Date: Wed, 15 Dec 2021 18:38:36 GMT From: Mark Murray <markm@FreeBSD.org> To: src-committers@FreeBSD.org, dev-commits-src-all@FreeBSD.org, dev-commits-src-main@FreeBSD.org Subject: git: 455b2ccda3df - main - * lib/msun/Makefile: . Disconnect b_exp.c and b_log.c from the build. Message-ID: <202112151838.1BFIcabs072155@gitrepo.freebsd.org>
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The branch main has been updated by markm: URL: https://cgit.FreeBSD.org/src/commit/?id=455b2ccda3df35f31a167f8a35f11ec31fac89bc commit 455b2ccda3df35f31a167f8a35f11ec31fac89bc Author: Mark Murray <markm@FreeBSD.org> AuthorDate: 2021-12-14 09:02:45 +0000 Commit: Mark Murray <markm@FreeBSD.org> CommitDate: 2021-12-15 18:36:19 +0000 * lib/msun/Makefile: . Disconnect b_exp.c and b_log.c from the build. * lib/msun/bsdsrc/b_exp.c: . Replace scalb() usage with C99's ldexp(). . Replace finite(x) usage with C99's isfinite(). . Whitespace changes towards style(9). . Remove include of "mathimpl.h". It is no longer needed. . Remove #if 0 ... #endif code, which has been present since svn r93211 (2002-03-26). . New minimax polynomial coefficients. . Add comments to explain origins of some constants. . Use ansi-C prototype. Remove K&R prototype. Add static to prototype. * lib/msun/bsdsrc/b_log.c: . Remove include of "mathimpl.h". It is no longer needed. . Fix comments to actually describe the code. . Reduce minimax polynomial from degree 4 to degree 3. This uses newly computed coefficients. . Use ansi-C prototype. Remove K&R prototype. Add static to prototype. . Remove volatile in declaration of u1. . Alphabetize decalaration list. . Whitespace changes towards style(9). . In argument reduction of x to g and m, replace use of logb() and ldexp() with a single call to frexp(). Add code to get 1 <= g < 2. . Remove #if 0 ... #endif code, which has been present since svn r93211 (2002-03-26). . The special case m == -1022, replace logb() with ilogb(). * lib/msun/bsdsrc/b_tgamma.c: . Update comments. Fix comments where needed. . Add float.h to get LDBL_MANT_DIG for weak reference of tgammal to tgamma. . Remove include of "mathimpl.h". It is no longer needed. . Use "math.h" instead of <math.h>. . Add '#include math_private.h" . Add struct Double from mathimpl.h and include b_log.c and b_exp.c. . Remove forward declarations of neg_gam(), small_gam(), smaller_gam, large_gam() and ratfun_gam() by re-arranging the code to move these function above their first reference. . New minimax coefficients for polynomial in large_gam(). . New splitting of a0 into a0hi nd a0lo, which include additional bits of precision. . Use ansi-C prototype. Remove K&R prototype. . Replace the TRUNC() macro with a simple cast of a double entities to float before assignment (functional changes). . Replace sin(M_PI*z) with sinpi(z) and cos(M_PI*(0.5-z)) with cospi(0.5-z). Submitted by: Steve Kargl Differential Revision: https://reviews.freebsd.org/D33444 Reviewed by: pfg --- lib/msun/Makefile | 2 +- lib/msun/bsdsrc/b_exp.c | 143 +++++--------- lib/msun/bsdsrc/b_log.c | 166 +++++------------ lib/msun/bsdsrc/b_tgamma.c | 455 +++++++++++++++++++++++++++------------------ 4 files changed, 370 insertions(+), 396 deletions(-) diff --git a/lib/msun/Makefile b/lib/msun/Makefile index 6274c27c12c5..b8cf6662c492 100644 --- a/lib/msun/Makefile +++ b/lib/msun/Makefile @@ -59,7 +59,7 @@ SHLIBDIR?= /lib SHLIB_MAJOR= 5 WARNS?