/* * ==================================================== * 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_asin(x) * Method : * Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ... * we approximate asin(x) on [0,0.5] by * asin(x) = x + x*x^2*R(x^2) * Between .5 and .625 the approximation is * asin(0.5625 + x) = asin(0.5625) + x rS(x) / sS(x) * For x in [0.625,1] * asin(x) = pi/2-2*asin(sqrt((1-x)/2)) * Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2; * then for x>0.98 * asin(x) = pi/2 - 2*(s+s*z*R(z)) * = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo) * For x<=0.98, let pio4_hi = pio2_hi/2, then * f = hi part of s; * c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z) * and * asin(x) = pi/2 - 2*(s+s*z*R(z)) * = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo) * = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c)) * * Special cases: * if x is NaN, return x itself; * if |x|>1, return NaN with invalid signal. * */ #include "math.h" #include "math_private.h" long double sqrtl (long double); #ifdef __STDC__ static const long double #else static long double #endif one = 1.0L, huge = 1.0e+4932L, pio2_hi = 1.5707963267948966192313216916397514420986L, pio2_lo = 4.3359050650618905123985220130216759843812E-35L, pio4_hi = 7.8539816339744830961566084581987569936977E-1L, /* coefficient for R(x^2) */ /* 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 */ /* asin(0.5625 + x) = asin(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 */ asinr5625 = 5.9740641664535021430381036628424864397707E-1L; #ifdef __STDC__ long double __ieee754_asinl (long double x) #else double __ieee754_asinl (x) long double x; #endif { long double t, w, p, q, c, r, s; int32_t ix, sign, flag; ieee854_long_double_shape_type u; flag = 0; 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) /* asin(1)=+-pi/2 with inexact */ return x * pio2_hi + x * pio2_lo; return (x - x) / (x - x); /* asin(|x|>1) is NaN */ } else if (ix < 0x3ffe0000) /* |x| < 0.5 */ { if (ix < 0x3fc60000) /* |x| < 2**-57 */ { if (huge + x > one) return x; /* return x with inexact if x!=0 */ } else { t = x * x; /* Mark to use pS, qS later on. */ flag = 1; } } else if (ix < 0x3ffe4000) /* 0.625 */ { t = u.value - 0.5625; 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; t = asinr5625 + p / q; if ((sign & 0x80000000) == 0) return t; else return -t; } else { /* 1 > |x| >= 0.625 */ w = one - u.value; t = w * 0.5; } p = (((((((((pS9 * t + pS8) * t + pS7) * t + pS6) * t + pS5) * t + pS4) * t + pS3) * t + pS2) * t + pS1) * t + pS0) * t; q = (((((((( t + qS8) * t + qS7) * t + qS6) * t + qS5) * t + qS4) * t + qS3) * t + qS2) * t + qS1) * t + qS0; if (flag) /* 2^-57 < |x| < 0.5 */ { w = p / q; return x + x * w; } s = __ieee754_sqrtl (t); if (ix >= 0x3ffef333) /* |x| > 0.975 */ { w = p / q; t = pio2_hi - (2.0 * (s + s * w) - pio2_lo); } else { u.value = s; u.parts32.w3 = 0; u.parts32.w2 = 0; w = u.value; c = (t - w * w) / (s + w); r = p / q; p = 2.0 * s * r - (pio2_lo - 2.0 * c); q = pio4_hi - 2.0 * w; t = pio4_hi - (p - q); } if ((sign & 0x80000000) == 0) return t; else return -t; }