Author: das
Date: Sun Dec 5 22:11:03 2010
New Revision: 216210
URL: http://svn.freebsd.org/changeset/base/216210
Log:
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).
Added:
head/lib/msun/src/k_log.h
- copied, changed from r216174, head/lib/msun/src/e_log.c
head/lib/msun/src/k_logf.h
- copied, changed from r216174, head/lib/msun/src/e_logf.c
Copied and modified: head/lib/msun/src/k_log.h (from r216174,
head/lib/msun/src/e_log.c)
==============================================================================
--- head/lib/msun/src/e_log.c Sat Dec 4 02:42:52 2010 (r216174, copy
source)
+++ head/lib/msun/src/k_log.h Sun Dec 5 22:11:03 2010 (r216210)
@@ -14,8 +14,13 @@
#include <sys/cdefs.h>
__FBSDID("$FreeBSD$");
-/* __ieee754_log(x)
- * Return the logrithm of x
+/* __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
@@ -65,13 +70,7 @@ __FBSDID("$FreeBSD$");
* to produce the hexadecimal values shown.
*/
-#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 */
Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
@@ -80,48 +79,27 @@ Lg5 = 1.818357216161805012e-01, /* 3FC7
Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
-static const double zero = 0.0;
-
-double
-__ieee754_log(double x)
+/*
+ * 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,dk;
- int32_t k,hx,i,j;
+ double hfsq,f,s,z,R,w,t1,t2;
+ int32_t hx,i,j;
u_int32_t lx;
EXTRACT_WORDS(hx,lx,x);
- k=0;
- if (hx < 0x00100000) { /* x < 2**-1022 */
- if (((hx&0x7fffffff)|lx)==0)
- return -two54/zero; /* log(+-0)=-inf */
- if (hx<0) return (x-x)/zero; /* log(-#) = NaN */
- k -= 54; x *= two54; /* subnormal number, scale up x */
- GET_HIGH_WORD(hx,x);
- }
- if (hx >= 0x7ff00000) return x+x;
- k += (hx>>20)-1023;
- hx &= 0x000fffff;
- i = (hx+0x95f64)&0x100000;
- SET_HIGH_WORD(x,hx|(i^0x3ff00000)); /* normalize x or x/2 */
- k += (i>>20);
f = x-1.0;
if((0x000fffff&(2+hx))<3) { /* -2**-20 <= f < 2**-20 */
- if(f==zero) {
- if(k==0) {
- return zero;
- } else {
- dk=(double)k;
- return dk*ln2_hi+dk*ln2_lo;
- }
- }
- R = f*f*(0.5-0.33333333333333333*f);
- if(k==0) return f-R; else {dk=(double)k;
- return dk*ln2_hi-((R-dk*ln2_lo)-f);}
+ if(f==0.0) return 0.0;
+ return f*f*(0.33333333333333333*f-0.5);
}
s = f/(2.0+f);
- dk = (double)k;
z = s*s;
+ hx &= 0x000fffff;
i = hx-0x6147a;
w = z*z;
j = 0x6b851-hx;
@@ -129,12 +107,10 @@ __ieee754_log(double x)
t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
i |= j;
R = t2+t1;
- if(i>0) {
+ if (i>0) {
hfsq=0.5*f*f;
- if(k==0) return f-(hfsq-s*(hfsq+R)); else
- return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
+ return s*(hfsq+R) - hfsq;
} else {
- if(k==0) return f-s*(f-R); else
- return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
+ return s*(R-f);
}
}
Copied and modified: head/lib/msun/src/k_logf.h (from r216174,
head/lib/msun/src/e_logf.c)
==============================================================================
--- head/lib/msun/src/e_logf.c Sat Dec 4 02:42:52 2010 (r216174, copy
source)
+++ head/lib/msun/src/k_logf.h Sun Dec 5 22:11:03 2010 (r216210)
@@ -1,7 +1,3 @@
-/* e_logf.c -- float version of e_log.c.
- * Conversion to float by Ian Lance Taylor, Cygnus Support, i...@cygnus.com.
- */
-
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
@@ -16,60 +12,33 @@
#include <sys/cdefs.h>
__FBSDID("$FreeBSD$");
-#include "math.h"
-#include "math_private.h"
+/* __kernel_logf(x)
+ * Return log(x) - (x-1) for x in ~[sqrt(2)/2, sqrt(2)].
+ */
static const float
-ln2_hi = 6.9313812256e-01, /* 0x3f317180 */
-ln2_lo = 9.0580006145e-06, /* 0x3717f7d1 */
-two25 = 3.355443200e+07, /* 0x4c000000 */
/* |(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 const float zero = 0.0;
-
-float
-__ieee754_logf(float x)
+static inline float
+__kernel_logf(float x)
{
- float hfsq,f,s,z,R,w,t1,t2,dk;
- int32_t k,ix,i,j;
+ float hfsq,f,s,z,R,w,t1,t2;
+ int32_t ix,i,j;
GET_FLOAT_WORD(ix,x);
- k=0;
- if (ix < 0x00800000) { /* x < 2**-126 */
- if ((ix&0x7fffffff)==0)
- return -two25/zero; /* log(+-0)=-inf */
- if (ix<0) return (x-x)/zero; /* log(-#) = NaN */
- k -= 25; x *= two25; /* subnormal number, scale up x */
- GET_FLOAT_WORD(ix,x);
- }
- if (ix >= 0x7f800000) return x+x;
- k += (ix>>23)-127;
- ix &= 0x007fffff;
- i = (ix+(0x95f64<<3))&0x800000;
- SET_FLOAT_WORD(x,ix|(i^0x3f800000)); /* normalize x or x/2 */
- k += (i>>23);
f = x-(float)1.0;
if((0x007fffff&(0x8000+ix))<0xc000) { /* -2**-9 <= f < 2**-9 */
- if(f==zero) {
- if(k==0) {
- return zero;
- } else {
- dk=(float)k;
- return dk*ln2_hi+dk*ln2_lo;
- }
- }
- R = f*f*((float)0.5-(float)0.33333333333333333*f);
- if(k==0) return f-R; else {dk=(float)k;
- return dk*ln2_hi-((R-dk*ln2_lo)-f);}
+ if(f==0.0) return 0.0;
+ return f*f*((float)0.33333333333333333*f-(float)0.5);
}
s = f/((float)2.0+f);
- dk = (float)k;
z = s*s;
+ ix &= 0x007fffff;
i = ix-(0x6147a<<3);
w = z*z;
j = (0x6b851<<3)-ix;
@@ -79,10 +48,8 @@ __ieee754_logf(float x)
R = t2+t1;
if(i>0) {
hfsq=(float)0.5*f*f;
- if(k==0) return f-(hfsq-s*(hfsq+R)); else
- return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
+ return s*(hfsq+R) - hfsq;
} else {
- if(k==0) return f-s*(f-R); else
- return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
+ return s*(R-f);
}
}
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