mirror of
https://github.com/freebsd/freebsd-src
synced 2024-10-07 00:50:50 +00:00
* 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
This commit is contained in:
Mark Murray
parent
a46856c3f9
commit
455b2ccda3
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@ -59,7 +59,7 @@ SHLIBDIR?= /lib
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SHLIB_MAJOR= 5
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WARNS?= 1
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IGNORE_PRAGMA=
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COMMON_SRCS= b_exp.c b_log.c b_tgamma.c \
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COMMON_SRCS= b_tgamma.c \
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e_acos.c e_acosf.c e_acosh.c e_acoshf.c e_asin.c e_asinf.c \
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e_atan2.c e_atan2f.c e_atanh.c e_atanhf.c e_cosh.c e_coshf.c e_exp.c \
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e_expf.c e_fmod.c e_fmodf.c e_gamma.c e_gamma_r.c e_gammaf.c \
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@ -33,7 +33,6 @@
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#include <sys/cdefs.h>
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__FBSDID("$FreeBSD$");
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/* EXP(X)
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* RETURN THE EXPONENTIAL OF X
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* DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS)
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@ -41,14 +40,14 @@ __FBSDID("$FreeBSD$");
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* REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85, 6/14/86.
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*
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* Required system supported functions:
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* scalb(x,n)
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* ldexp(x,n)
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* copysign(x,y)
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* finite(x)
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* isfinite(x)
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*
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* Method:
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* 1. Argument Reduction: given the input x, find r and integer k such
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* that
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* x = k*ln2 + r, |r| <= 0.5*ln2 .
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* x = k*ln2 + r, |r| <= 0.5*ln2.
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* r will be represented as r := z+c for better accuracy.
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*
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* 2. Compute exp(r) by
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@ -69,105 +68,59 @@ __FBSDID("$FreeBSD$");
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* with 1,156,000 random arguments on a VAX, the maximum observed
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* error was 0.869 ulps (units in the last place).
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*/
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static const double
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p1 = 1.6666666666666660e-01, /* 0x3fc55555, 0x55555553 */
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p2 = -2.7777777777564776e-03, /* 0xbf66c16c, 0x16c0ac3c */
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p3 = 6.6137564717940088e-05, /* 0x3f11566a, 0xb5c2ba0d */
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p4 = -1.6534060280704225e-06, /* 0xbebbbd53, 0x273e8fb7 */
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p5 = 4.1437773411069054e-08; /* 0x3e663f2a, 0x09c94b6c */
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#include "mathimpl.h"
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static const double
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ln2hi = 0x1.62e42fee00000p-1, /* High 32 bits round-down. */
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ln2lo = 0x1.a39ef35793c76p-33; /* Next 53 bits round-to-nearst. */
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static const double p1 = 0x1.555555555553ep-3;
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static const double p2 = -0x1.6c16c16bebd93p-9;
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static const double p3 = 0x1.1566aaf25de2cp-14;
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static const double p4 = -0x1.bbd41c5d26bf1p-20;
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static const double p5 = 0x1.6376972bea4d0p-25;
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static const double ln2hi = 0x1.62e42fee00000p-1;
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static const double ln2lo = 0x1.a39ef35793c76p-33;
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static const double lnhuge = 0x1.6602b15b7ecf2p9;
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static const double lntiny = -0x1.77af8ebeae354p9;
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static const double invln2 = 0x1.71547652b82fep0;
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#if 0
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double exp(x)
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double x;
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{
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double z,hi,lo,c;
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int k;
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#if !defined(vax)&&!defined(tahoe)
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if(x!=x) return(x); /* x is NaN */
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#endif /* !defined(vax)&&!defined(tahoe) */
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if( x <= lnhuge ) {
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if( x >= lntiny ) {
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/* argument reduction : x --> x - k*ln2 */
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k=invln2*x+copysign(0.5,x); /* k=NINT(x/ln2) */
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/* express x-k*ln2 as hi-lo and let x=hi-lo rounded */
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hi=x-k*ln2hi;
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x=hi-(lo=k*ln2lo);
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/* return 2^k*[1+x+x*c/(2+c)] */
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z=x*x;
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c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5))));
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return scalb(1.0+(hi-(lo-(x*c)/(2.0-c))),k);
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}
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/* end of x > lntiny */
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else
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/* exp(-big#) underflows to zero */
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if(finite(x)) return(scalb(1.0,-5000));
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/* exp(-INF) is zero */
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else return(0.0);
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}
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/* end of x < lnhuge */
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else
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/* exp(INF) is INF, exp(+big#) overflows to INF */
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return( finite(x) ? scalb(1.0,5000) : x);
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}
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#endif
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static const double
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lnhuge = 0x1.6602b15b7ecf2p9, /* (DBL_MAX_EXP + 9) * log(2.) */
