osprey/libm/mips/cos.c

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/*

  Copyright (C) 2000, 2001 Silicon Graphics, Inc.  All Rights Reserved.

  This program is free software; you can redistribute it and/or modify it
  under the terms of version 2.1 of the GNU Lesser General Public License 
  as published by the Free Software Foundation.

  This program is distributed in the hope that it would be useful, but
  WITHOUT ANY WARRANTY; without even the implied warranty of
  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  

  Further, this software is distributed without any warranty that it is
  free of the rightful claim of any third person regarding infringement 
  or the like.  Any license provided herein, whether implied or 
  otherwise, applies only to this software file.  Patent licenses, if
  any, provided herein do not apply to combinations of this program with 
  other software, or any other product whatsoever.  

  You should have received a copy of the GNU Lesser General Public 
  License along with this program; if not, write the Free Software 
  Foundation, Inc., 59 Temple Place - Suite 330, Boston MA 02111-1307, 
  USA.

  Contact information:  Silicon Graphics, Inc., 1600 Amphitheatre Pky,
  Mountain View, CA 94043, or:

  http://www.sgi.com

  For further information regarding this notice, see:

  http://oss.sgi.com/projects/GenInfo/NoticeExplan

*/

/* ====================================================================
 * ====================================================================
 *
 * Module: cos.c
 * $Revision$
 * $Date$
 * $Author$
 * $Source$
 *
 * Revision history:
 *  09-Jun-93 - Original Version
 *
 * Description: source code for cos function
 *
 * ====================================================================
 * ====================================================================
 */

static char *rcs_id = "$Source$ $Revision$";

#ifdef _CALL_MATHERR
#include <stdio.h>
#include <math.h>
#include <errno.h>
#endif

#include "libm.h"

#ifdef mips
extern  double  cos(double);

#pragma weak cos = __cos
#endif

#if defined(BUILD_OS_DARWIN) /* Mach-O doesn't support aliases */
extern double __cos(double);
#pragma weak cos
double cos(double x) {
  return __cos(x);
}
#elif defined(__GNUC__)
extern  double  __cos(double);

double    cos() __attribute__ ((weak, alias ("__cos")));

#endif

static const  du  Qnan =
{D(QNANHI, QNANLO)};

static const  du  Inf =
{D(0x7ff00000, 0x00000000)};

static const du half =
{D(0x3fe00000, 0x00000000)};

static const du one =
{D(0x3ff00000, 0x00000000)};

static const du rpiby2 =
{D(0x3fe45f30, 0x6dc9c883)};

static const du piby2hi =
{D(0x3ff921fb, 0x54400000)};

static const du piby2lo =
{D(0x3dd0b461, 0x1a600000)};

static const du piby2tiny =
{D(0x3ba3198a, 0x2e037073)};

static const du ph =
{D(0x3ff921fb, 0x50000000)};

static const du pl =
{D(0x3e5110b4, 0x60000000)};

static const du pt =
{D(0x3c91a626, 0x30000000)};

static const du pe =
{D(0x3ae8a2e0, 0x30000000)};

static const du pe2 =
{D(0x394c1cd1, 0x29024e09)};

static const du Ph =
{D(0x3ff921fb, 0x54000000)};

static const du Pl =
{D(0x3e110b46, 0x10000000)};

static const du Pt =
{D(0x3c5a6263, 0x3145c06e)};

/* coefficients for polynomial approximation of sin on +/- pi/4 */

static const du P[] =
{
{D(0x3ff00000, 0x00000000)},
{D(0xbfc55555, 0x55555548)},
{D(0x3f811111, 0x1110f7d0)},
{D(0xbf2a01a0, 0x19bfdf03)},
{D(0x3ec71de3, 0x567d4896)},
{D(0xbe5ae5e5, 0xa9291691)},
{D(0x3de5d8fd, 0x1fcf0ec1)},
};

