1 /** @file 2 Compute the logrithm of x. 3 4 Copyright (c) 2010 - 2011, Intel Corporation. All rights reserved.<BR> 5 This program and the accompanying materials are licensed and made available under 6 the terms and conditions of the BSD License that accompanies this distribution. 7 The full text of the license may be found at 8 http://opensource.org/licenses/bsd-license. 9 10 THE PROGRAM IS DISTRIBUTED UNDER THE BSD LICENSE ON AN "AS IS" BASIS, 11 WITHOUT WARRANTIES OR REPRESENTATIONS OF ANY KIND, EITHER EXPRESS OR IMPLIED. 12 13 * ==================================================== 14 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 15 * 16 * Developed at SunPro, a Sun Microsystems, Inc. business. 17 * Permission to use, copy, modify, and distribute this 18 * software is freely granted, provided that this notice 19 * is preserved. 20 * ==================================================== 21 22 e_log.c 5.1 93/09/24 23 NetBSD: e_log.c,v 1.12 2002/05/26 22:01:51 wiz Exp 24 **/ 25 #include <LibConfig.h> 26 #include <sys/EfiCdefs.h> 27 28 #if defined(_MSC_VER) /* Handle Microsoft VC++ compiler specifics. */ 29 // potential divide by 0 -- near line 118, (x-x)/zero is on purpose 30 #pragma warning ( disable : 4723 ) 31 #endif 32 33 /* __ieee754_log(x) 34 * Return the logrithm of x 35 * 36 * Method : 37 * 1. Argument Reduction: find k and f such that 38 * x = 2^k * (1+f), 39 * where sqrt(2)/2 < 1+f < sqrt(2) . 40 * 41 * 2. Approximation of log(1+f). 42 * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) 43 * = 2s + 2/3 s**3 + 2/5 s**5 + ....., 44 * = 2s + s*R 45 * We use a special Reme algorithm on [0,0.1716] to generate 46 * a polynomial of degree 14 to approximate R The maximum error 47 * of this polynomial approximation is bounded by 2**-58.45. In 48 * other words, 49 * 2 4 6 8 10 12 14 50 * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s 51 * (the values of Lg1 to Lg7 are listed in the program) 52 * and 53 * | 2 14 | -58.45 54 * | Lg1*s +...+Lg7*s - R(z) | <= 2 55 * | | 56 * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. 57 * In order to guarantee error in log below 1ulp, we compute log 58 * by 59 * log(1+f) = f - s*(f - R) (if f is not too large) 60 * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy) 61 * 62 * 3. Finally, log(x) = k*ln2 + log(1+f). 63 * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) 64 * Here ln2 is split into two floating point number: 65 * ln2_hi + ln2_lo, 66 * where n*ln2_hi is always exact for |n| < 2000. 67 * 68 * Special cases: 69 * log(x) is NaN with signal if x < 0 (including -INF) ; 70 * log(+INF) is +INF; log(0) is -INF with signal; 71 * log(NaN) is that NaN with no signal. 72 * 73 * Accuracy: 74 * according to an error analysis, the error is always less than 75 * 1 ulp (unit in the last place). 76 * 77 * Constants: 78 * The hexadecimal values are the intended ones for the following 79 * constants. The decimal values may be used, provided that the 80 * compiler will convert from decimal to binary accurately enough 81 * to produce the hexadecimal values shown. 82 */ 83 84 #include "math.h" 85 #include "math_private.h" 86 #include <errno.h> 87 88 static const double 89 ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ 90 ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ 91 two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */ 92 Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ 93 Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ 94 Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ 95 Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ 96 Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ 97 Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ 98 Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ 99 100 static const double zero = 0.0; 101 102 double 103 __ieee754_log(double x) 104 { 105 double hfsq,f,s,z,R,w,t1,t2,dk; 106 int32_t k,hx,i,j; 107 u_int32_t lx; 108 109 EXTRACT_WORDS(hx,lx,x); 110 111 k=0; 112 if (hx < 0x00100000) { /* x < 2**-1022 */ 113 if (((hx&0x7fffffff)|lx)==0) 114 return -two54/zero; /* log(+-0)=-inf */ 115 if (hx<0) { 116 errno = EDOM; 117 return (x-x)/zero; /* log(-#) = NaN */ 118 } 119 k -= 54; x *= two54; /* subnormal number, scale up x */ 120 GET_HIGH_WORD(hx,x); 121 } 122 if (hx >= 0x7ff00000) return x+x; 123 k += (hx>>20)-1023; 124 hx &= 0x000fffff; 125 i = (hx+0x95f64)&0x100000; 126 SET_HIGH_WORD(x,hx|(i^0x3ff00000)); /* normalize x or x/2 */ 127 k += (i>>20); 128 f = x-1.0; 129 if((0x000fffff&(2+hx))<3) { /* |f| < 2**-20 */ 130 if(f==zero) { if(k==0) return zero; else {dk=(double)k; 131 return dk*ln2_hi+dk*ln2_lo;} 132 } 133 R = f*f*(0.5-0.33333333333333333*f); 134 if(k==0) return f-R; else {dk=(double)k; 135 return dk*ln2_hi-((R-dk*ln2_lo)-f);} 136 } 137 s = f/(2.0+f); 138 dk = (double)k; 139 z = s*s; 140 i = hx-0x6147a; 141 w = z*z; 142 j = 0x6b851-hx; 143 t1= w*(Lg2+w*(Lg4+w*Lg6)); 144 t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7))); 145 i |= j; 146 R = t2+t1; 147 if(i>0) { 148 hfsq=0.5*f*f; 149 if(k==0) return f-(hfsq-s*(hfsq+R)); else 150 return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f); 151 } else { 152 if(k==0) return f-s*(f-R); else 153 return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f); 154 } 155 } 156
