glibc/sysdeps/ieee754/ldbl-128/e_acosl.c

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/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunPro, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/*
2002-08-26 Brian Youmans <3diff@gnu.org> * crypt/crypt.c: Changed copying permission notice to Lesser GPL from Library GPL. * crypt/crypt_util.c: Likewise. * crypt/ufc.c: Likewise. * elf/dl-conflict.c: Likewise. * elf/dl-iteratephdr.c: Likewise. * iconv/iconvconfig.h: Likewise. * linuxthreads/Examples/ex10.c: Likewise. * linuxthreads/Examples/ex11.c: Likewise. * linuxthreads/Examples/ex13.c: Likewise. * linuxthreads/Examples/ex8.c: Likewise. * linuxthreads/Examples/ex9.c: Likewise. * linuxthreads/barrier.c: Likewise. * linuxthreads/events.c: Likewise. * linuxthreads/lockfile.c: Likewise. * linuxthreads/no-tsd.c: Likewise. * linuxthreads/pt-machine.c: Likewise. * linuxthreads/ptclock_gettime.c: Likewise. * linuxthreads/ptclock_settime.c: Likewise. * linuxthreads/rwlock.c: Likewise. * linuxthreads/sysdeps/alpha/pspinlock.c: Likewise. * linuxthreads/sysdeps/alpha/pt-machine.h: Likewise. * linuxthreads/sysdeps/arm/pspinlock.c: Likewise. * linuxthreads/sysdeps/arm/pt-machine.h: Likewise. * linuxthreads/sysdeps/cris/pspinlock.c: Likewise. * linuxthreads/sysdeps/cris/pt-machine.h: Likewise. * linuxthreads/sysdeps/hppa/pspinlock.c: Likewise. * linuxthreads/sysdeps/hppa/pt-machine.h: Likewise. * linuxthreads/sysdeps/i386/i686/pt-machine.h: Likewise. * linuxthreads/sysdeps/i386/pspinlock.c: Likewise. * linuxthreads/sysdeps/i386/pt-machine.h: Likewise. * linuxthreads/sysdeps/i386/useldt.h: Likewise. * linuxthreads/sysdeps/ia64/pspinlock.c: Likewise. * linuxthreads/sysdeps/ia64/pt-machine.h: Likewise. * linuxthreads/sysdeps/m68k/pspinlock.c: Likewise. * linuxthreads/sysdeps/m68k/pt-machine.h: Likewise. * linuxthreads/sysdeps/mips/pspinlock.c: Likewise. * linuxthreads/sysdeps/mips/pt-machine.h: Likewise. * linuxthreads/sysdeps/powerpc/pspinlock.c: Likewise. * linuxthreads/sysdeps/powerpc/pt-machine.h: Likewise. * linuxthreads/sysdeps/pthread/bits/initspin.h: Likewise. * linuxthreads/sysdeps/pthread/bits/libc-lock.h: Likewise. * linuxthreads/sysdeps/pthread/bits/libc-tsd.h: Likewise. * linuxthreads/sysdeps/pthread/getcpuclockid.c: Likewise. * linuxthreads/sysdeps/pthread/posix-timer.h: Likewise. * linuxthreads/sysdeps/pthread/timer_create.c: Likewise. * linuxthreads/sysdeps/pthread/timer_delete.c: Likewise. * linuxthreads/sysdeps/pthread/timer_getoverr.c: Likewise. * linuxthreads/sysdeps/pthread/timer_gettime.c: Likewise. * linuxthreads/sysdeps/pthread/timer_routines.c: Likewise. * linuxthreads/sysdeps/pthread/timer_settime.c: Likewise. * linuxthreads/sysdeps/pthread/tst-timer.c: Likewise. * linuxthreads/sysdeps/s390/pspinlock.c: Likewise. * linuxthreads/sysdeps/s390/s390-32/pt-machine.h: Likewise. * linuxthreads/sysdeps/s390/s390-64/pt-machine.h: Likewise. * linuxthreads/sysdeps/sh/pspinlock.c: Likewise. * linuxthreads/sysdeps/sh/pt-machine.h: Likewise. * linuxthreads/sysdeps/sparc/sparc32/pspinlock.c: Likewise. * linuxthreads/sysdeps/sparc/sparc32/pt-machine.h: