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poly_sin.c

/*---------------------------------------------------------------------------+
 |  poly_sin.c                                                               |
 |                                                                           |
 |  Computation of an approximation of the sin function and the cosine       |
 |  function by a polynomial.                                                |
 |                                                                           |
 | Copyright (C) 1992,1993,1994,1997,1999                                    |
 |                  W. Metzenthen, 22 Parker St, Ormond, Vic 3163, Australia |
 |                  E-mail   billm@melbpc.org.au                             |
 |                                                                           |
 |                                                                           |
 +---------------------------------------------------------------------------*/


#include "exception.h"
#include "reg_constant.h"
#include "fpu_emu.h"
#include "fpu_system.h"
#include "control_w.h"
#include "poly.h"


#define     N_COEFF_P   4
#define     N_COEFF_N   4

static const unsigned long long pos_terms_l[N_COEFF_P] =
{
  0xaaaaaaaaaaaaaaabLL,
  0x00d00d00d00cf906LL,
  0x000006b99159a8bbLL,
  0x000000000d7392e6LL
};

static const unsigned long long neg_terms_l[N_COEFF_N] =
{
  0x2222222222222167LL,
  0x0002e3bc74aab624LL,
  0x0000000b09229062LL,
  0x00000000000c7973LL
};



#define     N_COEFF_PH  4
#define     N_COEFF_NH  4
static const unsigned long long pos_terms_h[N_COEFF_PH] =
{
  0x0000000000000000LL,
  0x05b05b05b05b0406LL,
  0x000049f93edd91a9LL,
  0x00000000c9c9ed62LL
};

static const unsigned long long neg_terms_h[N_COEFF_NH] =
{
  0xaaaaaaaaaaaaaa98LL,
  0x001a01a01a019064LL,
  0x0000008f76c68a77LL,
  0x0000000000d58f5eLL
};


/*--- poly_sine() -----------------------------------------------------------+
 |                                                                           |
 +---------------------------------------------------------------------------*/
void  poly_sine(FPU_REG *st0_ptr)
{
  int                 exponent, echange;
  Xsig                accumulator, argSqrd, argTo4;
  unsigned long       fix_up, adj;
  unsigned long long  fixed_arg;
  FPU_REG         result;

  exponent = exponent(st0_ptr);

  accumulator.lsw = accumulator.midw = accumulator.msw = 0;

  /* Split into two ranges, for arguments below and above 1.0 */
  /* The boundary between upper and lower is approx 0.88309101259 */
  if ( (exponent < -1) || ((exponent == -1) && (st0_ptr->sigh <= 0xe21240aa)) )
    {
      /* The argument is <= 0.88309101259 */

      argSqrd.msw = st0_ptr->sigh; argSqrd.midw = st0_ptr->sigl; argSqrd.lsw = 0;
      mul64_Xsig(&argSqrd, &significand(st0_ptr));
      shr_Xsig(&argSqrd, 2*(-1-exponent));
      argTo4.msw = argSqrd.msw; argTo4.midw = argSqrd.midw;
      argTo4.lsw = argSqrd.lsw;
      mul_Xsig_Xsig(&argTo4, &argTo4);

      polynomial_Xsig(&accumulator, &XSIG_LL(argTo4), neg_terms_l,
                  N_COEFF_N-1);
      mul_Xsig_Xsig(&accumulator, &argSqrd);
      negate_Xsig(&accumulator);

      polynomial_Xsig(&accumulator, &XSIG_LL(argTo4), pos_terms_l,
                  N_COEFF_P-1);

      shr_Xsig(&accumulator, 2);    /* Divide by four */
      accumulator.msw |= 0x80000000;  /* Add 1.0 */

      mul64_Xsig(&accumulator, &significand(st0_ptr));
      mul64_Xsig(&accumulator, &significand(st0_ptr));
      mul64_Xsig(&accumulator, &significand(st0_ptr));

      /* Divide by four, FPU_REG compatible, etc */
      exponent = 3*exponent;

      /* The minimum exponent difference is 3 */
      shr_Xsig(&accumulator, exponent(st0_ptr) - exponent);

      negate_Xsig(&accumulator);
      XSIG_LL(accumulator) += significand(st0_ptr);

      echange = round_Xsig(&accumulator);

      setexponentpos(&result, exponent(st0_ptr) + echange);
    }
  else
    {
      /* The argument is > 0.88309101259 */
      /* We use sin(st(0)) = cos(pi/2-st(0)) */

      fixed_arg = significand(st0_ptr);

      if ( exponent == 0 )
      {
        /* The argument is >= 1.0 */

        /* Put the binary point at the left. */
        fixed_arg <<= 1;
      }
      /* pi/2 in hex is: 1.921fb54442d18469 898CC51701B839A2 52049C1 */
      fixed_arg = 0x921fb54442d18469LL - fixed_arg;
      /* There is a special case which arises due to rounding, to fix here. */
      if ( fixed_arg == 0xffffffffffffffffLL )
      fixed_arg = 0;