= 1 IGNORE_PRAGMA= -COMMON_SRCS= b_exp.c b_log.c b_tgamma.c \ +COMMON_SRCS= b_tgamma.c \ e_acos.c e_acosf.c e_acosh.c e_acoshf.c e_asin.c e_asinf.c \ e_atan2.c e_atan2f.c e_atanh.c e_atanhf.c e_cosh.c e_coshf.c e_exp.c \ e_expf.c e_fmod.c e_fmodf.c e_gamma.c e_gamma_r.c e_gammaf.c \ diff --git a/lib/msun/bsdsrc/b_exp.c b/lib/msun/bsdsrc/b_exp.c index 89a290529da9..c667293ed7c6 100644 --- a/lib/msun/bsdsrc/b_exp.c +++ b/lib/msun/bsdsrc/b_exp.c @@ -33,7 +33,6 @@ #include <sys/cdefs.h> __FBSDID("$FreeBSD$"); - /* EXP(X) * RETURN THE EXPONENTIAL OF X * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS) @@ -41,14 +40,14 @@ __FBSDID("$FreeBSD$"); * REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85, 6/14/86. * * Required system supported functions: - * scalb(x,n) + * ldexp(x,n) * copysign(x,y) - * finite(x) + * isfinite(x) * * Method: * 1. Argument Reduction: given the input x, find r and integer k such * that - * x = k*ln2 + r, |r| <= 0.5*ln2 . + * x = k*ln2 + r, |r| <= 0.5*ln2. * r will be represented as r := z+c for better accuracy. * * 2. Compute exp(r) by @@ -69,105 +68,59 @@ __FBSDID("$FreeBSD$"); * with 1,156,000 random arguments on a VAX, the maximum observed * error was 0.869 ulps (units in the last place). */ - -#include "mathimpl.h" - -static const double p1 = 0x1.555555555553ep-3; -static const double p2 = -0x1.6c16c16bebd93p-9; -static const double p3 = 0x1.1566aaf25de2cp-14; -static const double p4 = -0x1.bbd41c5d26bf1p-20; -static const double p5 = 0x1.6376972bea4d0p-25; -static const double ln2hi = 0x1.62e42fee00000p-1; -static const double ln2lo = 0x1.a39ef35793c76p-33; -static const double lnhuge = 0x1.6602b15b7ecf2p9; -static const double lntiny = -0x1.77af8ebeae354p9; -static const double invln2 = 0x1.71547652b82fep0; - -#if 0 -double exp(x) -double x; -{ - double z,hi,lo,c; - int k; - -#if !defined(vax)&&!defined(tahoe) - if(x!=x) return(x); /* x is NaN */ -#endif /* !defined(vax)&&!defined(tahoe) */ - if( x <= lnhuge ) { - if( x >= lntiny ) { - - /* argument reduction : x --> x - k*ln2 */ - - k=invln2*x+copysign(0.5,x); /* k=NINT(x/ln2) */ - - /* express x-k*ln2 as hi-lo and let x=hi-lo rounded */ - - hi=x-k*ln2hi; - x=hi-(lo=k*ln2lo); - - /* return 2^k*[1+x+x*c/(2+c)] */ - z=x*x; - c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5)))); - return scalb(1.0+(hi-(lo-(x*c)/(2.0-c))),k); - - } - /* end of x > lntiny */ - - else - /* exp(-big#) underflows to zero */ - if(finite(x)) return(scalb(1.0,-5000)); - - /* exp(-INF) is zero */ - else return(0.0); - } - /* end of x < lnhuge */ - - else - /* exp(INF) is INF, exp(+big#) overflows to INF */ - return( finite(x) ? scalb(1.0,5000) : x); -} -#endif +static const double + p1 = 1.6666666666666660e-01, /* 0x3fc55555, 0x55555553 */ + p2 = -2.7777777777564776e-03, /* 0xbf66c16c, 0x16c0ac3c */ + p3 = 6.6137564717940088e-05, /* 0x3f11566a, 0xb5c2ba0d */ + p4 = -1.6534060280704225e-06, /* 0xbebbbd53, 0x273e8fb7 */ + p5 = 4.1437773411069054e-08; /* 0x3e663f2a, 0x09c94b6c */ + +static const double + ln2hi = 0x1.62e42fee00000p-1, /* High 32 bits round-down. */ + ln2lo = 0x1.a39ef35793c76p-33; /* Next 53 bits round-to-nearst. */ + +static const double + lnhuge = 0x1.6602b15b7ecf2p9, /* (DBL_MAX_EXP + 9) * log(2.) */ + lntiny = -0x1.77af8ebeae354p9, /* (DBL_MIN_EXP - 53 - 10) * log(2.) */ + invln2 = 0x1.71547652b82fep0; /* 1 / log(2.) */ /* returns exp(r = x + c) for |c| < |x| with no overlap. */ -double __exp__D(x, c) -double x, c; +static double +__exp__D(double x, double c) { - double z,hi,lo; + double hi, lo, z; int k; - if (x != x) /* x is NaN */ + if (x != x) /* x is NaN. */ return(x); - if ( x <= lnhuge ) { - if ( x >= lntiny ) { - /* argument reduction : x --> x - k*ln2 */ - z = invln2*x; - k = z + copysign(.5, x); - - /* express (x+c)-k*ln2 as hi-lo and let x=hi-lo rounded */ - - hi=(x-k*ln2hi); /* Exact. */ - x= hi - (lo = k*ln2lo-c); - /* return 2^k*[1+x+x*c/(2+c)] */ - z=x*x; - c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5)))); - c = (x*c)/(2.0-c); - - return scalb(1.+(hi-(lo - c)), k); + if (x <= lnhuge) { + if (x >= lntiny) { + /* argument reduction: x --> x - k*ln2 */ + z = invln2 * x; + k = z + copysign(0.5, x); + + /* + * Express (x + c) - k * ln2 as hi - lo. + * Let x = hi - lo rounded. + */ + hi = x - k * ln2hi; /* Exact. */ + lo = k * ln2lo - c; + x = hi - lo; + + /* Return 2^k*[1+x+x*c/(2+c)] */ + z = x * x; + c = x - z * (p1 + z * (p2 + z * (p3 + z * (p4 + + z * p5)))); + c = (x * c) / (2 - c); + + return (ldexp(1 + (hi - (lo - c)), k)); + } else { + /* exp(-INF) is 0. exp(-big) underflows to 0. */ + return (isfinite(x) ? ldexp(1., -5000) : 0); } - /* end of x > lntiny */ - - else - /* exp(-big#) underflows to zero */ - if(finite(x)) return(scalb(1.0,-5000)); - - /* exp(-INF) is zero */ - else return(0.0); - } - /* end of x < lnhuge */ - - else + } else /* exp(INF) is INF, exp(+big#) overflows to INF */ - return( finite(x) ? scalb(1.0,5000) : x); + return (isfinite(x) ? ldexp(1., 5000) : x); } diff --git a/lib/msun/bsdsrc/b_log.c b/lib/msun/bsdsrc/b_log.c index c164dfa5014c..9d09ac754706 100644 --- a/lib/msun/bsdsrc/b_log.c +++ b/lib/msun/bsdsrc/b_log.c @@ -33,10 +33,6 @@ #include <sys/cdefs.h> __FBSDID("$FreeBSD$"); -#include <math.h> - -#include "mathimpl.h" - /* Table-driven natural logarithm. * * This code was derived, with minor modifications, from: @@ -44,25 +40,27 @@ __FBSDID("$FreeBSD$"); * Logarithm in IEEE Floating-Point arithmetic." ACM Trans. * Math Software, vol 16. no 4, pp 378-400, Dec 1990). * - * Calculates log(2^m*F*(1+f/F)), |f/j| <= 1/256, + * Calculates log(2^m*F*(1+f/F)), |f/F| <= 1/256, * where F = j/128 for j an integer in [0, 128]. * * log(2^m) = log2_hi*m + log2_tail*m - * since m is an integer, the dominant term is exact. + * The leading term is exact, because m is an integer, * m has at most 10 digits (for subnormal numbers), * and log2_hi has 11 trailing zero bits. * - * log(F) = logF_hi[j] + logF_lo[j] is in tabular form in log_table.h + * log(F) = logF_hi[j] + logF_lo[j] is in table below. * logF_hi[] + 512 is exact. * * log(1+f/F) = 2*f/(2*F + f) + 1/12 * (2*f/(2*F + f))**3 + ... - * the leading term is calculated to extra precision in two + * + * The leading term is calculated to extra precision in two * parts, the larger of which adds exactly to the dominant * m and F terms. + * * There are two cases: - * 1. when m, j are non-zero (m | j), use absolute + * 1. When m and j are non-zero (m | j), use absolute * precision for the leading term. - * 2. when m = j = 0, |1-x| < 1/256, and log(x) ~= (x-1). + * 2. When m = j = 0, |1-x| < 1/256, and log(x) ~= (x-1). * In this case, use a relative precision of 24 bits. * (This is done differently in the original paper) * @@ -70,11 +68,21 @@ __FBSDID("$FreeBSD$"); * 0 return