| 1 | /* @(#)s_log1p.c 5.1 93/09/24 */
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| 2 | /*
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| 3 | * ====================================================
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| 4 | * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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| 5 | *
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| 6 | * Developed at SunPro, a Sun Microsystems, Inc. business.
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| 7 | * Permission to use, copy, modify, and distribute this
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| 8 | * software is freely granted, provided that this notice
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| 9 | * is preserved.
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| 10 | * ====================================================
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| 11 | */
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| 12 |
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| 13 | #ifndef lint
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| 14 | static char rcsid[] = "$FreeBSD: src/lib/msun/src/s_log1p.c,v 1.7 2002/05/28 18:15:04 alfred Exp $";
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| 15 | #endif
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| 16 |
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| 17 | /* double log1p(double x)
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| 18 | *
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| 19 | * Method :
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| 20 | * 1. Argument Reduction: find k and f such that
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| 21 | * 1+x = 2^k * (1+f),
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| 22 | * where sqrt(2)/2 < 1+f < sqrt(2) .
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| 23 | *
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| 24 | * Note. If k=0, then f=x is exact. However, if k!=0, then f
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| 25 | * may not be representable exactly. In that case, a correction
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| 26 | * term is need. Let u=1+x rounded. Let c = (1+x)-u, then
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| 27 | * log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
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| 28 | * and add back the correction term c/u.
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| 29 | * (Note: when x > 2**53, one can simply return log(x))
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| 30 | *
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| 31 | * 2. Approximation of log1p(f).
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| 32 | * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
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| 33 | * = 2s + 2/3 s**3 + 2/5 s**5 + .....,
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| 34 | * = 2s + s*R
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| 35 | * We use a special Reme algorithm on [0,0.1716] to generate
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| 36 | * a polynomial of degree 14 to approximate R The maximum error
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| 37 | * of this polynomial approximation is bounded by 2**-58.45. In
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| 38 | * other words,
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| 39 | * 2 4 6 8 10 12 14
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| 40 | * R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s
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| 41 | * (the values of Lp1 to Lp7 are listed in the program)
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| 42 | * and
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| 43 | * | 2 14 | -58.45
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| 44 | * | Lp1*s +...+Lp7*s - R(z) | <= 2
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| 45 | * | |
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| 46 | * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
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| 47 | * In order to guarantee error in log below 1ulp, we compute log
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| 48 | * by
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| 49 | * log1p(f) = f - (hfsq - s*(hfsq+R)).
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| 50 | *
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| 51 | * 3. Finally, log1p(x) = k*ln2 + log1p(f).
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| 52 | * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
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| 53 | * Here ln2 is split into two floating point number:
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| 54 | * ln2_hi + ln2_lo,
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| 55 | * where n*ln2_hi is always exact for |n| < 2000.
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| 56 | *
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| 57 | * Special cases:
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| 58 | * log1p(x) is NaN with signal if x < -1 (including -INF) ;
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| 59 | * log1p(+INF) is +INF; log1p(-1) is -INF with signal;
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| 60 | * log1p(NaN) is that NaN with no signal.
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| 61 | *
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| 62 | * Accuracy:
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| 63 | * according to an error analysis, the error is always less than
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| 64 | * 1 ulp (unit in the last place).
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| 65 | *
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| 66 | * Constants:
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| 67 | * The hexadecimal values are the intended ones for the following
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| 68 | * constants. The decimal values may be used, provided that the
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| 69 | * compiler will convert from decimal to binary accurately enough
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| 70 | * to produce the hexadecimal values shown.
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| 71 | *
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| 72 | * Note: Assuming log() return accurate answer, the following
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| 73 | * algorithm can be used to compute log1p(x) to within a few ULP:
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| 74 | *
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| 75 | * u = 1+x;
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| 76 | * if(u==1.0) return x ; else
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| 77 | * return log(u)*(x/(u-1.0));
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| 78 | *
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| 79 | * See HP-15C Advanced Functions Handbook, p.193.
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| 80 | */
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| 81 |
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| 82 | #include "math.h"
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| 83 | #include "math_private.h"
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| 84 |
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| 85 | static const double
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| 86 | ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
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| 87 | ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
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| 88 | two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
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| 89 | Lp1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
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| 90 | Lp2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
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| 91 | Lp3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
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| 92 | Lp4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
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| 93 | Lp5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
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| 94 | Lp6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
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| 95 | Lp7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
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| 96 |
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| 97 | static const double zero = 0.0;
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| 98 |
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| 99 | double
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| 100 | log1p(double x)
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| 101 | {
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| 102 | double hfsq,f,c,s,z,R,u;
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| 103 | int32_t k,hx,hu,ax;
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| 104 |
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| 105 | GET_HIGH_WORD(hx,x);
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| 106 | ax = hx&0x7fffffff;
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| 107 |
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| 108 | k = 1;
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| 109 | if (hx < 0x3FDA827A) { /* x < 0.41422 */
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| 110 | if(ax>=0x3ff00000) { /* x <= -1.0 */
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| 111 | if(x==-1.0) return -two54/zero; /* log1p(-1)=+inf */
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| 112 | else return (x-x)/(x-x); /* log1p(x<-1)=NaN */
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| 113 | }
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| 114 | if(ax<0x3e200000) { /* |x| < 2**-29 */
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| 115 | if(two54+x>zero /* raise inexact */
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| 116 | &&ax<0x3c900000) /* |x| < 2**-54 */
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| 117 | return x;
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| 118 | else
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| 119 | return x - x*x*0.5;
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| 120 | }
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| 121 | if(hx>0||hx<=((int32_t)0xbfd2bec3)) {
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| 122 | k=0;f=x;hu=1;} /* -0.2929<x<0.41422 */
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| 123 | }
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| 124 | if (hx >= 0x7ff00000) return x+x;
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| 125 | if(k!=0) {
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| 126 | if(hx<0x43400000) {
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| 127 | u = 1.0+x;
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| 128 | GET_HIGH_WORD(hu,u);
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| 129 | k = (hu>>20)-1023;
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| 130 | c = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */
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| 131 | c /= u;
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| 132 | } else {
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| 133 | u = x;
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| 134 | GET_HIGH_WORD(hu,u);
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| 135 | k = (hu>>20)-1023;
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| 136 | c = 0;
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| 137 | }
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| 138 | hu &= 0x000fffff;
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| 139 | if(hu<0x6a09e) {
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| 140 | SET_HIGH_WORD(u,hu|0x3ff00000); /* normalize u */
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| 141 | } else {
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| 142 | k += 1;
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| 143 | SET_HIGH_WORD(u,hu|0x3fe00000); /* normalize u/2 */
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| 144 | hu = (0x00100000-hu)>>2;
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| 145 | }
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| 146 | f = u-1.0;
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| 147 | }
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| 148 | hfsq=0.5*f*f;
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| 149 | if(hu==0) { /* |f| < 2**-20 */
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| 150 | if(f==zero) if(k==0) return zero;
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| 151 | else {c += k*ln2_lo; return k*ln2_hi+c;}
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| 152 | R = hfsq*(1.0-0.66666666666666666*f);
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| 153 | if(k==0) return f-R; else
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| 154 | return k*ln2_hi-((R-(k*ln2_lo+c))-f);
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| 155 | }
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| 156 | s = f/(2.0+f);
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| 157 | z = s*s;
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| 158 | R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7))))));
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| 159 | if(k==0) return f-(hfsq-s*(hfsq+R)); else
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| 160 | return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f);
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| 161 | }
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