libm: sync with upstream.
There's potential here to maybe lose some/all of builtins.cpp, but I'll look at that separately later. Test: treehugger Change-Id: I2c2bc1d0753affdd214daeb09fa1ac7cd73db347
This commit is contained in:
Elliott Hughes
parent
c79b02088b
commit
99ef447d0f
33 changed files with 1065 additions and 618 deletions
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@ -30,8 +30,6 @@ cc_library {
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whole_static_libs: ["libarm-optimized-routines-math"],
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srcs: [
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"upstream-freebsd/lib/msun/bsdsrc/b_exp.c",
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"upstream-freebsd/lib/msun/bsdsrc/b_log.c",
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"upstream-freebsd/lib/msun/bsdsrc/b_tgamma.c",
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"upstream-freebsd/lib/msun/src/catrig.c",
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"upstream-freebsd/lib/msun/src/catrigf.c",
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@ -112,6 +110,7 @@ cc_library {
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"upstream-freebsd/lib/msun/src/s_copysign.c",
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"upstream-freebsd/lib/msun/src/s_copysignf.c",
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"upstream-freebsd/lib/msun/src/s_cos.c",
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"upstream-freebsd/lib/msun/src/s_cospi.c",
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"upstream-freebsd/lib/msun/src/s_cpow.c",
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"upstream-freebsd/lib/msun/src/s_cpowf.c",
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"upstream-freebsd/lib/msun/src/s_cpowl.c",
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@ -177,6 +176,7 @@ cc_library {
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"upstream-freebsd/lib/msun/src/s_significand.c",
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"upstream-freebsd/lib/msun/src/s_significandf.c",
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"upstream-freebsd/lib/msun/src/s_sin.c",
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"upstream-freebsd/lib/msun/src/s_sinpi.c",
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"upstream-freebsd/lib/msun/src/s_sincos.c",
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"upstream-freebsd/lib/msun/src/s_tan.c",
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"upstream-freebsd/lib/msun/src/s_tanf.c",
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34
libm/NOTICE
34
libm/NOTICE
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@ -689,6 +689,14 @@ SUCH DAMAGE.
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-------------------------------------------------------------------
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Copyright (c) 2005-2020 Rich Felker, et al.
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Please see https://git.musl-libc.org/cgit/musl/tree/COPYRIGHT
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for all contributors to musl.
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-------------------------------------------------------------------
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Copyright (c) 2007 David Schultz
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All rights reserved.
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@ -1173,6 +1181,32 @@ SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
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-------------------------------------------------------------------
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Copyright (c) 2017 Steven G. Kargl
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All rights reserved.
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Redistribution and use in source and binary forms, with or without
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modification, are permitted provided that the following conditions
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are met:
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1. Redistributions of source code must retain the above copyright
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notice unmodified, this list of conditions, and the following
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disclaimer.
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2. Redistributions in binary form must reproduce the above copyright
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notice, this list of conditions and the following disclaimer in the
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documentation and/or other materials provided with the distribution.
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THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
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IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
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OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
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IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
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INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
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NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
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DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
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THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
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(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
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THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
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-------------------------------------------------------------------
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From: @(#)s_ilogb.c 5.1 93/09/24
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====================================================
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Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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@ -39,3 +39,7 @@ int digittoint(char ch);
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// Similarly rename _scan_nan.
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#define _scan_nan __libm_scan_nan
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// FreeBSD exports these in <math.h> but we don't.
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double cospi(double);
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double sinpi(double);
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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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@ -70,11 +68,21 @@ __FBSDID("$FreeBSD$");
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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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*/
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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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|
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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;
|
||||
volatile double u1;
|
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|
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/* Catch special cases */
|
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if (x <= 0)
|
||||
if (x == zero) /* log(0) = -Inf */
|
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return (-one/zero);
|
||||
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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|
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/* Argument reduction: 1 <= g < 2; x/2^m = g; */
|
||||
/* 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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|
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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;
|
||||
q = u*v*(A1 + v*(A2 + v*(A3 + v*A4)));
|
||||
|
||||
/* 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;
|
||||
* u1 = u to 24 bits.
|
||||
*/
