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author | das <das@FreeBSD.org> | 2010-12-05 22:11:03 +0000 |
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committer | das <das@FreeBSD.org> | 2010-12-05 22:11:03 +0000 |
commit | 225ceb18e8a1729edd452bb883adfdb72d75c769 (patch) | |
tree | 194be6588ef6450b9c4dc71ae44ad99e14dec9ec | |
parent | cec2e8182c77939ee3d4ff9ae865a9f36d437c57 (diff) | |
download | FreeBSD-src-225ceb18e8a1729edd452bb883adfdb72d75c769.zip FreeBSD-src-225ceb18e8a1729edd452bb883adfdb72d75c769.tar.gz |
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).
-rw-r--r-- | lib/msun/src/k_log.h | 116 | ||||
-rw-r--r-- | lib/msun/src/k_logf.h | 55 |
2 files changed, 171 insertions, 0 deletions
diff --git a/lib/msun/src/k_log.h b/lib/msun/src/k_log.h new file mode 100644 index 0000000..206355c --- /dev/null +++ b/lib/msun/src/k_log.h @@ -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); + } +} diff --git a/lib/msun/src/k_logf.h b/lib/msun/src/k_logf.h new file mode 100644 index 0000000..d11a563 --- /dev/null +++ b/lib/msun/src/k_logf.h @@ -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); + } +} |