mirror of
https://sourceware.org/git/glibc.git
synced 2024-11-30 08:40:07 +00:00
320 lines
9.7 KiB
C
320 lines
9.7 KiB
C
/*
|
|
* ====================================================
|
|
* 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.
|
|
* ====================================================
|
|
*/
|
|
|
|
/*
|
|
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:
|
|
|
|
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 >= 0x3ff00000) /* |x| >= 1 */
|
|
{
|
|
if (ix == 0x3ff00000
|
|
&& (u.parts32.w1 | (u.parts32.w2&0x7fffffff) | 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 < 0x3fe00000) /* |x| < 0.5 */
|
|
{
|
|
if (ix < 0x3c600000) /* |x| < 2**-57 */
|
|
return pio2_hi + pio2_lo;
|
|
if (ix < 0x3fde0000) /* |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 < 0x3fe40000) /* |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)
|