|
||||
else
|
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u1 = u, TRUNC(u1);
|
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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)) = */
|
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/* (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);
|
||||
}
|
||||
|
|
|
|||
|
|
@ -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
|
||||
|
|
|
|||
|
|
@ -21,8 +21,8 @@ __FBSDID("$FreeBSD$");
|
|||
#include "math_private.h"
|
||||
|
||||
/*
|
||||
* Domain [-0.7854, 0.7854], range ~[-1.80e-37, 1.79e-37]:
|
||||
* |cos(x) - c(x))| < 2**-122.0
|
||||
* Domain [-0.7854, 0.7854], range ~[-1.17e-39, 1.19e-39]:
|
||||
* |cos(x) - c(x))| < 2**-129.3
|
||||
*
|
||||
* 113-bit precision requires more care than 64-bit precision, since
|
||||
* simple methods give a minimax polynomial with coefficient for x^2
|
||||
|
|
@ -31,21 +31,19 @@ __FBSDID("$FreeBSD$");
|
|||
*/
|
||||
static const double
|
||||
one = 1.0;
|
||||
|
||||
static const long double
|
||||
C1 = 0.04166666666666666666666666666666658424671L,
|
||||
C2 = -0.001388888888888888888888888888863490893732L,
|
||||
C3 = 0.00002480158730158730158730158600795304914210L,
|
||||
C4 = -0.2755731922398589065255474947078934284324e-6L,
|
||||
C5 = 0.2087675698786809897659225313136400793948e-8L,
|
||||
C6 = -0.1147074559772972315817149986812031204775e-10L,
|
||||
C7 = 0.4779477332386808976875457937252120293400e-13L;
|
||||
|
||||
static const double
|
||||
C8 = -0.1561920696721507929516718307820958119868e-15,
|
||||
C9 = 0.4110317413744594971475941557607804508039e-18,
|
||||
C10 = -0.8896592467191938803288521958313920156409e-21,
|
||||
C11 = 0.1601061435794535138244346256065192782581e-23;
|
||||
C1 = 4.16666666666666666666666666666666667e-02L,
|
||||
C2 = -1.38888888888888888888888888888888834e-03L,
|
||||
C3 = 2.48015873015873015873015873015446795e-05L,
|
||||
C4 = -2.75573192239858906525573190949988493e-07L,
|
||||
C5 = 2.08767569878680989792098886701451072e-09L,
|
||||
C6 = -1.14707455977297247136657111139971865e-11L,
|
||||
C7 = 4.77947733238738518870113294139830239e-14L,
|
||||
C8 = -1.56192069685858079920640872925306403e-16L,
|
||||
C9 = 4.11031762320473354032038893429515732e-19L,
|
||||
C10= -8.89679121027589608738005163931958096e-22L,
|
||||
C11= 1.61171797801314301767074036661901531e-24L,
|
||||
C12= -2.46748624357670948912574279501044295e-27L;
|
||||
|
||||
long double
|
||||
__kernel_cosl(long double x, long double y)
|
||||
|
|
@ -54,7 +52,7 @@ __kernel_cosl(long double x, long double y)
|
|||
|
||||
z = x*x;
|
||||
r = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*(C6+z*(C7+
|
||||
z*(C8+z*(C9+z*(C10+z*C11))))))))));
|
||||
z*(C8+z*(C9+z*(C10+z*(C11+z*C12)))))))))));
|
||||
hz = 0.5*z;
|
||||
w = one-hz;
|
||||
return w + (((one-w)-hz) + (z*r-x*y));
|
||||
|
|
|
|||
|
|
@ -697,14 +697,15 @@ invln10_hi = 4.3429448176175356e-1, /* 0x1bcb7b15000000.0p-54 */
|
|||
invln2_hi = 1.4426950402557850e0; /* 0x17154765000000.0p-52 */
|
||||
static const long double
|
||||
invln10_lo = 1.41498268538580090791605082294397000e-10L, /* 0x137287195355baaafad33dc323ee3.0p-145L */
|
||||
invln2_lo = 6.33178418956604368501892137426645911e-10L; /* 0x15c17f0bbbe87fed0691d3e88eb57.0p-143L */
|
||||
invln2_lo = 6.33178418956604368501892137426645911e-10L, /* 0x15c17f0bbbe87fed0691d3e88eb57.0p-143L */
|
||||
invln10_lo_plus_hi = invln10_lo + invln10_hi,
|
||||
invln2_lo_plus_hi = invln2_lo + invln2_hi;
|
||||
|
||||
long double
|
||||
log10l(long double x)
|
||||
{
|
||||
struct ld r;
|
||||
long double lo;
|
||||
float hi;
|
||||
long double hi, lo;
|
||||
|
||||
ENTERI();
|
||||
DOPRINT_START(&x);
|
||||
|
|
@ -712,18 +713,17 @@ log10l(long double x)
|
|||
if (!r.lo_set)
|
||||
RETURNPI(r.hi);
|
||||
_2sumF(r.hi, r.lo);
|
||||
hi = r.hi;
|
||||
hi = (float)r.hi;
|
||||
lo = r.lo + (r.hi - hi);
|
||||
RETURN2PI(invln10_hi * hi,
|
||||
(invln10_lo + invln10_hi) * lo + invln10_lo * hi);
|
||||
invln10_lo_plus_hi * lo + invln10_lo * hi);
|
||||
}
|
||||
|
||||
long double
|
||||
log2l(long double x)
|
||||
{
|
||||
struct ld r;
|
||||
long double lo;
|
||||
float hi;
|
||||
long double hi, lo;
|
||||
|
||||
ENTERI();
|
||||
DOPRINT_START(&x);
|
||||
|
|
@ -731,10 +731,10 @@ log2l(long double x)
|
|||
if (!r.lo_set)
|
||||
RETURNPI(r.hi);
|
||||
_2sumF(r.hi, r.lo);
|
||||
hi = r.hi;
|
||||
hi = (float)r.hi;
|
||||
lo = r.lo + (r.hi - hi);
|
||||
RETURN2PI(invln2_hi * hi,
|
||||
(invln2_lo + invln2_hi) * lo + invln2_lo * hi);
|
||||
invln2_lo_plus_hi * lo + invln2_lo * hi);
|
||||
}
|
||||
|
||||
#endif /* STRUCT_RETURN */
|
||||
|
|
|
|||
|
|
@ -82,7 +82,7 @@ hypotl(long double x, long double y)
|
|||
man_t manh, manl;
|
||||
GET_LDBL_MAN(manh,manl,b);
|
||||
if((manh|manl)==0) return a;
|
||||
t1=0;
|
||||
t1=1;
|
||||
SET_HIGH_WORD(t1,ESW(MAX_EXP-2)); /* t1=2^(MAX_EXP-2) */
|
||||
b *= t1;
|
||||
a *= t1;
|
||||
|
|
|
|||
|
|
@ -136,7 +136,7 @@ __ieee754_powf(float x, float y)
|
|||
/* |y| is huge */
|
||||
if(iy>0x4d000000) { /* if |y| > 2**27 */
|
||||
/* over/underflow if x is not close to one */
|
||||
if(ix<0x3f7ffff8) return (hy<0)? sn*huge*huge:sn*tiny*tiny;
|
||||
if(ix<0x3f7ffff6) return (hy<0)? sn*huge*huge:sn*tiny*tiny;
|
||||
if(ix>0x3f800007) return (hy>0)? sn*huge*huge:sn*tiny*tiny;
|
||||
/* now |1-x| is tiny <= 2**-20, suffice to compute
|
||||
log(x) by x-x^2/2+x^3/3-x^4/4 */
|
||||
|
|
|
|||
|
|
@ -14,6 +14,18 @@
|
|||
#include <sys/cdefs.h>
|
||||
__FBSDID("$FreeBSD$");
|
||||
|
||||
#include <float.h>
|
||||
|
||||
#include "math.h"
|
||||
#include "math_private.h"
|
||||
|
||||
#ifdef USE_BUILTIN_SQRT
|
||||
double
|
||||
__ieee754_sqrt(double x)
|
||||
{
|
||||
return (__builtin_sqrt(x));
|
||||
}
|
||||
#else
|
||||
/* __ieee754_sqrt(x)
|
||||
* Return correctly rounded sqrt.
|
||||
* ------------------------------------------
|
||||
|
|
@ -84,11 +96,6 @@ __FBSDID("$FreeBSD$");
|
|||
*---------------
|
||||
*/
|
||||
|
||||
#include <float.h>
|
||||
|
||||
#include "math.h"
|
||||
#include "math_private.h"
|
||||
|
||||
static const double one = 1.0, tiny=1.0e-300;
|
||||
|
||||
double
|
||||
|
|
@ -187,6 +194,7 @@ __ieee754_sqrt(double x)
|
|||
INSERT_WORDS(z,ix0,ix1);
|
||||
return z;
|
||||
}
|
||||
#endif
|
||||
|
||||
#if (LDBL_MANT_DIG == 53)
|
||||
__weak_reference(sqrt, sqrtl);
|
||||
|
|
|
|||
|
|
@ -20,6 +20,13 @@ static char rcsid[] = "$FreeBSD$";
|
|||
#include "math.h"
|
||||
#include "math_private.h"
|
||||
|
||||
#ifdef USE_BUILTIN_SQRTF
|
||||
float
|
||||
__ieee754_sqrtf(float x)
|
||||
{
|
||||
return (__builtin_sqrtf(x));
|
||||
}
|
||||
#else
|
||||
static const float one = 1.0, tiny=1.0e-30;
|
||||
|
||||
float
|
||||
|
|
@ -87,3 +94,4 @@ __ieee754_sqrtf(float x)
|
|||
SET_FLOAT_WORD(z,ix);
|
||||
return z;
|
||||
}
|
||||
#endif
|
||||
|
|
|
|||
44
libm/upstream-freebsd/lib/msun/src/k_cospi.h
Normal file
44
libm/upstream-freebsd/lib/msun/src/k_cospi.h
Normal file
|
|
@ -0,0 +1,44 @@
|
|||
/*-
|
||||
* Copyright (c) 2017 Steven G. Kargl
|
||||
* All rights reserved.
|
||||
*
|
||||
* Redistribution and use in source and binary forms, with or without
|
||||
* modification, are permitted provided that the following conditions
|
||||
* are met:
|
||||
* 1. Redistributions of source code must retain the above copyright
|
||||
* notice unmodified, this list of conditions, and the following
|
||||
* disclaimer.
|
||||
* 2. Redistributions in binary form must reproduce the above copyright
|
||||
* notice, this list of conditions and the following disclaimer in the
|
||||
* documentation and/or other materials provided with the distribution.
|
||||
*
|
||||
* THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
|
||||
* IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
|
||||
* OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
|
||||
* IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
|
||||
* INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
|
||||
* NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
|
||||
* DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
|
||||
* THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
|
||||
* (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
|
||||
* THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
|
||||
*/
|
||||
|
||||
/*
|
||||
* The basic kernel for x in [0,0.25]. To use the kernel for cos(x), the
|
||||
* argument to __kernel_cospi() must be multiplied by pi.