/* coefficients for polynomial approximation of cos on +/- pi/4 */

static const du Q[] =
{
{D(0x3ff00000, 0x00000000)},
{D(0xbfdfffff, 0xffffff96)},
{D(0x3fa55555, 0x5554f0ab)},
{D(0xbf56c16c, 0x1640aaca)},
{D(0x3efa019f, 0x81cb6a1d)},
{D(0xbe927df4, 0x609cb202)},
{D(0x3e21b8b9, 0x947ab5c8)},
};

#ifdef _TABLE_BASED_REDUCTION

/* tables are used for argument reduction for args between +/- 16 radians;
   the sum of the high and low values of the kth entry is (k - 10)*pi/2
*/

static const du tblh[] =
{
{D(0xc02f6a7a, 0x2955385e)},
{D(0xc02c463a, 0xbeccb2bb)},
{D(0xc02921fb, 0x54442d18)},
{D(0xc025fdbb, 0xe9bba775)},
{D(0xc022d97c, 0x7f3321d2)},
{D(0xc01f6a7a, 0x2955385e)},
{D(0xc01921fb, 0x54442d18)},
{D(0xc012d97c, 0x7f3321d2)},
{D(0xc00921fb, 0x54442d18)},
{D(0xbff921fb, 0x54442d18)},
{D(0x00000000, 0x00000000)},
{D(0x3ff921fb, 0x54442d18)},
{D(0x400921fb, 0x54442d18)},
{D(0x4012d97c, 0x7f3321d2)},
{D(0x401921fb, 0x54442d18)},
{D(0x401f6a7a, 0x2955385e)},
{D(0x4022d97c, 0x7f3321d2)},
{D(0x4025fdbb, 0xe9bba775)},
{D(0x402921fb, 0x54442d18)},
{D(0x402c463a, 0xbeccb2bb)},
{D(0x402f6a7a, 0x2955385e)},
};

static const du tbll[] =
{
{D(0xbcc60faf, 0xbfd97309)},
{D(0xbcc3daea, 0xf976e788)},
{D(0xbcc1a626, 0x33145c07)},
{D(0xbcbee2c2, 0xd963a10c)},
{D(0xbcba7939, 0x4c9e8a0a)},
{D(0xbcb60faf, 0xbfd97309)},
{D(0xbcb1a626, 0x33145c07)},
{D(0xbcaa7939, 0x4c9e8a0a)},
{D(0xbca1a626, 0x33145c07)},
{D(0xbc91a626, 0x33145c07)},
{D(0x00000000, 0x00000000)},
{D(0x3c91a626, 0x33145c07)},
{D(0x3ca1a626, 0x33145c07)},
{D(0x3caa7939, 0x4c9e8a0a)},
{D(0x3cb1a626, 0x33145c07)},
{D(0x3cb60faf, 0xbfd97309)},
{D(0x3cba7939, 0x4c9e8a0a)},
{D(0x3cbee2c2, 0xd963a10c)},
{D(0x3cc1a626, 0x33145c07)},
{D(0x3cc3daea, 0xf976e788)},
{D(0x3cc60faf, 0xbfd97309)},
};
#endif


/* ====================================================================
 *
 * FunctionName   cos
 *
 * Description    computes cosine of arg
 *
 * Note:  Routine cis() computes cos(arg) + i*sin(arg) and should
 *    be kept in sync with this routine and sin() so that it
 *    returns the same result as one gets by calling cos() and
 *    sin() separately.
 *
 * ====================================================================
 */

double
__cos( x )
double  x;
{
#ifdef _32BIT_MACHINE
int ix, xpt, m, l;
#else
long long ix, xpt, m, l;
#endif

int n;
double  xsq;
double  poly;
double  result;
double  *p;
double  z, dn;
double  absx, dn1, dn2;
double  s, ss;
double  t, w, ww;
#ifdef _CALL_MATHERR
struct exception  exstruct;
#endif

