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 #include <sys/cdefs.h> 14 #if defined(LIBM_SCCS) && !defined(lint) 15 __RCSID("$NetBSD: e_exp.c,v 1.11 2002/05/26 22:01:49 wiz Exp $"); 16 #endif 17 18 /* __ieee754_exp(x) 19 * Returns the exponential of x. 20 * 21 * Method 22 * 1. Argument reduction: 23 * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658. 24 * Given x, find r and integer k such that 25 * 26 * x = k*ln2 + r, |r| <= 0.5*ln2. 27 * 28 * Here r will be represented as r = hi-lo for better 29 * accuracy. 30 * 31 * 2. Approximation of exp(r) by a special rational function on 32 * the interval [0,0.34658]: 33 * Write 34 * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ... 35 * We use a special Reme algorithm on [0,0.34658] to generate 36 * a polynomial of degree 5 to approximate R. The maximum error 37 * of this polynomial approximation is bounded by 2**-59. In 38 * other words, 39 * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5 40 * (where z=r*r, and the values of P1 to P5 are listed below) 41 * and 42 * | 5 | -59 43 * | 2.0+P1*z+...+P5*z - R(z) | <= 2 44 * | | 45 * The computation of exp(r) thus becomes 46 * 2*r 47 * exp(r) = 1 + ------- 48 * R - r 49 * r*R1(r) 50 * = 1 + r + ----------- (for better accuracy) 51 * 2 - R1(r) 52 * where 53 * 2 4 10 54 * R1(r) = r - (P1*r + P2*r + ... + P5*r ). 55 * 56 * 3. Scale back to obtain exp(x): 57 * From step 1, we have 58 * exp(x) = 2^k * exp(r) 59 * 60 * Special cases: 61 * exp(INF) is INF, exp(NaN) is NaN; 62 * exp(-INF) is 0, and 63 * for finite argument, only exp(0)=1 is exact. 64 * 65 * Accuracy: 66 * according to an error analysis, the error is always less than 67 * 1 ulp (unit in the last place). 68 * 69 * Misc. info. 70 * For IEEE double 71 * if x > 7.09782712893383973096e+02 then exp(x) overflow 72 * if x < -7.45133219101941108420e+02 then exp(x) underflow 73 * 74 * Constants: 75 * The hexadecimal values are the intended ones for the following 76 * constants. The decimal values may be used, provided that the 77 * compiler will convert from decimal to binary accurately enough 78 * to produce the hexadecimal values shown. 79 */ 80 81 #include "math.h" 82 #include "math_private.h" 83 84 static const double 85 one = 1.0, 86 halF[2] = {0.5,-0.5,}, 87 huge = 1.0e+300, 88 twom1000= 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/ 89 o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */ 90 u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */ 91 ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */ 92 -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */ 93 ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */ 94 -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */ 95 invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */ 96 P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ 97 P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ 98 P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ 99 P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ 100 P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */ 101 102 103 double 104 __ieee754_exp(double x) /* default IEEE double exp */ 105 { 106 double y,hi,lo,c,t; 107 int32_t k,xsb; 108 u_int32_t hx; 109 110 hi = lo = 0; 111 k = 0; 112 GET_HIGH_WORD(hx,x); 113 xsb = (hx>>31)&1; /* sign bit of x */ 114 hx &= 0x7fffffff; /* high word of |x| */ 115 116 /* filter out non-finite argument */ 117 if(hx >= 0x40862E42) { /* if |x|>=709.78... */ 118 if(hx>=0x7ff00000) { 119 u_int32_t lx; 120 GET_LOW_WORD(lx,x); 121 if(((hx&0xfffff)|lx)!=0) 122 return x+x; /* NaN */ 123 else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */ 124 } 125 if(x > o_threshold) return huge*huge; /* overflow */ 126 if(x < u_threshold) return twom1000*twom1000; /* underflow */ 127 } 128 129 /* argument reduction */ 130 if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ 131 if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ 132 hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb; 133 } else { 134 k = invln2*x+halF[xsb]; 135 t = k; 136 hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */ 137 lo = t*ln2LO[0]; 138 } 139 x = hi - lo; 140 } 141 else if(hx < 0x3e300000) { /* when |x|<2**-28 */ 142 if(huge+x>one) return one+x;/* trigger inexact */ 143 } 144 else k = 0; 145 146 /* x is now in primary range */ 147 t = x*x; 148 c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); 149 if(k==0) return one-((x*c)/(c-2.0)-x); 150 else y = one-((lo-(x*c)/(2.0-c))-hi); 151 if(k >= -1021) { 152 u_int32_t hy; 153 GET_HIGH_WORD(hy,y); 154 SET_HIGH_WORD(y,hy+(k<<20)); /* add k to y's exponent */ 155 return y; 156 } else { 157 u_int32_t hy; 158 GET_HIGH_WORD(hy,y); 159 SET_HIGH_WORD(y,hy+((k+1000)<<20)); /* add k to y's exponent */ 160 return y*twom1000; 161 } 162 } 163