|
||||
*/
|
||||
|
||||
static inline double
|
||||
__kernel_cospi(double x)
|
||||
{
|
||||
double_t hi, lo;
|
||||
|
||||
hi = (float)x;
|
||||
lo = x - hi;
|
||||
lo = lo * (pi_lo + pi_hi) + hi * pi_lo;
|
||||
hi *= pi_hi;
|
||||
_2sumF(hi, lo);
|
||||
return (__kernel_cos(hi, lo));
|
||||
}
|
||||
|
||||
|
|
@ -76,13 +76,6 @@ __kernel_sincosl(long double x, long double y, int iy, long double *sn,
|
|||
#elif LDBL_MANT_DIG == 113 /* ld128 version of k_sincosl.c. */
|
||||
|
||||
static const long double
|
||||
C1 = 0.04166666666666666666666666666666658424671L,
|
||||
C2 = -0.001388888888888888888888888888863490893732L,
|
||||
C3 = 0.00002480158730158730158730158600795304914210L,
|
||||
C4 = -0.2755731922398589065255474947078934284324e-6L,
|
||||
C5 = 0.2087675698786809897659225313136400793948e-8L,
|
||||
C6 = -0.1147074559772972315817149986812031204775e-10L,
|
||||
C7 = 0.4779477332386808976875457937252120293400e-13L,
|
||||
S1 = -0.16666666666666666666666666666666666606732416116558L,
|
||||
S2 = 0.0083333333333333333333333333333331135404851288270047L,
|
||||
S3 = -0.00019841269841269841269841269839935785325638310428717L,
|
||||
|
|
@ -93,15 +86,25 @@ S7 = -0.76471637318198151807063387954939213287488216303768e-12L,
|
|||
S8 = 0.28114572543451292625024967174638477283187397621303e-14L;
|
||||
|
||||
static const double
|
||||
C8 = -0.1561920696721507929516718307820958119868e-15,
|
||||
C9 = 0.4110317413744594971475941557607804508039e-18,
|
||||
C10 = -0.8896592467191938803288521958313920156409e-21,
|
||||
C11 = 0.1601061435794535138244346256065192782581e-23,
|
||||
S9 = -0.82206352458348947812512122163446202498005154296863e-17,
|
||||
S10 = 0.19572940011906109418080609928334380560135358385256e-19,
|
||||
S11 = -0.38680813379701966970673724299207480965452616911420e-22,
|
||||
S12 = 0.64038150078671872796678569586315881020659912139412e-25;
|
||||
|
||||
static const long double
|
||||
C1 = 4.16666666666666666666666666666666667e-02L,
|
||||
C2 = -1.38888888888888888888888888888888834e-03L,
|
||||
C3 = 2.48015873015873015873015873015446795e-05L,
|
||||
C4 = -2.75573192239858906525573190949988493e-07L,
|
||||
C5 = 2.08767569878680989792098886701451072e-09L,
|
||||
C6 = -1.14707455977297247136657111139971865e-11L,
|
||||
C7 = 4.77947733238738518870113294139830239e-14L,
|
||||
C8 = -1.56192069685858079920640872925306403e-16L,
|
||||
C9 = 4.11031762320473354032038893429515732e-19L,
|
||||
C10= -8.89679121027589608738005163931958096e-22L,
|
||||
C11= 1.61171797801314301767074036661901531e-24L,
|
||||
C12= -2.46748624357670948912574279501044295e-27L;
|
||||
|
||||
static inline void
|
||||
__kernel_sincosl(long double x, long double y, int iy, long double *sn,
|
||||
long double *cs)
|
||||
|
|
@ -120,12 +123,12 @@ __kernel_sincosl(long double x, long double y, int iy, long double *sn,
|
|||
if (iy == 0)
|
||||
*sn = x + v * (S1 + z * r);
|
||||
else
|
||||
*cs = x - ((z * (y / 2 - v * r) - y) - v * S1);
|
||||
*sn = x - ((z * (y / 2 - v * r) - y) - v * S1);
|
||||
|
||||
hz = z / 2;
|
||||
w = 1 - hz;
|
||||
r = z * (C1 + z * (C2 + z * (C3 + z * (C4 + z * (C5 + z * (C6 +
|
||||
z * (C7 + z * (C8 + z * (C9 + z * (C10 + z * C11))))))))));
|
||||
z * (C7 + z * (C8 + z * (C9 + z * (C10 + z * (C11+z*C12)))))))))));
|
||||
|
||||
*cs = w + (((1 - w) - hz) + (z * r - x * y));
|
||||
}
|
||||
|
|
|
|||
43
libm/upstream-freebsd/lib/msun/src/k_sinpi.h
Normal file
43
libm/upstream-freebsd/lib/msun/src/k_sinpi.h
Normal file
|
|
@ -0,0 +1,43 @@
|
|||
/*-
|
||||
* Copyright (c) 2017 Steven G. Kargl
|
||||
* All rights reserved.
|
||||
*
|
||||
* Redistribution and use in source and binary forms, with or without
|
||||
* modification, are permitted provided that the following conditions
|
||||
* are met:
|
||||
* 1. Redistributions of source code must retain the above copyright
|
||||
* notice unmodified, this list of conditions, and the following
|
||||
* disclaimer.
|
||||
* 2. Redistributions in binary form must reproduce the above copyright
|
||||
* notice, this list of conditions and the following disclaimer in the
|
||||
* documentation and/or other materials provided with the distribution.
|
||||
*
|
||||
* THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
|
||||
* IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
|
||||
* OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
|
||||
* IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
|
||||
* INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
|
||||
* NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
|
||||
* DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
|
||||
* THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
|
||||
* (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
|
||||
* THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
|
||||
*/
|
||||
|
||||
/*
|
||||
* The basic kernel for x in [0,0.25]. To use the kernel for sin(x), the
|
||||
* argument to __kernel_sinpi() must be multiplied by pi.
