xref: /third_party/jerryscript/jerry-libm/trig.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, 2004 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.
 *
 *     @(#)k_rem_pio2.c 1.3 95/01/18
 *     @(#)e_rem_pio2.c 1.4 95/01/18
 *     @(#)k_sin.c 1.3 95/01/18
 *     @(#)k_cos.c 1.3 95/01/18
 *     @(#)k_tan.c 1.5 04/04/22
 *     @(#)s_sin.c 1.3 95/01/18
 *     @(#)s_cos.c 1.3 95/01/18
 *     @(#)s_tan.c 1.3 95/01/18
 */

#include "jerry-libm-internal.h"

#define zero    0.00000000000000000000e+00 /* 0x00000000, 0x00000000 */
#define half    5.00000000000000000000e-01 /* 0x3FE00000, 0x00000000 */
#define one     1.00000000000000000000e+00 /* 0x3FF00000, 0x00000000 */
#define two24   1.67772160000000000000e+07 /* 0x41700000, 0x00000000 */
#define twon24  5.96046447753906250000e-08 /* 0x3E700000, 0x00000000 */

/* __kernel_rem_pio2(x,y,e0,nx,prec)
 * double x[],y[]; int e0,nx,prec;
 *
 * __kernel_rem_pio2 return the last three digits of N with
 *              y = x - N*pi/2
 * so that |y| < pi/2.
 *
 * The method is to compute the integer (mod 8) and fraction parts of
 * (2/pi)*x without doing the full multiplication. In general we
 * skip the part of the product that are known to be a huge integer (
 * more accurately, = 0 mod 8 ). Thus the number of operations are
 * independent of the exponent of the input.
 *
 * (2/pi) is represented by an array of 24-bit integers in ipio2[].
 *
 * Input parameters:
 *      x[]     The input value (must be positive) is broken into nx
 *              pieces of 24-bit integers in double precision format.
 *              x[i] will be the i-th 24 bit of x. The scaled exponent
 *              of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
 *              match x's up to 24 bits.
 *
 *              Example of breaking a double positive z into x[0]+x[1]+x[2]:
 *                      e0 = ilogb(z)-23
 *                      z  = scalbn(z,-e0)
 *              for i = 0,1,2
 *                      x[i] = floor(z)
 *                      z    = (z-x[i])*2**24
 *
 *      y[]     ouput result in an array of double precision numbers.
 *              The dimension of y[] is:
 *                      24-bit  precision       1
 *                      53-bit  precision       2
 *                      64-bit  precision       2
 *                      113-bit precision       3
 *              The actual value is the sum of them. Thus for 113-bit
 *              precison, one may have to do something like:
 *
 *              long double t,w,r_head, r_tail;
 *              t = (long double)y[2] + (long double)y[1];
 *              w = (long double)y[0];
 *              r_head = t+w;
 *              r_tail = w - (r_head - t);
 *
 *      e0      The exponent of x[0]
 *
 *      nx      dimension of x[]
 *
 *      prec    an integer indicating the precision:
 *                      0       24  bits (single)
 *                      1       53  bits (double)
 *                      2       64  bits (extended)
 *                      3       113 bits (quad)
 *
 * External function:
 *      double scalbn(), floor();
 *
 * Here is the description of some local variables:
 *
 *      ipio2[] integer array, contains the (24*i)-th to (24*i+23)-th
 *              bit of 2/pi after binary point. The corresponding
 *              floating value is
 *
 *                      ipio2[i] * 2^(-24(i+1)).
 *
 *      jk      jk+1 is the initial number of terms of ipio2[] needed
 *              in the computation. The recommended value is 2,3,4,
 *              6 for single, double, extended,and quad.
 *
 *      jz      local integer variable indicating the number of
 *              terms of ipio2[] used.
 *
 *      jx      nx - 1
 *
 *      jv      index for pointing to the suitable ipio2[] for the
 *              computation. In general, we want
 *                      ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
 *              is an integer. Thus
 *                      e0-3-24*jv >= 0 or (e0-3)/24 >= jv
 *              Hence jv = max(0,(e0-3)/24).
 *
 *      jp      jp+1 is the number of terms in PIo2[] needed, jp = jk.
 *
 *      q[]     double array with integral value, representing the
 *              24-bits chunk of the product of x and 2/pi.
 *
 *      q0      the corresponding exponent of q[0]. Note that the
 *              exponent for q[i] would be q0-24*i.
 *
 *      PIo2[]  double precision array, obtained by cutting pi/2
 *              into 24 bits chunks.
 *
 *      f[]     ipio2[] in floating point
 *
 *      iq[]    integer array by breaking up q[] in 24-bits chunk.
 *
 *      fq[]    final product of x*(2/pi) in fq[0],..,fq[jk]
 *
 *      ih      integer. If >0 it indicates q[] is >= 0.5, hence
 *              it also indicates the *sign* of the result.
 */

