = 1.0e+300, 84 twom1000= 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/ 85 o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */ 86 u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */ 87 ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */ 88 -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */ 89 ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */ 90 -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */ 91 invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */ 92 P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ 93 P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ 94 P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ 95 P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ 96 P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */ 97 98 __ieee754_exp(double x)99 double __ieee754_exp(double x) /* default IEEE double exp */ 100 { 101 double y,hi,lo,c,t; 102 int k,xsb; 103 unsigned hx; 104 105 hx = __HI(x); /* high word of x */ 106 xsb = (hx>>31)&1; /* sign bit of x */ 107 hx &= 0x7fffffff; /* high word of |x| */ 108 109 /* filter out non-finite argument */ 110 if(hx >= 0x40862E42) { /* if |x|>=709.78... */ 111 if(hx>=0x7ff00000) { 112 if(((hx&0xfffff)|__LO(x))!=0) 113 return x+x; /* NaN */ 114 else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */ 115 } 116 if(x > o_threshold) return Huge*Huge; /* overflow */ 117 if(x < u_threshold) return twom1000*twom1000; /* underflow */ 118 } 119 120 /* argument reduction */ 121 if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ 122 if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ 123 hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb; 124 } else { 125 k = invln2*x+halF[xsb]; 126 t = k; 127 hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */ 128 lo = t*ln2LO[0]; 129 } 130 x = hi - lo; 131 } 132 else if(hx < 0x3e300000) { /* when |x|<2**-28 */ 133 if(Huge+x>one) return one+x;/* trigger inexact */ 134 } 135 else k = 0; 136 137 /* x is now in primary range */ 138 t = x*x; 139 c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); 140 if(k==0) return one-((x*c)/(c-2.0)-x); 141 else y = one-((lo-(x*c)/(2.0-c))-hi); 142 if(k >= -1021) { 143 __HI(y) += (k<<20); /* add k to y's exponent */ 144 return y; 145 } else { 146 __HI(y) += ((k+1000)<<20);/* add k to y's exponent */ 147 return y*twom1000; 148 } 149 } 150