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lntiny = -0x1.77af8ebeae354p9, /* (DBL_MIN_EXP - 53 - 10) * log(2.) */
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invln2 = 0x1.71547652b82fep0; /* 1 / log(2.) */
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/* returns exp(r = x + c) for |c| < |x| with no overlap. */
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double __exp__D(x, c)
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double x, c;
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static double
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__exp__D(double x, double c)
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{
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double z,hi,lo;
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double hi, lo, z;
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int k;
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if (x != x) /* x is NaN */
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if (x != x) /* x is NaN. */
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return(x);
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if ( x <= lnhuge ) {
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if ( x >= lntiny ) {
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/* argument reduction : x --> x - k*ln2 */
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z = invln2*x;
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k = z + copysign(.5, x);
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if (x <= lnhuge) {
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if (x >= lntiny) {
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/* argument reduction: x --> x - k*ln2 */
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z = invln2 * x;
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k = z + copysign(0.5, x);
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/* express (x+c)-k*ln2 as hi-lo and let x=hi-lo rounded */
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/*
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* Express (x + c) - k * ln2 as hi - lo.
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* Let x = hi - lo rounded.
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*/
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hi = x - k * ln2hi; /* Exact. */
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lo = k * ln2lo - c;
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x = hi - lo;
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hi=(x-k*ln2hi); /* Exact. */
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x= hi - (lo = k*ln2lo-c);
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/* return 2^k*[1+x+x*c/(2+c)] */
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z=x*x;
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c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5))));
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c = (x*c)/(2.0-c);
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/* Return 2^k*[1+x+x*c/(2+c)] */
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z = x * x;
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c = x - z * (p1 + z * (p2 + z * (p3 + z * (p4 +
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z * p5))));
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c = (x * c) / (2 - c);
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return scalb(1.+(hi-(lo - c)), k);
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return (ldexp(1 + (hi - (lo - c)), k));
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} else {
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/* exp(-INF) is 0. exp(-big) underflows to 0. */
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return (isfinite(x) ? ldexp(1., -5000) : 0);
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}
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/* end of x > lntiny */
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else
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/* exp(-big#) underflows to zero */
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if(finite(x)) return(scalb(1.0,-5000));
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/* exp(-INF) is zero */
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else return(0.0);
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}
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/* end of x < lnhuge */
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else
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} else
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/* exp(INF) is INF, exp(+big#) overflows to INF */
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return( finite(x) ? scalb(1.0,5000) : x);
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return (isfinite(x) ? ldexp(1., 5000) : x);
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}
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@ -33,10 +33,6 @@
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#include <sys/cdefs.h>
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__FBSDID("$FreeBSD$");
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#include <math.h>
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#include "mathimpl.h"
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/* Table-driven natural logarithm.
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*
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* This code was derived, with minor modifications, from:
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@ -44,25 +40,27 @@ __FBSDID("$FreeBSD$");
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* Logarithm in IEEE Floating-Point arithmetic." ACM Trans.
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* Math Software, vol 16. no 4, pp 378-400, Dec 1990).
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*
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* Calculates log(2^m*F*(1+f/F)), |f/j| <= 1/256,
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* Calculates log(2^m*F*(1+f/F)), |f/F| <= 1/256,
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* where F = j/128 for j an integer in [0, 128].
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*
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* log(2^m) = log2_hi*m + log2_tail*m
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* since m is an integer, the dominant term is exact.
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* The leading term is exact, because m is an integer,
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* m has at most 10 digits (for subnormal numbers),
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* and log2_hi has 11 trailing zero bits.
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*
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* log(F) = logF_hi[j] + logF_lo[j] is in tabular form in log_table.h
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* log(F) = logF_hi[j] + logF_lo[j] is in table below.
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* logF_hi[] + 512 is exact.
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*
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* log(1+f/F) = 2*f/(2*F + f) + 1/12 * (2*f/(2*F + f))**3 + ...
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* the leading term is calculated to extra precision in two
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*
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* The leading term is calculated to extra precision in two
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* parts, the larger of which adds exactly to the dominant
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* m and F terms.
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*
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* There are two cases:
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* 1. when m, j are non-zero (m | j), use absolute
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* 1. When m and j are non-zero (m | j), use absolute
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* precision for the leading term.
