| e_exp.c (117912) | e_exp.c (141296) |
|---|---|
| 1/* @(#)e_exp.c 5.1 93/09/24 */ | 1 2/* @(#)e_exp.c 1.6 04/04/22 */ |
| 2/* 3 * ==================================================== | 3/* 4 * ==================================================== |
| 4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. | 5 * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved. |
| 5 * | 6 * |
| 6 * Developed at SunPro, a Sun Microsystems, Inc. business. | |
| 7 * Permission to use, copy, modify, and distribute this | 7 * Permission to use, copy, modify, and distribute this |
| 8 * software is freely granted, provided that this notice | 8 * software is freely granted, provided that this notice |
| 9 * is preserved. 10 * ==================================================== 11 */ 12 13#ifndef lint | 9 * is preserved. 10 * ==================================================== 11 */ 12 13#ifndef lint |
| 14static char rcsid[] = "$FreeBSD: head/lib/msun/src/e_exp.c 117912 2003-07-23 04:53:47Z peter $"; | 14static char rcsid[] = "$FreeBSD: head/lib/msun/src/e_exp.c 141296 2005-02-04 18:26:06Z das $"; |
| 15#endif 16 17/* __ieee754_exp(x) 18 * Returns the exponential of x. 19 * 20 * Method 21 * 1. Argument reduction: 22 * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658. 23 * Given x, find r and integer k such that 24 * | 15#endif 16 17/* __ieee754_exp(x) 18 * Returns the exponential of x. 19 * 20 * Method 21 * 1. Argument reduction: 22 * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658. 23 * Given x, find r and integer k such that 24 * |
| 25 * x = k*ln2 + r, |r| <= 0.5*ln2. | 25 * x = k*ln2 + r, |r| <= 0.5*ln2. |
| 26 * | 26 * |
| 27 * Here r will be represented as r = hi-lo for better | 27 * Here r will be represented as r = hi-lo for better |
| 28 * accuracy. 29 * 30 * 2. Approximation of exp(r) by a special rational function on 31 * the interval [0,0.34658]: 32 * Write 33 * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ... | 28 * accuracy. 29 * 30 * 2. Approximation of exp(r) by a special rational function on 31 * the interval [0,0.34658]: 32 * Write 33 * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ... |
| 34 * We use a special Reme algorithm on [0,0.34658] to generate 35 * a polynomial of degree 5 to approximate R. The maximum error | 34 * We use a special Remes algorithm on [0,0.34658] to generate 35 * a polynomial of degree 5 to approximate R. The maximum error |
| 36 * of this polynomial approximation is bounded by 2**-59. In 37 * other words, 38 * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5 39 * (where z=r*r, and the values of P1 to P5 are listed below) 40 * and 41 * | 5 | -59 | 36 * of this polynomial approximation is bounded by 2**-59. In 37 * other words, 38 * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5 39 * (where z=r*r, and the values of P1 to P5 are listed below) 40 * and 41 * | 5 | -59 |
| 42 * | 2.0+P1*z+...+P5*z - R(z) | <= 2 | 42 * | 2.0+P1*z+...+P5*z - R(z) | <= 2 |
| 43 * | | 44 * The computation of exp(r) thus becomes 45 * 2*r 46 * exp(r) = 1 + ------- 47 * R - r | 43 * | | 44 * The computation of exp(r) thus becomes 45 * 2*r 46 * exp(r) = 1 + ------- 47 * R - r |
| 48 * r*R1(r) | 48 * r*R1(r) |
| 49 * = 1 + r + ----------- (for better accuracy) 50 * 2 - R1(r) 51 * where 52 * 2 4 10 53 * R1(r) = r - (P1*r + P2*r + ... + P5*r ). | 49 * = 1 + r + ----------- (for better accuracy) 50 * 2 - R1(r) 51 * where 52 * 2 4 10 53 * R1(r) = r - (P1*r + P2*r + ... + P5*r ). |
| 54 * | 54 * |
| 55 * 3. Scale back to obtain exp(x): 56 * From step 1, we have 57 * exp(x) = 2^k * exp(r) 58 * 59 * Special cases: 60 * exp(INF) is INF, exp(NaN) is NaN; 61 * exp(-INF) is 0, and 62 * for finite argument, only exp(0)=1 is exact. 63 * 64 * Accuracy: 65 * according to an error analysis, the error is always less than 66 * 1 ulp (unit in the last place). 67 * 68 * Misc. info. | 55 * 3. Scale back to obtain exp(x): 56 * From step 1, we have 57 * exp(x) = 2^k * exp(r) 58 * 59 * Special cases: 60 * exp(INF) is INF, exp(NaN) is NaN; 61 * exp(-INF) is 0, and 62 * for finite argument, only exp(0)=1 is exact. 63 * 64 * Accuracy: 65 * according to an error analysis, the error is always less than 66 * 1 ulp (unit in the last place). 67 * 68 * Misc. info. |