|
||||
*/
|
||||
|
||||
static inline double
|
||||
__kernel_sinpi(double x)
|
||||
{
|
||||
double_t hi, lo;
|
||||
|
||||
hi = (float)x;
|
||||
lo = x - hi;
|
||||
lo = lo * (pi_lo + pi_hi) + hi * pi_lo;
|
||||
hi *= pi_hi;
|
||||
_2sumF(hi, lo);
|
||||
return (__kernel_sin(hi, lo, 1));
|
||||
}
|
||||
|
|
@ -30,6 +30,7 @@
|
|||
__FBSDID("$FreeBSD$");
|
||||
|
||||
#include <complex.h>
|
||||
#include <float.h>
|
||||
#include <math.h>
|
||||
|
||||
#include "math_private.h"
|
||||
|
|
@ -41,7 +42,7 @@ cexp_ovfl = 0x4096b8e4; /* (MAX_EXP - MIN_DENORM_EXP) * ln2 */
|
|||
double complex
|
||||
cexp(double complex z)
|
||||
{
|
||||
double x, y, exp_x;
|
||||
double c, exp_x, s, x, y;
|
||||
uint32_t hx, hy, lx, ly;
|
||||
|
||||
x = creal(z);
|
||||
|
|
@ -55,8 +56,10 @@ cexp(double complex z)
|
|||
return (CMPLX(exp(x), y));
|
||||
EXTRACT_WORDS(hx, lx, x);
|
||||
/* cexp(0 + I y) = cos(y) + I sin(y) */
|
||||
if (((hx & 0x7fffffff) | lx) == 0)
|
||||
return (CMPLX(cos(y), sin(y)));
|
||||
if (((hx & 0x7fffffff) | lx) == 0) {
|
||||
sincos(y, &s, &c);
|
||||
return (CMPLX(c, s));
|
||||
}
|
||||
|
||||
if (hy >= 0x7ff00000) {
|
||||
if (lx != 0 || (hx & 0x7fffffff) != 0x7ff00000) {
|
||||
|
|
@ -86,6 +89,11 @@ cexp(double complex z)
|
|||
* - x = NaN (spurious inexact exception from y)
|
||||
*/
|
||||
exp_x = exp(x);
|
||||
return (CMPLX(exp_x * cos(y), exp_x * sin(y)));
|
||||
sincos(y, &s, &c);
|
||||
return (CMPLX(exp_x * c, exp_x * s));
|
||||
}
|
||||
}
|
||||
|
||||
#if (LDBL_MANT_DIG == 53)
|
||||
__weak_reference(cexp, cexpl);
|
||||
#endif
|
||||
|
|
|
|||
|
|
@ -41,7 +41,7 @@ cexp_ovfl = 0x43400074; /* (MAX_EXP - MIN_DENORM_EXP) * ln2 */
|
|||
float complex
|
||||
cexpf(float complex z)
|
||||
{
|
||||
float x, y, exp_x;
|
||||
float c, exp_x, s, x, y;
|
||||
uint32_t hx, hy;
|
||||
|
||||
x = crealf(z);
|
||||
|
|
@ -55,8 +55,10 @@ cexpf(float complex z)
|
|||
return (CMPLXF(expf(x), y));
|
||||
GET_FLOAT_WORD(hx, x);
|
||||
/* cexp(0 + I y) = cos(y) + I sin(y) */
|
||||
if ((hx & 0x7fffffff) == 0)
|
||||
return (CMPLXF(cosf(y), sinf(y)));
|
||||
if ((hx & 0x7fffffff) == 0) {
|
||||
sincosf(y, &s, &c);
|
||||
return (CMPLXF(c, s));
|
||||
}
|
||||
|
||||
if (hy >= 0x7f800000) {
|
||||
if ((hx & 0x7fffffff) != 0x7f800000) {
|
||||
|
|
@ -86,6 +88,7 @@ cexpf(float complex z)
|
|||
* - x = NaN (spurious inexact exception from y)
|
||||
*/
|
||||
exp_x = expf(x);
|
||||
return (CMPLXF(exp_x * cosf(y), exp_x * sinf(y)));
|
||||
sincosf(y, &s, &c);
|
||||
return (CMPLXF(exp_x * c, exp_x * s));
|
||||
}
|
||||
}
|
||||
|
|
|
|||
|
|
@ -39,12 +39,17 @@ __FBSDID("$FreeBSD$");
|
|||
#include <ieeefp.h>
|
||||
#endif
|
||||
|
||||
#include "fpmath.h"
|
||||
#include "math.h"
|
||||
#include "math_private.h"
|
||||
#if LDBL_MANT_DIG == 64
|
||||
#include "../ld80/e_rem_pio2l.h"
|
||||
static const union IEEEl2bits
|
||||
pio4u = LD80C(0xc90fdaa22168c235, -00001, 7.85398163397448309628e-01L);
|
||||
#define pio4 (pio4u.e)
|
||||
#elif LDBL_MANT_DIG == 113
|
||||
#include "../ld128/e_rem_pio2l.h"
|
||||
long double pio4 = 7.85398163397448309615660845819875721e-1L;
|
||||
#else
|
||||
#error "Unsupported long double format"
|
||||
#endif
|
||||
|
|
@ -71,7 +76,7 @@ cosl(long double x)
|
|||
ENTERI();
|
||||
|
||||
/* Optimize the case where x is already within range. */
|
||||
if (z.e < M_PI_4)
|
||||
if (z.e < pio4)
|
||||
RETURNI(__kernel_cosl(z.e, 0));
|
||||
|
||||
e0 = __ieee754_rem_pio2l(x, y);
|
||||
|
|
|
|||
152
libm/upstream-freebsd/lib/msun/src/s_cospi.c
Normal file
152
libm/upstream-freebsd/lib/msun/src/s_cospi.c
Normal file
|
|
@ -0,0 +1,152 @@
|
|||
/*-
|
||||
* Copyright (c) 2017 Steven G. Kargl
|
||||
* All rights reserved.
|
||||
*
|
||||
* Redistribution and use in source and binary forms, with or without
|
||||
* modification, are permitted provided that the following conditions
|
||||
* are met:
|
||||
* 1. Redistributions of source code must retain the above copyright
|
||||
* notice unmodified, this list of conditions, and the following
|
||||
* disclaimer.
|
||||
* 2. Redistributions in binary form must reproduce the above copyright
|
||||
* notice, this list of conditions and the following disclaimer in the
|
||||
* documentation and/or other materials provided with the distribution.
|
||||
*
|
||||
* THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
|
||||
* IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
|
||||
* OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
|
||||
* IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
|
||||
* INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
|
||||
* NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
|
||||
* DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
|
||||
* THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
|
||||
* (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
|
||||
* THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
|
||||
*/
|
||||
|
||||
/**
|
||||
* cospi(x) computes cos(pi*x) without multiplication by pi (almost). First,
|
||||
* note that cospi(-x) = cospi(x), so the algorithm considers only |x|. The
|
||||
* method used depends on the magnitude of x.
|
||||
*
|
||||
* 1. For small |x|, cospi(x) = 1 with FE_INEXACT raised where a sloppy
|
||||
* threshold is used. The threshold is |x| < 0x1pN with N = -(P/2+M).
|
||||
* P is the precision of the floating-point type and M = 2 to 4.
|
||||
*
|
||||
* 2. For |x| < 1, argument reduction is not required and sinpi(x) is
|
||||
* computed by calling a kernel that leverages the kernels for sin(x)
|
||||
* ans cos(x). See k_sinpi.c and k_cospi.c for details.
|
||||
*
|
||||
* 3. For 1 <= |x| < 0x1p(P-1), argument reduction is required where
|
||||
* |x| = j0 + r with j0 an integer and the remainder r satisfies
|
||||
* 0 <= r < 1. With the given domain, a simplified inline floor(x)
|
||||
* is used. Also, note the following identity
|
||||
*
|
||||
* cospi(x) = cos(pi*(j0+r))
|
||||
* = cos(pi*j0) * cos(pi*r) - sin(pi*j0) * sin(pi*r)
|
||||
* = cos(pi*j0) * cos(pi*r)
|
||||
* = +-cospi(r)
|
||||
*
|
||||
* If j0 is even, then cos(pi*j0) = 1. If j0 is odd, then cos(pi*j0) = -1.
|
||||
* cospi(r) is then computed via an appropriate kernel.
|
||||
*
|
||||
* 4. For |x| >= 0x1p(P-1), |x| is integral and cospi(x) = 1.
|
||||
*
|
||||
* 5. Special cases:
|
||||
*
|
||||
* cospi(+-0) = 1.
|
||||
* cospi(n.5) = 0 for n an integer.
|
||||
* cospi(+-inf) = nan. Raises the "invalid" floating-point exception.
|
||||
* cospi(nan) = nan. Raises the "invalid" floating-point exception.