Likewise. * linuxthreads/sysdeps/sparc/sparc32/sparcv9/pspinlock.c: Likewise. * linuxthreads/sysdeps/sparc/sparc64/pspinlock.c: Likewise. * linuxthreads/sysdeps/sparc/sparc64/pt-machine.h: Likewise. * linuxthreads/sysdeps/unix/sysv/linux/bits/local_lim.h: Likewise. * linuxthreads/sysdeps/unix/sysv/linux/bits/posix_opt.h: Likewise. * linuxthreads/sysdeps/unix/sysv/linux/bits/sigthread.h: Likewise. * linuxthreads/sysdeps/unix/sysv/linux/hppa/bits/initspin.h: Likewise. * linuxthreads/sysdeps/unix/sysv/linux/i386/bits/posix_opt.h: Likewise. * linuxthreads/tststack.c: Likewise. * linuxthreads/unload.c: Likewise. * linuxthreads/weaks.c: Likewise. * linuxthreads/wrapsyscall.c: Likewise. * malloc/arena.c: Likewise. * malloc/hooks.c: Likewise. * malloc/malloc.c: Likewise. * posix/glob/Makefile.ami: Likewise. * posix/glob/Makefile.in: Likewise. * stdlib/gmp-impl.h: Likewise. * stdlib/gmp.h: Likewise. * sysdeps/generic/dl-iteratephdr-static.c: Likewise. * sysdeps/generic/strnlen.c: Likewise. * sysdeps/mach/hurd/powerpc/bits/sigcontext.h: Likewise. * sysdeps/mach/hurd/recvmsg.c: Likewise. * sysdeps/mach/hurd/sendmsg.c: Likewise. * sysdeps/mach/hurd/spawni.c: Likewise. * sysdeps/mach/powerpc/machine-sp.h: Likewise. * sysdeps/mach/powerpc/sysdep.h: Likewise. * sysdeps/mach/powerpc/thread_state.h: Likewise. * sysdeps/unix/bsd/bsd4.4/bits/socket.h: Likewise. * sysdeps/unix/sysv/linux/ia64/dl-iteratephdr-static.c: Likewise. * sysdeps/x86_64/gmp-mparam.h: Likewise.
2002-08-26 22:40:48 +00:00
Long double expansions are
Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov>
and are incorporated herein by permission of the author. The author
reserves the right to distribute this material elsewhere under different
copying permissions. These modifications are distributed here under
the following terms:
2002-08-26 Brian Youmans <3diff@gnu.org> * crypt/crypt.c: Changed copying permission notice to Lesser GPL from Library GPL. * crypt/crypt_util.c: Likewise. * crypt/ufc.c: Likewise. * elf/dl-conflict.c: Likewise. * elf/dl-iteratephdr.c: Likewise. * iconv/iconvconfig.h: Likewise. * linuxthreads/Examples/ex10.c: Likewise. * linuxthreads/Examples/ex11.c: Likewise. * linuxthreads/Examples/ex13.c: Likewise. * linuxthreads/Examples/ex8.c: Likewise. * linuxthreads/Examples/ex9.c: Likewise. * linuxthreads/barrier.c: Likewise. * linuxthreads/events.c: Likewise. * linuxthreads/lockfile.c: Likewise. * linuxthreads/no-tsd.c: Likewise. * linuxthreads/pt-machine.c: Likewise. * linuxthreads/ptclock_gettime.c: Likewise. * linuxthreads/ptclock_settime.c: Likewise. * linuxthreads/rwlock.c: Likewise. * linuxthreads/sysdeps/alpha/pspinlock.c: Likewise. * linuxthreads/sysdeps/alpha/pt-machine.h: Likewise. * linuxthreads/sysdeps/arm/pspinlock.c: Likewise. * linuxthreads/sysdeps/arm/pt-machine.h: Likewise. * linuxthreads/sysdeps/cris/pspinlock.c: Likewise. * linuxthreads/sysdeps/cris/pt-machine.h: Likewise. * linuxthreads/sysdeps/hppa/pspinlock.c: Likewise. * linuxthreads/sysdeps/hppa/pt-machine.h: Likewise. * linuxthreads/sysdeps/i386/i686/pt-machine.h: Likewise. * linuxthreads/sysdeps/i386/pspinlock.c: Likewise. * linuxthreads/sysdeps/i386/pt-machine.h: Likewise. * linuxthreads/sysdeps/i386/useldt.h: Likewise. * linuxthreads/sysdeps/ia64/pspinlock.c: Likewise. * linuxthreads/sysdeps/ia64/pt-machine.h: Likewise. * linuxthreads/sysdeps/m68k/pspinlock.c: Likewise. * linuxthreads/sysdeps/m68k/pt-machine.h: Likewise. * linuxthreads/sysdeps/mips/pspinlock.c: Likewise. * linuxthreads/sysdeps/mips/pt-machine.h: Likewise. * linuxthreads/sysdeps/powerpc/pspinlock.c: Likewise. * linuxthreads/sysdeps/powerpc/pt-machine.h: Likewise. * linuxthreads/sysdeps/pthread/bits/initspin.h: Likewise. * linuxthreads/sysdeps/pthread/bits/libc-lock.h: Likewise. * linuxthreads/sysdeps/pthread/bits/libc-tsd.h: Likewise. * linuxthreads/sysdeps/pthread/getcpuclockid.c: Likewise. * linuxthreads/sysdeps/pthread/posix-timer.h: Likewise. * linuxthreads/sysdeps/pthread/timer_create.c: Likewise. * linuxthreads/sysdeps/pthread/timer_delete.c: Likewise. * linuxthreads/sysdeps/pthread/timer_getoverr.c: Likewise. * linuxthreads/sysdeps/pthread/timer_gettime.c: Likewise. * linuxthreads/sysdeps/pthread/timer_routines.c: Likewise. * linuxthreads/sysdeps/pthread/timer_settime.c: Likewise. * linuxthreads/sysdeps/pthread/tst-timer.c: Likewise. * linuxthreads/sysdeps/s390/pspinlock.c: Likewise. * linuxthreads/sysdeps/s390/s390-32/pt-machine.h: Likewise. * linuxthreads/sysdeps/s390/s390-64/pt-machine.h: Likewise. * linuxthreads/sysdeps/sh/pspinlock.c: Likewise. * linuxthreads/sysdeps/sh/pt-machine.h: Likewise. * linuxthreads/sysdeps/sparc/sparc32/pspinlock.c: Likewise. * linuxthreads/sysdeps/sparc/sparc32/pt-machine.h: Likewise. * linuxthreads/sysdeps/sparc/sparc32/sparcv9/pspinlock.c: Likewise. * linuxthreads/sysdeps/sparc/sparc64/pspinlock.c: Likewise. * linuxthreads/sysdeps/sparc/sparc64/pt-machine.h: Likewise. * linuxthreads/sysdeps/unix/sysv/linux/bits/local_lim.h: Likewise. * linuxthreads/sysdeps/unix/sysv/linux/bits/posix_opt.h: Likewise. * linuxthreads/sysdeps/unix/sysv/linux/bits/sigthread.h: Likewise. * linuxthreads/sysdeps/unix/sysv/linux/hppa/bits/initspin.h: Likewise. * linuxthreads/sysdeps/unix/sysv/linux/i386/bits/posix_opt.h: Likewise. * linuxthreads/tststack.c: Likewise. * linuxthreads/unload.c: Likewise. * linuxthreads/weaks.c: Likewise. * linuxthreads/wrapsyscall.c: Likewise. * malloc/arena.c: Likewise. * malloc/hooks.c: Likewise. * malloc/malloc.c: Likewise. * posix/glob/Makefile.ami: Likewise. * posix/glob/Makefile.in: Likewise. * stdlib/gmp-impl.h: Likewise. * stdlib/gmp.h: Likewise. * sysdeps/generic/dl-iteratephdr-static.c: Likewise. * sysdeps/generic/strnlen.c: Likewise. * sysdeps/mach/hurd/powerpc/bits/sigcontext.h: Likewise. * sysdeps/mach/hurd/recvmsg.c: Likewise. * sysdeps/mach/hurd/sendmsg.c: Likewise. * sysdeps/mach/hurd/spawni.c: Likewise. * sysdeps/mach/powerpc/machine-sp.h: Likewise. * sysdeps/mach/powerpc/sysdep.h: Likewise. * sysdeps/mach/powerpc/thread_state.h: Likewise. * sysdeps/unix/bsd/bsd4.4/bits/socket.h: Likewise. * sysdeps/unix/sysv/linux/ia64/dl-iteratephdr-static.c: Likewise. * sysdeps/x86_64/gmp-mparam.h: Likewise.
