Date: Fri, 19 Oct 2012 22:46:48 +0000 (UTC) From: Warner Losh <imp@FreeBSD.org> To: src-committers@freebsd.org, svn-src-all@freebsd.org, svn-src-head@freebsd.org Subject: svn commit: r241755 - head/lib/msun/src Message-ID: <201210192246.q9JMkm4R092929@svn.freebsd.org>
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Author: imp Date: Fri Oct 19 22:46:48 2012 New Revision: 241755 URL: http://svn.freebsd.org/changeset/base/241755 Log: Document the methods used to compute logf. Taken and edited from the double version, with adaptations for the differences between it and the float version. Modified: head/lib/msun/src/e_logf.c Modified: head/lib/msun/src/e_logf.c ============================================================================== --- head/lib/msun/src/e_logf.c Fri Oct 19 22:21:01 2012 (r241754) +++ head/lib/msun/src/e_logf.c Fri Oct 19 22:46:48 2012 (r241755) @@ -19,6 +19,57 @@ __FBSDID("$FreeBSD$"); #include "math.h" #include "math_private.h" +/* __ieee754_log(x) + * Return the logrithm of x + * + * 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 8 to approximate R The maximum error + * of this polynomial approximation is bounded by 2**-34.24. In + * other words, + * 2 4 6 8 + * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s + * (the values of Lg1 to Lg7 are listed in the program) + * and + * | 2 8 | -34.24 + * | Lg1*s +...+Lg4*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 float ln2_hi = 6.9313812256e-01, /* 0x3f317180 */ ln2_lo = 9.0580006145e-06, /* 0x3717f7d1 */
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