      XSIG_LL(argSqrd) = fixed_arg; argSqrd.lsw = 0;
      mul64_Xsig(&argSqrd, &fixed_arg);

      XSIG_LL(argTo4) = XSIG_LL(argSqrd); argTo4.lsw = argSqrd.lsw;
      mul_Xsig_Xsig(&argTo4, &argTo4);

      polynomial_Xsig(&accumulator, &XSIG_LL(argTo4), neg_terms_h,
                  N_COEFF_NH-1);
      mul_Xsig_Xsig(&accumulator, &argSqrd);
      negate_Xsig(&accumulator);

      polynomial_Xsig(&accumulator, &XSIG_LL(argTo4), pos_terms_h,
                  N_COEFF_PH-1);
      negate_Xsig(&accumulator);

      mul64_Xsig(&accumulator, &fixed_arg);
      mul64_Xsig(&accumulator, &fixed_arg);

      shr_Xsig(&accumulator, 3);
      negate_Xsig(&accumulator);

      add_Xsig_Xsig(&accumulator, &argSqrd);

      shr_Xsig(&accumulator, 1);

      accumulator.lsw |= 1;  /* A zero accumulator here would cause problems */
      negate_Xsig(&accumulator);

      /* The basic computation is complete. Now fix the answer to
       compensate for the error due to the approximation used for
       pi/2
       */

      /* This has an exponent of -65 */
      fix_up = 0x898cc517;
      /* The fix-up needs to be improved for larger args */
      if ( argSqrd.msw & 0xffc00000 )
      {
        /* Get about 32 bit precision in these: */
        fix_up -= mul_32_32(0x898cc517, argSqrd.msw) / 6;
      }
      fix_up = mul_32_32(fix_up, LL_MSW(fixed_arg));

      adj = accumulator.lsw;    /* temp save */
      accumulator.lsw -= fix_up;
      if ( accumulator.lsw > adj )
      XSIG_LL(accumulator) --;

      echange = round_Xsig(&accumulator);

      setexponentpos(&result, echange - 1);
    }

  significand(&result) = XSIG_LL(accumulator);
  setsign(&result, getsign(st0_ptr));
  FPU_copy_to_reg0(&result, TAG_Valid);

#ifdef PARANOID
  if ( (exponent(&result) >= 0)
      && (significand(&result) > 0x8000000000000000LL) )
    {
      EXCEPTION(EX_INTERNAL|0x150);
    }
#endif /* PARANOID */

}



/*--- poly_cos() ------------------------------------------------------------+
 |                                                                           |
 +---------------------------------------------------------------------------*/
void  poly_cos(FPU_REG *st0_ptr)
{
  FPU_REG         result;
  long int            exponent, exp2, echange;
  Xsig                accumulator, argSqrd, fix_up, argTo4;
  unsigned long long  fixed_arg;

#ifdef PARANOID
  if ( (exponent(st0_ptr) > 0)
      || ((exponent(st0_ptr) == 0)
        && (significand(st0_ptr) > 0xc90fdaa22168c234LL)) )
    {
      EXCEPTION(EX_Invalid);
      FPU_copy_to_reg0(&CONST_QNaN, TAG_Special);
      return;
    }
#endif /* PARANOID */

  exponent = exponent(st0_ptr);

  accumulator.lsw = accumulator.midw = accumulator.msw = 0;

  if ( (exponent < -1) || ((exponent == -1) && (st0_ptr->sigh <= 0xb00d6f54)) )
    {
      /* arg is < 0.687705 */

      argSqrd.msw = st0_ptr->sigh; argSqrd.midw = st0_ptr->sigl;
      argSqrd.lsw = 0;
      mul64_Xsig(&argSqrd, &significand(st0_ptr));

      if ( exponent < -1 )
      {
        /* shift the argument right by the required places */
        shr_Xsig(&argSqrd, 2*(-1-exponent));
      }

      argTo4.msw = argSqrd.msw; argTo4.midw = argSqrd.midw;
      argTo4.lsw = argSqrd.lsw;
      mul_Xsig_Xsig(&argTo4, &argTo4);

      polynomial_Xsig(&accumulator, &XSIG_LL(argTo4), neg_terms_h,
                  N_COEFF_NH-1);
      mul_Xsig_Xsig(&accumulator, &argSqrd);
      negate_Xsig(&accumulator);

      polynomial_Xsig(&accumulator, &XSIG_LL(argTo4), pos_terms_h,
                  N_COEFF_PH-1);
      negate_Xsig(&accumulator);

      mul64_Xsig(&accumulator, &significand(st0_ptr));
      mul64_Xsig(&accumulator, &significand(st0_ptr));
      shr_Xsig(&accumulator, -2*(1+exponent));

      shr_Xsig(&accumulator, 3);
      negate_Xsig(&accumulator);

      add_Xsig_Xsig(&accumulator, &argSqrd);

      shr_Xsig(&accumulator, 1);

