tan.c

/*******************************************************************************
MIT License
-----------

Copyright (c) 2002-2019 Advanced Micro Devices, Inc.

Permission is hereby granted, free of charge, to any person obtaining a copy
of this Software and associated documentaon files (the "Software"), to deal
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THE SOFTWARE.
*******************************************************************************/

#include "libm.h"
#include "libm_util.h"

#define USE_NAN_WITH_FLAGS
#define USE_VAL_WITH_FLAGS
#define USE_HANDLE_ERROR
#include "libm_inlines.h"
#undef USE_NAN_WITH_FLAGS
#undef USE_VAL_WITH_FLAGS
#undef USE_HANDLE_ERROR

#include "libm_errno.h"

/* tan(x + xx) approximation valid on the interval [-pi/4,pi/4].
   If recip is true return -1/tan(x + xx) instead. */
static inline double tan_piby4(double x, double xx, int recip)
{
  double r, t1, t2, xl;
  int transform = 0;
  static const double
     piby4_lead = 7.85398163397448278999e-01, /* 0x3fe921fb54442d18 */
     piby4_tail = 3.06161699786838240164e-17; /* 0x3c81a62633145c06 */

  /* In order to maintain relative precision transform using the identity:
     tan(pi/4-x) = (1-tan(x))/(1+tan(x)) for arguments close to pi/4.
     Similarly use tan(x-pi/4) = (tan(x)-1)/(tan(x)+1) close to -pi/4. */

  if (x > 0.68)
    {
      transform = 1;
      x = piby4_lead - x;
      xl = piby4_tail - xx;
      x += xl;
      xx = 0.0;
    }
  else if (x < -0.68)
    {
      transform = -1;
      x = piby4_lead + x;
      xl = piby4_tail + xx;
      x += xl;
      xx = 0.0;
    }

  /* Core Remez [2,3] approximation to tan(x+xx) on the
     interval [0,0.68]. */

  r = x*x + 2.0 * x * xx;
  t1 = x;
  t2 = xx + x*r*
    (0.372379159759792203640806338901e0 +
     (-0.229345080057565662883358588111e-1 +
      0.224044448537022097264602535574e-3*r)*r)/
    (0.111713747927937668539901657944e1 +
     (-0.515658515729031149329237816945e0 +
      (0.260656620398645407524064091208e-1 -
       0.232371494088563558304549252913e-3*r)*r)*r);

  /* Reconstruct tan(x) in the transformed case. */

  if (transform)
    {
      double t;
      t = t1 + t2;
      if (recip)
         return transform*(2*t/(t-1) - 1.0);
      else
         return transform*(1.0 - 2*t/(1+t));
    }

  if (recip)
    {
      /* Compute -1.0/(t1 + t2) accurately */
      double trec, trec_top, z1, z2, t;
      unsigned long u;
      t = t1 + t2;
      GET_BITS_DP64(t, u);
      u &= 0xffffffff00000000;
      PUT_BITS_DP64(u, z1);
      z2 = t2 - (z1 - t1);
      trec = -1.0 / t;
      GET_BITS_DP64(trec, u);
      u &= 0xffffffff00000000;
      PUT_BITS_DP64(u, trec_top);
      return trec_top + trec * ((1.0 + trec_top * z1) + trec_top * z2);

    }
  else
    return t1 + t2;
}

#pragma function(tan)

double tan(double x)
{
  double r, rr;
  int region, xneg;