  /* extract exponent of x and 1 bit of mantissa for some quick screening */

#ifdef _32BIT_MACHINE

  DBLHI2INT(x, ix); /* copy MSW of x to ix  */
#else
  DBL2LL(x, ix);    /* copy x to ix */
#endif
  xpt = (ix >> (DMANTWIDTH-1));
  xpt &= 0xfff;

  if ( xpt < 0x7fd )
  {
    /*   |x| < 0.75   */

    if ( xpt >= 0x7c2 )
    {
      /*   |x| >= 2^(-30)   */

      /* just compute cos(x) */

      xsq = x*x;

      poly = (((((Q[6].d*xsq + Q[5].d)*xsq + Q[4].d)*xsq +
        Q[3].d)*xsq + Q[2].d)*xsq + Q[1].d)*xsq + one.d;

      return ( poly );
    }

    return ( one.d );
  }

#ifdef _TABLE_BASED_REDUCTION

  if ( xpt < 0x806 )
  {
    /*   |x| < 16.0   */

    /*  do a table based argument reduction to +/- pi/4  */

    dn = x*rpiby2.d;
    n = ROUND(dn);

    /*  compute z = x - n*pi/2  */

    x = x - tblh[n+10].d;
    x = x - tbll[n+10].d;

    xsq = x*x;

    n++;
  
    p = (double *)&P[0].d;

    /* if ( n&1 )
      compute cos(x)
       else
      compute sin(x)

       if ( n&2 )
      negate result
    */

    if ( n&1 )
    {
      p = (double *)&Q[0].d;
      x = 1.0;
    }
  
    result = (((((p[6]*xsq + p[5])*xsq + p[4])*xsq 
      + p[3])*xsq + p[2])*xsq + p[1])*(xsq*x) + x;
  
    if ( n&2 )
      result = -result;
  
    return ( result );
  }
#endif

  if ( xpt < 0x827 )
  {
    /*  |x| < 1.5*2^20  */

    dn = x*rpiby2.d;
    n = ROUND(dn);
    dn = n;

    x = x - dn*piby2hi.d;
    x = x - dn*piby2lo.d;
    x = x - dn*piby2tiny.d; /* x = x - n*pi/2 */

    xsq = x*x;
  
    n++;

    p = (double *)&P[0].d;

    /* if ( n&1 )
      compute cos(x)
       else
      compute sin(x)

       if ( n&2 )
      negate result
    */

    if ( n&1 )
    {
      p = (double *)&Q[0].d;
      x = 1.0;
    }
  
    result = (((((p[6]*xsq + p[5])*xsq + p[4])*xsq 
      + p[3])*xsq + p[2])*xsq + p[1])*(xsq*x) + x;
  
    if ( n&2 )
      result = -result;
  
    return ( result );
  }

  if ( xpt < 0x836 )
  {
    /*  |x| < 2^28  */

    dn = x*rpiby2.d;
    n = ROUND(dn);
    dn = n;

    x = x - dn*ph.d;
    x = x - dn*pl.d;
    x = x - dn*pt.d;
    x = x - dn*pe.d;
    x = x - dn*pe2.d;

    xsq = x*x;
  
    n++;

    p = (double *)&P[0].d;

    /* if ( n&1 )
      compute cos(x)
       else
      compute sin(x)

       if ( n&2 )
      negate result
    */

    if ( n&1 )
    {
      p = (double *)&Q[0].d;
      x = 1.0;
    }
  
    result = (((((p[6]*xsq + p[5])*xsq + p[4])*xsq 
      + p[3])*xsq + p[2])*xsq + p[1])*(xsq*x) + x;
  
    if ( n&2 )
      result = -result;
  
    return ( result );
  }

  if ( xpt < 0x862 )
  {
    /*  |x| < 2^50  */

    absx = fabs(x);

    dn = z = absx*rpiby2.d;

    /* round dn to the nearest integer */

#ifdef _32BIT_MACHINE

    DBLHI2INT(dn, l);
    m = (l >> DMANTWIDTH);
    m &= 0x7ff;

    /* shift off fractional bits of dn */

    DBLLO2INT(dn, l);

    l >>= (0x433 - m);
    n = l;
    l <<= (0x433 - m);
    INT2DBLLO(l, dn);
#else
    DBL2LL(dn, l);
    m = (l >> DMANTWIDTH);
    m &= 0x7ff;