|
||||
*/
|
||||
|
||||
#include <float.h>
|
||||
#include "math.h"
|
||||
#include "math_private.h"
|
||||
|
||||
static const double
|
||||
pi_hi = 3.1415926814079285e+00, /* 0x400921fb 0x58000000 */
|
||||
pi_lo =-2.7818135228334233e-08; /* 0xbe5dde97 0x3dcb3b3a */
|
||||
|
||||
#include "k_cospi.h"
|
||||
#include "k_sinpi.h"
|
||||
|
||||
volatile static const double vzero = 0;
|
||||
|
||||
double
|
||||
cospi(double x)
|
||||
{
|
||||
double ax, c;
|
||||
uint32_t hx, ix, j0, lx;
|
||||
|
||||
EXTRACT_WORDS(hx, lx, x);
|
||||
ix = hx & 0x7fffffff;
|
||||
INSERT_WORDS(ax, ix, lx);
|
||||
|
||||
if (ix < 0x3ff00000) { /* |x| < 1 */
|
||||
if (ix < 0x3fd00000) { /* |x| < 0.25 */
|
||||
if (ix < 0x3e200000) { /* |x| < 0x1p-29 */
|
||||
if ((int)ax == 0)
|
||||
return (1);
|
||||
}
|
||||
return (__kernel_cospi(ax));
|
||||
}
|
||||
|
||||
if (ix < 0x3fe00000) /* |x| < 0.5 */
|
||||
c = __kernel_sinpi(0.5 - ax);
|
||||
else if (ix < 0x3fe80000){ /* |x| < 0.75 */
|
||||
if (ax == 0.5)
|
||||
return (0);
|
||||
c = -__kernel_sinpi(ax - 0.5);
|
||||
} else
|
||||
c = -__kernel_cospi(1 - ax);
|
||||
return (c);
|
||||
}
|
||||
|
||||
if (ix < 0x43300000) { /* 1 <= |x| < 0x1p52 */
|
||||
/* Determine integer part of ax. */
|
||||
j0 = ((ix >> 20) & 0x7ff) - 0x3ff;
|
||||
if (j0 < 20) {
|
||||
ix &= ~(0x000fffff >> j0);
|
||||
lx = 0;
|
||||
} else {
|
||||
lx &= ~((uint32_t)0xffffffff >> (j0 - 20));
|
||||
}
|
||||
INSERT_WORDS(x, ix, lx);
|
||||
|
||||
ax -= x;
|
||||
EXTRACT_WORDS(ix, lx, ax);
|
||||
|
||||
|
||||
if (ix < 0x3fe00000) { /* |x| < 0.5 */
|
||||
if (ix < 0x3fd00000) /* |x| < 0.25 */
|
||||
c = ix == 0 ? 1 : __kernel_cospi(ax);
|
||||
else
|
||||
c = __kernel_sinpi(0.5 - ax);
|
||||
} else {
|
||||
if (ix < 0x3fe80000) { /* |x| < 0.75 */
|
||||
if (ax == 0.5)
|
||||
return (0);
|
||||
c = -__kernel_sinpi(ax - 0.5);
|
||||
} else
|
||||
c = -__kernel_cospi(1 - ax);
|
||||
}
|
||||
|
||||
if (j0 > 30)
|
||||
x -= 0x1p30;
|
||||
j0 = (uint32_t)x;
|
||||
return (j0 & 1 ? -c : c);
|
||||
}
|
||||
|
||||
if (ix >= 0x7f800000)
|
||||
return (vzero / vzero);
|
||||
|
||||
/*
|
||||
* |x| >= 0x1p52 is always an even integer, so return 1.
|
||||
*/
|
||||
return (1);
|
||||
}
|
||||
|
||||
#if LDBL_MANT_DIG == 53
|
||||
__weak_reference(cospi, cospil);
|
||||
#endif
|
||||
|
|
@ -111,11 +111,13 @@ ctanh(double complex z)
|
|||
}
|
||||
|
||||
/*
|
||||
* ctanh(x + I NaN) = d(NaN) + I d(NaN)
|
||||
* ctanh(x +- I Inf) = dNaN + I dNaN
|
||||
* ctanh(+-0 + i NAN) = +-0 + i NaN
|
||||
* ctanh(+-0 +- i Inf) = +-0 + i NaN
|
||||
* ctanh(x + i NAN) = NaN + i NaN
|
||||
* ctanh(x +- i Inf) = NaN + i NaN
|
||||
*/
|
||||
if (!isfinite(y))
|
||||
return (CMPLX(y - y, y - y));
|
||||
return (CMPLX(x ? y - y : x, y - y));
|
||||
|
||||
/*
|
||||
* ctanh(+-huge +- I y) ~= +-1 +- I 2sin(2y)/exp(2x), using the
|
||||
|
|
|
|||
|
|
@ -61,7 +61,7 @@ ctanhf(float complex z)
|
|||
}
|
||||
|
||||
if (!isfinite(y))
|
||||
return (CMPLXF(y - y, y - y));
|
||||
return (CMPLXF(ix ? y - y : x, y - y));
|
||||
|
||||
if (ix >= 0x41300000) { /* |x| >= 11 */
|
||||
float exp_mx = expf(-fabsf(x));
|
||||
|
|
|
|||
|
|
@ -35,6 +35,13 @@ __FBSDID("$FreeBSD$");
|
|||
|
||||
#include "math_private.h"
|
||||
|
||||
#ifdef USE_BUILTIN_FMA
|
||||
double
|
||||
fma(double x, double y, double z)
|
||||
{
|
||||
return (__builtin_fma(x, y, z));
|
||||
}
|
||||
#else
|
||||
/*
|
||||
* A struct dd represents a floating-point number with twice the precision
|
||||
* of a double. We maintain the invariant that "hi" stores the 53 high-order
|
||||
|
|
@ -284,6 +291,7 @@ fma(double x, double y, double z)
|
|||
else
|
||||
return (add_and_denormalize(r.hi, adj, spread));
|
||||
}
|
||||
#endif /* !USE_BUILTIN_FMA */
|
||||
|
||||
#if (LDBL_MANT_DIG == 53)
|
||||
__weak_reference(fma, fmal);
|
||||
|
|
|
|||
|
|
@ -34,6 +34,13 @@ __FBSDID("$FreeBSD$");
|
|||
#include "math.h"
|
||||
#include "math_private.h"
|
||||
|
||||
#ifdef USE_BUILTIN_FMAF
|
||||
float
|
||||
fmaf(float x, float y, float z)
|
||||
{
|
||||
return (__builtin_fmaf(x, y, z));
|
||||
}
|
||||
#else
|
||||
/*
|
||||
* Fused multiply-add: Compute x * y + z with a single rounding error.