  unsigned long ux, ax;
  GET_BITS_DP64(x, ux);
  ax = (ux & ~SIGNBIT_DP64);
  if (ax <= 0x3fe921fb54442d18) /* abs(x) <= pi/4 */
    {
      if (ax < 0x3f20000000000000) /* abs(x) < 2.0^(-13) */
        {
          if (ax < 0x3e40000000000000) /* abs(x) < 2.0^(-27) */
	    {
	      if (ax == 0x0000000000000000) return x;
              else return val_with_flags(x, AMD_F_INEXACT);
	    }
          else
            {
              /* Using a temporary variable prevents 64-bit VC++ from
                 rearranging
                    x + x*x*x*0.333333333333333333;
                 into
                    x * (1 + x*x*0.333333333333333333);
                 The latter results in an incorrectly rounded answer. */
              double tmp;
              tmp = x*x*x*0.333333333333333333;
              return x + tmp;
            }
        }
      else
        return tan_piby4(x, 0.0, 0);
    }
  else if ((ux & EXPBITS_DP64) == EXPBITS_DP64)
    {
      /* x is either NaN or infinity */
      if (ux & MANTBITS_DP64)
        /* x is NaN */
        return _handle_error("tan", OP_TAN, ux|0x0008000000000000, _DOMAIN, 0,
                            EDOM, x, 0.0, 1);
      else
        /* x is infinity. Return a NaN */
        return _handle_error("tan", OP_TAN, INDEFBITPATT_DP64, _DOMAIN, AMD_F_INVALID,
                            EDOM, x, 0.0, 1);
    }
  xneg = (ax != ux);


  if (xneg)
    x = -x;

  if (x < 5.0e5)
    {
      /* For these size arguments we can just carefully subtract the
         appropriate multiple of pi/2, using extra precision where
         x is close to an exact multiple of pi/2 */
      static const double
        twobypi =  6.36619772367581382433e-01, /* 0x3fe45f306dc9c883 */
        piby2_1  =  1.57079632673412561417e+00, /* 0x3ff921fb54400000 */
        piby2_1tail =  6.07710050650619224932e-11, /* 0x3dd0b4611a626331 */
        piby2_2  =  6.07710050630396597660e-11, /* 0x3dd0b4611a600000 */
        piby2_2tail =  2.02226624879595063154e-21, /* 0x3ba3198a2e037073 */
        piby2_3  =  2.02226624871116645580e-21, /* 0x3ba3198a2e000000 */
        piby2_3tail =  8.47842766036889956997e-32; /* 0x397b839a252049c1 */
      double t, rhead, rtail;
      int npi2;
      unsigned long uy, xexp, expdiff;
      xexp  = ax >> EXPSHIFTBITS_DP64;
      /* How many pi/2 is x a multiple of? */
      if (ax <= 0x400f6a7a2955385e) /* 5pi/4 */
        {
          if (ax <= 0x4002d97c7f3321d2) /* 3pi/4 */
            npi2 = 1;
          else
            npi2 = 2;
        }
      else if (ax <= 0x401c463abeccb2bb) /* 9pi/4 */
        {
          if (ax <= 0x4015fdbbe9bba775) /* 7pi/4 */
            npi2 = 3;
          else
            npi2 = 4;
        }
      else
        npi2  = (int)(x * twobypi + 0.5);
      /* Subtract the multiple from x to get an extra-precision remainder */
      rhead  = x - npi2 * piby2_1;
      rtail  = npi2 * piby2_1tail;
      GET_BITS_DP64(rhead, uy);
      expdiff = xexp - ((uy & EXPBITS_DP64) >> EXPSHIFTBITS_DP64);
      if (expdiff > 15)
        {
          /* The remainder is pretty small compared with x, which
             implies that x is a near multiple of pi/2
             (x matches the multiple to at least 15 bits) */
          t  = rhead;
          rtail  = npi2 * piby2_2;
          rhead  = t - rtail;
          rtail  = npi2 * piby2_2tail - ((t - rhead) - rtail);
          if (expdiff > 48)
            {
              /* x matches a pi/2 multiple to at least 48 bits */
              t  = rhead;
              rtail  = npi2 * piby2_3;
              rhead  = t - rtail;
              rtail  = npi2 * piby2_3tail - ((t - rhead) - rtail);
            }
        }
      r = rhead - rtail;
      rr = (rhead - r) - rtail;
      region = npi2 & 3;
    }
  else
    {
      /* Reduce x into range [-pi/4,pi/4] */
      __remainder_piby2(x, &r, &rr, &region);
    }

  if (xneg)
    return -tan_piby4(r, rr, region & 1);
  else
    return tan_piby4(r, rr, region & 1);
}