      /* It doesn't matter if accumulator is all zero here, the
       following code will work ok */
      negate_Xsig(&accumulator);

      if ( accumulator.lsw & 0x80000000 )
      XSIG_LL(accumulator) ++;
      if ( accumulator.msw == 0 )
      {
        /* The result is 1.0 */
        FPU_copy_to_reg0(&CONST_1, TAG_Valid);
        return;
      }
      else
      {
        significand(&result) = XSIG_LL(accumulator);
      
        /* will be a valid positive nr with expon = -1 */
        setexponentpos(&result, -1);
      }
    }
  else
    {
      fixed_arg = significand(st0_ptr);

      if ( exponent == 0 )
      {
        /* The argument is >= 1.0 */

        /* Put the binary point at the left. */
        fixed_arg <<= 1;
      }
      /* pi/2 in hex is: 1.921fb54442d18469 898CC51701B839A2 52049C1 */
      fixed_arg = 0x921fb54442d18469LL - fixed_arg;
      /* There is a special case which arises due to rounding, to fix here. */
      if ( fixed_arg == 0xffffffffffffffffLL )
      fixed_arg = 0;

      exponent = -1;
      exp2 = -1;

      /* A shift is needed here only for a narrow range of arguments,
       i.e. for fixed_arg approx 2^-32, but we pick up more... */
      if ( !(LL_MSW(fixed_arg) & 0xffff0000) )
      {
        fixed_arg <<= 16;
        exponent -= 16;
        exp2 -= 16;
      }

      XSIG_LL(argSqrd) = fixed_arg; argSqrd.lsw = 0;
      mul64_Xsig(&argSqrd, &fixed_arg);

      if ( exponent < -1 )
      {
        /* shift the argument right by the required places */
        shr_Xsig(&argSqrd, 2*(-1-exponent));
      }

      argTo4.msw = argSqrd.msw; argTo4.midw = argSqrd.midw;
      argTo4.lsw = argSqrd.lsw;
      mul_Xsig_Xsig(&argTo4, &argTo4);

      polynomial_Xsig(&accumulator, &XSIG_LL(argTo4), neg_terms_l,
                  N_COEFF_N-1);
      mul_Xsig_Xsig(&accumulator, &argSqrd);
      negate_Xsig(&accumulator);

      polynomial_Xsig(&accumulator, &XSIG_LL(argTo4), pos_terms_l,
                  N_COEFF_P-1);

      shr_Xsig(&accumulator, 2);    /* Divide by four */
      accumulator.msw |= 0x80000000;  /* Add 1.0 */

      mul64_Xsig(&accumulator, &fixed_arg);
      mul64_Xsig(&accumulator, &fixed_arg);
      mul64_Xsig(&accumulator, &fixed_arg);

      /* Divide by four, FPU_REG compatible, etc */
      exponent = 3*exponent;

      /* The minimum exponent difference is 3 */
      shr_Xsig(&accumulator, exp2 - exponent);

      negate_Xsig(&accumulator);
      XSIG_LL(accumulator) += fixed_arg;

      /* The basic computation is complete. Now fix the answer to
       compensate for the error due to the approximation used for
       pi/2
       */

      /* This has an exponent of -65 */
      XSIG_LL(fix_up) = 0x898cc51701b839a2ll;
      fix_up.lsw = 0;

      /* The fix-up needs to be improved for larger args */
      if ( argSqrd.msw & 0xffc00000 )
      {
        /* Get about 32 bit precision in these: */
        fix_up.msw -= mul_32_32(0x898cc517, argSqrd.msw) / 2;
        fix_up.msw += mul_32_32(0x898cc517, argTo4.msw) / 24;
      }

      exp2 += norm_Xsig(&accumulator);
      shr_Xsig(&accumulator, 1); /* Prevent overflow */
      exp2++;
      shr_Xsig(&fix_up, 65 + exp2);

      add_Xsig_Xsig(&accumulator, &fix_up);

      echange = round_Xsig(&accumulator);

      setexponentpos(&result, exp2 + echange);
      significand(&result) = XSIG_LL(accumulator);
    }

  FPU_copy_to_reg0(&result, TAG_Valid);

#ifdef PARANOID
  if ( (exponent(&result) >= 0)
      && (significand(&result) > 0x8000000000000000LL) )
    {
      EXCEPTION(EX_INTERNAL|0x151);
    }
#endif /* PARANOID */

}

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