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jeremybenn |
/*
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* ====================================================
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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*
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* Developed at SunPro, a Sun Microsystems, Inc. business.
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* Permission to use, copy, modify, and distribute this
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* software is freely granted, provided that this notice
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* is preserved.
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* ====================================================
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*/
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/*
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__float128 expansions are
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Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov>
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and are incorporated herein by permission of the author. The author
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reserves the right to distribute this material elsewhere under different
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copying permissions. These modifications are distributed here under
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the following terms:
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This library is free software; you can redistribute it and/or
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modify it under the terms of the GNU Lesser General Public
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License as published by the Free Software Foundation; either
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version 2.1 of the License, or (at your option) any later version.
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This library is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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Lesser General Public License for more details.
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You should have received a copy of the GNU Lesser General Public
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License along with this library; if not, write to the Free Software
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Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */
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/* __ieee754_acosl(x)
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* Method :
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* acos(x) = pi/2 - asin(x)
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* acos(-x) = pi/2 + asin(x)
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* For |x| <= 0.375
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* acos(x) = pi/2 - asin(x)
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* Between .375 and .5 the approximation is
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* acos(0.4375 + x) = acos(0.4375) + x P(x) / Q(x)
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* Between .5 and .625 the approximation is
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* acos(0.5625 + x) = acos(0.5625) + x rS(x) / sS(x)
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* For x > 0.625,
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* acos(x) = 2 asin(sqrt((1-x)/2))
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* computed with an extended precision square root in the leading term.
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* For x < -0.625
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* acos(x) = pi - 2 asin(sqrt((1-|x|)/2))
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*
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* Special cases:
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* if x is NaN, return x itself;
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* if |x|>1, return NaN with invalid signal.
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*
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* Functions needed: __ieee754_sqrtl.
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*/
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#include "quadmath-imp.h"
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static const __float128
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one = 1.0Q,
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pio2_hi = 1.5707963267948966192313216916397514420986Q,
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pio2_lo = 4.3359050650618905123985220130216759843812E-35Q,
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/* acos(0.5625 + x) = acos(0.5625) + x rS(x) / sS(x)
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-0.0625 <= x <= 0.0625
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peak relative error 3.3e-35 */
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rS0 = 5.619049346208901520945464704848780243887E0Q,
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rS1 = -4.460504162777731472539175700169871920352E1Q,
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rS2 = 1.317669505315409261479577040530751477488E2Q,
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rS3 = -1.626532582423661989632442410808596009227E2Q,
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rS4 = 3.144806644195158614904369445440583873264E1Q,
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rS5 = 9.806674443470740708765165604769099559553E1Q,
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rS6 = -5.708468492052010816555762842394927806920E1Q,
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rS7 = -1.396540499232262112248553357962639431922E1Q,
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rS8 = 1.126243289311910363001762058295832610344E1Q,
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rS9 = 4.956179821329901954211277873774472383512E-1Q,
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rS10 = -3.313227657082367169241333738391762525780E-1Q,
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sS0 = -4.645814742084009935700221277307007679325E0Q,
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sS1 = 3.879074822457694323970438316317961918430E1Q,
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sS2 = -1.221986588013474694623973554726201001066E2Q,
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sS3 = 1.658821150347718105012079876756201905822E2Q,
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sS4 = -4.804379630977558197953176474426239748977E1Q,
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sS5 = -1.004296417397316948114344573811562952793E2Q,
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sS6 = 7.530281592861320234941101403870010111138E1Q,
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sS7 = 1.270735595411673647119592092304357226607E1Q,
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sS8 = -1.815144839646376500705105967064792930282E1Q,
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sS9 = -7.821597334910963922204235247786840828217E-2Q,
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/* 1.000000000000000000000000000000000000000E0 */
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acosr5625 = 9.7338991014954640492751132535550279812151E-1Q,
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pimacosr5625 = 2.1682027434402468335351320579240000860757E0Q,
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/* acos(0.4375 + x) = acos(0.4375) + x rS(x) / sS(x)
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-0.0625 <= x <= 0.0625
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peak relative error 2.1e-35 */
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P0 = 2.177690192235413635229046633751390484892E0Q,
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P1 = -2.848698225706605746657192566166142909573E1Q,
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P2 = 1.040076477655245590871244795403659880304E2Q,
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P3 = -1.400087608918906358323551402881238180553E2Q,
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P4 = 2.221047917671449176051896400503615543757E1Q,
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P5 = 9.643714856395587663736110523917499638702E1Q,