/*
 * 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.
 */

/* initial value for jk */
static const int init_jk[] =
{
  2, 3, 4, 6
};

static const double PIo2[] =
{
  1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
  7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
  5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
  3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
  1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
  1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
  2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
  2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
};

/*
 * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi
 */
static const int ipio2[] =
{
  0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62,
  0x95993C, 0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A,
  0x424DD2, 0xE00649, 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129,
  0xA73EE8, 0x8235F5, 0x2EBB44, 0x84E99C, 0x7026B4, 0x5F7E41,
  0x3991D6, 0x398353, 0x39F49C, 0x845F8B, 0xBDF928, 0x3B1FF8,
  0x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D, 0x367ECF,
  0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5,
  0xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08,
  0x560330, 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3,
  0x91615E, 0xE61B08, 0x659985, 0x5F14A0, 0x68408D, 0xFFD880,
  0x4D7327, 0x310606, 0x1556CA, 0x73A8C9, 0x60E27B, 0xC08C6B,
};

static int
__kernel_rem_pio2 (double *x, double *y, int e0, int nx, int prec)
{
  int jz, jx, jv, jp, jk, carry, n, iq[20], i, j, k, m, q0, ih;
  double z, fw, f[20], fq[20], q[20];

  /* initialize jk */
  jk = init_jk[prec];
  jp = jk;

  /* determine jx, jv, q0, note that 3 > q0 */
  jx = nx - 1;
  jv = (e0 - 3) / 24;
  if (jv < 0)
  {
    jv = 0;
  }
  q0 = e0 - 24 * (jv + 1);

  /* set up f[0] to f[jx + jk] where f[jx + jk] = ipio2[jv + jk] */
  j = jv - jx;
  m = jx + jk;
  for (i = 0; i <= m; i++, j++)
  {
    f[i] = (j < 0) ? zero : (double) ipio2[j];
  }

  /* compute q[0], q[1], ... q[jk] */
  for (i = 0; i <= jk; i++)
  {
    for (j = 0, fw = 0.0; j <= jx; j++)
    {
      fw += x[j] * f[jx + i - j];
    }
    q[i] = fw;
  }

  jz = jk;
recompute:
  /* distill q[] into iq[] reversingly */
  for (i = 0, j = jz, z = q[jz]; j > 0; i++, j--)
  {
    fw = (double) ((int) (twon24 * z));
    iq[i] = (int) (z - two24 * fw);
    z = q[j - 1] + fw;
  }

  /* compute n */
  z = scalbn (z, q0); /* actual value of z */
  z -= 8.0 * floor (z * 0.125); /* trim off integer >= 8 */
  n = (int) z;
  z -= (double) n;
  ih = 0;
  if (q0 > 0) /* need iq[jz - 1] to determine n */
  {
    i = (iq[jz - 1] >> (24 - q0));
    n += i;
    iq[jz - 1] -= i << (24 - q0);
    ih = iq[jz - 1] >> (23 - q0);
  }
  else if (q0 == 0)
  {
    ih = iq[jz - 1] >> 23;
  }
  else if (z >= 0.5)
  {
    ih = 2;
  }

  if (ih > 0) /* q > 0.5 */
  {
    n += 1;
    carry = 0;
    for (i = 0; i < jz; i++) /* compute 1 - q */
    {
      j = iq[i];
      if (carry == 0)
      {
        if (j != 0)
        {
          carry = 1;
          iq[i] = 0x1000000 - j;
        }
      }
      else
      {
        iq[i] = 0xffffff - j;
      }
    }
    if (q0 > 0) /* rare case: chance is 1 in 12 */
    {
      switch (q0)
      {
        case 1:
        {
          iq[jz - 1] &= 0x7fffff;
          break;
        }
        case 2:
        {
          iq[jz - 1] &= 0x3fffff;
          break;
        }
      }
    }
    if (ih == 2)
    {
      z = one - z;
      if (carry != 0)
      {
        z -= scalbn (one, q0);
      }
    }
  }