2002-08-26 22:40:48 +00:00
This library is free software; you can redistribute it and/or
modify it under the terms of the GNU Lesser General Public
License as published by the Free Software Foundation; either
version 2.1 of the License, or (at your option) any later version.
This library is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
Lesser General Public License for more details.
You should have received a copy of the GNU Lesser General Public
License along with this library; if not, see
<http://www.gnu.org/licenses/>. */
/* __ieee754_acosl(x)
* Method :
* acos(x) = pi/2 - asin(x)
* acos(-x) = pi/2 + asin(x)
* For |x| <= 0.375
* acos(x) = pi/2 - asin(x)
* Between .375 and .5 the approximation is
* acos(0.4375 + x) = acos(0.4375) + x P(x) / Q(x)
* Between .5 and .625 the approximation is
* acos(0.5625 + x) = acos(0.5625) + x rS(x) / sS(x)
* For x > 0.625,
* acos(x) = 2 asin(sqrt((1-x)/2))
* computed with an extended precision square root in the leading term.
* For x < -0.625
* acos(x) = pi - 2 asin(sqrt((1-|x|)/2))
*
* Special cases:
* if x is NaN, return x itself;
* if |x|>1, return NaN with invalid signal.
*
* Functions needed: __ieee754_sqrtl.
*/
#include <math.h>
#include <math_private.h>
static const long double
one = 1.0L,
pio2_hi = 1.5707963267948966192313216916397514420986L,
pio2_lo = 4.3359050650618905123985220130216759843812E-35L,
/* acos(0.5625 + x) = acos(0.5625) + x rS(x) / sS(x)
-0.0625 <= x <= 0.0625
peak relative error 3.3e-35 */
rS0 = 5.619049346208901520945464704848780243887E0L,
rS1 = -4.460504162777731472539175700169871920352E1L,
rS2 = 1.317669505315409261479577040530751477488E2L,
rS3 = -1.626532582423661989632442410808596009227E2L,
rS4 = 3.144806644195158614904369445440583873264E1L,
rS5 = 9.806674443470740708765165604769099559553E1L,
rS6 = -5.708468492052010816555762842394927806920E1L,
rS7 = -1.396540499232262112248553357962639431922E1L,
rS8 = 1.126243289311910363001762058295832610344E1L,
rS9 = 4.956179821329901954211277873774472383512E-1L,
rS10 = -3.313227657082367169241333738391762525780E-1L,
sS0 = -4.645814742084009935700221277307007679325E0L,
sS1 = 3.879074822457694323970438316317961918430E1L,
sS2 = -1.221986588013474694623973554726201001066E2L,
sS3 = 1.658821150347718105012079876756201905822E2L,
sS4 = -4.804379630977558197953176474426239748977E1L,
sS5 = -1.004296417397316948114344573811562952793E2L,
sS6 = 7.530281592861320234941101403870010111138E1L,
sS7 = 1.270735595411673647119592092304357226607E1L,
sS8 = -1.815144839646376500705105967064792930282E1L,
sS9 = -7.821597334910963922204235247786840828217E-2L,
/* 1.000000000000000000000000000000000000000E0 */
acosr5625 = 9.7338991014954640492751132535550279812151E-1L,
