Add a "kernel" log function, based on e_log.c, which is useful for
implementing accurate logarithms in different bases. This is based on an approach bde coded up years ago. This function should always be inlined; it will be used in only a few places, and rudimentary tests show a 40% performance improvement in implementations of log2() and log10() on amd64. The kernel takes a reduced argument x and returns the same polynomial approximation as e_log.c, but omitting the low-order term. The low-order term is much larger than the rest of the approximation, so the caller of the kernel function can scale it to the appropriate base in extra precision and obtain a much more accurate answer than by using log(x)/log(b).
This commit is contained in:
parent
1c40e1f66c
commit
e7780530fa
Notes:
svn2git
2020-12-20 02:59:44 +00:00
svn path=/head/; revision=216210
116
lib/msun/src/k_log.h
Normal file
116
lib/msun/src/k_log.h
Normal file
@ -0,0 +1,116 @@
|
||||
|
||||
/* @(#)e_log.c 1.3 95/01/18 */
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunSoft, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
#include <sys/cdefs.h>
|
||||
__FBSDID("$FreeBSD$");
|
||||
|
||||
/* __kernel_log(x)
|
||||
* Return log(x) - (x-1) for x in ~[sqrt(2)/2, sqrt(2)].
|
||||
*
|
||||
* The following describes the overall strategy for computing
|
||||
* logarithms in base e. The argument reduction and adding the final
|
||||
* term of the polynomial are done by the caller for increased accuracy
|
||||
* when different bases are used.
|
||||
*
|
||||
* Method :
|
||||
* 1. Argument Reduction: find k and f such that
|
||||
* x = 2^k * (1+f),
|
||||
* where sqrt(2)/2 < 1+f < sqrt(2) .
|
||||
*
|
||||
* 2. Approximation of log(1+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) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
|
||||
* (the values of Lg1 to Lg7 are listed in the program)
|
||||
* and
|
||||
* | 2 14 | -58.45
|
||||
* | Lg1*s +...+Lg7*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
|
||||
* log(1+f) = f - s*(f - R) (if f is not too large)
|
||||
* log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
|
||||
*
|
||||
* 3. Finally, log(x) = k*ln2 + log(1+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:
|
||||
* log(x) is NaN with signal if x < 0 (including -INF) ;
|
||||
* log(+INF) is +INF; log(0) is -INF with signal;
|
||||
* log(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.
|
||||
*/
|
||||
|
||||
static const double
|
||||
Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
|
||||
Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
|
||||
Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
|
||||
Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
|
||||
Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
|
||||
Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
|
||||
Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
|
||||
|
||||
/*
|
||||
* We always inline __kernel_log(), since doing so produces a
|
||||
* substantial performance improvement (~40% on amd64).
|
||||
*/
|
||||
static inline double
|
||||
__kernel_log(double x)
|
||||
{
|
||||
double hfsq,f,s,z,R,w,t1,t2;
|
||||
int32_t hx,i,j;
|
||||
u_int32_t lx;
|
||||
|
||||
EXTRACT_WORDS(hx,lx,x);
|
||||
|
||||
f = x-1.0;
|
||||
if((0x000fffff&(2+hx))<3) { /* -2**-20 <= f < 2**-20 */
|
||||
if(f==0.0) return 0.0;
|
||||
return f*f*(0.33333333333333333*f-0.5);
|
||||
}
|
||||
s = f/(2.0+f);
|
||||
z = s*s;
|
||||
hx &= 0x000fffff;
|
||||
i = hx-0x6147a;
|
||||
w = z*z;
|
||||
j = 0x6b851-hx;
|
||||
t1= w*(Lg2+w*(Lg4+w*Lg6));
|
||||
t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
|
||||
i |= j;
|
||||
R = t2+t1;
|
||||
if (i>0) {
|
||||
hfsq=0.5*f*f;
|
||||
return s*(hfsq+R) - hfsq;
|
||||
} else {
|
||||
return s*(R-f);
|
||||
}
|
||||
}
|
55
lib/msun/src/k_logf.h
Normal file
55
lib/msun/src/k_logf.h
Normal file
@ -0,0 +1,55 @@
|
||||
/*
|
||||
* ====================================================
|
||||
* 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.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
#include <sys/cdefs.h>
|
||||
__FBSDID("$FreeBSD$");
|
||||
|
||||
/* __kernel_logf(x)
|
||||
* Return log(x) - (x-1) for x in ~[sqrt(2)/2, sqrt(2)].
|
||||
*/
|
||||
|
||||
static const float
|
||||
/* |(log(1+s)-log(1-s))/s - Lg(s)| < 2**-34.24 (~[-4.95e-11, 4.97e-11]). */
|
||||
Lg1 = 0xaaaaaa.0p-24, /* 0.66666662693 */
|
||||
Lg2 = 0xccce13.0p-25, /* 0.40000972152 */
|
||||
Lg3 = 0x91e9ee.0p-25, /* 0.28498786688 */
|
||||
Lg4 = 0xf89e26.0p-26; /* 0.24279078841 */
|
||||
|
||||
static inline float
|
||||
__kernel_logf(float x)
|
||||
{
|
||||
float hfsq,f,s,z,R,w,t1,t2;
|
||||
int32_t ix,i,j;
|
||||
|
||||
GET_FLOAT_WORD(ix,x);
|
||||
|
||||
f = x-(float)1.0;
|
||||
if((0x007fffff&(0x8000+ix))<0xc000) { /* -2**-9 <= f < 2**-9 */
|
||||
if(f==0.0) return 0.0;
|
||||
return f*f*((float)0.33333333333333333*f-(float)0.5);
|
||||
}
|
||||
s = f/((float)2.0+f);
|
||||
z = s*s;
|
||||
ix &= 0x007fffff;
|
||||
i = ix-(0x6147a<<3);
|
||||
w = z*z;
|
||||
j = (0x6b851<<3)-ix;
|
||||
t1= w*(Lg2+w*Lg4);
|
||||
t2= z*(Lg1+w*Lg3);
|
||||
i |= j;
|
||||
R = t2+t1;
|
||||
if(i>0) {
|
||||
hfsq=(float)0.5*f*f;
|
||||
return s*(hfsq+R) - hfsq;
|
||||
} else {
|
||||
return s*(R-f);
|
||||
}
|
||||
}
|
Loading…
Reference in New Issue
Block a user