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 #include <sys/cdefs.h> 14 __FBSDID("$FreeBSD$"); 15 16 /* double log1p(double x) 17 * 18 * Method : 19 * 1. Argument Reduction: find k and f such that 20 * 1+x = 2^k * (1+f), 21 * where sqrt(2)/2 < 1+f < sqrt(2) . 22 * 23 * Note. If k=0, then f=x is exact. However, if k!=0, then f 24 * may not be representable exactly. In that case, a correction 25 * term is need. Let u=1+x rounded. Let c = (1+x)-u, then 26 * log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u), 27 * and add back the correction term c/u. 28 * (Note: when x > 2**53, one can simply return log(x)) 29 * 30 * 2. Approximation of log1p(f). 31 * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) 32 * = 2s + 2/3 s**3 + 2/5 s**5 + ....., 33 * = 2s + s*R 34 * We use a special Reme algorithm on [0,0.1716] to generate 35 * a polynomial of degree 14 to approximate R The maximum error 36 * of this polynomial approximation is bounded by 2**-58.45. In 37 * other words, 38 * 2 4 6 8 10 12 14 39 * R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s 40 * (the values of Lp1 to Lp7 are listed in the program) 41 * and 42 * | 2 14 | -58.45 43 * | Lp1*s +...+Lp7*s - R(z) | <= 2 44 * | | 45 * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. 46 * In order to guarantee error in log below 1ulp, we compute log 47 * by 48 * log1p(f) = f - (hfsq - s*(hfsq+R)). 49 * 50 * 3. Finally, log1p(x) = k*ln2 + log1p(f). 51 * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) 52 * Here ln2 is split into two floating point number: 53 * ln2_hi + ln2_lo, 54 * where n*ln2_hi is always exact for |n| < 2000. 55 * 56 * Special cases: 57 * log1p(x) is NaN with signal if x < -1 (including -INF) ; 58 * log1p(+INF) is +INF; log1p(-1) is -INF with signal; 59 * log1p(NaN) is that NaN with no signal. 60 * 61 * Accuracy: 62 * according to an error analysis, the error is always less than 63 * 1 ulp (unit in the last place). 64 * 65 * Constants: 66 * The hexadecimal values are the intended ones for the following 67 * constants. The decimal values may be used, provided that the 68 * compiler will convert from decimal to binary accurately enough 69 * to produce the hexadecimal values shown. 70 * 71 * Note: Assuming log() return accurate answer, the following 72 * algorithm can be used to compute log1p(x) to within a few ULP: 73 * 74 * u = 1+x; 75 * if(u==1.0) return x ; else 76 * return log(u)*(x/(u-1.0)); 77 * 78 * See HP-15C Advanced Functions Handbook, p.193. 79 */ 80 81 #include <float.h> 82 83 #include "math.h" 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) { /* 1+x < sqrt(2)+ */ 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)0xbfd2bec4)) { 123 k=0;f=x;hu=1;} /* sqrt(2)/2- <= 1+x < sqrt(2)+ */ 124 } 125 if (hx >= 0x7ff00000) return x+x; 126 if(k!=0) { 127 if(hx<0x43400000) { 128 STRICT_ASSIGN(double,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 /* 141 * The approximation to sqrt(2) used in thresholds is not 142 * critical. However, the ones used above must give less 143 * strict bounds than the one here so that the k==0 case is 144 * never reached from here, since here we have committed to 145 * using the correction term but don't use it if k==0. 146 */ 147 if(hu<0x6a09e) { /* u ~< sqrt(2) */ 148 SET_HIGH_WORD(u,hu|0x3ff00000); /* normalize u */ 149 } else { 150 k += 1; 151 SET_HIGH_WORD(u,hu|0x3fe00000); /* normalize u/2 */ 152 hu = (0x00100000-hu)>>2; 153 } 154 f = u-1.0; 155 } 156 hfsq=0.5*f*f; 157 if(hu==0) { /* |f| < 2**-20 */ 158 if(f==zero) { 159 if(k==0) { 160 return zero; 161 } else { 162 c += k*ln2_lo; 163 return k*ln2_hi+c; 164 } 165 } 166 R = hfsq*(1.0-0.66666666666666666*f); 167 if(k==0) return f-R; else 168 return k*ln2_hi-((R-(k*ln2_lo+c))-f); 169 } 170 s = f/(2.0+f); 171 z = s*s; 172 R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7)))))); 173 if(k==0) return f-(hfsq-s*(hfsq+R)); else 174 return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f); 175 } 176