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 #include <sys/cdefs.h> 13 /* double log1p(double x) 14 * 15 * Method : 16 * 1. Argument Reduction: find k and f such that 17 * 1+x = 2^k * (1+f), 18 * where sqrt(2)/2 < 1+f < sqrt(2) . 19 * 20 * Note. If k=0, then f=x is exact. However, if k!=0, then f 21 * may not be representable exactly. In that case, a correction 22 * term is need. Let u=1+x rounded. Let c = (1+x)-u, then 23 * log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u), 24 * and add back the correction term c/u. 25 * (Note: when x > 2**53, one can simply return log(x)) 26 * 27 * 2. Approximation of log1p(f). 28 * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) 29 * = 2s + 2/3 s**3 + 2/5 s**5 + ....., 30 * = 2s + s*R 31 * We use a special Reme algorithm on [0,0.1716] to generate 32 * a polynomial of degree 14 to approximate R The maximum error 33 * of this polynomial approximation is bounded by 2**-58.45. In 34 * other words, 35 * 2 4 6 8 10 12 14 36 * R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s 37 * (the values of Lp1 to Lp7 are listed in the program) 38 * and 39 * | 2 14 | -58.45 40 * | Lp1*s +...+Lp7*s - R(z) | <= 2 41 * | | 42 * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. 43 * In order to guarantee error in log below 1ulp, we compute log 44 * by 45 * log1p(f) = f - (hfsq - s*(hfsq+R)). 46 * 47 * 3. Finally, log1p(x) = k*ln2 + log1p(f). 48 * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) 49 * Here ln2 is split into two floating point number: 50 * ln2_hi + ln2_lo, 51 * where n*ln2_hi is always exact for |n| < 2000. 52 * 53 * Special cases: 54 * log1p(x) is NaN with signal if x < -1 (including -INF) ; 55 * log1p(+INF) is +INF; log1p(-1) is -INF with signal; 56 * log1p(NaN) is that NaN with no signal. 57 * 58 * Accuracy: 59 * according to an error analysis, the error is always less than 60 * 1 ulp (unit in the last place). 61 * 62 * Constants: 63 * The hexadecimal values are the intended ones for the following 64 * constants. The decimal values may be used, provided that the 65 * compiler will convert from decimal to binary accurately enough 66 * to produce the hexadecimal values shown. 67 * 68 * Note: Assuming log() return accurate answer, the following 69 * algorithm can be used to compute log1p(x) to within a few ULP: 70 * 71 * u = 1+x; 72 * if(u==1.0) return x ; else 73 * return log(u)*(x/(u-1.0)); 74 * 75 * See HP-15C Advanced Functions Handbook, p.193. 76 */ 77 78 #include <float.h> 79 80 #include "math.h" 81 #include "math_private.h" 82 83 static const double 84 ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ 85 ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ 86 two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */ 87 Lp1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ 88 Lp2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ 89 Lp3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ 90 Lp4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ 91 Lp5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ 92 Lp6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ 93 Lp7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ 94 95 static const double zero = 0.0; 96 static volatile double vzero = 0.0; 97 98 double 99 log1p(double x) 100 { 101 double hfsq,f,c,s,z,R,u; 102 int32_t k,hx,hu,ax; 103 104 GET_HIGH_WORD(hx,x); 105 ax = hx&0x7fffffff; 106 107 k = 1; 108 if (hx < 0x3FDA827A) { /* 1+x < sqrt(2)+ */ 109 if(ax>=0x3ff00000) { /* x <= -1.0 */ 110 if(x==-1.0) return -two54/vzero; /* log1p(-1)=+inf */ 111 else return (x-x)/(x-x); /* log1p(x<-1)=NaN */ 112 } 113 if(ax<0x3e200000) { /* |x| < 2**-29 */ 114 if(two54+x>zero /* raise inexact */ 115 &&ax<0x3c900000) /* |x| < 2**-54 */ 116 return x; 117 else 118 return x - x*x*0.5; 119 } 120 if(hx>0||hx<=((int32_t)0xbfd2bec4)) { 121 k=0;f=x;hu=1;} /* sqrt(2)/2- <= 1+x < sqrt(2)+ */ 122 } 123 if (hx >= 0x7ff00000) return x+x; 124 if(k!=0) { 125 if(hx<0x43400000) { 126 STRICT_ASSIGN(double,u,1.0+x); 127 GET_HIGH_WORD(hu,u); 128 k = (hu>>20)-1023; 129 c = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */ 130 c /= u; 131 } else { 132 u = x; 133 GET_HIGH_WORD(hu,u); 134 k = (hu>>20)-1023; 135 c = 0; 136 } 137 hu &= 0x000fffff; 138 /* 139 * The approximation to sqrt(2) used in thresholds is not 140 * critical. However, the ones used above must give less 141 * strict bounds than the one here so that the k==0 case is 142 * never reached from here, since here we have committed to 143 * using the correction term but don't use it if k==0. 144 */ 145 if(hu<0x6a09e) { /* u ~< sqrt(2) */ 146 SET_HIGH_WORD(u,hu|0x3ff00000); /* normalize u */ 147 } else { 148 k += 1; 149 SET_HIGH_WORD(u,hu|0x3fe00000); /* normalize u/2 */ 150 hu = (0x00100000-hu)>>2; 151 } 152 f = u-1.0; 153 } 154 hfsq=0.5*f*f; 155 if(hu==0) { /* |f| < 2**-20 */ 156 if(f==zero) { 157 if(k==0) { 158 return zero; 159 } else { 160 c += k*ln2_lo; 161 return k*ln2_hi+c; 162 } 163 } 164 R = hfsq*(1.0-0.66666666666666666*f); 165 if(k==0) return f-R; else 166 return k*ln2_hi-((R-(k*ln2_lo+c))-f); 167 } 168 s = f/(2.0+f); 169 z = s*s; 170 R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7)))))); 171 if(k==0) return f-(hfsq-s*(hfsq+R)); else 172 return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f); 173 } 174 175 #if (LDBL_MANT_DIG == 53) 176 __weak_reference(log1p, log1pl); 177 #endif 178