atan.c

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/*-
 * Copyright (c) 2014-2018 Carsten Sonne Larsen <cs@innolan.net>
 * All rights reserved.
 *
 * Redistribution and use in source and binary forms, with or without
 * modification, are permitted provided that the following conditions
 * are met:
 * 1. Redistributions of source code must retain the above copyright
 *    notice, this list of conditions and the following disclaimer.
 * 2. Redistributions in binary form must reproduce the above copyright
 *    notice, this list of conditions and the following disclaimer in the
 *    documentation and/or other materials provided with the distribution.
 *
 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
 * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
 * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
 * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
 * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
 * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
 * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
 * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
 *
 * Project homepage:
 * https://amath.innolan.net
 *
 * The original source code can be obtained from:
 * http://www.netlib.org/fdlibm/s_atan.c
 * 
 * =================================================================
 * 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.
 * =================================================================
 */

/**
 * @file  atan.c
 * @brief Inverse tangent function
 */

#include "prim.h"

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 */
};

static const double aT[] = {
    3.33333333333329318027e-01,  /* 0x3FD55555, 0x5555550D */
    -1.99999999998764832476e-01, /* 0xBFC99999, 0x9998EBC4 */
    1.42857142725034663711e-01,  /* 0x3FC24924, 0x920083FF */
    -1.11111104054623557880e-01, /* 0xBFBC71C6, 0xFE231671 */
    9.09088713343650656196e-02,  /* 0x3FB745CD, 0xC54C206E */
    -7.69187620504482999495e-02, /* 0xBFB3B0F2, 0xAF749A6D */
    6.66107313738753120669e-02,  /* 0x3FB10D66, 0xA0D03D51 */
    -5.83357013379057348645e-02, /* 0xBFADDE2D, 0x52DEFD9A */
    4.97687799461593236017e-02,  /* 0x3FA97B4B, 0x24760DEB */
    -3.65315727442169155270e-02, /* 0xBFA2B444, 0x2C6A6C2F */
    1.62858201153657823623e-02,  /* 0x3F90AD3A, 0xE322DA11 */
};

static const double
    one = 1.0,
    huge = 1.0e300;

/**
 * @brief   Inverse tangent function
 * @details
 * <pre>
 * 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.
 * </pre>
 */
double atan(double x)
{
    double w, s1, s2, z;
    int32_t ix, hx, id;

    GET_HIGH_WORD(hx, x);
    ix = hx & 0x7fffffff;
    if (ix >= 0x44100000)
    { /* if |x| >= 2^66 */
        uint32_t low;

        GET_LOW_WORD(low, x);
        if (ix > 0x7ff00000 || (ix == 0x7ff00000 && (low != 0)))
            return 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)
                return x; /* raise inexact */
        }
        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 * (aT[0] + w * (aT[2] + w * (aT[4] + w * (aT[6] + w * (aT[8] + w * aT[10])))));
    s2 = w * (aT[1] + w * (aT[3] + w * (aT[5] + w * (aT[7] + w * aT[9]))));
    if (id < 0)
    {
        return x - x * (s1 + s2);
    }

    z = atanhi[id] - ((x * (s1 + s2) - atanlo[id]) - x);
    return (hx < 0) ? -z : z;
}