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* 2. when m = j = 0, |1-x| < 1/256, and log(x) ~= (x-1).
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* 2. When m = j = 0, |1-x| < 1/256, and log(x) ~= (x-1).
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* In this case, use a relative precision of 24 bits.
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* (This is done differently in the original paper)
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*
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* 0 return signalling -Inf
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* neg return signalling NaN
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* +Inf return +Inf
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*/
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*/
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#define N 128
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/* Table of log(Fj) = logF_head[j] + logF_tail[j], for Fj = 1+j/128.
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/*
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* Coefficients in the polynomial approximation of log(1+f/F).
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* Domain of x is [0,1./256] with 2**(-64.187) precision.
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*/
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static const double
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A1 = 8.3333333333333329e-02, /* 0x3fb55555, 0x55555555 */
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A2 = 1.2499999999943598e-02, /* 0x3f899999, 0x99991a98 */
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A3 = 2.2321527525957776e-03; /* 0x3f624929, 0xe24e70be */
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/*
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* Table of log(Fj) = logF_head[j] + logF_tail[j], for Fj = 1+j/128.
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* Used for generation of extend precision logarithms.
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* The constant 35184372088832 is 2^45, so the divide is exact.
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* It ensures correct reading of logF_head, even for inaccurate
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@ -82,12 +90,7 @@ __FBSDID("$FreeBSD$");
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* right answer for integers less than 2^53.)
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* Values for log(F) were generated using error < 10^-57 absolute
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* with the bc -l package.
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*/
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static double A1 = .08333333333333178827;
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static double A2 = .01250000000377174923;
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static double A3 = .002232139987919447809;
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static double A4 = .0004348877777076145742;
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*/
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static double logF_head[N+1] = {
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0.,
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.007782140442060381246,
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@ -351,118 +354,51 @@ static double logF_tail[N+1] = {
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.00000000000025144230728376072,
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-.00000000000017239444525614834
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};
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#if 0
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double
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#ifdef _ANSI_SOURCE
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log(double x)
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#else
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log(x) double x;
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#endif
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{
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int m, j;
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double F, f, g, q, u, u2, v, zero = 0.0, one = 1.0;
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volatile double u1;
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/* Catch special cases */
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if (x <= 0)
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if (x == zero) /* log(0) = -Inf */
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return (-one/zero);
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else /* log(neg) = NaN */
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return (zero/zero);
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else if (!finite(x))
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return (x+x); /* x = NaN, Inf */
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/* Argument reduction: 1 <= g < 2; x/2^m = g; */
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/* y = F*(1 + f/F) for |f| <= 2^-8 */
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m = logb(x);
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g = ldexp(x, -m);
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if (m == -1022) {
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j = logb(g), m += j;
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g = ldexp(g, -j);
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}
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j = N*(g-1) + .5;
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F = (1.0/N) * j + 1; /* F*128 is an integer in [128, 512] */
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f = g - F;
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/* Approximate expansion for log(1+f/F) ~= u + q */
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g = 1/(2*F+f);
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u = 2*f*g;
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v = u*u;
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q = u*v*(A1 + v*(A2 + v*(A3 + v*A4)));
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/* case 1: u1 = u rounded to 2^-43 absolute. Since u < 2^-8,
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* u1 has at most 35 bits, and F*u1 is exact, as F has < 8 bits.
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* It also adds exactly to |m*log2_hi + log_F_head[j] | < 750
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*/
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if (m | j)
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u1 = u + 513, u1 -= 513;
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/* case 2: |1-x| < 1/256. The m- and j- dependent terms are zero;
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* u1 = u to 24 bits.
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*/
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else
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u1 = u, TRUNC(u1);
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u2 = (2.0*(f - F*u1) - u1*f) * g;
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/* u1 + u2 = 2f/(2F+f) to extra precision. */
|
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/* log(x) = log(2^m*F*(1+f/F)) = */
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/* (m*log2_hi+logF_head[j]+u1) + (m*log2_lo+logF_tail[j]+q); */
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/* (exact) + (tiny) */
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u1 += m*logF_head[N] + logF_head[j]; /* exact */
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u2 = (u2 + logF_tail[j]) + q; /* tiny */
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u2 += logF_tail[N]*m;
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return (u1 + u2);
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}
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#endif
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|
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/*
|
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* Extra precision variant, returning struct {double a, b;};
|
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* log(x) = a+b to 63 bits, with a rounded to 26 bits.