signalling -Inf * neg return signalling NaN * +Inf return +Inf -*/ + */ #define N 128 -/* Table of log(Fj) = logF_head[j] + logF_tail[j], for Fj = 1+j/128. +/* + * Coefficients in the polynomial approximation of log(1+f/F). + * Domain of x is [0,1./256] with 2**(-64.187) precision. + */ +static const double + A1 = 8.3333333333333329e-02, /* 0x3fb55555, 0x55555555 */ + A2 = 1.2499999999943598e-02, /* 0x3f899999, 0x99991a98 */ + A3 = 2.2321527525957776e-03; /* 0x3f624929, 0xe24e70be */ + +/* + * Table of log(Fj) = logF_head[j] + logF_tail[j], for Fj = 1+j/128. * Used for generation of extend precision logarithms. * The constant 35184372088832 is 2^45, so the divide is exact. * It ensures correct reading of logF_head, even for inaccurate @@ -82,12 +90,7 @@ __FBSDID("$FreeBSD$"); * right answer for integers less than 2^53.) * Values for log(F) were generated using error < 10^-57 absolute * with the bc -l package. -*/ -static double A1 = .08333333333333178827; -static double A2 = .01250000000377174923; -static double A3 = .002232139987919447809; -static double A4 = .0004348877777076145742; - + */ static double logF_head[N+1] = { 0., .007782140442060381246, @@ -351,118 +354,51 @@ static double logF_tail[N+1] = { .00000000000025144230728376072, -.00000000000017239444525614834 }; - -#if 0 -double -#ifdef _ANSI_SOURCE -log(double x) -#else -log(x) double x; -#endif -{ - int m, j; - double F, f, g, q, u, u2, v, zero = 0.0, one = 1.0; - volatile double u1; - - /* Catch special cases */ - if (x <= 0) - if (x == zero) /* log(0) = -Inf */ - return (-one/zero); - else /* log(neg) = NaN */ - return (zero/zero); - else if (!finite(x)) - return (x+x); /* x = NaN, Inf */ - - /* Argument reduction: 1 <= g < 2; x/2^m = g; */ - /* y = F*(1 + f/F) for |f| <= 2^-8 */ - - m = logb(x); - g = ldexp(x, -m); - if (m == -1022) { - j = logb(g), m += j; - g = ldexp(g, -j); - } - j = N*(g-1) + .5; - F = (1.0/N) * j + 1; /* F*128 is an integer in [128, 512] */ - f = g - F; - - /* Approximate expansion for log(1+f/F) ~= u + q */ - g = 1/(2*F+f); - u = 2*f*g; - v = u*u; - q = u*v*(A1 + v*(A2 + v*(A3 + v*A4))); - - /* case 1: u1 = u rounded to 2^-43 absolute. Since u < 2^-8, - * u1 has at most 35 bits, and F*u1 is exact, as F has < 8 bits. - * It also adds exactly to |m*log2_hi + log_F_head[j] | < 750 - */ - if (m | j) - u1 = u + 513, u1 -= 513; - - /* case 2: |1-x| < 1/256. The m- and j- dependent terms are zero; - * u1 = u to 24 bits. - */ - else - u1 = u, TRUNC(u1); - u2 = (2.0*(f - F*u1) - u1*f) * g; - /* u1 + u2 = 2f/(2F+f) to extra precision. */ - - /* log(x) = log(2^m*F*(1+f/F)) = */ - /* (m*log2_hi+logF_head[j]+u1) + (m*log2_lo+logF_tail[j]+q); */ - /* (exact) + (tiny) */ - - u1 += m*logF_head[N] + logF_head[j]; /* exact */ - u2 = (u2 + logF_tail[j]) + q; /* tiny */ - u2 += logF_tail[N]*m; - return (u1 + u2); -} -#endif - /* * Extra precision variant, returning struct {double a, b;}; - * log(x) = a+b to 63 bits, with a rounded to 26 bits. + * log(x) = a+b to 63 bits, with 'a' rounded to 24 bits. */ -struct Double -#ifdef _ANSI_SOURCE +static struct Double __log__D(double x) -#else -__log__D(x) double x; -#endif { int m, j; - double F, f, g, q, u, v, u2; - volatile double u1; + double F, f, g, q, u, v, u1, u2; struct Double r; - /* Argument reduction: 1 <= g < 2; x/2^m = g; */ - /* y = F*(1 + f/F) for |f| <= 2^-8 */ - - m = logb(x); - g = ldexp(x, -m); + /* + * Argument