  /* check if recomputation is needed */
  if (z == zero)
  {
    j = 0;
    for (i = jz - 1; i >= jk; i--)
    {
      j |= iq[i];
    }
    if (j == 0) /* need recomputation */
    {
      for (k = 1; iq[jk - k] == 0; k++) /* k = no. of terms needed */
      {
      }

      for (i = jz + 1; i <= jz + k; i++) /* add q[jz + 1] to q[jz + k] */
      {
        f[jx + i] = (double) ipio2[jv + i];
        for (j = 0, fw = 0.0; j <= jx; j++)
        {
          fw += x[j] * f[jx + i - j];
        }
        q[i] = fw;
      }
      jz += k;
      goto recompute;
    }
  }

  /* chop off zero terms */
  if (z == 0.0)
  {
    jz -= 1;
    q0 -= 24;
    while (iq[jz] == 0)
    {
      jz--;
      q0 -= 24;
    }
  }
  else
  { /* break z into 24-bit if necessary */
    z = scalbn (z, -q0);
    if (z >= two24)
    {
      fw = (double) ((int) (twon24 * z));
      iq[jz] = (int) (z - two24 * fw);
      jz += 1;
      q0 += 24;
      iq[jz] = (int) fw;
    }
    else
    {
      iq[jz] = (int) z;
    }
  }

  /* convert integer "bit" chunk to floating-point value */
  fw = scalbn (one, q0);
  for (i = jz; i >= 0; i--)
  {
    q[i] = fw * (double) iq[i];
    fw *= twon24;
  }

  /* compute PIo2[0, ..., jp] * q[jz, ..., 0] */
  for (i = jz; i >= 0; i--)
  {
    for (fw = 0.0, k = 0; k <= jp && k <= jz - i; k++)
    {
      fw += PIo2[k] * q[i + k];
    }
    fq[jz - i] = fw;
  }

  /* compress fq[] into y[] */
  switch (prec)
  {
    case 0:
    {
      fw = 0.0;
      for (i = jz; i >= 0; i--)
      {
        fw += fq[i];
      }
      y[0] = (ih == 0) ? fw : -fw;
      break;
    }
    case 1:
    case 2:
    {
      fw = 0.0;
      for (i = jz; i >= 0; i--)
      {
        fw += fq[i];
      }
      y[0] = (ih == 0) ? fw : -fw;
      fw = fq[0] - fw;
      for (i = 1; i <= jz; i++)
      {
        fw += fq[i];
      }
      y[1] = (ih == 0) ? fw : -fw;
      break;
    }
    case 3: /* painful */
    {
      for (i = jz; i > 0; i--)
      {
        fw = fq[i - 1] + fq[i];
        fq[i] += fq[i - 1] - fw;
        fq[i - 1] = fw;
      }
      for (i = jz; i > 1; i--)
      {
        fw = fq[i - 1] + fq[i];
        fq[i] += fq[i - 1] - fw;
        fq[i - 1] = fw;
      }
      for (fw = 0.0, i = jz; i >= 2; i--)
      {
        fw += fq[i];
      }
      if (ih == 0)
      {
        y[0] = fq[0];
        y[1] = fq[1];
        y[2] = fw;
      }
      else
      {
        y[0] = -fq[0];
        y[1] = -fq[1];
        y[2] = -fw;
      }
    }
  }
  return n & 7;
} /* __kernel_rem_pio2 */

/* __ieee754_rem_pio2(x,y)
 * return the remainder of x rem pi/2 in y[0]+y[1]
 * use __kernel_rem_pio2()
 */

static const int npio2_hw[] =
{
  0x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C,
  0x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C,
  0x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A,
  0x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C,
  0x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB,
  0x404858EB, 0x404921FB,
};