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P6 = -5.158406639829833829027457284942389079196E1Q,
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P7 = -1.578651828337585944715290382181219741813E1Q,
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P8 = 1.093632715903802870546857764647931045906E1Q,
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P9 = 5.448925479898460003048760932274085300103E-1Q,
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P10 = -3.315886001095605268470690485170092986337E-1Q,
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Q0 = -1.958219113487162405143608843774587557016E0Q,
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Q1 = 2.614577866876185080678907676023269360520E1Q,
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Q2 = -9.990858606464150981009763389881793660938E1Q,
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Q3 = 1.443958741356995763628660823395334281596E2Q,
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Q4 = -3.206441012484232867657763518369723873129E1Q,
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Q5 = -1.048560885341833443564920145642588991492E2Q,
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Q6 = 6.745883931909770880159915641984874746358E1Q,
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Q7 = 1.806809656342804436118449982647641392951E1Q,
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Q8 = -1.770150690652438294290020775359580915464E1Q,
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Q9 = -5.659156469628629327045433069052560211164E-1Q,
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/* 1.000000000000000000000000000000000000000E0 */
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acosr4375 = 1.1179797320499710475919903296900511518755E0Q,
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pimacosr4375 = 2.0236129215398221908706530535894517323217E0Q,
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/* asin(x) = x + x^3 pS(x^2) / qS(x^2)
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peak relative error 1.9e-35 */
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pS0 = -8.358099012470680544198472400254596543711E2Q,
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pS1 = 3.674973957689619490312782828051860366493E3Q,
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pS2 = -6.730729094812979665807581609853656623219E3Q,
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pS3 = 6.643843795209060298375552684423454077633E3Q,
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pS4 = -3.817341990928606692235481812252049415993E3Q,
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pS5 = 1.284635388402653715636722822195716476156E3Q,
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pS6 = -2.410736125231549204856567737329112037867E2Q,
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pS7 = 2.219191969382402856557594215833622156220E1Q,
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pS8 = -7.249056260830627156600112195061001036533E-1Q,
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pS9 = 1.055923570937755300061509030361395604448E-3Q,
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qS0 = -5.014859407482408326519083440151745519205E3Q,
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qS1 = 2.430653047950480068881028451580393430537E4Q,
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qS2 = -4.997904737193653607449250593976069726962E4Q,
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qS3 = 5.675712336110456923807959930107347511086E4Q,
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qS4 = -3.881523118339661268482937768522572588022E4Q,
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qS5 = 1.634202194895541569749717032234510811216E4Q,
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qS6 = -4.151452662440709301601820849901296953752E3Q,
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qS7 = 5.956050864057192019085175976175695342168E2Q,
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qS8 = -4.175375777334867025769346564600396877176E1Q;
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/* 1.000000000000000000000000000000000000000E0 */
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__float128
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acosq (__float128 x)
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{
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__float128 z, r, w, p, q, s, t, f2;
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int32_t ix, sign;
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ieee854_float128 u;
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u.value = x;
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sign = u.words32.w0;
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ix = sign & 0x7fffffff;
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u.words32.w0 = ix; /* |x| */
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if (ix >= 0x3fff0000) /* |x| >= 1 */
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{
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if (ix == 0x3fff0000
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&& (u.words32.w1 | u.words32.w2 | u.words32.w3) == 0)
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{ /* |x| == 1 */
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if ((sign & 0x80000000) == 0)
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return 0.0; /* acos(1) = 0 */
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else
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return (2.0 * pio2_hi) + (2.0 * pio2_lo); /* acos(-1)= pi */
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}
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return (x - x) / (x - x); /* acos(|x| > 1) is NaN */
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}
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else if (ix < 0x3ffe0000) /* |x| < 0.5 */
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{
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if (ix < 0x3fc60000) /* |x| < 2**-57 */
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return pio2_hi + pio2_lo;
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if (ix < 0x3ffde000) /* |x| < .4375 */
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{
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/* Arcsine of x. */
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z = x * x;
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p = (((((((((pS9 * z
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+ pS8) * z
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+ pS7) * z
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+ pS6) * z
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+ pS5) * z
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+ pS4) * z
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+ pS3) * z
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+ pS2) * z
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+ pS1) * z
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+ pS0) * z;
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q = (((((((( z
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+ qS8) * z
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+ qS7) * z
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+ qS6) * z
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+ qS5) * z
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+ qS4) * z
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+ qS3) * z
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+ qS2) * z
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+ qS1) * z
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+ qS0;
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r = x + x * p / q;
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z = pio2_hi - (r - pio2_lo);
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return z;
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}
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/* .4375 <= |x| < .5 */
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t = u.value - 0.4375Q;
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p = ((((((((((P10 * t
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+ P9) * t
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+ P8) * t
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+ P7) * t
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+ P6) * t
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+ P5) * t