reduction: 1 <= g < 2; x/2^m = g; + * y = F*(1 + f/F) for |f| <= 2^-8 + */ + g = frexp(x, &m); + g *= 2; + m--; if (m == -1022) { - j = logb(g), m += j; + j = ilogb(g); + m += j; g = ldexp(g, -j); } - j = N*(g-1) + .5; - F = (1.0/N) * j + 1; + j = N * (g - 1) + 0.5; + F = (1. / N) * j + 1; f = g - F; - g = 1/(2*F+f); - u = 2*f*g; - v = u*u; - q = u*v*(A1 + v*(A2 + v*(A3 + v*A4))); - if (m | j) - u1 = u + 513, u1 -= 513; - else - u1 = u, TRUNC(u1); - u2 = (2.0*(f - F*u1) - u1*f) * g; + g = 1 / (2 * F + f); + u = 2 * f * g; + v = u * u; + q = u * v * (A1 + v * (A2 + v * A3)); + if (m | j) { + u1 = u + 513; + u1 -= 513; + } else { + u1 = (float)u; + } + u2 = (2 * (f - F * u1) - u1 * f) * g; - u1 += m*logF_head[N] + logF_head[j]; + u1 += m * logF_head[N] + logF_head[j]; - u2 += logF_tail[j]; u2 += q; - u2 += logF_tail[N]*m; - r.a = u1 + u2; /* Only difference is here */ - TRUNC(r.a); + u2 += logF_tail[j]; + u2 += q; + u2 += logF_tail[N] * m; + r.a = (float)(u1 + u2); /* Only difference is here. */ r.b = (u1 - r.a) + u2; return (r); } diff --git a/lib/msun/bsdsrc/b_tgamma.c b/lib/msun/bsdsrc/b_tgamma.c index 5cb1f93f25ed..493ced3769c7 100644 --- a/lib/msun/bsdsrc/b_tgamma.c +++ b/lib/msun/bsdsrc/b_tgamma.c @@ -29,37 +29,46 @@ * SUCH DAMAGE. */ +/* + * The original code, FreeBSD's old svn r93211, contained the following + * attribution: + * + * This code by P. McIlroy, Oct 1992; + * + * The financial support of UUNET Communications Services is greatfully + * acknowledged. + * + * The algorithm remains, but the code has been re-arranged to facilitate + * porting to other precisions. + */ + /* @(#)gamma.c 8.1 (Berkeley) 6/4/93 */ #include <sys/cdefs.h> __FBSDID("$FreeBSD$"); -/* - * This code by P. McIlroy, Oct 1992; - * - * The financial support of UUNET Communications Services is greatfully - * acknowledged. - */ +#include <float.h> -#include <math.h> -#include "mathimpl.h" +#include "math.h" +#include "math_private.h" -/* METHOD: - * x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x)) - * At negative integers, return NaN and raise invalid. - * - * x < 6.5: - * Use argument reduction G(x+1) = xG(x) to reach the - * range [1.066124,2.066124]. Use a rational - * approximation centered at the minimum (x0+1) to - * ensure monotonicity. - * - * x >= 6.5: Use the asymptotic approximation (Stirling's formula) - * adjusted for equal-ripples: - * - * log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + 1/x*P(1/(x*x)) +/* Used in b_log.c and below. */ +struct Double { + double a; + double b; +}; + +#include "b_log.c" +#include "b_exp.c" + +/* + * The range is broken into several subranges. Each is handled by its + * helper functions. * - * Keep extra precision in multiplying (x-.5)(log(x)-1), to - * avoid premature round-off. + * x >= 6.0: large_gam(x) + * 6.0 > x >= xleft: small_gam(x) where xleft = 1 + left + x0. + * xleft > x > iota: smaller_gam(x) where iota = 1e-17. + * iota > x > -itoa: Handle x near 0. + * -iota > x : neg_gam * * Special values: * -Inf: return NaN and raise invalid; @@ -77,201 +86,224 @@ __FBSDID("$FreeBSD$"); * Maximum observed error < 4ulp in 1,000,000 trials. */ -static double neg_gam(double); -static double small_gam(double); -static double smaller_gam(double); -static struct Double large_gam(double); -static struct Double ratfun_gam(double, double); - -/* - * Rational approximation, A0 + x*x*P(x)/Q(x), on