|
||||
*
|
||||
|
|
@ -69,3 +76,4 @@ fmaf(float x, float y, float z)
|
|||
SET_LOW_WORD(adjusted_result, lr + 1);
|
||||
return (adjusted_result);
|
||||
}
|
||||
#endif /* !USE_BUILTIN_FMAF */
|
||||
|
|
|
|||
|
|
@ -34,6 +34,13 @@ __FBSDID("$FreeBSD$");
|
|||
|
||||
#include "fpmath.h"
|
||||
|
||||
#ifdef USE_BUILTIN_FMAX
|
||||
double
|
||||
fmax(double x, double y)
|
||||
{
|
||||
return (__builtin_fmax(x, y));
|
||||
}
|
||||
#else
|
||||
double
|
||||
fmax(double x, double y)
|
||||
{
|
||||
|
|
@ -54,6 +61,7 @@ fmax(double x, double y)
|
|||
|
||||
return (x > y ? x : y);
|
||||
}
|
||||
#endif
|
||||
|
||||
#if (LDBL_MANT_DIG == 53)
|
||||
__weak_reference(fmax, fmaxl);
|
||||
|
|
|
|||
|
|
@ -33,6 +33,13 @@ __FBSDID("$FreeBSD$");
|
|||
|
||||
#include "fpmath.h"
|
||||
|
||||
#ifdef USE_BUILTIN_FMAXF
|
||||
float
|
||||
fmaxf(float x, float y)
|
||||
{
|
||||
return (__builtin_fmaxf(x, y));
|
||||
}
|
||||
#else
|
||||
float
|
||||
fmaxf(float x, float y)
|
||||
{
|
||||
|
|
@ -53,3 +60,4 @@ fmaxf(float x, float y)
|
|||
|
||||
return (x > y ? x : y);
|
||||
}
|
||||
#endif
|
||||
|
|
|
|||
|
|
@ -34,6 +34,13 @@ __FBSDID("$FreeBSD$");
|
|||
|
||||
#include "fpmath.h"
|
||||
|
||||
#ifdef USE_BUILTIN_FMIN
|
||||
double
|
||||
fmin(double x, double y)
|
||||
{
|
||||
return (__builtin_fmin(x, y));
|
||||
}
|
||||
#else
|
||||
double
|
||||
fmin(double x, double y)
|
||||
{
|
||||
|
|
@ -54,6 +61,7 @@ fmin(double x, double y)
|
|||
|
||||
return (x < y ? x : y);
|
||||
}
|
||||
#endif
|
||||
|
||||
#if (LDBL_MANT_DIG == 53)
|
||||
__weak_reference(fmin, fminl);
|
||||
|
|
|
|||
|
|
@ -33,6 +33,13 @@ __FBSDID("$FreeBSD$");
|
|||
|
||||
#include "fpmath.h"
|
||||
|
||||
#ifdef USE_BUILTIN_FMINF
|
||||
float
|
||||
fminf(float x, float y)
|
||||
{
|
||||
return (__builtin_fminf(x, y));
|
||||
}
|
||||
#else
|
||||
float
|
||||
fminf(float x, float y)
|
||||
{
|
||||
|
|
@ -53,3 +60,4 @@ fminf(float x, float y)
|
|||
|
||||
return (x < y ? x : y);
|
||||
}
|
||||
#endif
|
||||
|
|
|
|||
|
|
@ -49,9 +49,11 @@ __FBSDID("$FreeBSD$");
|
|||
* that everything is in range. At compile time, INRANGE(x) should reduce to
|
||||
* two floating-point comparisons in the former case, or TRUE otherwise.
|
||||
*/
|
||||
static const type type_min = (type)DTYPE_MIN;
|
||||
static const type type_max = (type)DTYPE_MAX;
|
||||
static const type dtype_min = (type)DTYPE_MIN - 0.5;
|
||||
static const type dtype_max = (type)DTYPE_MAX + 0.5;
|
||||
#define INRANGE(x) (dtype_max - (type)DTYPE_MAX != 0.5 || \
|
||||
#define INRANGE(x) (dtype_max - type_max != 0.5 || \
|
||||
((x) > dtype_min && (x) < dtype_max))
|
||||
|
||||
dtype
|
||||
|
|
|
|||
|
|
@ -1,66 +1,47 @@
|
|||
/* @(#)s_scalbn.c 5.1 93/09/24 */
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
* Copyright (c) 2005-2020 Rich Felker, et al.
|
||||
*
|
||||
* 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.
|
||||
* ====================================================
|
||||
* SPDX-License-Identifier: MIT
|
||||
*
|
||||
* Please see https://git.musl-libc.org/cgit/musl/tree/COPYRIGHT
|
||||
* for all contributors to musl.
|
||||
*/
|
||||
|
||||
#include <sys/cdefs.h>
|
||||
__FBSDID("$FreeBSD$");
|
||||
|
||||
/*
|
||||
* scalbn (double x, int n)
|
||||
* scalbn(x,n) returns x* 2**n computed by exponent
|
||||
* manipulation rather than by actually performing an
|
||||
* exponentiation or a multiplication.
|
||||
*/
|
||||
|
||||
#include <float.h>
|
||||
#include <math.h>
|
||||
#include <stdint.h>
|
||||
|
||||
#include "math.h"
|
||||
#include "math_private.h"
|
||||
|
||||
static const double
|
||||
two54 = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */
|
||||
twom54 = 5.55111512312578270212e-17, /* 0x3C900000, 0x00000000 */
|
||||
huge = 1.0e+300,
|
||||
tiny = 1.0e-300;
|
||||
|
||||
double
|
||||
scalbn (double x, int n)
|
||||
double scalbn(double x, int n)
|
||||
{
|
||||
int32_t k,hx,lx;
|
||||
EXTRACT_WORDS(hx,lx,x);
|
||||
k = (hx&0x7ff00000)>>20; /* extract exponent */
|
||||
if (k==0) { /* 0 or subnormal x */
|
||||
if ((lx|(hx&0x7fffffff))==0) return x; /* +-0 */
|
||||
x *= two54;
|
||||
GET_HIGH_WORD(hx,x);
|
||||
k = ((hx&0x7ff00000)>>20) - 54;
|
||||
if (n< -50000) return tiny*x; /*underflow*/
|
||||
}
|
||||
if (k==0x7ff) return x+x; /* NaN or Inf */
|
||||
k = k+n;
|
||||
if (k > 0x7fe) return huge*copysign(huge,x); /* overflow */
|
||||
if (k > 0) /* normal result */
|
||||
{SET_HIGH_WORD(x,(hx&0x800fffff)|(k<<20)); return x;}
|
||||
if (k <= -54) {
|
||||
if (n > 50000) /* in case integer overflow in n+k */
|
||||
return huge*copysign(huge,x); /*overflow*/
|
||||
else
|
||||
return tiny*copysign(tiny,x); /*underflow*/
|
||||
union {double f; uint64_t i;} u;
|
||||
double_t y = x;
|
||||
|
||||
if (n > 1023) {
|
||||
y *= 0x1p1023;
|
||||
n -= 1023;
|
||||
if (n > 1023) {
|
||||
y *= 0x1p1023;
|
||||
n -= 1023;
|
||||
if (n > 1023)
|
||||
n = 1023;
|
||||
}
|
||||
} else if (n < -1022) {
|
||||
/* make sure final n < -53 to avoid double
|
||||
rounding in the subnormal range */
|
||||
y *= 0x1p-1022 * 0x1p53;
|
||||
n += 1022 - 53;
|
||||
if (n < -1022) {
|
||||
y *= 0x1p-1022 * 0x1p53;
|
||||
n += 1022 - 53;
|
||||
if (n < -1022)
|
||||
n = -1022;
|
||||
}
|
||||
}
|
||||
k += 54; /* subnormal result */
|
||||
SET_HIGH_WORD(x,(hx&0x800fffff)|(k<<20));
|
||||
return x*twom54;
|
||||
u.i = (uint64_t)(0x3ff+n)<<52;
|
||||
x = y * u.f;
|
||||
return x;
|
||||
}
|
||||
|
||||
#if (LDBL_MANT_DIG == 53)
|
||||
#if (LDBL_MANT_DIG == 53) && !defined(scalbn)
|
||||
__weak_reference(scalbn, ldexpl);
|
||||
__weak_reference(scalbn, scalbnl);
|
||||
#endif
|
||||
|
|
|
|||
|
|
@ -1,57 +1,41 @@
|
|||
/* s_scalbnf.c -- float version of s_scalbn.c.
|
||||
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
|
||||
*/
|
||||
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
* Copyright (c) 2005-2020 Rich Felker, et al.
|
||||
*
|
||||
* 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.
|
||||
* ====================================================
|
||||
* SPDX-License-Identifier: MIT
|
||||
*
|
||||
* Please see https://git.musl-libc.org/cgit/musl/tree/COPYRIGHT
|
||||
* for all contributors to musl.
