1 /* 2 * ==================================================== 3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 4 * 5 * Developed at SunPro, a Sun Microsystems, Inc. business. 6 * Permission to use, copy, modify, and distribute this 7 * software is freely granted, provided that this notice 8 * is preserved. 9 * ==================================================== 10 */ 11 12 /* double log1p(double x) 13 * 14 * Method : 15 * 1. Argument Reduction: find k and f such that 16 * 1+x = 2^k * (1+f), 17 * where sqrt(2)/2 < 1+f < sqrt(2) . 18 * 19 * Note. If k=0, then f=x is exact. However, if k!=0, then f 20 * may not be representable exactly. In that case, a correction 21 * term is need. Let u=1+x rounded. Let c = (1+x)-u, then 22 * log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u), 23 * and add back the correction term c/u. 24 * (Note: when x > 2**53, one can simply return log(x)) 25 * 26 * 2. Approximation of log1p(f). 27 * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) 28 * = 2s + 2/3 s**3 + 2/5 s**5 + ....., 29 * = 2s + s*R 30 * We use a special Reme algorithm on [0,0.1716] to generate 31 * a polynomial of degree 14 to approximate R The maximum error 32 * of this polynomial approximation is bounded by 2**-58.45. In 33 * other words, 34 * 2 4 6 8 10 12 14 35 * R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s 36 * (the values of Lp1 to Lp7 are listed in the program) 37 * and 38 * | 2 14 | -58.45 39 * | Lp1*s +...+Lp7*s - R(z) | <= 2 40 * | | 41 * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. 42 * In order to guarantee error in log below 1ulp, we compute log 43 * by 44 * log1p(f) = f - (hfsq - s*(hfsq+R)). 45 * 46 * 3. Finally, log1p(x) = k*ln2 + log1p(f). 47 * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) 48 * Here ln2 is split into two floating point number: 49 * ln2_hi + ln2_lo, 50 * where n*ln2_hi is always exact for |n| < 2000. 51 * 52 * Special cases: 53 * log1p(x) is NaN with signal if x < -1 (including -INF) ; 54 * log1p(+INF) is +INF; log1p(-1) is -INF with signal; 55 * log1p(NaN) is that NaN with no signal. 56 * 57 * Accuracy: 58 * according to an error analysis, the error is always less than 59 * 1 ulp (unit in the last place). 60 * 61 * Constants: 62 * The hexadecimal values are the intended ones for the following 63 * constants. The decimal values may be used, provided that the 64 * compiler will convert from decimal to binary accurately enough 65 * to produce the hexadecimal values shown. 66 * 67 * Note: Assuming log() return accurate answer, the following 68 * algorithm can be used to compute log1p(x) to within a few ULP: 69 * 70 * u = 1+x; 71 * if(u==1.0) return x ; else 72 * return log(u)*(x/(u-1.0)); 73 * 74 * See HP-15C Advanced Functions Handbook, p.193. 75 */ 76 77 #include <float.h> 78 79 #include "math.h" 80 #include "math_private.h" 81 82 static const double 83 ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ 84 ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ 85 two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */ 86 Lp1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ 87 Lp2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ 88 Lp3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ 89 Lp4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ 90 Lp5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ 91 Lp6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ 92 Lp7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ 93 94 static const double zero = 0.0; 95 static volatile double vzero = 0.0; 96 97 double 98 log1p(double x) 99 { 100 double hfsq,f,c,s,z,R,u; 101 int32_t k,hx,hu,ax; 102 103 GET_HIGH_WORD(hx,x); 104 ax = hx&0x7fffffff; 105 106 k = 1; 107 if (hx < 0x3FDA827A) { /* 1+x < sqrt(2)+ */ 108 if(ax>=0x3ff00000) { /* x <= -1.0 */ 109 if(x==-1.0) return -two54/vzero; /* log1p(-1)=+inf */ 110 else return (x-x)/(x-x); /* log1p(x<-1)=NaN */ 111 } 112 if(ax<0x3e200000) { /* |x| < 2**-29 */ 113 if(two54+x>zero /* raise inexact */ 114 &&ax<0x3c900000) /* |x| < 2**-54 */ 115 return x; 116 else 117 return x - x*x*0.5; 118 } 119 if(hx>0||hx<=((int32_t)0xbfd2bec4)) { 120 k=0;f=x;hu=1;} /* sqrt(2)/2- <= 1+x < sqrt(2)+ */ 121 } 122 if (hx >= 0x7ff00000) return x+x; 123 if(k!=0) { 124 if(hx<0x43400000) { 125 STRICT_ASSIGN(double,u,1.0+x); 126 GET_HIGH_WORD(hu,u); 127 k = (hu>>20)-1023; 128 c = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */ 129 c /= u; 130 } else { 131 u = x; 132 GET_HIGH_WORD(hu,u); 133 k = (hu>>20)-1023; 134 c = 0; 135 } 136 hu &= 0x000fffff; 137 /* 138 * The approximation to sqrt(2) used in thresholds is not 139 * critical. However, the ones used above must give less 140 * strict bounds than the one here so that the k==0 case is 141 * never reached from here, since here we have committed to 142 * using the correction term but don't use it if k==0. 143 */ 144 if(hu<0x6a09e) { /* u ~< sqrt(2) */ 145 SET_HIGH_WORD(u,hu|0x3ff00000); /* normalize u */ 146 } else { 147 k += 1; 148 SET_HIGH_WORD(u,hu|0x3fe00000); /* normalize u/2 */ 149 hu = (0x00100000-hu)>>2; 150 } 151 f = u-1.0; 152 } 153 hfsq=0.5*f*f; 154 if(hu==0) { /* |f| < 2**-20 */ 155 if(f==zero) { 156 if(k==0) { 157 return zero; 158 } else { 159 c += k*ln2_lo; 160 return k*ln2_hi+c; 161 } 162 } 163 R = hfsq*(1.0-0.66666666666666666*f); 164 if(k==0) return f-R; else 165 return k*ln2_hi-((R-(k*ln2_lo+c))-f); 166 } 167 s = f/(2.0+f); 168 z = s*s; 169 R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7)))))); 170 if(k==0) return f-(hfsq-s*(hfsq+R)); else 171 return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f); 172 } 173 174 #if (LDBL_MANT_DIG == 53) 175 __weak_reference(log1p, log1pl); 176 #endif 177