the interval - * [1.066.., 2.066..] accurate to 4.25e-19. - */ -#define LEFT -.3955078125 /* left boundary for rat. approx */ -#define x0 .461632144968362356785 /* xmin - 1 */ - -#define a0_hi 0.88560319441088874992 -#define a0_lo -.00000000000000004996427036469019695 -#define P0 6.21389571821820863029017800727e-01 -#define P1 2.65757198651533466104979197553e-01 -#define P2 5.53859446429917461063308081748e-03 -#define P3 1.38456698304096573887145282811e-03 -#define P4 2.40659950032711365819348969808e-03 -#define Q0 1.45019531250000000000000000000e+00 -#define Q1 1.06258521948016171343454061571e+00 -#define Q2 -2.07474561943859936441469926649e-01 -#define Q3 -1.46734131782005422506287573015e-01 -#define Q4 3.07878176156175520361557573779e-02 -#define Q5 5.12449347980666221336054633184e-03 -#define Q6 -1.76012741431666995019222898833e-03 -#define Q7 9.35021023573788935372153030556e-05 -#define Q8 6.13275507472443958924745652239e-06 /* * Constants for large x approximation (x in [6, Inf]) * (Accurate to 2.8*10^-19 absolute) */ -#define lns2pi_hi 0.418945312500000 -#define lns2pi_lo -.000006779295327258219670263595 -#define Pa0 8.33333333333333148296162562474e-02 -#define Pa1 -2.77777777774548123579378966497e-03 -#define Pa2 7.93650778754435631476282786423e-04 -#define Pa3 -5.95235082566672847950717262222e-04 -#define Pa4 8.41428560346653702135821806252e-04 -#define Pa5 -1.89773526463879200348872089421e-03 -#define Pa6 5.69394463439411649408050664078e-03 -#define Pa7 -1.44705562421428915453880392761e-02 - -static const double zero = 0., one = 1.0, tiny = 1e-300; -double -tgamma(x) - double x; +static const double zero = 0.; +static const volatile double tiny = 1e-300; +/* + * x >= 6 + * + * Use the asymptotic approximation (Stirling's formula) adjusted fof + * equal-ripples: + * + * log(G(x)) ~= (x-0.5)*(log(x)-1) + 0.5(log(2*pi)-1) + 1/x*P(1/(x*x)) + * + * Keep extra precision in multiplying (x-.5)(log(x)-1), to avoid + * premature round-off. + * + * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error. + */ +static const double + ln2pi_hi = 0.41894531250000000, + ln2pi_lo = -6.7792953272582197e-6, + Pa0 = 8.3333333333333329e-02, /* 0x3fb55555, 0x55555555 */ + Pa1 = -2.7777777777735404e-03, /* 0xbf66c16c, 0x16c145ec */ + Pa2 = 7.9365079044114095e-04, /* 0x3f4a01a0, 0x183de82d */ + Pa3 = -5.9523715464225254e-04, /* 0xbf438136, 0x0e681f62 */ + Pa4 = 8.4161391899445698e-04, /* 0x3f4b93f8, 0x21042a13 */ + Pa5 = -1.9065246069191080e-03, /* 0xbf5f3c8b, 0x357cb64e */ + Pa6 = 5.9047708485785158e-03, /* 0x3f782f99, 0xdaf5d65f */ + Pa7 = -1.6484018705183290e-02; /* 0xbf90e12f, 0xc4fb4df0 */ + +static struct Double +large_gam(double x) { + double p, z, thi, tlo, xhi, xlo; struct Double u; - if (x >= 6) { - if(x > 171.63) - return (x / zero); - u = large_gam(x); - return(__exp__D(u.a, u.b)); - } else if (x >= 1.0 + LEFT + x0) - return (small_gam(x)); - else if (x > 1.e-17) - return (smaller_gam(x)); - else if (x > -1.e-17) { - if (x != 0.0) - u.a = one - tiny; /* raise inexact */ - return (one/x); - } else if (!finite(x)) - return (x - x); /* x is NaN or -Inf */ - else - return (neg_gam(x)); + z = 1 / (x * x); + p = Pa0 + z * (Pa1 + z * (Pa2 + z * (Pa3 + z * (Pa4 + z * (Pa5 + + z * (Pa6 + z * Pa7)))))); + p = p / x; + + u = __log__D(x); + u.a -= 1; + + /* Split (x - 0.5) in high and low parts. */ + x -= 0.5; + xhi = (float)x; + xlo = x - xhi; + + /* Compute t = (x-.5)*(log(x)-1) in extra precision. */ + thi = xhi * u.a; + tlo = xlo * u.a + x * u.b; + + /* Compute thi + tlo + ln2pi_hi + ln2pi_lo + p. */ + tlo += ln2pi_lo; + tlo += p; + u.a = ln2pi_hi + tlo; + u.a += thi; + u.b = thi - u.a; + u.b += ln2pi_hi; + u.b += tlo; + return (u); } /* - * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error. + * Rational approximation, A0 + x * x * P(x) / Q(x), on the interval + * [1.066.., 2.066..] accurate to 4.25e-19. + * + * Returns r.a + r.b = a0 + (z + c)^2 * p / q, with r.a truncated. */ +static const double +#if 0 + a0_hi = 8.8560319441088875e-1, + a0_lo = -4.9964270364690197e-17, +#else + a0_hi = 8.8560319441088875e-01, /* 0x3fec56dc, 0x82a74aef */ + a0_lo = -4.9642368725563397e-17, /* 0xbc8c9deb, 0xaa64afc3 */ +#endif + P0 = 6.2138957182182086e-1, + P1 = 2.6575719865153347e-1, + P2 = 5.5385944642991746e-3, + P3 = 1.3845669830409657e-3, + P4 = 2.4065995003271137e-3, + Q0 = 1.4501953125000000e+0, + Q1 = 1.0625852194801617e+0, + Q2 = -2.0747456194385994e-1, + Q3 = -1.4673413178200542e-1, + Q4 = 3.0787817615617552e-2, + Q5 = 5.1244934798066622e-3, + Q6 = -1.7601274143166700e-3, + Q7 = 9.3502102357378894e-5, + Q8 = 6.1327550747244396e-6; + static struct Double -large_gam(x) - double x; +ratfun_gam(double z, double c) { - double z, p; - struct Double t, u, v; + double p, q, thi, tlo; + struct Double r; - z = one/(x*x); - p = Pa0+z*(Pa1+z*(Pa2+z*(Pa3+z*(Pa4+z*(Pa5+z*(Pa6+z*Pa7)))))); - p = p/x; + q = Q0 + z * (Q1 + z * (Q2 + z * (Q3 + z * (Q4 + z * (Q5 + + z * (Q6 + z * (Q7 + z * Q8))))))); + p = P0 + z * (P1 + z * (P2 + z * (P3 + z * P4))); + p = p / q; - u = __log__D(x); - u.a -= one; - v.a = (x -= .5); - TRUNC(v.a); - v.b = x - v.a; - t.a = v.a*u.a; /* t = (x-.5)*(log(x)-1) */ - t.b = v.b*u.a + x*u.b; - /* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */ - t.b += lns2pi_lo; t.b += p; - u.a = lns2pi_hi + t.b; u.a += t.a; - u.b = t.a - u.a; - u.b += lns2pi_hi; u.b += t.b; - return (u); + /* Split z into high and low parts. */ + thi = (float)z; + tlo = (z - thi) + c; + tlo *= (thi + z); + + /* Split (z+c)^2 into high and low parts. */ + thi *= thi; + q = thi; + thi = (float)thi; + tlo += (q - thi); + + /* Split p/q into high and low parts. */ + r.a = (float)p; + r.b = p - r.a; + + tlo = tlo * p + thi * r.b + a0_lo; + thi *= r.a; /* t = (z+c)^2*(P/Q) */ + r.a = (float)(thi + a0_hi); + r.b = ((a0_hi - r.a) + thi) + tlo; + return (r); /* r = a0 + t */ } /* + * x < 6 + * + * Use argument reduction G(x+1) = xG(x) to reach the range [1.066124, + * 2.066124]. Use a rational approximation centered at the minimum + * (x0+1) to ensure monotonicity. + * * Good to < 1 ulp. (provably .90 ulp; .87 ulp on 1,000,000 runs.) * It also has correct monotonicity. */ +static const double + left = -0.3955078125, /* left boundary for rat. approx */ + x0 = 4.6163214496836236e-1; /* xmin - 1 */ + static double -small_gam(x) - double x; +small_gam(double x) { - double y, ym1, t; + double t, y, ym1; struct Double yy, r; - y = x - one; - ym1 = y - one; - if (y <= 1.0 + (LEFT + x0)) { + + y = x - 1; + if (y <= 1 + (left + x0)) { yy = ratfun_gam(y - x0, 0); return (yy.a + yy.b); } - r.a = y; - TRUNC(r.a); - yy.a = r.a - one; - y = ym1; - yy.b = r.b = y - yy.a; + + r.a = (float)y; + yy.a = r.a - 1; + y = y - 1 ; + r.b = yy.b = y - yy.a; + /* Argument reduction: G(x+1) = x*G(x) */ - for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) { - t = r.a*yy.a; - r.b = r.a*yy.b + y*r.b; - r.a = t; - TRUNC(r.a); + for (ym1 = y - 1; ym1 > left + x0; y = ym1--, yy.a--) { + t = r.a * yy.a; + r.b = r.a * yy.b + y * r.b; + r.a = (float)t; r.b += (t - r.a); } + /* Return r*tgamma(y). */ yy = ratfun_gam(y - x0, 0); - y = r.b*(yy.a + yy.b) + r.a*yy.b; - y += yy.a*r.a; + y = r.b * (yy.a + yy.b) + r.a * yy.b; + y += yy.a * r.a; return (y); } /* - * Good on (0, 1+x0+LEFT]. Accurate to 1ulp. + * Good on (0, 1+x0+left]. Accurate to 1 ulp. */ static double -smaller_gam(x) - double x; +smaller_gam(double x) { - double t, d; - struct Double r, xx; - if (x < x0 + LEFT) { - t = x, TRUNC(t); - d = (t+x)*(x-t); + double d, rhi, rlo, t, xhi, xlo; + struct Double r; + + if (x < x0 + left) { + t = (float)x; + d = (t + x) * (x - t); t *= t; - xx.a = (t + x), TRUNC(xx.a); - xx.b = x - xx.a; xx.b += t; xx.b += d; - t = (one-x0); t += x; - d = (one-x0); d -= t; d += x; - x = xx.a + xx.b; + xhi = (float)(t + x); + xlo = x - xhi; + xlo += t; + xlo += d; + t = 1 - x0; + t += x; + d = 1 - x0; + d -= t; + d += x; + x = xhi + xlo; } else { - xx.a = x, TRUNC(xx.a); - xx.b = x - xx.a; + xhi = (float)x; + xlo = x - xhi; t = x - x0; - d = (-x0 -t); d += x; + d = - x0 - t; + d += x; } + r = ratfun_gam(t, d); - d = r.a/x, TRUNC(d); - r.a -= d*xx.a; r.a -= d*xx.b; r.a += r.b; - return (d + r.a/x); + d = (float)(r.a / x); + r.a -= d * xhi; + r.a -= d * xlo; + r.a += r.b; + + return (d + r.a / x); } /* - * returns (z+c)^2 * P(z)/Q(z) + a0 + * x < 0 + * + * Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x)). + * At negative integers, return NaN and raise invalid. */ -static struct Double -ratfun_gam(z, c) - double z, c; -{ - double p, q; - struct Double r, t; - - q = Q0 +z*(Q1+z*(Q2+z*(Q3+z*(Q4+z*(Q5+z*(Q6+z*(Q7+z*Q8))))))); - p = P0 + z*(P1 + z*(P2 + z*(P3 + z*P4))); - - /* return r.a + r.b = a0 + (z+c)^2*p/q, with r.a truncated to 26 bits. */ - p = p/q; - t.a = z, TRUNC(t.a); /* t ~= z + c */ - t.b = (z - t.a) + c; - t.b *= (t.a + z); - q = (t.a *= t.a); /* t = (z+c)^2 */ - TRUNC(t.a); - t.b += (q - t.a); - r.a = p, TRUNC(r.a); /* r = P/Q */ - r.b = p - r.a; - t.b = t.b*p + t.a*r.b + a0_lo; - t.a *= r.a; /* t = (z+c)^2*(P/Q) */ - r.a = t.a + a0_hi, TRUNC(r.a); - r.b = ((a0_hi-r.a) + t.a) + t.b; - return (r); /* r = a0 + t */ -} - static double -neg_gam(x) - double x; +neg_gam(double x) { int sgn = 1; struct Double lg, lsine; @@ -280,23 +312,29 @@ neg_gam(x) y = ceil(x); if (y == x) /* Negative integer. */ return ((x - x) / zero); + z = y - x; if (z > 0.5) - z = one - z; - y = 0.5 * y; + z = 1 - z; + + y = y / 2; if (y == ceil(y)) sgn = -1; - if (z < .25) - z = sin(M_PI*z); + + if (z < 0.25) + z = sinpi(z); else - z = cos(M_PI*(0.5-z)); + z = cospi(0.5 - z); + /* Special case: G(1-x) = Inf; G(x) may be nonzero. */ if (x < -170) { + if (x < -190) - return ((double)sgn*tiny*tiny); - y = one - x; /* exact: 128 < |x| < 255 */ + return (sgn * tiny * tiny); + + y = 1 - x; /* exact: 128 < |x| < 255 */ lg = large_gam(y); - lsine = __log__D(M_PI/z); /* = TRUNC(log(u)) + small */ + lsine = __log__D(M_PI / z); /* = TRUNC(log(u)) + small */ lg.a -= lsine.a; /* exact (opposite signs) */ lg.b -= lsine.b; y = -(lg.a + lg.b); @@ -305,11 +343,58 @@ neg_gam(x) if (sgn < 0) y = -y; return (y); *** 60 LINES SKIPPED ***
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