|
||||
*/
|
||||
#include <math.h>
|
||||
#include <stdint.h>
|
||||
|
||||
#include <sys/cdefs.h>
|
||||
__FBSDID("$FreeBSD$");
|
||||
|
||||
#include "math.h"
|
||||
#include "math_private.h"
|
||||
|
||||
static const float
|
||||
two25 = 3.355443200e+07, /* 0x4c000000 */
|
||||
twom25 = 2.9802322388e-08, /* 0x33000000 */
|
||||
huge = 1.0e+30,
|
||||
tiny = 1.0e-30;
|
||||
|
||||
float
|
||||
scalbnf (float x, int n)
|
||||
float scalbnf(float x, int n)
|
||||
{
|
||||
int32_t k,ix;
|
||||
GET_FLOAT_WORD(ix,x);
|
||||
k = (ix&0x7f800000)>>23; /* extract exponent */
|
||||
if (k==0) { /* 0 or subnormal x */
|
||||
if ((ix&0x7fffffff)==0) return x; /* +-0 */
|
||||
x *= two25;
|
||||
GET_FLOAT_WORD(ix,x);
|
||||
k = ((ix&0x7f800000)>>23) - 25;
|
||||
if (n< -50000) return tiny*x; /*underflow*/
|
||||
}
|
||||
if (k==0xff) return x+x; /* NaN or Inf */
|
||||
k = k+n;
|
||||
if (k > 0xfe) return huge*copysignf(huge,x); /* overflow */
|
||||
if (k > 0) /* normal result */
|
||||
{SET_FLOAT_WORD(x,(ix&0x807fffff)|(k<<23)); return x;}
|
||||
if (k <= -25) {
|
||||
if (n > 50000) /* in case integer overflow in n+k */
|
||||
return huge*copysignf(huge,x); /*overflow*/
|
||||
else
|
||||
return tiny*copysignf(tiny,x); /*underflow*/
|
||||
union {float f; uint32_t i;} u;
|
||||
float_t y = x;
|
||||
|
||||
if (n > 127) {
|
||||
y *= 0x1p127f;
|
||||
n -= 127;
|
||||
if (n > 127) {
|
||||
y *= 0x1p127f;
|
||||
n -= 127;
|
||||
if (n > 127)
|
||||
n = 127;
|
||||
}
|
||||
} else if (n < -126) {
|
||||
y *= 0x1p-126f * 0x1p24f;
|
||||
n += 126 - 24;
|
||||
if (n < -126) {
|
||||
y *= 0x1p-126f * 0x1p24f;
|
||||
n += 126 - 24;
|
||||
if (n < -126)
|
||||
n = -126;
|
||||
}
|
||||
}
|
||||
k += 25; /* subnormal result */
|
||||
SET_FLOAT_WORD(x,(ix&0x807fffff)|(k<<23));
|
||||
return x*twom25;
|
||||
u.i = (uint32_t)(0x7f+n)<<23;
|
||||
x = y * u.f;
|
||||
return x;
|
||||
}
|
||||
|
||||
__strong_reference(scalbnf, ldexpf);
|
||||
|
|
|
|||
|
|
@ -1,71 +1,49 @@
|
|||
/* @(#)s_scalbn.c 5.1 93/09/24 */
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
* Copyright (c) 2005-2020 Rich Felker, et al.
|
||||
*
|
||||
* 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.
|
||||
* ====================================================
|
||||
* SPDX-License-Identifier: MIT
|
||||
*
|
||||
* Please see https://git.musl-libc.org/cgit/musl/tree/COPYRIGHT
|
||||
* for all contributors to musl.
|
||||
*/
|
||||
|
||||
#include <sys/cdefs.h>
|
||||
__FBSDID("$FreeBSD$");
|
||||
|
||||
#include <math.h>
|
||||
#include <float.h>
|
||||
#include "math_private.h"
|
||||
#include "fpmath.h"
|
||||
/*
|
||||
* scalbnl (long double x, int n)
|
||||
* scalbnl(x,n) returns x* 2**n computed by exponent
|
||||
* manipulation rather than by actually performing an
|
||||
* exponentiation or a multiplication.
|
||||
*/
|
||||
|
||||
/*
|
||||
* We assume that a long double has a 15-bit exponent. On systems
|
||||
* where long double is the same as double, scalbnl() is an alias
|
||||
* for scalbn(), so we don't use this routine.
|
||||
*/
|
||||
|
||||
#include <float.h>
|
||||
#include <math.h>
|
||||
|
||||
#include "fpmath.h"
|
||||
|
||||
#if LDBL_MAX_EXP != 0x4000
|
||||
#error "Unsupported long double format"
|
||||
#endif
|
||||
|
||||
static const long double
|
||||
huge = 0x1p16000L,
|
||||
tiny = 0x1p-16000L;
|
||||
|
||||
long double
|
||||
scalbnl (long double x, int n)
|
||||
#if (LDBL_MANT_DIG == 64 || LDBL_MANT_DIG == 113) && LDBL_MAX_EXP == 16384
|
||||
long double scalbnl(long double x, int n)
|
||||
{
|
||||
union IEEEl2bits u;
|
||||
int k;
|
||||
u.e = x;
|
||||
k = u.bits.exp; /* extract exponent */
|
||||
if (k==0) { /* 0 or subnormal x */
|
||||
if ((u.bits.manh|u.bits.manl)==0) return x; /* +-0 */
|
||||
u.e *= 0x1p+128;
|
||||
k = u.bits.exp - 128;
|
||||
if (n< -50000) return tiny*x; /*underflow*/
|
||||
}
|
||||
if (k==0x7fff) return x+x; /* NaN or Inf */
|
||||
k = k+n;
|
||||
if (k >= 0x7fff) return huge*copysignl(huge,x); /* overflow */
|
||||
if (k > 0) /* normal result */
|
||||
{u.bits.exp = k; return u.e;}
|
||||
if (k <= -128) {
|
||||
if (n > 50000) /* in case integer overflow in n+k */
|
||||
return huge*copysign(huge,x); /*overflow*/
|
||||
else
|
||||
return tiny*copysign(tiny,x); /*underflow*/
|
||||
}
|
||||
k += 128; /* subnormal result */
|
||||
u.bits.exp = k;
|
||||
return u.e*0x1p-128;
|
||||
}
|
||||
|
||||
if (n > 16383) {
|
||||
x *= 0x1p16383L;
|
||||
n -= 16383;
|
||||
if (n > 16383) {
|
||||
x *= 0x1p16383L;
|
||||
n -= 16383;
|
||||
if (n > 16383)
|
||||
n = 16383;
|
||||
}
|
||||
} else if (n < -16382) {
|
||||
x *= 0x1p-16382L * 0x1p113L;
|
||||
n += 16382 - 113;
|
||||
if (n < -16382) {
|
||||
x *= 0x1p-16382L * 0x1p113L;
|
||||
n += 16382 - 113;
|
||||
if (n < -16382)
|
||||
n = -16382;
|
||||
}
|
||||
}
|
||||
u.e = 1.0;
|
||||
u.xbits.expsign = 0x3fff + n;
|
||||
return x * u.e;
|
||||
}
|
||||
__strong_reference(scalbnl, ldexpl);
|
||||
#endif
|
||||
|
||||
|
|
|
|||
|
|
@ -50,11 +50,10 @@ void
|
|||
sincosl(long double x, long double *sn, long double *cs)
|
||||
{
|
||||
union IEEEl2bits z;
|
||||
int e0, sgn;
|
||||
int e0;
|
||||
long double y[2];
|
||||
|
||||
z.e = x;
|
||||
sgn = z.bits.sign;
|
||||
z.bits.sign = 0;
|
||||
|
||||
ENTERV();
|
||||
|
|
|
|||
169
libm/upstream-freebsd/lib/msun/src/s_sinpi.c
Normal file
169
libm/upstream-freebsd/lib/msun/src/s_sinpi.c
Normal file
|
|
@ -0,0 +1,169 @@
|
|||
/*-
|
||||
* Copyright (c) 2017 Steven G. Kargl
|
||||
* All rights reserved.