pimacosr5625 = 2.1682027434402468335351320579240000860757E0L,
/* acos(0.4375 + x) = acos(0.4375) + x rS(x) / sS(x)
-0.0625 <= x <= 0.0625
peak relative error 2.1e-35 */
P0 = 2.177690192235413635229046633751390484892E0L,
P1 = -2.848698225706605746657192566166142909573E1L,
P2 = 1.040076477655245590871244795403659880304E2L,
P3 = -1.400087608918906358323551402881238180553E2L,
P4 = 2.221047917671449176051896400503615543757E1L,
P5 = 9.643714856395587663736110523917499638702E1L,
P6 = -5.158406639829833829027457284942389079196E1L,
P7 = -1.578651828337585944715290382181219741813E1L,
P8 = 1.093632715903802870546857764647931045906E1L,
P9 = 5.448925479898460003048760932274085300103E-1L,
P10 = -3.315886001095605268470690485170092986337E-1L,
Q0 = -1.958219113487162405143608843774587557016E0L,
Q1 = 2.614577866876185080678907676023269360520E1L,
Q2 = -9.990858606464150981009763389881793660938E1L,
Q3 = 1.443958741356995763628660823395334281596E2L,
Q4 = -3.206441012484232867657763518369723873129E1L,
Q5 = -1.048560885341833443564920145642588991492E2L,
Q6 = 6.745883931909770880159915641984874746358E1L,
Q7 = 1.806809656342804436118449982647641392951E1L,
Q8 = -1.770150690652438294290020775359580915464E1L,
Q9 = -5.659156469628629327045433069052560211164E-1L,
/* 1.000000000000000000000000000000000000000E0 */
acosr4375 = 1.1179797320499710475919903296900511518755E0L,
pimacosr4375 = 2.0236129215398221908706530535894517323217E0L,
/* asin(x) = x + x^3 pS(x^2) / qS(x^2)
0 <= x <= 0.5
peak relative error 1.9e-35 */
pS0 = -8.358099012470680544198472400254596543711E2L,
pS1 = 3.674973957689619490312782828051860366493E3L,
pS2 = -6.730729094812979665807581609853656623219E3L,
pS3 = 6.643843795209060298375552684423454077633E3L,
pS4 = -3.817341990928606692235481812252049415993E3L,
pS5 = 1.284635388402653715636722822195716476156E3L,
pS6 = -2.410736125231549204856567737329112037867E2L,
pS7 = 2.219191969382402856557594215833622156220E1L,
pS8 = -7.249056260830627156600112195061001036533E-1L,
pS9 = 1.055923570937755300061509030361395604448E-3L,
qS0 = -5.014859407482408326519083440151745519205E3L,
qS1 = 2.430653047950480068881028451580393430537E4L,
qS2 = -4.997904737193653607449250593976069726962E4L,
qS3 = 5.675712336110456923807959930107347511086E4L,
qS4 = -3.881523118339661268482937768522572588022E4L,
qS5 = 1.634202194895541569749717032234510811216E4L,
qS6 = -4.151452662440709301601820849901296953752E3L,
qS7 = 5.956050864057192019085175976175695342168E2L,
qS8 = -4.175375777334867025769346564600396877176E1L;
/* 1.000000000000000000000000000000000000000E0 */
long double
__ieee754_acosl (long double x)
{
long double z, r, w, p, q, s, t, f2;
int32_t ix, sign;