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+ P4) * t
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+ P3) * t
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+ P2) * t
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| 216 |
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+ P1) * t
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+ P0) * t;
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q = (((((((((t
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+ Q9) * t
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+ Q8) * t
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+ Q7) * t
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+ Q6) * t
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+ Q5) * t
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+ Q4) * t
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+ Q3) * t
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| 227 |
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+ Q2) * t
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| 228 |
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+ Q1) * t
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+ Q0;
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r = p / q;
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if (sign & 0x80000000)
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r = pimacosr4375 - r;
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else
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| 234 |
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r = acosr4375 + r;
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return r;
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}
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else if (ix < 0x3ffe4000) /* |x| < 0.625 */
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{
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| 239 |
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t = u.value - 0.5625Q;
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| 240 |
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p = ((((((((((rS10 * t
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| 241 |
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+ rS9) * t
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| 242 |
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+ rS8) * t
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+ rS7) * t
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| 244 |
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+ rS6) * t
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| 245 |
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+ rS5) * t
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| 246 |
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+ rS4) * t
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| 247 |
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+ rS3) * t
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| 248 |
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+ rS2) * t
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| 249 |
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+ rS1) * t
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| 250 |
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+ rS0) * t;
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| 251 |
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| 252 |
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q = (((((((((t
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| 253 |
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+ sS9) * t
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| 254 |
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+ sS8) * t
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| 255 |
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+ sS7) * t
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| 256 |
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+ sS6) * t
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| 257 |
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+ sS5) * t
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| 258 |
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+ sS4) * t
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| 259 |
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+ sS3) * t
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| 260 |
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+ sS2) * t
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| 261 |
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+ sS1) * t
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| 262 |
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+ sS0;
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| 263 |
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if (sign & 0x80000000)
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| 264 |
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r = pimacosr5625 - p / q;
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| 265 |
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else
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| 266 |
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r = acosr5625 + p / q;
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return r;
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| 268 |
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}
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| 269 |
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else
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{ /* |x| >= .625 */
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| 271 |
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z = (one - u.value) * 0.5;
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| 272 |
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s = sqrtq (z);
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| 273 |
|
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/* Compute an extended precision square root from
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| 274 |
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the Newton iteration s -> 0.5 * (s + z / s).
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| 275 |
|
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The change w from s to the improved value is
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| 276 |
|
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w = 0.5 * (s + z / s) - s = (s^2 + z)/2s - s = (z - s^2)/2s.
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| 277 |
|
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Express s = f1 + f2 where f1 * f1 is exactly representable.
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| 278 |
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w = (z - s^2)/2s = (z - f1^2 - 2 f1 f2 - f2^2)/2s .
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| 279 |
|
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s + w has extended precision. */
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| 280 |
|
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u.value = s;
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| 281 |
|
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u.words32.w2 = 0;
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| 282 |
|
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u.words32.w3 = 0;
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| 283 |
|
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f2 = s - u.value;
|
| 284 |
|
|
w = z - u.value * u.value;
|
| 285 |
|
|
w = w - 2.0 * u.value * f2;
|
| 286 |
|
|
w = w - f2 * f2;
|
| 287 |
|
|
w = w / (2.0 * s);
|
| 288 |
|
|
/* Arcsine of s. */
|
| 289 |
|
|
p = (((((((((pS9 * z
|
| 290 |
|
|
+ pS8) * z
|
| 291 |
|
|
+ pS7) * z
|
| 292 |
|
|
+ pS6) * z
|
| 293 |
|
|
+ pS5) * z
|
| 294 |
|
|
+ pS4) * z
|
| 295 |
|
|
+ pS3) * z
|
| 296 |
|
|
+ pS2) * z
|
| 297 |
|
|
+ pS1) * z
|
| 298 |
|
|
+ pS0) * z;
|
| 299 |
|
|
q = (((((((( z
|
| 300 |
|
|
+ qS8) * z
|
| 301 |
|
|
+ qS7) * z
|
| 302 |
|
|
+ qS6) * z
|
| 303 |
|
|
+ qS5) * z
|
| 304 |
|
|
+ qS4) * z
|
| 305 |
|
|
+ qS3) * z
|
| 306 |
|
|
+ qS2) * z
|
| 307 |
|
|
+ qS1) * z
|
| 308 |
|
|
+ qS0;
|
| 309 |
|
|
r = s + (w + s * p / q);
|
| 310 |
|
|
|
| 311 |
|
|
if (sign & 0x80000000)
|
| 312 |
|
|
w = pio2_hi + (pio2_lo - r);
|
| 313 |
|
|
else
|
| 314 |
|
|
w = r;
|
| 315 |
|
|
return 2.0 * w;
|
| 316 |
|
|
}
|
| 317 |
|
|
}
|
