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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. * ==================================================== */ /* Long double expansions are Copyright (C) 2001 Stephen L. Moshier 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 . */ /* acosq(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: sqrtq. */ #include "quadmath-imp.h" static const __float128 one = 1, pio2_hi = 1.5707963267948966192313216916397514420986Q, pio2_lo = 4.3359050650618905123985220130216759843812E-35Q, /* acos(0.5625 + x) = acos(0.5625) + x rS(x) / sS(x) -0.0625 <= x <= 0.0625 peak relative error 3.3e-35 */ rS0 = 5.619049346208901520945464704848780243887E0Q, rS1 = -4.460504162777731472539175700169871920352E1Q, rS2 = 1.317669505315409261479577040530751477488E2Q, rS3 = -1.626532582423661989632442410808596009227E2Q, rS4 = 3.144806644195158614904369445440583873264E1Q, rS5 = 9.806674443470740708765165604769099559553E1Q, rS6 = -5.708468492052010816555762842394927806920E1Q, rS7 = -1.396540499232262112248553357962639431922E1Q, rS8 = 1.126243289311910363001762058295832610344E1Q, rS9 = 4.956179821329901954211277873774472383512E-1Q, rS10 = -3.313227657082367169241333738391762525780E-1Q, sS0 = -4.645814742084009935700221277307007679325E0Q, sS1 = 3.879074822457694323970438316317961918430E1Q, sS2 = -1.221986588013474694623973554726201001066E2Q, sS3 = 1.658821150347718105012079876756201905822E2Q, sS4 = -4.804379630977558197953176474426239748977E1Q, sS5 = -1.004296417397316948114344573811562952793E2Q, sS6 = 7.530281592861320234941101403870010111138E1Q, sS7 = 1.270735595411673647119592092304357226607E1Q, sS8 = -1.815144839646376500705105967064792930282E1Q, sS9 = -7.821597334910963922204235247786840828217E-2Q, /* 1.000000000000000000000000000000000000000E0 */ acosr5625 = 9.7338991014954640492751132535550279812151E-1Q, pimacosr5625 = 2.1682027434402468335351320579240000860757E0Q, /* acos(0.4375 + x) = acos(0.4375) + x rS(x) / sS(x) -0.0625 <= x <= 0.0625 peak relative error 2.1e-35 */ P0 = 2.177690192235413635229046633751390484892E0Q, P1 = -2.848698225706605746657192566166142909573E1Q, P2 = 1.040076477655245590871244795403659880304E2Q, P3 = -1.400087608918906358323551402881238180553E2Q, P4 = 2.221047917671449176051896400503615543757E1Q, P5 = 9.643714856395587663736110523917499638702E1Q, P6 = -5.158406639829833829027457284942389079196E1Q, P7 = -1.578651828337585944715290382181219741813E1Q, P8 = 1.093632715903802870546857764647931045906E1Q, P9 = 5.448925479898460003048760932274085300103E-1Q, P10 = -3.315886001095605268470690485170092986337E-1Q, Q0 = -1.958219113487162405143608843774587557016E0Q, Q1 = 2.614577866876185080678907676023269360520E1Q, Q2 = -9.990858606464150981009763389881793660938E1Q, Q3 = 1.443958741356995763628660823395334281596E2Q, Q4 = -3.206441012484232867657763518369723873129E1Q, Q5 = -1.048560885341833443564920145642588991492E2Q, Q6 = 6.745883931909770880159915641984874746358E1Q, Q7 = 1.806809656342804436118449982647641392951E1Q, Q8 = -1.770150690652438294290020775359580915464E1Q, Q9 = -5.659156469628629327045433069052560211164E-1Q, /* 1.000000000000000000000000000000000000000E0 */ acosr4375 = 1.1179797320499710475919903296900511518755E0Q, pimacosr4375 = 2.0236129215398221908706530535894517323217E0Q, /* asin(x) = x + x^3 pS(x^2) / qS(x^2) 0 <= x <= 0.5 peak relative error 1.9e-35 */ pS0 = -8.358099012470680544198472400254596543711E2Q, pS1 = 3.674973957689619490312782828051860366493E3Q, pS2 = -6.730729094812979665807581609853656623219E3Q, pS3 = 6.643843795209060298375552684423454077633E3Q, pS4 = -3.817341990928606692235481812252049415993E3Q, pS5 = 1.284635388402653715636722822195716476156E3Q, pS6 = -2.410736125231549204856567737329112037867E2Q, pS7 = 2.219191969382402856557594215833622156220E1Q, pS8 = -7.249056260830627156600112195061001036533E-1Q, pS9 = 1.055923570937755300061509030361395604448E-3Q, qS0 = -5.014859407482408326519083440151745519205E3Q, qS1 = 2.430653047950480068881028451580393430537E4Q, qS2 = -4.997904737193653607449250593976069726962E4Q, qS3 = 5.675712336110456923807959930107347511086E4Q, qS4 = -3.881523118339661268482937768522572588022E4Q, qS5 = 1.634202194895541569749717032234510811216E4Q, qS6 = -4.151452662440709301601820849901296953752E3Q, qS7 = 5.956050864057192019085175976175695342168E2Q, qS8 = -4.175375777334867025769346564600396877176E1Q; /* 1.000000000000000000000000000000000000000E0 */ __float128 acosq (__float128 x) { __float128 z, r, w, p, q, s, t, f2; int32_t ix, sign; ieee854_float128 u; u.value = x; sign = u.words32.w0; ix = sign & 0x7fffffff; u.words32.w0 = ix; /* |x| */ if (ix >= 0x3fff0000) /* |x| >= 1 */ { if (ix == 0x3fff0000 && (u.words32.w1 | u.words32.w2 | u.words32.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 - 0.4375Q; 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 - 0.5625Q; 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 = sqrtq (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.words32.w2 = 0; u.words32.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; } }