    /* shift off fractional bits of dn */

    l >>= (0x433 - m);
    n = l;
    l <<= (0x433 - m);
    LL2DBL(l, dn);
#endif
    /* adjust dn and n if the fractional part of dn 
       was >= 0.5
    */

    n &= 3;

    if ( (z - dn) >= half.d )
    {
      dn += one.d;
      n += 1;
    }

  /* compute x - dn*Ph - dn*Pl - dn*Pt by dividing dn into
     two parts and using the Kahan summation formula
  */

    /* split dn into 2 parts */

    dn1 = dn;

#ifdef _32BIT_MACHINE

    DBLLO2INT(dn1, m);
    m >>= 28;
    m <<= 28;
    INT2DBLLO(m, dn1);
#else
    DBL2LL(dn1, m);
    m >>= 28;
    m <<= 28;
    LL2DBL(m, dn1);
#endif
    dn2 = dn - dn1;

    z = absx - dn1*Ph.d;  /* this operation is exact */

    t = dn2*Ph.d;
    s = z - t;
    ss = z - s - t;   /* correction term */

    t = ss - dn1*Pl.d;
    w = s + t;
    ww = s - w + t;   /* correction term */

    t = ww - dn2*Pl.d;
    s = w + t;
    ss = w - s + t;   /* correction term */

    t = ss - dn1*Pt.d;
    w = s + t;
    ww = s - w + t;   /* correction term */

    t = ww - dn2*Pt.d;
    z = w + t;    /* z = reduced arg */

    if ( x < 0.0 )
    { /* adjust for sign of arg */

      z = -z;
      n = -n;
    }

    xsq = z*z;
  
    n++;

    p = (double *)&P[0].d;

    /* if ( n&1 )
      compute cos(z)
       else
      compute sin(z)

       if ( n&2 )
      negate result
    */

    if ( n&1 )
    {
      p = (double *)&Q[0].d;
      z = 1.0;
    }
  
    result = (((((p[6]*xsq + p[5])*xsq + p[4])*xsq 
      + p[3])*xsq + p[2])*xsq + p[1])*(xsq*z) + z;
  
    if ( n&2 )
      result = -result;
  
    return ( result );
  }

  if ( (x != x) || (fabs(x) == Inf.d) )
  {
    /* x is a NaN or +/-inf; return a quiet NaN */

#ifdef _CALL_MATHERR

    exstruct.type = DOMAIN;
    exstruct.name = "cos";
    exstruct.arg1 = x;
    exstruct.retval = Qnan.d;

    if ( matherr( &exstruct ) == 0 )
    {
      fprintf(stderr, "domain error in cos\n");
      SETERRNO(EDOM);
    }

    return ( exstruct.retval );
#else
    NAN_SETERRNO(EDOM);

    return ( Qnan.d );
#endif
  }

  /* just give up and return 0.0, setting errno = ERANGE */

#ifdef _CALL_MATHERR

    exstruct.type = TLOSS;
    exstruct.name = "cos";
    exstruct.arg1 = x;
    exstruct.retval = 0.0;

    if ( matherr( &exstruct ) == 0 )
    {
      fprintf(stderr, "range error in cos (total loss \
of significance)\n");
      SETERRNO(ERANGE);
    }

    return ( exstruct.retval );
#else
    SETERRNO(ERANGE);

    return ( 0.0 );
#endif
}

#ifdef NO_LONG_DOUBLE

#if defined(BUILD_OS_DARWIN) /* Mach-O doesn't support aliases */
extern long double __cosl(long double);
long double cosl( long double x ) {
  return ( (long double)__cos((double)x) );
}
#elif defined(__GNUC__)
extern  long double  __cosl(long double);

long double    cosl() __attribute__ ((weak, alias ("__cosl")));

#endif

long double
__cosl( long double x )
{ 
  return ( (long double)__cos((double)x) );
}

#endif

---