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[/] [openrisc/] [trunk/] [gnu-dev/] [or1k-gcc/] [libquadmath/] [math/] [acosq.c] - Blame information for rev 742

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Line No. Rev Author Line
1 740 jeremybenn 
/*
2 
 * ====================================================
3 
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
4 
 *
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
8 
 * is preserved.
9 
 * ====================================================
10 
 */
11 
 
12 
/*
13 
   __float128 expansions are
14 
   Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov>
15 
   and are incorporated herein by permission of the author.  The author
16 
   reserves the right to distribute this material elsewhere under different
17 
   copying permissions.  These modifications are distributed here under
18 
   the following terms:
19 
 
20 
    This library is free software; you can redistribute it and/or
21 
    modify it under the terms of the GNU Lesser General Public
22 
    License as published by the Free Software Foundation; either
23 
    version 2.1 of the License, or (at your option) any later version.
24 
 
25 
    This library is distributed in the hope that it will be useful,
26 
    but WITHOUT ANY WARRANTY; without even the implied warranty of
27 
    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
28 
    Lesser General Public License for more details.
29 
 
30 
    You should have received a copy of the GNU Lesser General Public
31 
    License along with this library; if not, write to the Free Software
32 
    Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307  USA */
33 
 
34 
/* __ieee754_acosl(x)
35 
 * Method :
36 
 *      acos(x)  = pi/2 - asin(x)
37 
 *      acos(-x) = pi/2 + asin(x)
38 
 * For |x| <= 0.375
39 
 *      acos(x) = pi/2 - asin(x)
40 
 * Between .375 and .5 the approximation is
41 
 *      acos(0.4375 + x) = acos(0.4375) + x P(x) / Q(x)
42 
 * Between .5 and .625 the approximation is
43 
 *      acos(0.5625 + x) = acos(0.5625) + x rS(x) / sS(x)
44 
 * For x > 0.625,
45 
 *      acos(x) = 2 asin(sqrt((1-x)/2))
46 
 *      computed with an extended precision square root in the leading term.
47 
 * For x < -0.625
48 
 *      acos(x) = pi - 2 asin(sqrt((1-|x|)/2))
49 
 *
50 
 * Special cases:
51 
 *      if x is NaN, return x itself;
52 
 *      if |x|>1, return NaN with invalid signal.
53 
 *
54 
 * Functions needed: __ieee754_sqrtl.
55 
 */
56 
 
57 
#include "quadmath-imp.h"
58 
 
59 
static const __float128
60 
  one = 1.0Q,
61 
  pio2_hi = 1.5707963267948966192313216916397514420986Q,
62 
  pio2_lo = 4.3359050650618905123985220130216759843812E-35Q,
63 
 
64 
  /* acos(0.5625 + x) = acos(0.5625) + x rS(x) / sS(x)
65 
     -0.0625 <= x <= 0.0625
66 
     peak relative error 3.3e-35  */
67 
 
68 
  rS0 =  5.619049346208901520945464704848780243887E0Q,
69 
  rS1 = -4.460504162777731472539175700169871920352E1Q,
70 
  rS2 =  1.317669505315409261479577040530751477488E2Q,
71 
  rS3 = -1.626532582423661989632442410808596009227E2Q,
72 
  rS4 =  3.144806644195158614904369445440583873264E1Q,
73 
  rS5 =  9.806674443470740708765165604769099559553E1Q,
74 
  rS6 = -5.708468492052010816555762842394927806920E1Q,
75 
  rS7 = -1.396540499232262112248553357962639431922E1Q,
76 
  rS8 =  1.126243289311910363001762058295832610344E1Q,
77 
  rS9 =  4.956179821329901954211277873774472383512E-1Q,
78 
  rS10 = -3.313227657082367169241333738391762525780E-1Q,
79 
 
80 
  sS0 = -4.645814742084009935700221277307007679325E0Q,
81 
  sS1 =  3.879074822457694323970438316317961918430E1Q,
82 
  sS2 = -1.221986588013474694623973554726201001066E2Q,
83 
  sS3 =  1.658821150347718105012079876756201905822E2Q,
84 
  sS4 = -4.804379630977558197953176474426239748977E1Q,
85 
  sS5 = -1.004296417397316948114344573811562952793E2Q,
86 
  sS6 =  7.530281592861320234941101403870010111138E1Q,
87 
  sS7 =  1.270735595411673647119592092304357226607E1Q,
88 
  sS8 = -1.815144839646376500705105967064792930282E1Q,
89 
  sS9 = -7.821597334910963922204235247786840828217E-2Q,
90 
  /* 1.000000000000000000000000000000000000000E0 */
91 
 