/*
 * invpio2:  53 bits of 2/pi
 * pio2_1:   first  33 bit of pi/2
 * pio2_1t:  pi/2 - pio2_1
 * pio2_2:   second 33 bit of pi/2
 * pio2_2t:  pi/2 - (pio2_1 + pio2_2)
 * pio2_3:   third  33 bit of pi/2
 * pio2_3t:  pi/2 - (pio2_1 + pio2_2 + pio2_3)
 */
#define invpio2 6.36619772367581382433e-01 /* 0x3FE45F30, 0x6DC9C883 */
#define pio2_1  1.57079632673412561417e+00 /* 0x3FF921FB, 0x54400000 */
#define pio2_1t 6.07710050650619224932e-11 /* 0x3DD0B461, 0x1A626331 */
#define pio2_2  6.07710050630396597660e-11 /* 0x3DD0B461, 0x1A600000 */
#define pio2_2t 2.02226624879595063154e-21 /* 0x3BA3198A, 0x2E037073 */
#define pio2_3  2.02226624871116645580e-21 /* 0x3BA3198A, 0x2E000000 */
#define pio2_3t 8.47842766036889956997e-32 /* 0x397B839A, 0x252049C1 */

static int
__ieee754_rem_pio2 (double x, double *y)
{
  double_accessor z;
  double w, t, r, fn;
  double tx[3];
  int e0, i, j, nx, n, ix, hx;

  hx = __HI (x); /* high word of x */
  ix = hx & 0x7fffffff;
  if (ix <= 0x3fe921fb) /* |x| ~<= pi/4 , no need for reduction */
  {
    y[0] = x;
    y[1] = 0;
    return 0;
  }
  if (ix < 0x4002d97c) /* |x| < 3pi/4, special case with n = +-1 */
  {
    if (hx > 0)
    {
      z.dbl = x - pio2_1;
      if (ix != 0x3ff921fb) /* 33 + 53 bit pi is good enough */
      {
        y[0] = z.dbl - pio2_1t;
        y[1] = (z.dbl - y[0]) - pio2_1t;
      }
      else /* near pi/2, use 33 + 33 + 53 bit pi */
      {
        z.dbl -= pio2_2;
        y[0] = z.dbl - pio2_2t;
        y[1] = (z.dbl - y[0]) - pio2_2t;
      }
      return 1;
    }
    else /* negative x */
    {
      z.dbl = x + pio2_1;
      if (ix != 0x3ff921fb) /* 33 + 53 bit pi is good enough */
      {
        y[0] = z.dbl + pio2_1t;
        y[1] = (z.dbl - y[0]) + pio2_1t;
      }
      else /* near pi/2, use 33 + 33 + 53 bit pi */
      {
        z.dbl += pio2_2;
        y[0] = z.dbl + pio2_2t;
        y[1] = (z.dbl - y[0]) + pio2_2t;
      }
      return -1;
    }
  }
  if (ix <= 0x413921fb) /* |x| ~<= 2^19 * (pi/2), medium size */
  {
    t = fabs (x);
    n = (int) (t * invpio2 + half);
    fn = (double) n;
    r = t - fn * pio2_1;
    w = fn * pio2_1t; /* 1st round good to 85 bit */
    if (n < 32 && ix != npio2_hw[n - 1])
    {
      y[0] = r - w; /* quick check no cancellation */
    }
    else
    {
      j = ix >> 20;
      y[0] = r - w;
      i = j - (((__HI (y[0])) >> 20) & 0x7ff);
      if (i > 16) /* 2nd iteration needed, good to 118 */
      {
        t = r;
        w = fn * pio2_2;
        r = t - w;
        w = fn * pio2_2t - ((t - r) - w);
        y[0] = r - w;
        i = j - (((__HI (y[0])) >> 20) & 0x7ff);
        if (i > 49) /* 3rd iteration need, 151 bits acc, will cover all possible cases */
        {
          t = r;
          w = fn * pio2_3;
          r = t - w;
          w = fn * pio2_3t - ((t - r) - w);
          y[0] = r - w;
        }
      }
    }
    y[1] = (r - y[0]) - w;
    if (hx < 0)
    {
      y[0] = -y[0];
      y[1] = -y[1];
      return -n;
    }
    else
    {
      return n;
    }
  }
  /*
   * all other (large) arguments
   */
  if (ix >= 0x7ff00000) /* x is inf or NaN */
  {
    y[0] = y[1] = x - x;
    return 0;
  }
  /* set z = scalbn(|x|, ilogb(x) - 23) */
  z.as_int.lo = __LO (x);
  e0 = (ix >> 20) - 1046; /* e0 = ilogb(z) - 23; */
  z.as_int.hi = ix - (e0 << 20);
  for (i = 0; i < 2; i++)
  {
    tx[i] = (double) ((int) (z.dbl));
    z.dbl = (z.dbl - tx[i]) * two24;
  }
  tx[2] = z.dbl;
  nx = 3;
  while (tx[nx - 1] == zero) /* skip zero term */
  {
    nx--;
  }
  n = __kernel_rem_pio2 (tx, y, e0, nx, 2);
  if (hx < 0)
  {
    y[0] = -y[0];
    y[1] = -y[1];
    return -n;
  }
  return n;
} /* __ieee754_rem_pio2 */

