| t2ex/bsd_source/lib/libc/src_bsd/math/s_log1p.c | bare source | permlink (0.01 seconds) |
1: /* @(#)s_log1p.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: /* double log1p(double x) 16: * 17: * Method : 18: * 1. Argument Reduction: find k and f such that 19: * 1+x = 2^k * (1+f), 20: * where sqrt(2)/2 < 1+f < sqrt(2) . 21: * 22: * Note. If k=0, then f=x is exact. However, if k!=0, then f 23: * may not be representable exactly. In that case, a correction 24: * term is need. Let u=1+x rounded. Let c = (1+x)-u, then 25: * log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u), 26: * and add back the correction term c/u. 27: * (Note: when x > 2**53, one can simply return log(x)) 28: * 29: * 2. Approximation of log1p(f). 30: * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) 31: * = 2s + 2/3 s**3 + 2/5 s**5 + ....., 32: * = 2s + s*R 33: * We use a special Remes algorithm on [0,0.1716] to generate 34: * a polynomial of degree 14 to approximate R The maximum error 35: * of this polynomial approximation is bounded by 2**-58.45. In 36: * other words, 37: * 2 4 6 8 10 12 14 38: * R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s 39: * (the values of Lp1 to Lp7 are listed in the program) 40: * and 41: * | 2 14 | -58.45 42: * | Lp1*s +...+Lp7*s - R(z) | <= 2 43: * | | 44: * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. 45: * In order to guarantee error in log below 1ulp, we compute log 46: * by 47: * log1p(f) = f - (hfsq - s*(hfsq+R)). 48: * 49: * 3. Finally, log1p(x) = k*ln2 + log1p(f). 50: * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) 51: * Here ln2 is split into two floating point number: 52: * ln2_hi + ln2_lo, 53: * where n*ln2_hi is always exact for |n| < 2000. 54: * 55: * Special cases: 56: * log1p(x) is NaN with signal if x < -1 (including -INF) ; 57: * log1p(+INF) is +INF; log1p(-1) is -INF with signal; 58: * log1p(NaN) is that NaN with no signal. 59: * 60: * Accuracy: 61: * according to an error analysis, the error is always less than 62: * 1 ulp (unit in the last place). 63: * 64: * Constants: 65: * The hexadecimal values are the intended ones for the following 66: * constants. The decimal values may be used, provided that the 67: * compiler will convert from decimal to binary accurately enough 68: * to produce the hexadecimal values shown. 69: * 70: * Note: Assuming log() return accurate answer, the following 71: * algorithm can be used to compute log1p(x) to within a few ULP: 72: * 73: * u = 1+x; 74: * if(u==1.0) return x ; else 75: * return log(u)*(x/(u-1.0)); 76: * 77: * See HP-15C Advanced Functions Handbook, p.193. 78: */ 79: 80: #include <sys/cdefs.h> 81: #include <float.h> 82: #include <math.h> 83: 84: #include "math_private.h" 85: 86: static const double 87: ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ 88: ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ 89: two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */ 90: Lp1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ 91: Lp2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ 92: Lp3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ 93: Lp4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ 94: Lp5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ 95: Lp6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ 96: Lp7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ 97: 98: static const double zero = 0.0; 99: 100: double 101: log1p(double x) 102: { 103: double hfsq,f,c,s,z,R,u; 104: int32_t k,hx,hu,ax; 105: 106: GET_HIGH_WORD(hx,x); 107: ax = hx&0x7fffffff; 108: 109: k = 1; 110: if (hx < 0x3FDA827A) { /* x < 0.41422 */ 111: if(ax>=0x3ff00000) { /* x <= -1.0 */ 112: if(x==-1.0) return -two54/zero; /* log1p(-1)=+inf */ 113: else return (x-x)/(x-x); /* log1p(x<-1)=NaN */ 114: } 115: if(ax<0x3e200000) { /* |x| < 2**-29 */ 116: if(two54+x>zero /* raise inexact */ 117: &&ax<0x3c900000) /* |x| < 2**-54 */ 118: return x; 119: else 120: return x - x*x*0.5; 121: } 122: if(hx>0||hx<=((int32_t)0xbfd2bec3)) { 123: k=0;f=x;hu=1;} /* -0.2929<x<0.41422 */ 124: } 125: if (hx >= 0x7ff00000) return x+x; 126: if(k!=0) { 127: if(hx<0x43400000) { 128: u = 1.0+x; 129: GET_HIGH_WORD(hu,u); 130: k = (hu>>20)-1023; 131: c = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */ 132: c /= u; 133: } else { 134: u = x; 135: GET_HIGH_WORD(hu,u); 136: k = (hu>>20)-1023; 137: c = 0; 138: } 139: hu &= 0x000fffff; 140: if(hu<0x6a09e) { 141: SET_HIGH_WORD(u,hu|0x3ff00000); /* normalize u */ 142: } else { 143: k += 1; 144: SET_HIGH_WORD(u,hu|0x3fe00000); /* normalize u/2 */ 145: hu = (0x00100000-hu)>>2; 146: } 147: f = u-1.0; 148: } 149: hfsq=0.5*f*f; 150: if(hu==0) { /* |f| < 2**-20 */ 151: if(f==zero) if(k==0) return zero; 152: else {c += k*ln2_lo; return k*ln2_hi+c;} 153: R = hfsq*(1.0-0.66666666666666666*f); 154: if(k==0) return f-R; else 155: return k*ln2_hi-((R-(k*ln2_lo+c))-f); 156: } 157: s = f/(2.0+f); 158: z = s*s; 159: R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7)))))); 160: if(k==0) return f-(hfsq-s*(hfsq+R)); else 161: return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f); 162: } 163: 164: #if LDBL_MANT_DIG == 53 165: #ifdef lint 166: /* PROTOLIB1 */ 167: long double log1pl(long double); 168: #else /* lint */ 169: __weak_alias(log1pl, log1p); 170: #endif /* lint */ 171: #endif /* LDBL_MANT_DIG == 53 */