92 
  acosr5625 = 9.7338991014954640492751132535550279812151E-1Q,
93 
  pimacosr5625 = 2.1682027434402468335351320579240000860757E0Q,
94 
 
95 
  /* acos(0.4375 + x) = acos(0.4375) + x rS(x) / sS(x)
96 
     -0.0625 <= x <= 0.0625
97 
     peak relative error 2.1e-35  */
98 
 
99 
  P0 =  2.177690192235413635229046633751390484892E0Q,
100 
  P1 = -2.848698225706605746657192566166142909573E1Q,
101 
  P2 =  1.040076477655245590871244795403659880304E2Q,
102 
  P3 = -1.400087608918906358323551402881238180553E2Q,
103 
  P4 =  2.221047917671449176051896400503615543757E1Q,
104 
  P5 =  9.643714856395587663736110523917499638702E1Q,
105 
  P6 = -5.158406639829833829027457284942389079196E1Q,
106 
  P7 = -1.578651828337585944715290382181219741813E1Q,
107 
  P8 =  1.093632715903802870546857764647931045906E1Q,
108 
  P9 =  5.448925479898460003048760932274085300103E-1Q,
109 
  P10 = -3.315886001095605268470690485170092986337E-1Q,
110 
  Q0 = -1.958219113487162405143608843774587557016E0Q,
111 
  Q1 =  2.614577866876185080678907676023269360520E1Q,
112 
  Q2 = -9.990858606464150981009763389881793660938E1Q,
113 
  Q3 =  1.443958741356995763628660823395334281596E2Q,
114 
  Q4 = -3.206441012484232867657763518369723873129E1Q,
115 
  Q5 = -1.048560885341833443564920145642588991492E2Q,
116 
  Q6 =  6.745883931909770880159915641984874746358E1Q,
117 
  Q7 =  1.806809656342804436118449982647641392951E1Q,
118 
  Q8 = -1.770150690652438294290020775359580915464E1Q,
119 
  Q9 = -5.659156469628629327045433069052560211164E-1Q,
120 
  /* 1.000000000000000000000000000000000000000E0 */
121 
 
122 
  acosr4375 = 1.1179797320499710475919903296900511518755E0Q,
123 
  pimacosr4375 = 2.0236129215398221908706530535894517323217E0Q,
124 
 
125 
  /* asin(x) = x + x^3 pS(x^2) / qS(x^2)
126 
 
127 
     peak relative error 1.9e-35  */
128 
  pS0 = -8.358099012470680544198472400254596543711E2Q,
129 
  pS1 =  3.674973957689619490312782828051860366493E3Q,
130 
  pS2 = -6.730729094812979665807581609853656623219E3Q,
131 
  pS3 =  6.643843795209060298375552684423454077633E3Q,
132 
  pS4 = -3.817341990928606692235481812252049415993E3Q,
133 
  pS5 =  1.284635388402653715636722822195716476156E3Q,
134 
  pS6 = -2.410736125231549204856567737329112037867E2Q,
135 
  pS7 =  2.219191969382402856557594215833622156220E1Q,
136 
  pS8 = -7.249056260830627156600112195061001036533E-1Q,
137 
  pS9 =  1.055923570937755300061509030361395604448E-3Q,
138 
 
139 
  qS0 = -5.014859407482408326519083440151745519205E3Q,
140 
  qS1 =  2.430653047950480068881028451580393430537E4Q,
141 
  qS2 = -4.997904737193653607449250593976069726962E4Q,
142 
  qS3 =  5.675712336110456923807959930107347511086E4Q,
143 
  qS4 = -3.881523118339661268482937768522572588022E4Q,
144 
  qS5 =  1.634202194895541569749717032234510811216E4Q,
145 
  qS6 = -4.151452662440709301601820849901296953752E3Q,
146 
  qS7 =  5.956050864057192019085175976175695342168E2Q,
147 
  qS8 = -4.175375777334867025769346564600396877176E1Q;
148 
  /* 1.000000000000000000000000000000000000000E0 */
149 
 
150 
__float128
151 
acosq (__float128 x)
152 
{
153 
  __float128 z, r, w, p, q, s, t, f2;
154 
  int32_t ix, sign;
155 
  ieee854_float128 u;
156 
 