/* __kernel_sin( x, y, iy)
 * kernel sin function on [-pi/4, pi/4], pi/4 ~ 0.7854
 * Input x is assumed to be bounded by ~pi/4 in magnitude.
 * Input y is the tail of x.
 * Input iy indicates whether y is 0. (if iy=0, y assume to be 0).
 *
 * Algorithm
 *      1. Since sin(-x) = -sin(x), we need only to consider positive x.
 *      2. if x < 2^-27 (hx<0x3e400000 0), return x with inexact if x!=0.
 *      3. sin(x) is approximated by a polynomial of degree 13 on
 *         [0,pi/4]
 *                               3            13
 *              sin(x) ~ x + S1*x + ... + S6*x
 *         where
 *
 *      |sin(x)         2     4     6     8     10     12  |     -58
 *      |----- - (1+S1*x +S2*x +S3*x +S4*x +S5*x  +S6*x   )| <= 2
 *      |  x                                               |
 *
 *      4. sin(x+y) = sin(x) + sin'(x')*y
 *                  ~ sin(x) + (1-x*x/2)*y
 *         For better accuracy, let
 *                   3      2      2      2      2
 *              r = x *(S2+x *(S3+x *(S4+x *(S5+x *S6))))
 *         then                   3    2
 *              sin(x) = x + (S1*x + (x *(r-y/2)+y))
 */

#define S1   -1.66666666666666324348e-01 /* 0xBFC55555, 0x55555549 */
#define S2    8.33333333332248946124e-03 /* 0x3F811111, 0x1110F8A6 */
#define S3   -1.98412698298579493134e-04 /* 0xBF2A01A0, 0x19C161D5 */
#define S4    2.75573137070700676789e-06 /* 0x3EC71DE3, 0x57B1FE7D */
#define S5   -2.50507602534068634195e-08 /* 0xBE5AE5E6, 0x8A2B9CEB */
#define S6    1.58969099521155010221e-10 /* 0x3DE5D93A, 0x5ACFD57C */

static double
__kernel_sin (double x, double y, int iy)
{
  double z, r, v;
  int ix;

  ix = __HI (x) & 0x7fffffff; /* high word of x */
  if (ix < 0x3e400000) /* |x| < 2**-27 */
  {
    if ((int) x == 0)
    {
      return x; /* generate inexact */
    }
  }
  z = x * x;
  v = z * x;
  r = S2 + z * (S3 + z * (S4 + z * (S5 + z * S6)));
  if (iy == 0)
  {
    return x + v * (S1 + z * r);
  }
  else
  {
    return x - ((z * (half * y - v * r) - y) - v * S1);
  }
} /* __kernel_sin */

