atan.c

/* Copyright JS Foundation and other contributors, http://js.foundation
 *
 * Licensed under the Apache License, Version 2.0 (the "License");
 * you may not use this file except in compliance with the License.
 * You may obtain a copy of the License at
 *
 *     http://www.apache.org/licenses/LICENSE-2.0
 *
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 * See the License for the specific language governing permissions and
 * limitations under the License.
 *
 * This file is based on work under the following copyright and permission
 * notice:
 *
 *     Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 *     Developed at SunSoft, a Sun Microsystems, Inc. business.
 *     Permission to use, copy, modify, and distribute this
 *     software is freely granted, provided that this notice
 *     is preserved.
 *
 *     @(#)s_atan.c 1.3 95/01/18
 */

#include "jerry-math-internal.h"

/* atan(x)
 *
 * Method:
 *   1. Reduce x to positive by atan(x) = -atan(-x).
 *   2. According to the integer k=4t+0.25 chopped, t=x, the argument
 *      is further reduced to one of the following intervals and the
 *      arctangent of t is evaluated by the corresponding formula:
 *
 *      [0,7/16]      atan(x) = t-t^3*(a1+t^2*(a2+...(a10+t^2*a11)...)
 *      [7/16,11/16]  atan(x) = atan(1/2) + atan( (t-0.5)/(1+t/2) )
 *      [11/16.19/16] atan(x) = atan( 1 ) + atan( (t-1)/(1+t) )
 *      [19/16,39/16] atan(x) = atan(3/2) + atan( (t-1.5)/(1+1.5t) )
 *      [39/16,INF]   atan(x) = atan(INF) + atan( -1/t )
 *
 * Constants:
 * The hexadecimal values are the intended ones for the following
 * constants. The decimal values may be used, provided that the
 * compiler will convert from decimal to binary accurately enough
 * to produce the hexadecimal values shown.
 */

static const double atanhi[] = {
  4.63647609000806093515e-01, /* atan(0.5)hi 0x3FDDAC67, 0x0561BB4F */
  7.85398163397448278999e-01, /* atan(1.0)hi 0x3FE921FB, 0x54442D18 */
  9.82793723247329054082e-01, /* atan(1.5)hi 0x3FEF730B, 0xD281F69B */
  1.57079632679489655800e+00, /* atan(inf)hi 0x3FF921FB, 0x54442D18 */
};

static const double atanlo[] = {
  2.26987774529616870924e-17, /* atan(0.5)lo 0x3C7A2B7F, 0x222F65E2 */
  3.06161699786838301793e-17, /* atan(1.0)lo 0x3C81A626, 0x33145C07 */
  1.39033110312309984516e-17, /* atan(1.5)lo 0x3C700788, 0x7AF0CBBD */
  6.12323399573676603587e-17, /* atan(inf)lo 0x3C91A626, 0x33145C07 */
};

#define aT0  3.33333333333329318027e-01 /* 0x3FD55555, 0x5555550D */
#define aT1  -1.99999999998764832476e-01 /* 0xBFC99999, 0x9998EBC4 */
#define aT2  1.42857142725034663711e-01 /* 0x3FC24924, 0x920083FF */
#define aT3  -1.11111104054623557880e-01 /* 0xBFBC71C6, 0xFE231671 */
#define aT4  9.09088713343650656196e-02 /* 0x3FB745CD, 0xC54C206E */
#define aT5  -7.69187620504482999495e-02 /* 0xBFB3B0F2, 0xAF749A6D */
#define aT6  6.66107313738753120669e-02 /* 0x3FB10D66, 0xA0D03D51 */
#define aT7  -5.83357013379057348645e-02 /* 0xBFADDE2D, 0x52DEFD9A */
#define aT8  4.97687799461593236017e-02 /* 0x3FA97B4B, 0x24760DEB */
#define aT9  -3.65315727442169155270e-02 /* 0xBFA2B444, 0x2C6A6C2F */
#define aT10 1.62858201153657823623e-02 /* 0x3F90AD3A, 0xE322DA11 */

#define one  1.0
#define huge 1.0e300

double
atan (double x)
{
  double w, s1, s2, z;
  int ix, hx, id;

  hx = __HI (x);
  ix = hx & 0x7fffffff;
  if (ix >= 0x44100000) /* if |x| >= 2^66 */
  {
    if (ix > 0x7ff00000 || (ix == 0x7ff00000 && (__LO (x) != 0)))
    {
      return x + x; /* NaN */
    }
    if (hx > 0)
    {
      return atanhi[3] + atanlo[3];
    }
    else
    {
      return -atanhi[3] - atanlo[3];
    }
  }
  if (ix < 0x3fdc0000) /* |x| < 0.4375 */
  {
    if (ix < 0x3e200000) /* |x| < 2^-29 */
    {
      if (huge + x > one) /* raise inexact */
      {
        return x;
      }
    }
    id = -1;
  }
  else
  {
    x = fabs (x);
    if (ix < 0x3ff30000) /* |x| < 1.1875 */
    {
      if (ix < 0x3fe60000) /* 7/16 <= |x| < 11/16 */
      {
        id = 0;
        x = (2.0 * x - one) / (2.0 + x);
      }
      else /* 11/16 <= |x| < 19/16 */
      {
        id = 1;
        x = (x - one) / (x + one);
      }
    }
    else
    {
      if (ix < 0x40038000) /* |x| < 2.4375 */
      {
        id = 2;
        x = (x - 1.5) / (one + 1.5 * x);
      }
      else /* 2.4375 <= |x| < 2^66 */
      {
        id = 3;
        x = -1.0 / x;
      }
    }
  }
  /* end of argument reduction */
  z = x * x;
  w = z * z;
  /* break sum from i=0 to 10 aT[i] z**(i+1) into odd and even poly */
  s1 = z * (aT0 + w * (aT2 + w * (aT4 + w * (aT6 + w * (aT8 + w * aT10)))));
  s2 = w * (aT1 + w * (aT3 + w * (aT5 + w * (aT7 + w * aT9))));
  if (id < 0)
  {
    return x - x * (s1 + s2);
  }
  else
  {
    z = atanhi[id] - ((x * (s1 + s2) - atanlo[id]) - x);
    return (hx < 0) ? -z : z;
  }
} /* atan */

#undef aT0
#undef aT1
#undef aT2
#undef aT3
#undef aT4
#undef aT5
#undef aT6
#undef aT7
#undef aT8
#undef aT9
#undef aT10
#undef one
#undef huge