|
||||
*
|
||||
* Redistribution and use in source and binary forms, with or without
|
||||
* modification, are permitted provided that the following conditions
|
||||
* are met:
|
||||
* 1. Redistributions of source code must retain the above copyright
|
||||
* notice unmodified, this list of conditions, and the following
|
||||
* disclaimer.
|
||||
* 2. Redistributions in binary form must reproduce the above copyright
|
||||
* notice, this list of conditions and the following disclaimer in the
|
||||
* documentation and/or other materials provided with the distribution.
|
||||
*
|
||||
* THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
|
||||
* IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
|
||||
* OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
|
||||
* IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
|
||||
* INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
|
||||
* NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
|
||||
* DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
|
||||
* THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
|
||||
* (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
|
||||
* THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
|
||||
*/
|
||||
|
||||
/**
|
||||
* sinpi(x) computes sin(pi*x) without multiplication by pi (almost). First,
|
||||
* note that sinpi(-x) = -sinpi(x), so the algorithm considers only |x| and
|
||||
* includes reflection symmetry by considering the sign of x on output. The
|
||||
* method used depends on the magnitude of x.
|
||||
*
|
||||
* 1. For small |x|, sinpi(x) = pi * x where a sloppy threshold is used. The
|
||||
* threshold is |x| < 0x1pN with N = -(P/2+M). P is the precision of the
|
||||
* floating-point type and M = 2 to 4. To achieve high accuracy, pi is
|
||||
* decomposed into high and low parts with the high part containing a
|
||||
* number of trailing zero bits. x is also split into high and low parts.
|
||||
*
|
||||
* 2. For |x| < 1, argument reduction is not required and sinpi(x) is
|
||||
* computed by calling a kernel that leverages the kernels for sin(x)
|
||||
* ans cos(x). See k_sinpi.c and k_cospi.c for details.
|
||||
*
|
||||
* 3. For 1 <= |x| < 0x1p(P-1), argument reduction is required where
|
||||
* |x| = j0 + r with j0 an integer and the remainder r satisfies
|
||||
* 0 <= r < 1. With the given domain, a simplified inline floor(x)
|
||||
* is used. Also, note the following identity
|
||||
*
|
||||
* sinpi(x) = sin(pi*(j0+r))
|
||||
* = sin(pi*j0) * cos(pi*r) + cos(pi*j0) * sin(pi*r)
|
||||
* = cos(pi*j0) * sin(pi*r)
|
||||
* = +-sinpi(r)
|
||||
*
|
||||
* If j0 is even, then cos(pi*j0) = 1. If j0 is odd, then cos(pi*j0) = -1.
|
||||
* sinpi(r) is then computed via an appropriate kernel.
|
||||
*
|
||||
* 4. For |x| >= 0x1p(P-1), |x| is integral and sinpi(x) = copysign(0,x).
|
||||
*
|
||||
* 5. Special cases:
|
||||
*
|
||||
* sinpi(+-0) = +-0
|
||||
* sinpi(+-n) = +-0, for positive integers n.
|
||||
* sinpi(+-inf) = nan. Raises the "invalid" floating-point exception.
|
||||
* sinpi(nan) = nan. Raises the "invalid" floating-point exception.
|
||||
*/
|
||||
|
||||
#include <float.h>
|
||||
#include "math.h"
|
||||
#include "math_private.h"
|
||||
|
||||
static const double
|
||||
pi_hi = 3.1415926814079285e+00, /* 0x400921fb 0x58000000 */
|
||||
pi_lo =-2.7818135228334233e-08; /* 0xbe5dde97 0x3dcb3b3a */
|
||||
|
||||
#include "k_cospi.h"
|
||||
#include "k_sinpi.h"
|
||||
|
||||
volatile static const double vzero = 0;
|
||||
|
||||
double
|
||||
sinpi(double x)
|
||||
{
|
||||
double ax, hi, lo, s;
|
||||
uint32_t hx, ix, j0, lx;
|
||||
|
||||
EXTRACT_WORDS(hx, lx, x);
|
||||
ix = hx & 0x7fffffff;
|
||||
INSERT_WORDS(ax, ix, lx);
|
||||
|
||||
if (ix < 0x3ff00000) { /* |x| < 1 */
|
||||
if (ix < 0x3fd00000) { /* |x| < 0.25 */
|
||||
if (ix < 0x3e200000) { /* |x| < 0x1p-29 */
|
||||
if (x == 0)
|
||||
return (x);
|
||||
/*
|
||||
* To avoid issues with subnormal values,
|
||||
* scale the computation and rescale on
|
||||
* return.
|
||||
*/
|
||||
INSERT_WORDS(hi, hx, 0);
|
||||
hi *= 0x1p53;
|
||||
lo = x * 0x1p53 - hi;
|
||||
s = (pi_lo + pi_hi) * lo + pi_lo * hi +
|
||||
pi_hi * hi;
|
||||
return (s * 0x1p-53);
|
||||
}
|
||||
|
||||
s = __kernel_sinpi(ax);
|
||||
return ((hx & 0x80000000) ? -s : s);
|
||||
}
|
||||
|
||||
if (ix < 0x3fe00000) /* |x| < 0.5 */
|
||||
s = __kernel_cospi(0.5 - ax);
|
||||
else if (ix < 0x3fe80000) /* |x| < 0.75 */
|
||||
s = __kernel_cospi(ax - 0.5);
|
||||
else
|
||||
s = __kernel_sinpi(1 - ax);
|
||||
return ((hx & 0x80000000) ? -s : s);
|
||||
}
|
||||
|
||||
if (ix < 0x43300000) { /* 1 <= |x| < 0x1p52 */
|
||||
/* Determine integer part of ax. */
|
||||
j0 = ((ix >> 20) & 0x7ff) - 0x3ff;
|
||||
if (j0 < 20) {
|
||||
ix &= ~(0x000fffff >> j0);
|
||||
lx = 0;
|
||||
} else {
|
||||
lx &= ~((uint32_t)0xffffffff >> (j0 - 20));
|
||||
}
|
||||
INSERT_WORDS(x, ix, lx);
|
||||
|
||||
ax -= x;
|
||||
EXTRACT_WORDS(ix, lx, ax);
|
||||
|
||||
if (ix == 0)
|
||||
s = 0;
|
||||
else {
|
||||
if (ix < 0x3fe00000) { /* |x| < 0.5 */
|
||||
if (ix < 0x3fd00000) /* |x| < 0.25 */
|
||||
s = __kernel_sinpi(ax);
|
||||
else
|
||||
s = __kernel_cospi(0.5 - ax);
|
||||
} else {
|
||||
if (ix < 0x3fe80000) /* |x| < 0.75 */
|
||||
s = __kernel_cospi(ax - 0.5);
|
||||
else
|
||||
s = __kernel_sinpi(1 - ax);
|
||||
}
|
||||
|
||||
if (j0 > 30)
|
||||
x -= 0x1p30;
|
||||
j0 = (uint32_t)x;
|
||||
if (j0 & 1) s = -s;
|
||||
}
|
||||
|
||||
return ((hx & 0x80000000) ? -s : s);
|
||||
}
|
||||
|
||||
if (ix >= 0x7f800000)
|
||||
return (vzero / vzero);
|
||||
|
||||
/*
|
||||
* |x| >= 0x1p52 is always an integer, so return +-0.
|
||||
*/
|
||||
return (copysign(0, x));
|
||||
}
|
||||
|
||||
#if LDBL_MANT_DIG == 53
|
||||
__weak_reference(sinpi, sinpil);
|
||||
#endif
|
||||
Loading…
Reference in a new issue