/*
 * __kernel_cos( x,  y )
 * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
 * Input x is assumed to be bounded by ~pi/4 in magnitude.
 * Input y is the tail of x.
 *
 * Algorithm
 *      1. Since cos(-x) = cos(x), we need only to consider positive x.
 *      2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0.
 *      3. cos(x) is approximated by a polynomial of degree 14 on
 *         [0,pi/4]
 *                                       4            14
 *              cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
 *         where the remez error is
 *
 *      |              2     4     6     8     10    12     14 |     -58
 *      |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  )| <= 2
 *      |                                                      |
 *
 *                     4     6     8     10    12     14
 *      4. let r = C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  , then
 *             cos(x) = 1 - x*x/2 + r
 *         since cos(x+y) ~ cos(x) - sin(x)*y
 *                        ~ cos(x) - x*y,
 *         a correction term is necessary in cos(x) and hence
 *              cos(x+y) = 1 - (x*x/2 - (r - x*y))
 *         For better accuracy when x > 0.3, let qx = |x|/4 with
 *         the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125.
 *         Then
 *              cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)).
 *         Note that 1-qx and (x*x/2-qx) is EXACT here, and the
 *         magnitude of the latter is at least a quarter of x*x/2,
 *         thus, reducing the rounding error in the subtraction.
 */

#define C1   4.16666666666666019037e-02 /* 0x3FA55555, 0x5555554C */
#define C2  -1.38888888888741095749e-03 /* 0xBF56C16C, 0x16C15177 */
#define C3   2.48015872894767294178e-05 /* 0x3EFA01A0, 0x19CB1590 */
#define C4  -2.75573143513906633035e-07 /* 0xBE927E4F, 0x809C52AD */
#define C5   2.08757232129817482790e-09 /* 0x3E21EE9E, 0xBDB4B1C4 */
#define C6  -1.13596475577881948265e-11 /* 0xBDA8FAE9, 0xBE8838D4 */

static double
__kernel_cos (double x, double y)
{
  double a, hz, z, r;
  int ix;

  ix = __HI (x) & 0x7fffffff; /* ix = |x|'s high word */
  if (ix < 0x3e400000) /* if x < 2**27 */
  {
    if (((int) x) == 0)
    {
      return one; /* generate inexact */
    }
  }
  z = x * x;
  r = z * (C1 + z * (C2 + z * (C3 + z * (C4 + z * (C5 + z * C6)))));
  if (ix < 0x3FD33333) /* if |x| < 0.3 */
  {
    return one - (0.5 * z - (z * r - x * y));
  }
  else
  {
    double_accessor qx;
    if (ix > 0x3fe90000) /* x > 0.78125 */
    {
      qx.dbl = 0.28125;
    }
    else
    {
      qx.as_int.hi = ix - 0x00200000; /* x / 4 */
      qx.as_int.lo = 0;
    }
    hz = 0.5 * z - qx.dbl;
    a = one - qx.dbl;
    return a - (hz - (z * r - x * y));
  }
} /* __kernel_cos */

/* __kernel_tan( x, y, k )
 * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
 * Input x is assumed to be bounded by ~pi/4 in magnitude.
 * Input y is the tail of x.
 * Input k indicates whether tan (if k = 1) or -1/tan (if k = -1) is returned.
 *
 * Algorithm
 *      1. Since tan(-x) = -tan(x), we need only to consider positive x.
 *      2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
 *      3. tan(x) is approximated by a odd polynomial of degree 27 on
 *         [0,0.67434]
 *                               3             27
 *              tan(x) ~ x + T1*x + ... + T13*x
 *         where
 *
 *              |tan(x)         2     4            26   |     -59.2
 *              |----- - (1+T1*x +T2*x +.... +T13*x    )| <= 2
 *              |  x                                    |
 *
 *         Note: tan(x+y) = tan(x) + tan'(x)*y
 *                        ~ tan(x) + (1+x*x)*y
 *         Therefore, for better accuracy in computing tan(x+y), let
 *                   3      2      2       2       2
 *              r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
 *         then
 *                                  3    2
 *              tan(x+y) = x + (T1*x + (x *(r+y)+y))
 *
 *      4. For x in [0.67434,pi/4],  let y = pi/4 - x, then
 *              tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
 *                     = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
 */

