1 2 /* @(#)e_exp.c 1.6 04/04/22 */ 3 /* 4 * ==================================================== 5 * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved. 6 * 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 __FBSDID("$FreeBSD$"); 15 16 /* __ieee754_exp(x) 17 * Returns the exponential of x. 18 * 19 * Method 20 * 1. Argument reduction: 21 * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658. 22 * Given x, find r and integer k such that 23 * 24 * x = k*ln2 + r, |r| <= 0.5*ln2. 25 * 26 * Here r will be represented as r = hi-lo for better 27 * accuracy. 28 * 29 * 2. Approximation of exp(r) by a special rational function on 30 * the interval [0,0.34658]: 31 * Write 32 * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ... 33 * We use a special Remes algorithm on [0,0.34658] to generate 34 * a polynomial of degree 5 to approximate R. The maximum error 35 * of this polynomial approximation is bounded by 2**-59. In 36 * other words, 37 * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5 38 * (where z=r*r, and the values of P1 to P5 are listed below) 39 * and 40 * | 5 | -59 41 * | 2.0+P1*z+...+P5*z - R(z) | <= 2 42 * | | 43 * The computation of exp(r) thus becomes 44 * 2*r 45 * exp(r) = 1 + ------- 46 * R - r 47 * r*R1(r) 48 * = 1 + r + ----------- (for better accuracy) 49 * 2 - R1(r) 50 * where 51 * 2 4 10 52 * R1(r) = r - (P1*r + P2*r + ... + P5*r ). 53 * 54 * 3. Scale back to obtain exp(x): 55 * From step 1, we have 56 * exp(x) = 2^k * exp(r) 57 * 58 * Special cases: 59 * exp(INF) is INF, exp(NaN) is NaN; 60 * exp(-INF) is 0, and 61 * for finite argument, only exp(0)=1 is exact. 62 * 63 * Accuracy: 64 * according to an error analysis, the error is always less than 65 * 1 ulp (unit in the last place). 66 * 67 * Misc. info. 68 * For IEEE double 69 * if x > 7.09782712893383973096e+02 then exp(x) overflow 70 * if x < -7.45133219101941108420e+02 then exp(x) underflow 71 * 72 * Constants: 73 * The hexadecimal values are the intended ones for the following 74 * constants. The decimal values may be used, provided that the 75 * compiler will convert from decimal to binary accurately enough 76 * to produce the hexadecimal values shown. 77 */ 78 79 #include <float.h> 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 o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */ 88 u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */ 89 ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */ 90 -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */ 91 ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */ 92 -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */ 93 invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */ 94 P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ 95 P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ 96 P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ 97 P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ 98 P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */ 99 100 static volatile double 101 huge = 1.0e+300, 102 twom1000= 9.33263618503218878990e-302; /* 2**-1000=0x01700000,0*/ 103 104 double 105 __ieee754_exp(double x) /* default IEEE double exp */ 106 { 107 double y,hi=0.0,lo=0.0,c,t,twopk; 108 int32_t k=0,xsb; 109 u_int32_t hx; 110 111 GET_HIGH_WORD(hx,x); 112 xsb = (hx>>31)&1; /* sign bit of x */ 113 hx &= 0x7fffffff; /* high word of |x| */ 114 115 /* filter out non-finite argument */ 116 if(hx >= 0x40862E42) { /* if |x|>=709.78... */ 117 if(hx>=0x7ff00000) { 118 u_int32_t lx; 119 GET_LOW_WORD(lx,x); 120 if(((hx&0xfffff)|lx)!=0) 121 return x+x; /* NaN */ 122 else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */ 123 } 124 if(x > o_threshold) return huge*huge; /* overflow */ 125 if(x < u_threshold) return twom1000*twom1000; /* underflow */ 126 } 127 128 /* argument reduction */ 129 if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ 130 if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ 131 hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb; 132 } else { 133 k = (int)(invln2*x+halF[xsb]); 134 t = k; 135 hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */ 136 lo = t*ln2LO[0]; 137 } 138 STRICT_ASSIGN(double, x, hi - lo); 139 } 140 else if(hx < 0x3e300000) { /* when |x|<2**-28 */ 141 if(huge+x>one) return one+x;/* trigger inexact */ 142 } 143 else k = 0; 144 145 /* x is now in primary range */ 146 t = x*x; 147 if(k >= -1021) 148 INSERT_WORDS(twopk,0x3ff00000+(k<<20), 0); 149 else 150 INSERT_WORDS(twopk,0x3ff00000+((k+1000)<<20), 0); 151 c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); 152 if(k==0) return one-((x*c)/(c-2.0)-x); 153 else y = one-((lo-(x*c)/(2.0-c))-hi); 154 if(k >= -1021) { 155 if (k==1024) return y*2.0*0x1p1023; 156 return y*twopk; 157 } else { 158 return y*twopk*twom1000; 159 } 160 } 161 162 #if (LDBL_MANT_DIG == 53) 163 __weak_reference(exp, expl); 164 #endif 165