| /* @(#)s_log1p.c 5.1 93/09/24 */ |
| /* |
| * ==================================================== |
| * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
| * |
| * Developed at SunPro, a Sun Microsystems, Inc. business. |
| * Permission to use, copy, modify, and distribute this |
| * software is freely granted, provided that this notice |
| * is preserved. |
| * ==================================================== |
| */ |
| |
| /* double log1p(double x) |
| * |
| * Method : |
| * 1. Argument Reduction: find k and f such that |
| * 1+x = 2^k * (1+f), |
| * where sqrt(2)/2 < 1+f < sqrt(2) . |
| * |
| * Note. If k=0, then f=x is exact. However, if k!=0, then f |
| * may not be representable exactly. In that case, a correction |
| * term is need. Let u=1+x rounded. Let c = (1+x)-u, then |
| * log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u), |
| * and add back the correction term c/u. |
| * (Note: when x > 2**53, one can simply return log(x)) |
| * |
| * 2. Approximation of log1p(f). |
| * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) |
| * = 2s + 2/3 s**3 + 2/5 s**5 + ....., |
| * = 2s + s*R |
| * We use a special Reme algorithm on [0,0.1716] to generate |
| * a polynomial of degree 14 to approximate R The maximum error |
| * of this polynomial approximation is bounded by 2**-58.45. In |
| * other words, |
| * 2 4 6 8 10 12 14 |
| * R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s |
| * (the values of Lp1 to Lp7 are listed in the program) |
| * and |
| * | 2 14 | -58.45 |
| * | Lp1*s +...+Lp7*s - R(z) | <= 2 |
| * | | |
| * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. |
| * In order to guarantee error in log below 1ulp, we compute log |
| * by |
| * log1p(f) = f - (hfsq - s*(hfsq+R)). |
| * |
| * 3. Finally, log1p(x) = k*ln2 + log1p(f). |
| * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) |
| * Here ln2 is split into two floating point number: |
| * ln2_hi + ln2_lo, |
| * where n*ln2_hi is always exact for |n| < 2000. |
| * |
| * Special cases: |
| * log1p(x) is NaN with signal if x < -1 (including -INF) ; |
| * log1p(+INF) is +INF; log1p(-1) is -INF with signal; |
| * log1p(NaN) is that NaN with no signal. |
| * |
| * Accuracy: |
| * according to an error analysis, the error is always less than |
| * 1 ulp (unit in the last place). |
| * |
| * Constants: |
| * The hexadecimal values are the intended ones for the following |
| * constants. The decimal values may be used, provided that the |
| * compiler will convert from decimal to binary accurately enough |
| * to produce the hexadecimal values shown. |
| * |
| * Note: Assuming log() return accurate answer, the following |
| * algorithm can be used to compute log1p(x) to within a few ULP: |
| * |
| * u = 1+x; |
| * if(u==1.0) return x ; else |
| * return log(u)*(x/(u-1.0)); |
| * |
| * See HP-15C Advanced Functions Handbook, p.193. |
| */ |
| |
| #include <math.h> |
| #include "math_private.h" |
| |
| static const double |
| ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ |
| ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ |
| two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */ |
| Lp1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ |
| Lp2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ |
| Lp3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ |
| Lp4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ |
| Lp5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ |
| Lp6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ |
| Lp7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ |
| |
| static const double zero = 0.0; |
| |
| double |
| log1p(double x) |
| { |
| double hfsq,f=0,c=0,s,z,R,u; |
| int32_t k,hx,hu=0,ax; |
| |
| GET_HIGH_WORD(hx,x); |
| ax = hx&0x7fffffff; |
| |
| k = 1; |
| if (hx < 0x3FDA827A) { /* x < 0.41422 */ |
| if(ax>=0x3ff00000) { /* x <= -1.0 */ |
| if(x==-1.0) return -two54/zero; /* log1p(-1)=+inf */ |
| else return (x-x)/(x-x); /* log1p(x<-1)=NaN */ |
| } |
| if(ax<0x3e200000) { /* |x| < 2**-29 */ |
| if(two54+x>zero /* raise inexact */ |
| &&ax<0x3c900000) /* |x| < 2**-54 */ |
| return x; |
| else |
| return x - x*x*0.5; |
| } |
| if(hx>0||hx<=((int32_t)0xbfd2bec3)) { |
| k=0;f=x;hu=1;} /* -0.2929<x<0.41422 */ |
| } |
| if (hx >= 0x7ff00000) return x+x; |
| if(k!=0) { |
| if(hx<0x43400000) { |
| u = 1.0+x; |
| GET_HIGH_WORD(hu,u); |
| k = (hu>>20)-1023; |
| c = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */ |
| c /= u; |
| } else { |
| u = x; |
| GET_HIGH_WORD(hu,u); |
| k = (hu>>20)-1023; |
| c = 0; |
| } |
| hu &= 0x000fffff; |
| if(hu<0x6a09e) { |
| SET_HIGH_WORD(u,hu|0x3ff00000); /* normalize u */ |
| } else { |
| k += 1; |
| SET_HIGH_WORD(u,hu|0x3fe00000); /* normalize u/2 */ |
| hu = (0x00100000-hu)>>2; |
| } |
| f = u-1.0; |
| } |
| hfsq=0.5*f*f; |
| if(hu==0) { /* |f| < 2**-20 */ |
| if(f==zero) { if(k==0) return zero; |
| else {c += k*ln2_lo; return k*ln2_hi+c;} } |
| R = hfsq*(1.0-0.66666666666666666*f); |
| if(k==0) return f-R; else |
| return k*ln2_hi-((R-(k*ln2_lo+c))-f); |
| } |
| s = f/(2.0+f); |
| z = s*s; |
| R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7)))))); |
| if(k==0) return f-(hfsq-s*(hfsq+R)); else |
| return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f); |
| } |