#define T0      3.33333333333334091986e-01 /* 3FD55555, 55555563 */
#define T1      1.33333333333201242699e-01 /* 3FC11111, 1110FE7A */
#define T2      5.39682539762260521377e-02 /* 3FABA1BA, 1BB341FE */
#define T3      2.18694882948595424599e-02 /* 3F9664F4, 8406D637 */
#define T4      8.86323982359930005737e-03 /* 3F8226E3, E96E8493 */
#define T5      3.59207910759131235356e-03 /* 3F6D6D22, C9560328 */
#define T6      1.45620945432529025516e-03 /* 3F57DBC8, FEE08315 */
#define T7      5.88041240820264096874e-04 /* 3F4344D8, F2F26501 */
#define T8      2.46463134818469906812e-04 /* 3F3026F7, 1A8D1068 */
#define T9      7.81794442939557092300e-05 /* 3F147E88, A03792A6 */
#define T10     7.14072491382608190305e-05 /* 3F12B80F, 32F0A7E9 */
#define T11    -1.85586374855275456654e-05 /* BEF375CB, DB605373 */
#define T12     2.59073051863633712884e-05 /* 3EFB2A70, 74BF7AD4 */
#define pio4    7.85398163397448278999e-01 /* 3FE921FB, 54442D18 */
#define pio4lo  3.06161699786838301793e-17 /* 3C81A626, 33145C07 */

static double
__kernel_tan (double x, double y, int iy)
{
  double_accessor z;
  double r, v, w, s;
  int ix, hx;

  hx = __HI (x); /* high word of x */
  ix = hx & 0x7fffffff; /* high word of |x| */
  if (ix < 0x3e300000) /* x < 2**-28 */
  {
    if ((int) x == 0) /* generate inexact */
    {
      if (((ix | __LO (x)) | (iy + 1)) == 0)
      {
        return one / fabs (x);
      }
      else
      {
        if (iy == 1)
        {
          return x;
        }
        else /* compute -1 / (x + y) carefully */
        {
          double a;
          double_accessor t;

          z.dbl = w = x + y;
          z.as_int.lo = 0;
          v = y - (z.dbl - x);
          t.dbl = a = -one / w;
          t.as_int.lo = 0;
          s = one + t.dbl * z.dbl;
          return t.dbl + a * (s + t.dbl * v);
        }
      }
    }
  }
  if (ix >= 0x3FE59428) /* |x| >= 0.6744 */
  {
    if (hx < 0)
    {
      x = -x;
      y = -y;
    }
    z.dbl = pio4 - x;
    w = pio4lo - y;
    x = z.dbl + w;
    y = 0.0;
  }
  z.dbl = x * x;
  w = z.dbl * z.dbl;
  /*
   * Break x^5 * (T[1] + x^2 * T[2] + ...) into
   * x^5 (T[1] + x^4 * T[3] + ... + x^20 * T[11]) +
   * x^5 (x^2 * (T[2] + x^4 * T[4] + ... + x^22 * [T12]))
   */
  r = T1 + w * (T3 + w * (T5 + w * (T7 + w * (T9 + w * T11))));
  v = z.dbl * (T2 + w * (T4 + w * (T6 + w * (T8 + w * (T10 + w * T12)))));
  s = z.dbl * x;
  r = y + z.dbl * (s * (r + v) + y);
  r += T0 * s;
  w = x + r;
  if (ix >= 0x3FE59428)
  {
    v = (double) iy;
    return (double) (1 - ((hx >> 30) & 2)) * (v - 2.0 * (x - (w * w / (w + v) - r)));
  }
  if (iy == 1)
  {
    return w;
  }
  else
  {
    /*
     * if allow error up to 2 ulp, simply return
     * -1.0 / (x + r) here
     */
    /* compute -1.0 / (x + r) accurately */
    double a;
    double_accessor t;

    z.dbl = w;
    z.as_int.lo = 0;
    v = r - (z.dbl - x); /* z + v = r + x */
    t.dbl = a = -1.0 / w; /* a = -1.0 / w */
    t.as_int.lo = 0;
    s = 1.0 + t.dbl * z.dbl;
    return t.dbl + a * (s + t.dbl * v);
  }
} /* __kernel_tan */

/* Method:
 *      Let S,C and T denote the sin, cos and tan respectively on
 *      [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
 *      in [-pi/4 , +pi/4], and let n = k mod 4.
 *      We have
 *
 *          n        sin(x)      cos(x)        tan(x)
 *     ----------------------------------------------------------
 *          0          S           C             T
 *          1          C          -S            -1/T
 *          2         -S          -C             T
 *          3         -C           S            -1/T
 *     ----------------------------------------------------------
 *
 * Special cases:
 *      Let trig be any of sin, cos, or tan.
 *      trig(+-INF)  is NaN, with signals;
 *      trig(NaN)    is that NaN;
 *
 * Accuracy:
 *      TRIG(x) returns trig(x) nearly rounded
 */

