glibc/sysdeps/ieee754/ldbl-128/k_tanl.c

/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunPro, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice
 * is preserved.
 * ====================================================
 */

/*
  Long double expansions contributed by
  Stephen L. Moshier <moshier@na-net.ornl.gov>
*/

/* __kernel_tanl( 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^-57, return x with inexact if x!=0.
 *	3. tan(x) is approximated by a rational form x + x^3 / 3 + x^5 R(x^2)
 *          on [0,0.67433].
 *
 *	   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
 *		r = x^3 * R(x^2)
 *	   then
 *		tan(x+y) = x + (x^3 / 3 + (x^2 *(r+y)+y))
 *
 *      4. For x in [0.67433,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)))
 */

#include "math.h"
#include "math_private.h"
#ifdef __STDC__
static const long double
#else
static long double
#endif
  one = 1.0L,
  pio4hi = 7.8539816339744830961566084581987569936977E-1L,
  pio4lo = 2.1679525325309452561992610065108379921906E-35L,

  /* tan x = x + x^3 / 3 + x^5 T(x^2)/U(x^2)
     0 <= x <= 0.6743316650390625
     Peak relative error 8.0e-36  */
 TH =  3.333333333333333333333333333333333333333E-1L,
 T0 = -1.813014711743583437742363284336855889393E7L,
 T1 =  1.320767960008972224312740075083259247618E6L,
 T2 = -2.626775478255838182468651821863299023956E4L,
 T3 =  1.764573356488504935415411383687150199315E2L,
 T4 = -3.333267763822178690794678978979803526092E-1L,

 U0 = -1.359761033807687578306772463253710042010E8L,
 U1 =  6.494370630656893175666729313065113194784E7L,
 U2 = -4.180787672237927475505536849168729386782E6L,
 U3 =  8.031643765106170040139966622980914621521E4L,
 U4 = -5.323131271912475695157127875560667378597E2L;
  /* 1.000000000000000000000000000000000000000E0 */


#ifdef __STDC__
long double
__kernel_tanl (long double x, long double y, int iy)
#else
long double
__kernel_tanl (x, y, iy)
     long double x, y;
     int iy;
#endif
{
  long double z, r, v, w, s;
  int32_t ix, sign;
  ieee854_long_double_shape_type u, u1;

  u.value = x;
  ix = u.parts32.w0 & 0x7fffffff;
  if (ix < 0x3fc60000)		/* x < 2**-57 */
    {
      if ((int) x == 0)
	{			/* generate inexact */
	  if ((ix | u.parts32.w1 | u.parts32.w2 | u.parts32.w3
	       | (iy + 1)) == 0)
	    return one / fabs (x);
	  else
	    return (iy == 1) ? x : -one / x;
	}
    }
  if (ix >= 0x3ffe5942) /* |x| >= 0.6743316650390625 */
    {
      if ((u.parts32.w0 & 0x80000000) != 0)
	{
	  x = -x;
	  y = -y;
	  sign = -1;
	}
      else
	sign = 1;
      z = pio4hi - x;
      w = pio4lo - y;
      x = z + w;
      y = 0.0;
    }
  z = x * x;
  r = T0 + z * (T1 + z * (T2 + z * (T3 + z * T4)));
  v = U0 + z * (U1 + z * (U2 + z * (U3 + z * (U4 + z))));
  r = r / v;

  s = z * x;
  r = y + z * (s * r + y);
  r += TH * s;
  w = x + r;
  if (ix >= 0x3ffe5942)
    {
      v = (long double) iy;
      w = (v - 2.0 * (x - (w * w / (w + v) - r)));
      if (sign < 0)
	w = -w;
      return w;
    }
  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 */
      u1.value = w;
      u1.parts32.w2 = 0;
      u1.parts32.w3 = 0;
      v = r - (u1.value - x);		/* u1+v = r+x */
      z = -1.0 / w;
      u.value = z;
      u.parts32.w2 = 0;
      u.parts32.w3 = 0;
      s = 1.0 + u.value * u1.value;
      return u.value + z * (s + u.value * v);
    }
}