/* * ==================================================== * 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 contributed by Stephen L. Moshier */ /* __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" #ifdef __STDC__ static const long double #else static long double #endif 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 */ #ifdef __STDC__ long double __ieee754_acosl (long double x) #else long double __ieee754_acosl (x) long double x; #endif { 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 < 0x3fc60000) /* |x| < 2**-57 */ 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; } }