2 * ====================================================
3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 * Developed at SunPro, a Sun Microsystems, Inc. business.
6 * Permission to use, copy, modify, and distribute this
7 * software is freely granted, provided that this notice
9 * ====================================================
18 Long double expansions contributed by
19 Stephen L. Moshier <moshier@na-net.ornl.gov>
24 * Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
25 * we approximate asin(x) on [0,0.5] by
26 * asin(x) = x + x*x^2*R(x^2)
27 * Between .5 and .625 the approximation is
28 * asin(0.5625 + x) = asin(0.5625) + x rS(x) / sS(x)
30 * asin(x) = pi/2-2*asin(sqrt((1-x)/2))
33 * if x is NaN, return x itself;
34 * if |x|>1, return NaN with invalid signal.
39 static const long double
42 pio2_hi = 1.5707963267948966192313216916397514420986L,
43 pio2_lo = 4.3359050650618905123985220130216759843812E-35L,
44 pio4_hi = 7.8539816339744830961566084581987569936977E-1L,
46 /* coefficient for R(x^2) */
48 /* asin(x) = x + x^3 pS(x^2) / qS(x^2)
50 peak relative error 1.9e-35 */
51 pS0 = -8.358099012470680544198472400254596543711E2L,
52 pS1 = 3.674973957689619490312782828051860366493E3L,
53 pS2 = -6.730729094812979665807581609853656623219E3L,
54 pS3 = 6.643843795209060298375552684423454077633E3L,
55 pS4 = -3.817341990928606692235481812252049415993E3L,
56 pS5 = 1.284635388402653715636722822195716476156E3L,
57 pS6 = -2.410736125231549204856567737329112037867E2L,
58 pS7 = 2.219191969382402856557594215833622156220E1L,
59 pS8 = -7.249056260830627156600112195061001036533E-1L,
60 pS9 = 1.055923570937755300061509030361395604448E-3L,
62 qS0 = -5.014859407482408326519083440151745519205E3L,
63 qS1 = 2.430653047950480068881028451580393430537E4L,
64 qS2 = -4.997904737193653607449250593976069726962E4L,
65 qS3 = 5.675712336110456923807959930107347511086E4L,
66 qS4 = -3.881523118339661268482937768522572588022E4L,
67 qS5 = 1.634202194895541569749717032234510811216E4L,
68 qS6 = -4.151452662440709301601820849901296953752E3L,
69 qS7 = 5.956050864057192019085175976175695342168E2L,
70 qS8 = -4.175375777334867025769346564600396877176E1L,
71 /* 1.000000000000000000000000000000000000000E0 */
73 /* asin(0.5625 + x) = asin(0.5625) + x rS(x) / sS(x)
74 -0.0625 <= x <= 0.0625
75 peak relative error 3.3e-35 */
76 rS0 = -5.619049346208901520945464704848780243887E0L,
77 rS1 = 4.460504162777731472539175700169871920352E1L,
78 rS2 = -1.317669505315409261479577040530751477488E2L,
79 rS3 = 1.626532582423661989632442410808596009227E2L,
80 rS4 = -3.144806644195158614904369445440583873264E1L,
81 rS5 = -9.806674443470740708765165604769099559553E1L,
82 rS6 = 5.708468492052010816555762842394927806920E1L,
83 rS7 = 1.396540499232262112248553357962639431922E1L,
84 rS8 = -1.126243289311910363001762058295832610344E1L,
85 rS9 = -4.956179821329901954211277873774472383512E-1L,
86 rS10 = 3.313227657082367169241333738391762525780E-1L,
88 sS0 = -4.645814742084009935700221277307007679325E0L,
89 sS1 = 3.879074822457694323970438316317961918430E1L,
90 sS2 = -1.221986588013474694623973554726201001066E2L,
91 sS3 = 1.658821150347718105012079876756201905822E2L,
92 sS4 = -4.804379630977558197953176474426239748977E1L,
93 sS5 = -1.004296417397316948114344573811562952793E2L,
94 sS6 = 7.530281592861320234941101403870010111138E1L,
95 sS7 = 1.270735595411673647119592092304357226607E1L,
96 sS8 = -1.815144839646376500705105967064792930282E1L,
97 sS9 = -7.821597334910963922204235247786840828217E-2L,
98 /* 1.000000000000000000000000000000000000000E0 */
100 asinr5625 = 5.9740641664535021430381036628424864397707E-1L;
104 asinl (long double x)
106 long double y, t, p, q;
117 if (y >= 1.0L) /* |x|>= 1 */
120 /* asin(1)=+-pi/2 with inexact */
121 return x * pio2_hi + x * pio2_lo;
123 return (x - x) / (x - x); /* asin(|x|>1) is NaN */
125 else if (y < 0.5L) /* |x| < 0.5 */
127 if (y < 0.000000000000000006938893903907228377647697925567626953125L) /* |x| < 2**-57 */
129 return y; /* return x with inexact if x!=0 */
154 return x + x * (p / q);
157 else if (y < 0.625) /* 0.625 */
160 p = ((((((((((rS10 * t
183 t = asinr5625 + p / q;
186 t = pio2_hi + pio2_lo - 2 * asinl (sqrtl ((1 - y) / 2));
195 printf ("%.18Lg %.18Lg\n",
197 1.5707963267948966192313216916397514420984L);
198 printf ("%.18Lg %.18Lg\n",
199 asinl (0.7071067811865475244008443621048490392848L),
200 0.7853981633974483096156608458198757210492L);
201 printf ("%.18Lg %.18Lg\n",
203 0.5235987755982988730771072305465838140328L);
204 printf ("%.18Lg %.18Lg\n",
205 asinl (0.3090169943749474241022934171828190588600L),
206 0.3141592653589793238462643383279502884196L);
207 printf ("%.18Lg %.18Lg\n",
209 -1.5707963267948966192313216916397514420984L);
210 printf ("%.18Lg %.18Lg\n",
211 asinl (-0.7071067811865475244008443621048490392848L),
212 -0.7853981633974483096156608458198757210492L);
213 printf ("%.18Lg %.18Lg\n",
215 -0.5235987755982988730771072305465838140328L);
216 printf ("%.18Lg %.18Lg\n",
217 asinl (-0.3090169943749474241022934171828190588600L),
218 -0.3141592653589793238462643383279502884196L);