| t2ex/bsd_source/lib/libc/src_bsd/math/e_log.c | bare source | permlink (0.01 seconds) |
1: /* @(#)e_log.c 5.1 93/09/24 */ 2: /* 3: * ==================================================== 4: * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 5: * 6: * Developed at SunPro, a Sun Microsystems, Inc. business. 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: /* LINTLIBRARY */ 14: 15: /* log(x) 16: * Return the logarithm of x 17: * 18: * Method : 19: * 1. Argument Reduction: find k and f such that 20: * x = 2^k * (1+f), 21: * where sqrt(2)/2 < 1+f < sqrt(2) . 22: * 23: * 2. Approximation of log(1+f). 24: * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) 25: * = 2s + 2/3 s**3 + 2/5 s**5 + ....., 26: * = 2s + s*R 27: * We use a special Remes algorithm on [0,0.1716] to generate 28: * a polynomial of degree 14 to approximate R The maximum error 29: * of this polynomial approximation is bounded by 2**-58.45. In 30: * other words, 31: * 2 4 6 8 10 12 14 32: * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s 33: * (the values of Lg1 to Lg7 are listed in the program) 34: * and 35: * | 2 14 | -58.45 36: * | Lg1*s +...+Lg7*s - R(z) | <= 2 37: * | | 38: * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. 39: * In order to guarantee error in log below 1ulp, we compute log 40: * by 41: * log(1+f) = f - s*(f - R) (if f is not too large) 42: * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy) 43: * 44: * 3. Finally, log(x) = k*ln2 + log(1+f). 45: * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) 46: * Here ln2 is split into two floating point number: 47: * ln2_hi + ln2_lo, 48: * where n*ln2_hi is always exact for |n| < 2000. 49: * 50: * Special cases: 51: * log(x) is NaN with signal if x < 0 (including -INF) ; 52: * log(+INF) is +INF; log(0) is -INF with signal; 53: * log(NaN) is that NaN with no signal. 54: * 55: * Accuracy: 56: * according to an error analysis, the error is always less than 57: * 1 ulp (unit in the last place). 58: * 59: * Constants: 60: * The hexadecimal values are the intended ones for the following 61: * constants. The decimal values may be used, provided that the 62: * compiler will convert from decimal to binary accurately enough 63: * to produce the hexadecimal values shown. 64: */ 65: 66: #include <sys/cdefs.h> 67: #include <float.h> 68: #include <math.h> 69: 70: #include "math_private.h" 71: 72: static const double 73: ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ 74: ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ 75: two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */ 76: Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ 77: Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ 78: Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ 79: Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ 80: Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ 81: Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ 82: Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ 83: 84: static const double zero = 0.0; 85: 86: double 87: log(double x) 88: { 89: double hfsq,f,s,z,R,w,t1,t2,dk; 90: int32_t k,hx,i,j; 91: u_int32_t lx; 92: 93: EXTRACT_WORDS(hx,lx,x); 94: 95: k=0; 96: if (hx < 0x00100000) { /* x < 2**-1022 */ 97: if (((hx&0x7fffffff)|lx)==0) 98: return -two54/zero; /* log(+-0)=-inf */ 99: if (hx<0) return (x-x)/zero; /* log(-#) = NaN */ 100: k -= 54; x *= two54; /* subnormal number, scale up x */ 101: GET_HIGH_WORD(hx,x); 102: } 103: if (hx >= 0x7ff00000) return x+x; 104: k += (hx>>20)-1023; 105: hx &= 0x000fffff; 106: i = (hx+0x95f64)&0x100000; 107: SET_HIGH_WORD(x,hx|(i^0x3ff00000)); /* normalize x or x/2 */ 108: k += (i>>20); 109: f = x-1.0; 110: if((0x000fffff&(2+hx))<3) { /* |f| < 2**-20 */ 111: if(f==zero) if(k==0) return zero; else {dk=(double)k; 112: return dk*ln2_hi+dk*ln2_lo;} 113: R = f*f*(0.5-0.33333333333333333*f); 114: if(k==0) return f-R; else {dk=(double)k; 115: return dk*ln2_hi-((R-dk*ln2_lo)-f);} 116: } 117: s = f/(2.0+f); 118: dk = (double)k; 119: z = s*s; 120: i = hx-0x6147a; 121: w = z*z; 122: j = 0x6b851-hx; 123: t1= w*(Lg2+w*(Lg4+w*Lg6)); 124: t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7))); 125: i |= j; 126: R = t2+t1; 127: if(i>0) { 128: hfsq=0.5*f*f; 129: if(k==0) return f-(hfsq-s*(hfsq+R)); else 130: return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f); 131: } else { 132: if(k==0) return f-s*(f-R); else 133: return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f); 134: } 135: } 136: 137: #if LDBL_MANT_DIG == 53 138: #ifdef lint 139: /* PROTOLIB1 */ 140: long double logl(long double); 141: #else /* lint */ 142: __weak_alias(logl, log); 143: #endif /* lint */ 144: #endif /* LDBL_MANT_DIG == 53 */