|
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* log(x) = a+b to 63 bits, with 'a' rounded to 24 bits.
|
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*/
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struct Double
|
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#ifdef _ANSI_SOURCE
|
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static struct Double
|
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__log__D(double x)
|
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#else
|
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__log__D(x) double x;
|
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#endif
|
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{
|
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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;
|
||||
|
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/* 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
|
||||
*/
|
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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);
|
||||
}
|
||||
|
|
|
|||
|
|
@ -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("$FreeBSD$");
|
||||
|
||||
#include <float.h>
|
||||
|
||||
#include "math.h"
|
||||
#include "math_private.h"
|
||||
|
||||
/* Used in b_log.c and below. */
|
||||
struct Double {
|
||||
double a;
|
||||
double b;
|
||||
};
|
||||
|
||||
#include "b_log.c"
|
||||
#include "b_exp.c"
|
||||
|
||||
/*
|
||||
* This code by P. McIlroy, Oct 1992;
|
||||
* The range is broken into several subranges. Each is handled by its
|
||||
* helper functions.
|
||||
*
|
||||
* The financial support of UUNET Communications Services is greatfully
|
||||
* acknowledged.
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "mathimpl.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))
|
||||
*
|
||||
* 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;
|
||||
{
|
||||
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));
|
||||
}
|
||||
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 struct Double
|
||||
large_gam(x)
|
||||
double x;
|
||||
{
|
||||
double z, p;
|
||||
struct Double t, u, v;
|
||||
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 */
|
||||
|
||||
z = one/(x*x);
|
||||
p = Pa0+z*(Pa1+z*(Pa2+z*(Pa3+z*(Pa4+z*(Pa5+z*(Pa6+z*Pa7))))));
|
||||
p = p/x;
|
||||
static struct Double
|
||||
large_gam(double x)
|
||||
{
|
||||
double p, z, thi, tlo, xhi, xlo;
|
||||
struct Double u;
|
||||
|
||||
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 -= 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;
|
||||
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);
|
||||
}
|
||||
/*
|
||||
* 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
|
||||
ratfun_gam(double z, double c)
|
||||
{
|
||||
double p, q, thi, tlo;
|
||||
struct Double r;
|
||||
|
||||
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;
|
||||
|
||||
/* 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);
|
||||
}
|
||||
y = one-x;
|
||||
if (one-y == x)
|
||||
|
||||
y = 1 - x;
|
||||
if (1 - y == x)
|
||||
y = tgamma(y);
|
||||
else /* 1-x is inexact */
|
||||
y = -x*tgamma(-x);
|
||||
y = - x * tgamma(-x);
|
||||
|
||||
if (sgn < 0) y = -y;
|
||||
return (M_PI / (y*z));
|
||||
return (M_PI / (y * z));
|
||||
}
|
||||
/*
|
||||
* xmax comes from lgamma(xmax) - emax * log(2) = 0.
|
||||
* static const float xmax = 35.040095f
|
||||
* static const double xmax = 171.624376956302725;
|
||||
* ld80: LD80C(0xdb718c066b352e20, 10, 1.75554834290446291689e+03L),
|
||||
* ld128: 1.75554834290446291700388921607020320e+03L,
|
||||
*
|
||||
* iota is a sloppy threshold to isolate x = 0.
|
||||
*/
|
||||
static const double xmax = 171.624376956302725;
|
||||
static const double iota = 0x1p-56;
|
||||
|
||||
double
|
||||
tgamma(double x)
|
||||
{
|
||||
struct Double u;
|
||||
|
||||
if (x >= 6) {
|
||||
if (x > xmax)
|
||||
return (x / zero);
|
||||
u = large_gam(x);
|
||||
return (__exp__D(u.a, u.b));
|
||||
}
|
||||
|
||||
if (x >= 1 + left + x0)
|
||||
return (small_gam(x));
|
||||
|
||||
if (x > iota)
|
||||
return (smaller_gam(x));
|
||||
|
||||
if (x > -iota) {
|
||||
if (x != 0.)
|
||||
u.a = 1 - tiny; /* raise inexact */
|
||||
return (1 / x);
|
||||
}
|
||||
|
||||
if (!isfinite(x))
|
||||
return (x - x); /* x is NaN or -Inf */
|
||||
|
||||
return (neg_gam(x));
|
||||
}
|
||||
|
||||
#if (LDBL_MANT_DIG == 53)
|
||||
__weak_reference(tgamma, tgammal);
|
||||
#endif
|
||||
|
|
|
|||
Loading…
Reference in a new issue