1 2 /* 3 * ==================================================== 4 * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved. 5 * 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 /* 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 Remes 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 <float.h> 76 77 #include "math.h" 78 #include "math_private.h" 79 80 static const double 81 one = 1.0, 82 halF[2] = {0.5,-0.5,}, 83 o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */ 84 u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */ 85 ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */ 86 -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */ 87 ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */ 88 -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */ 89 invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */ 90 P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ 91 P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ 92 P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ 93 P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ 94 P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */ 95 96 static volatile double 97 huge = 1.0e+300, 98 twom1000= 9.33263618503218878990e-302; /* 2**-1000=0x01700000,0*/ 99 100 double 101 exp(double x) /* default IEEE double exp */ 102 { 103 double y,hi=0.0,lo=0.0,c,t,twopk; 104 int32_t k=0,xsb; 105 u_int32_t hx; 106 107 GET_HIGH_WORD(hx,x); 108 xsb = (hx>>31)&1; /* sign bit of x */ 109 hx &= 0x7fffffff; /* high word of |x| */ 110 111 /* filter out non-finite argument */ 112 if(hx >= 0x40862E42) { /* if |x|>=709.78... */ 113 if(hx>=0x7ff00000) { 114 u_int32_t lx; 115 GET_LOW_WORD(lx,x); 116 if(((hx&0xfffff)|lx)!=0) 117 return x+x; /* NaN */ 118 else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */ 119 } 120 if(x > o_threshold) return huge*huge; /* overflow */ 121 if(x < u_threshold) return twom1000*twom1000; /* underflow */ 122 } 123 124 /* argument reduction */ 125 if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ 126 if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ 127 hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb; 128 } else { 129 k = (int)(invln2*x+halF[xsb]); 130 t = k; 131 hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */ 132 lo = t*ln2LO[0]; 133 } 134 STRICT_ASSIGN(double, x, hi - lo); 135 } 136 else if(hx < 0x3e300000) { /* when |x|<2**-28 */ 137 if(huge+x>one) return one+x;/* trigger inexact */ 138 } 139 else k = 0; 140 141 /* x is now in primary range */ 142 t = x*x; 143 if(k >= -1021) 144 INSERT_WORDS(twopk,((u_int32_t)(0x3ff+k))<<20, 0); 145 else 146 INSERT_WORDS(twopk,((u_int32_t)(0x3ff+(k+1000)))<<20, 0); 147 c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); 148 if(k==0) return one-((x*c)/(c-2.0)-x); 149 else y = one-((lo-(x*c)/(2.0-c))-hi); 150 if(k >= -1021) { 151 if (k==1024) return y*2.0*0x1p1023; 152 return y*twopk; 153 } else { 154 return y*twopk*twom1000; 155 } 156 } 157 158 #if (LDBL_MANT_DIG == 53) 159 __weak_reference(exp, expl); 160 #endif 161