157 
  u.value = x;
158 
  sign = u.words32.w0;
159 
  ix = sign & 0x7fffffff;
160 
  u.words32.w0 = ix;            /* |x| */
161 
  if (ix >= 0x3fff0000)         /* |x| >= 1 */
162 
    {
163 
      if (ix == 0x3fff0000
164 
          && (u.words32.w1 | u.words32.w2 | u.words32.w3) == 0)
165 
        {                       /* |x| == 1 */
166 
          if ((sign & 0x80000000) == 0)
167 
            return 0.0;         /* acos(1) = 0  */
168 
          else
169 
            return (2.0 * pio2_hi) + (2.0 * pio2_lo);   /* acos(-1)= pi */
170 
        }
171 
      return (x - x) / (x - x); /* acos(|x| > 1) is NaN */
172 
    }
173 
  else if (ix < 0x3ffe0000)     /* |x| < 0.5 */
174 
    {
175 
      if (ix < 0x3fc60000)      /* |x| < 2**-57 */
176 
        return pio2_hi + pio2_lo;
177 
      if (ix < 0x3ffde000)      /* |x| < .4375 */
178 
        {
179 
          /* Arcsine of x.  */
180 
          z = x * x;
181 
          p = (((((((((pS9 * z
182 
                       + pS8) * z
183 
                      + pS7) * z
184 
                     + pS6) * z
185 
                    + pS5) * z
186 
                   + pS4) * z
187 
                  + pS3) * z
188 
                 + pS2) * z
189 
                + pS1) * z
190 
               + pS0) * z;
191 
          q = (((((((( z
192 
                       + qS8) * z
193 
                     + qS7) * z
194 
                    + qS6) * z
195 
                   + qS5) * z
196 
                  + qS4) * z
197 
                 + qS3) * z
198 
                + qS2) * z
199 
               + qS1) * z
200 
            + qS0;
201 
          r = x + x * p / q;
202 
          z = pio2_hi - (r - pio2_lo);
203 
          return z;
204 
        }
205 
      /* .4375 <= |x| < .5 */
206 
      t = u.value - 0.4375Q;
207 
      p = ((((((((((P10 * t
208 
                    + P9) * t
209 
                   + P8) * t
210 
                  + P7) * t
211 
                 + P6) * t
212 
                + P5) * t
213 
               + P4) * t
214 
              + P3) * t
215 
             + P2) * t
216 
            + P1) * t
217 
           + P0) * t;
218 
 
219 
      q = (((((((((t
220 
                   + Q9) * t
221 
                  + Q8) * t
222 
                 + Q7) * t
223 
                + Q6) * t
224 
               + Q5) * t
225 
              + Q4) * t
226 
             + Q3) * t
227 
            + Q2) * t
228 
           + Q1) * t
229 
        + Q0;
230 
      r = p / q;
231 
      if (sign & 0x80000000)
232 
        r = pimacosr4375 - r;
233 
      else
234 
        r = acosr4375 + r;
235 
      return r;
236 
    }
237 
  else if (ix < 0x3ffe4000)     /* |x| < 0.625 */
238 
    {
239 
      t = u.value - 0.5625Q;
240 
      p = ((((((((((rS10 * t
241 
                    + rS9) * t
242 
                   + rS8) * t
243 
                  + rS7) * t
244 
                 + rS6) * t
245 
                + rS5) * t
246 
               + rS4) * t
247 
              + rS3) * t
248 
             + rS2) * t
249 
            + rS1) * t
250 
           + rS0) * t;
251 
 
252 
      q = (((((((((t
253 
                   + sS9) * t
254 
                  + sS8) * t
255 
                 + sS7) * t
256 
                + sS6) * t
257 
               + sS5) * t
258 
              + sS4) * t
259 
             + sS3) * t
260 
            + sS2) * t
261 
           + sS1) * t
262 
        + sS0;
263 
      if (sign & 0x80000000)
264 
        r = pimacosr5625 - p / q;
265 
      else
266 
        r = acosr5625 + p / q;
267 
      return r;
268 
    }
269 
  else
270 
    {                           /* |x| >= .625 */
271 
      z = (one - u.value) * 0.5;
272 
      s = sqrtq (z);
273 
      /* Compute an extended precision square root from
274 
         the Newton iteration  s -> 0.5 * (s + z / s).
275 
         The change w from s to the improved value is
276 
            w = 0.5 * (s + z / s) - s  = (s^2 + z)/2s - s = (z - s^2)/2s.
277 
          Express s = f1 + f2 where f1 * f1 is exactly representable.
278 
          w = (z - s^2)/2s = (z - f1^2 - 2 f1 f2 - f2^2)/2s .
279 
          s + w has extended precision.  */
280 
      u.value = s;
281 
      u.words32.w2 = 0;
282 
      u.words32.w3 = 0;
283 
      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 
}

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