0,0 → 1,217 |
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/* @(#)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. |
* ==================================================== |
*/ |
|
/* |
FUNCTION |
<<log1p>>, <<log1pf>>---log of <<1 + <[x]>>> |
|
INDEX |
log1p |
INDEX |
log1pf |
|
ANSI_SYNOPSIS |
#include <math.h> |
double log1p(double <[x]>); |
float log1pf(float <[x]>); |
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TRAD_SYNOPSIS |
#include <math.h> |
double log1p(<[x]>) |
double <[x]>; |
|
float log1pf(<[x]>) |
float <[x]>; |
|
DESCRIPTION |
<<log1p>> calculates |
@tex |
$ln(1+x)$, |
@end tex |
the natural logarithm of <<1+<[x]>>>. You can use <<log1p>> rather |
than `<<log(1+<[x]>)>>' for greater precision when <[x]> is very |
small. |
|
<<log1pf>> calculates the same thing, but accepts and returns |
<<float>> values rather than <<double>>. |
|
RETURNS |
<<log1p>> returns a <<double>>, the natural log of <<1+<[x]>>>. |
<<log1pf>> returns a <<float>>, the natural log of <<1+<[x]>>>. |
|
PORTABILITY |
Neither <<log1p>> nor <<log1pf>> is required by ANSI C or by the System V |
Interface Definition (Issue 2). |
|
*/ |
|
/* 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 "fdlibm.h" |
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#ifndef _DOUBLE_IS_32BITS |
|
#ifdef __STDC__ |
static const double |
#else |
static double |
#endif |
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 */ |
|
#ifdef __STDC__ |
static const double zero = 0.0; |
#else |
static double zero = 0.0; |
#endif |
|
#ifdef __STDC__ |
double log1p(double x) |
#else |
double log1p(x) |
double x; |
#endif |
{ |
double hfsq,f,c,s,z,R,u; |
__int32_t k,hx,hu,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); |
} |
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#endif /* _DOUBLE_IS_32BITS */ |