ieee854_long_double_shape_type u;
u.value = x;
sign = u.parts32.w0;
ix = sign & 0x7fffffff;
u.parts32.w0 = ix; /* |x| */
if (ix >= 0x3fff0000) /* |x| >= 1 */
{
if (ix == 0x3fff0000
&& (u.parts32.w1 | u.parts32.w2 | u.parts32.w3) == 0)
{ /* |x| == 1 */
if ((sign & 0x80000000) == 0)
return 0.0; /* acos(1) = 0 */
else
return (2.0 * pio2_hi) + (2.0 * pio2_lo); /* acos(-1)= pi */
}
return (x - x) / (x - x); /* acos(|x| > 1) is NaN */
}
else if (ix < 0x3ffe0000) /* |x| < 0.5 */
{
if (ix < 0x3f8e0000) /* |x| < 2**-113 */
return pio2_hi + pio2_lo;
if (ix < 0x3ffde000) /* |x| < .4375 */
{
/* Arcsine of x. */
z = x * x;
p = (((((((((pS9 * z
+ pS8) * z
+ pS7) * z
+ pS6) * z
+ pS5) * z
+ pS4) * z
+ pS3) * z
+ pS2) * z
+ pS1) * z
+ pS0) * z;
q = (((((((( z
+ qS8) * z
+ qS7) * z
+ qS6) * z
+ qS5) * z
+ qS4) * z
+ qS3) * z
+ qS2) * z
+ qS1) * z
+ qS0;
r = x + x * p / q;
z = pio2_hi - (r - pio2_lo);
return z;
}
/* .4375 <= |x| < .5 */
t = u.value - 0.4375L;
p = ((((((((((P10 * t
+ P9) * t
+ P8) * t
+ P7) * t
+ P6) * t
+ P5) * t
+ P4) * t
+ P3) * t
+ P2) * t
+ P1) * t
+ P0) * t;
q = (((((((((t
+ Q9) * t
+ Q8) * t
+ Q7) * t
+ Q6) * t
+ Q5) * t
+ Q4) * t
+ Q3) * t
+ Q2) * t
+ Q1) * t
+ Q0;
r = p / q;
if (sign & 0x80000000)
r = pimacosr4375 - r;
else
r = acosr4375 + r;
return r;
}
else if (ix < 0x3ffe4000) /* |x| < 0.625 */
{
t = u.value - 0.5625L;
p = ((((((((((rS10 * t
+ rS9) * t
+ rS8) * t
+ rS7) * t
+ rS6) * t
+ rS5) * t
+ rS4) * t
+ rS3) * t
+ rS2) * t
+ rS1) * t
+ rS0) * t;
q = (((((((((t
+ sS9) * t
+ sS8) * t
+ sS7) * t
+ sS6) * t
+ sS5) * t
+ sS4) * t
+ sS3) * t
+ sS2) * t
+ sS1) * t
+ sS0;
if (sign & 0x80000000)
r = pimacosr5625 - p / q;
else
r = acosr5625 + p / q;
return r;
}
else
{ /* |x| >= .625 */
z = (one - u.value) * 0.5;
s = __ieee754_sqrtl (z);
/* Compute an extended precision square root from
the Newton iteration s -> 0.5 * (s + z / s).
The change w from s to the improved value is
w = 0.5 * (s + z / s) - s = (s^2 + z)/2s - s = (z - s^2)/2s.
Express s = f1 + f2 where f1 * f1 is exactly representable.
w = (z - s^2)/2s = (z - f1^2 - 2 f1 f2 - f2^2)/2s .
s + w has extended precision. */
u.value = s;
u.parts32.w2 = 0;
u.parts32.w3 = 0;
f2 = s - u.value;
w = z - u.value * u.value;
w = w - 2.0 * u.value * f2;
w = w - f2 * f2;
w = w / (2.0 * s);
/* Arcsine of s. */
p = (((((((((pS9 * z
+ pS8) * z
+ pS7) * z
+ pS6) * z
+ pS5) * z
+ pS4) * z
+ pS3) * z
+ pS2) * z
+ pS1) * z
+ pS0) * z;
q = (((((((( z
+ qS8) * z
+ qS7) * z
+ qS6) * z
+ qS5) * z
+ qS4) * z
+ qS3) * z
+ qS2) * z
+ qS1) * z
+ qS0;
r = s + (w + s * p / q);
if (sign & 0x80000000)
w = pio2_hi + (pio2_lo - r);
else
w = r;
return 2.0 * w;
}
}
strong_alias (__ieee754_acosl, __acosl_finite)