| 69 * For IEEE double | 69 * For IEEE double |
| 70 * if x > 7.09782712893383973096e+02 then exp(x) overflow 71 * if x < -7.45133219101941108420e+02 then exp(x) underflow 72 * 73 * Constants: | 70 * if x > 7.09782712893383973096e+02 then exp(x) overflow 71 * if x < -7.45133219101941108420e+02 then exp(x) underflow 72 * 73 * Constants: |
| 74 * The hexadecimal values are the intended ones for the following 75 * constants. The decimal values may be used, provided that the | 74 * The hexadecimal values are the intended ones for the following 75 * constants. The decimal values may be used, provided that the |
| 76 * compiler will convert from decimal to binary accurately enough 77 * to produce the hexadecimal values shown. 78 */ 79 80#include "math.h" 81#include "math_private.h" 82 83static const double --- 35 unchanged lines hidden (view full) --- 119 return x+x; /* NaN */ 120 else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */ 121 } 122 if(x > o_threshold) return huge*huge; /* overflow */ 123 if(x < u_threshold) return twom1000*twom1000; /* underflow */ 124 } 125 126 /* argument reduction */ | 76 * compiler will convert from decimal to binary accurately enough 77 * to produce the hexadecimal values shown. 78 */ 79 80#include "math.h" 81#include "math_private.h" 82 83static const double --- 35 unchanged lines hidden (view full) --- 119 return x+x; /* NaN */ 120 else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */ 121 } 122 if(x > o_threshold) return huge*huge; /* overflow */ 123 if(x < u_threshold) return twom1000*twom1000; /* underflow */ 124 } 125 126 /* argument reduction */ |
| 127 if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ | 127 if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ |
| 128 if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ 129 hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb; 130 } else { | 128 if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ 129 hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb; 130 } else { |
| 131 k = invln2*x+halF[xsb]; | 131 k = (int)(invln2*x+halF[xsb]); |
| 132 t = k; 133 hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */ 134 lo = t*ln2LO[0]; 135 } 136 x = hi - lo; | 132 t = k; 133 hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */ 134 lo = t*ln2LO[0]; 135 } 136 x = hi - lo; |
| 137 } | 137 } |
| 138 else if(hx < 0x3e300000) { /* when |x|<2**-28 */ 139 if(huge+x>one) return one+x;/* trigger inexact */ 140 } 141 else k = 0; 142 143 /* x is now in primary range */ 144 t = x*x; 145 c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); | 138 else if(hx < 0x3e300000) { /* when |x|<2**-28 */ 139 if(huge+x>one) return one+x;/* trigger inexact */ 140 } 141 else k = 0; 142 143 /* x is now in primary range */ 144 t = x*x; 145 c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); |
| 146 if(k==0) return one-((x*c)/(c-2.0)-x); | 146 if(k==0) return one-((x*c)/(c-2.0)-x); |
| 147 else y = one-((lo-(x*c)/(2.0-c))-hi); 148 if(k >= -1021) { 149 u_int32_t hy; 150 GET_HIGH_WORD(hy,y); 151 SET_HIGH_WORD(y,hy+(k<<20)); /* add k to y's exponent */ 152 return y; 153 } else { 154 u_int32_t hy; 155 GET_HIGH_WORD(hy,y); 156 SET_HIGH_WORD(y,hy+((k+1000)<<20)); /* add k to y's exponent */ 157 return y*twom1000; 158 } 159} | 147 else y = one-((lo-(x*c)/(2.0-c))-hi); 148 if(k >= -1021) { 149 u_int32_t hy; 150 GET_HIGH_WORD(hy,y); 151 SET_HIGH_WORD(y,hy+(k<<20)); /* add k to y's exponent */ 152 return y; 153 } else { 154 u_int32_t hy; 155 GET_HIGH_WORD(hy,y); 156 SET_HIGH_WORD(y,hy+((k+1000)<<20)); /* add k to y's exponent */ 157 return y*twom1000; 158 } 159} |