/* sin(x)
 * Return sine function of x.
 *
 * kernel function:
 *      __kernel_sin            ... sine function on [-pi/4,pi/4]
 *      __kernel_cos            ... cose function on [-pi/4,pi/4]
 *      __ieee754_rem_pio2      ... argument reduction routine
 */
double
sin (double x)
{
  double y[2], z = 0.0;
  int n, ix;

  /* High word of x. */
  ix = __HI (x);

  /* |x| ~< pi/4 */
  ix &= 0x7fffffff;
  if (ix <= 0x3fe921fb)
  {
    return __kernel_sin (x, z, 0);
  }

  /* sin(Inf or NaN) is NaN */
  else if (ix >= 0x7ff00000)
  {
    return x - x;
  }

  /* argument reduction needed */
  else
  {
    n = __ieee754_rem_pio2 (x, y);
    switch (n & 3)
    {
      case 0:
      {
        return __kernel_sin (y[0], y[1], 1);
      }
      case 1:
      {
        return __kernel_cos (y[0], y[1]);
      }
      case 2:
      {
        return -__kernel_sin (y[0], y[1], 1);
      }
      default:
      {
        return -__kernel_cos (y[0], y[1]);
      }
    }
  }
} /* sin */

/* cos(x)
 * Return cosine function of x.
 *
 * kernel function:
 *      __kernel_sin            ... sine function on [-pi/4,pi/4]
 *      __kernel_cos            ... cosine function on [-pi/4,pi/4]
 *      __ieee754_rem_pio2      ... argument reduction routine
 */

double
cos (double x)
{
  double y[2], z = 0.0;
  int n, ix;

  /* High word of x. */
  ix = __HI (x);

  /* |x| ~< pi/4 */
  ix &= 0x7fffffff;
  if (ix <= 0x3fe921fb)
  {
    return __kernel_cos (x, z);
  }

  /* cos(Inf or NaN) is NaN */
  else if (ix >= 0x7ff00000)
  {
    return x - x;
  }

  /* argument reduction needed */
  else
  {
    n = __ieee754_rem_pio2 (x, y);
    switch (n & 3)
    {
      case 0:
      {
        return __kernel_cos (y[0], y[1]);
      }
      case 1:
      {
        return -__kernel_sin (y[0], y[1], 1);
      }
      case 2:
      {
        return -__kernel_cos (y[0], y[1]);
      }
      default:
      {
        return __kernel_sin (y[0], y[1], 1);
      }
    }
  }
} /* cos */

/* tan(x)
 * Return tangent function of x.
 *
 * kernel function:
 *      __kernel_tan            ... tangent function on [-pi/4,pi/4]
 *      __ieee754_rem_pio2      ... argument reduction routine
 */

double
tan (double x)
{
  double y[2], z = 0.0;
  int n, ix;

  /* High word of x. */
  ix = __HI (x);

  /* |x| ~< pi/4 */
  ix &= 0x7fffffff;
  if (ix <= 0x3fe921fb)
  {
    return __kernel_tan (x, z, 1);
  }

  /* tan(Inf or NaN) is NaN */
  else if (ix >= 0x7ff00000)
  {
    return x - x; /* NaN */
  }

  /* argument reduction needed */
  else
  {
    n = __ieee754_rem_pio2 (x, y);
    return __kernel_tan (y[0], y[1], 1 - ((n & 1) << 1)); /*   1 -- n even, -1 -- n odd */
  }
} /* tan */

#undef zero
#undef half
#undef one
#undef two24
#undef twon24
#undef invpio2
#undef pio2_1
#undef pio2_1t
#undef pio2_2
#undef pio2_2t
#undef pio2_3
#undef pio2_3t
#undef S1
#undef S2
#undef S3
#undef S4
#undef S5
#undef S6
#undef C1
#undef C2
#undef C3
#undef C4
#undef C5
#undef C6
#undef T0
#undef T1
#undef T2
#undef T3
#undef T4
#undef T5
#undef T6
#undef T7
#undef T8
#undef T9
#undef T10
#undef T11
#undef T12
#undef pio4
#undef pio4lo