e_exp.c (5935B)
1 /* 2 * ==================================================== 3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 4 * 5 * Developed at SunPro, a Sun Microsystems, Inc. business. 6 * Permission to use, copy, modify, and distribute this 7 * software is freely granted, provided that this notice 8 * is preserved. 9 * ==================================================== 10 */ 11 12 /* __ieee754_exp(x) 13 * Returns the exponential of x. 14 * 15 * Method 16 * 1. Argument reduction: 17 * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658. 18 * Given x, find r and integer k such that 19 * 20 * x = k*ln2 + r, |r| <= 0.5*ln2. 21 * 22 * Here r will be represented as r = hi-lo for better 23 * accuracy. 24 * 25 * 2. Approximation of exp(r) by a special rational function on 26 * the interval [0,0.34658]: 27 * Write 28 * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ... 29 * We use a special Reme algorithm on [0,0.34658] to generate 30 * a polynomial of degree 5 to approximate R. The maximum error 31 * of this polynomial approximation is bounded by 2**-59. In 32 * other words, 33 * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5 34 * (where z=r*r, and the values of P1 to P5 are listed below) 35 * and 36 * | 5 | -59 37 * | 2.0+P1*z+...+P5*z - R(z) | <= 2 38 * | | 39 * The computation of exp(r) thus becomes 40 * 2*r 41 * exp(r) = 1 + ------- 42 * R - r 43 * r*R1(r) 44 * = 1 + r + ----------- (for better accuracy) 45 * 2 - R1(r) 46 * where 47 * 2 4 10 48 * R1(r) = r - (P1*r + P2*r + ... + P5*r ). 49 * 50 * 3. Scale back to obtain exp(x): 51 * From step 1, we have 52 * exp(x) = 2^k * exp(r) 53 * 54 * Special cases: 55 * exp(INF) is INF, exp(NaN) is NaN; 56 * exp(-INF) is 0, and 57 * for finite argument, only exp(0)=1 is exact. 58 * 59 * Accuracy: 60 * according to an error analysis, the error is always less than 61 * 1 ulp (unit in the last place). 62 * 63 * Misc. info. 64 * For IEEE double 65 * if x > 7.09782712893383973096e+02 then exp(x) overflow 66 * if x < -7.45133219101941108420e+02 then exp(x) underflow 67 * 68 * Constants: 69 * The hexadecimal values are the intended ones for the following 70 * constants. The decimal values may be used, provided that the 71 * compiler will convert from decimal to binary accurately enough 72 * to produce the hexadecimal values shown. 73 */ 74 75 #include "math_libm.h" 76 #include "math_private.h" 77 78 #ifdef __WATCOMC__ /* Watcom defines huge=__huge */ 79 #undef huge 80 #endif 81 82 static const double 83 one = 1.0, 84 halF[2] = {0.5,-0.5,}, 85 huge = 1.0e+300, 86 twom1000= 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/ 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 double __ieee754_exp(double x) /* default IEEE double exp */ 101 { 102 double y; 103 double hi = 0.0; 104 double lo = 0.0; 105 double c; 106 double t; 107 int32_t k=0; 108 int32_t 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 1 125 if(x > o_threshold) return huge*huge; /* overflow */ 126 #else /* !!! FIXME: check this: "huge * huge" is a compiler warning, maybe they wanted +Inf? */ 127 if(x > o_threshold) return INFINITY; /* overflow */ 128 #endif 129 130 if(x < u_threshold) return twom1000*twom1000; /* underflow */ 131 } 132 133 /* argument reduction */ 134 if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ 135 if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ 136 hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb; 137 } else { 138 k = (int32_t) (invln2*x+halF[xsb]); 139 t = k; 140 hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */ 141 lo = t*ln2LO[0]; 142 } 143 x = hi - lo; 144 } 145 else if(hx < 0x3e300000) { /* when |x|<2**-28 */ 146 if(huge+x>one) return one+x;/* trigger inexact */ 147 } 148 else k = 0; 149 150 /* x is now in primary range */ 151 t = x*x; 152 c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); 153 if(k==0) return one-((x*c)/(c-2.0)-x); 154 else y = one-((lo-(x*c)/(2.0-c))-hi); 155 if(k >= -1021) { 156 u_int32_t hy; 157 GET_HIGH_WORD(hy,y); 158 SET_HIGH_WORD(y,hy+(k<<20)); /* add k to y's exponent */ 159 return y; 160 } else { 161 u_int32_t hy; 162 GET_HIGH_WORD(hy,y); 163 SET_HIGH_WORD(y,hy+((k+1000)<<20)); /* add k to y's exponent */ 164 return y*twom1000; 165 } 166 } 167 168 /* 169 * wrapper exp(x) 170 */ 171 #ifndef _IEEE_LIBM 172 double exp(double x) 173 { 174 static const double o_threshold = 7.09782712893383973096e+02; /* 0x40862E42, 0xFEFA39EF */ 175 static const double u_threshold = -7.45133219101941108420e+02; /* 0xc0874910, 0xD52D3051 */ 176 177 double z = __ieee754_exp(x); 178 if (_LIB_VERSION == _IEEE_) 179 return z; 180 if (isfinite(x)) { 181 if (x > o_threshold) 182 return __kernel_standard(x, x, 6); /* exp overflow */ 183 if (x < u_threshold) 184 return __kernel_standard(x, x, 7); /* exp underflow */ 185 } 186 return z; 187 } 188 #else 189 strong_alias(__ieee754_exp, exp) 190 #endif 191 libm_hidden_def(exp)