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 #ifndef lint 14 static char rcsid[] = "$FreeBSD$"; 15 #endif 16 17 /* double log1p(double x) 18 * 19 * Method : 20 * 1. Argument Reduction: find k and f such that 21 * 1+x = 2^k * (1+f), 22 * where sqrt(2)/2 < 1+f < sqrt(2) . 23 * 24 * Note. If k=0, then f=x is exact. However, if k!=0, then f 25 * may not be representable exactly. In that case, a correction 26 * term is need. Let u=1+x rounded. Let c = (1+x)-u, then 27 * log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u), 28 * and add back the correction term c/u. 29 * (Note: when x > 2**53, one can simply return log(x)) 30 * 31 * 2. Approximation of log1p(f). 32 * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) 33 * = 2s + 2/3 s**3 + 2/5 s**5 + ....., 34 * = 2s + s*R 35 * We use a special Reme algorithm on [0,0.1716] to generate 36 * a polynomial of degree 14 to approximate R The maximum error 37 * of this polynomial approximation is bounded by 2**-58.45. In 38 * other words, 39 * 2 4 6 8 10 12 14 40 * R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s 41 * (the values of Lp1 to Lp7 are listed in the program) 42 * and 43 * | 2 14 | -58.45 44 * | Lp1*s +...+Lp7*s - R(z) | <= 2 45 * | | 46 * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. 47 * In order to guarantee error in log below 1ulp, we compute log 48 * by 49 * log1p(f) = f - (hfsq - s*(hfsq+R)). 50 * 51 * 3. Finally, log1p(x) = k*ln2 + log1p(f). 52 * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) 53 * Here ln2 is split into two floating point number: 54 * ln2_hi + ln2_lo, 55 * where n*ln2_hi is always exact for |n| < 2000. 56 * 57 * Special cases: 58 * log1p(x) is NaN with signal if x < -1 (including -INF) ; 59 * log1p(+INF) is +INF; log1p(-1) is -INF with signal; 60 * log1p(NaN) is that NaN with no signal. 61 * 62 * Accuracy: 63 * according to an error analysis, the error is always less than 64 * 1 ulp (unit in the last place). 65 * 66 * Constants: 67 * The hexadecimal values are the intended ones for the following 68 * constants. The decimal values may be used, provided that the 69 * compiler will convert from decimal to binary accurately enough 70 * to produce the hexadecimal values shown. 71 * 72 * Note: Assuming log() return accurate answer, the following 73 * algorithm can be used to compute log1p(x) to within a few ULP: 74 * 75 * u = 1+x; 76 * if(u==1.0) return x ; else 77 * return log(u)*(x/(u-1.0)); 78 * 79 * See HP-15C Advanced Functions Handbook, p.193. 80 */ 81 82 #include "math.h" 83 #include "math_private.h" 84 85 static const double 86 ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ 87 ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ 88 two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */ 89 Lp1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ 90 Lp2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ 91 Lp3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ 92 Lp4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ 93 Lp5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ 94 Lp6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ 95 Lp7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ 96 97 static const double zero = 0.0; 98 99 double 100 log1p(double x) 101 { 102 double hfsq,f,c,s,z,R,u; 103 int32_t k,hx,hu,ax; 104 105 GET_HIGH_WORD(hx,x); 106 ax = hx&0x7fffffff; 107 108 k = 1; 109 if (hx < 0x3FDA827A) { /* 1+x < sqrt(2)+ */ 110 if(ax>=0x3ff00000) { /* x <= -1.0 */ 111 if(x==-1.0) return -two54/zero; /* log1p(-1)=+inf */ 112 else return (x-x)/(x-x); /* log1p(x<-1)=NaN */ 113 } 114 if(ax<0x3e200000) { /* |x| < 2**-29 */ 115 if(two54+x>zero /* raise inexact */ 116 &&ax<0x3c900000) /* |x| < 2**-54 */ 117 return x; 118 else 119 return x - x*x*0.5; 120 } 121 if(hx>0||hx<=((int32_t)0xbfd2bec4)) { 122 k=0;f=x;hu=1;} /* sqrt(2)/2- <= 1+x < sqrt(2)+ */ 123 } 124 if (hx >= 0x7ff00000) return x+x; 125 if(k!=0) { 126 if(hx<0x43400000) { 127 u = 1.0+x; 128 GET_HIGH_WORD(hu,u); 129 k = (hu>>20)-1023; 130 c = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */ 131 c /= u; 132 } else { 133 u = x; 134 GET_HIGH_WORD(hu,u); 135 k = (hu>>20)-1023; 136 c = 0; 137 } 138 hu &= 0x000fffff; 139 /* 140 * The approximation to sqrt(2) used in thresholds is not 141 * critical. However, the ones used above must give less 142 * strict bounds than the one here so that the k==0 case is 143 * never reached from here, since here we have committed to 144 * using the correction term but don't use it if k==0. 145 */ 146 if(hu<0x6a09e) { /* u ~< sqrt(2) */ 147 SET_HIGH_WORD(u,hu|0x3ff00000); /* normalize u */ 148 } else { 149 k += 1; 150 SET_HIGH_WORD(u,hu|0x3fe00000); /* normalize u/2 */ 151 hu = (0x00100000-hu)>>2; 152 } 153 f = u-1.0; 154 } 155 hfsq=0.5*f*f; 156 if(hu==0) { /* |f| < 2**-20 */ 157 if(f==zero) if(k==0) return zero; 158 else {c += k*ln2_lo; return k*ln2_hi+c;} 159 R = hfsq*(1.0-0.66666666666666666*f); 160 if(k==0) return f-R; else 161 return k*ln2_hi-((R-(k*ln2_lo+c))-f); 162 } 163 s = f/(2.0+f); 164 z = s*s; 165 R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7)))))); 166 if(k==0) return f-(hfsq-s*(hfsq+R)); else 167 return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f); 168 } 169