path: root/libc/sysdeps/ieee754/dbl-64/mpa.c

/*
 * IBM Accurate Mathematical Library
 * written by International Business Machines Corp.
 * Copyright (C) 2001-2013 Free Software Foundation, Inc.
 *
 * This program is free software; you can redistribute it and/or modify
 * it under the terms of the GNU Lesser General Public License as published by
 * the Free Software Foundation; either version 2.1 of the License, or
 * (at your option) any later version.
 *
 * This program is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 * GNU Lesser General Public License for more details.
 *
 * You should have received a copy of the GNU Lesser General Public License
 * along with this program; if not, see <http://www.gnu.org/licenses/>.
 */
/************************************************************************/
/*  MODULE_NAME: mpa.c                                                  */
/*                                                                      */
/*  FUNCTIONS:                                                          */
/*               mcr                                                    */
/*               acr                                                    */
/*               cr                                                     */
/*               cpy                                                    */
/*               cpymn                                                  */
/*               norm                                                   */
/*               denorm                                                 */
/*               mp_dbl                                                 */
/*               dbl_mp                                                 */
/*               add_magnitudes                                         */
/*               sub_magnitudes                                         */
/*               add                                                    */
/*               sub                                                    */
/*               mul                                                    */
/*               inv                                                    */
/*               dvd                                                    */
/*                                                                      */
/* Arithmetic functions for multiple precision numbers.                 */
/* Relative errors are bounded                                          */
/************************************************************************/


#include "endian.h"
#include "mpa.h"
#include "mpa2.h"
#include <sys/param.h>

#ifndef SECTION
# define SECTION
#endif

#ifndef NO__CONST
const mp_no mpone = {1, {1.0, 1.0}};
const mp_no mptwo = {1, {1.0, 2.0}};
#endif

#ifndef NO___ACR
/* Compare mantissa of two multiple precision numbers regardless of the sign
   and exponent of the numbers.  */
static int
mcr(const mp_no *x, const mp_no *y, int p) {
  int i;
  for (i=1; i<=p; i++) {
    if      (X[i] == Y[i])  continue;
    else if (X[i] >  Y[i])  return  1;
    else                    return -1; }
  return 0;
}

/* Compare the absolute values of two multiple precision numbers.  */
int
__acr(const mp_no *x, const mp_no *y, int p) {
  int i;

  if      (X[0] == ZERO) {
    if    (Y[0] == ZERO) i= 0;
    else                 i=-1;
  }
  else if (Y[0] == ZERO) i= 1;
  else {
    if      (EX >  EY)   i= 1;
    else if (EX <  EY)   i=-1;
    else                 i= mcr(x,y,p);
  }

  return i;
}
#endif

#ifndef NO___CPY
/* Copy multiple precision number X into Y.  They could be the same
   number.  */
void __cpy(const mp_no *x, mp_no *y, int p) {
  EY = EX;
  for (int i=0; i <= p; i++)    Y[i] = X[i];
}
#endif

#ifndef NO___MP_DBL
/* Convert a multiple precision number *X into a double precision
   number *Y, normalized case  (|x| >= 2**(-1022))).  */
static void norm(const mp_no *x, double *y, int p)
{
  #define R  RADIXI
  int i;
  double a,c,u,v,z[5];
  if (p<5) {
    if      (p==1) c = X[1];
    else if (p==2) c = X[1] + R* X[2];
    else if (p==3) c = X[1] + R*(X[2]  +   R* X[3]);
    else if (p==4) c =(X[1] + R* X[2]) + R*R*(X[3] + R*X[4]);
  }
  else {
    for (a=ONE, z[1]=X[1]; z[1] < TWO23; )
	{a *= TWO;   z[1] *= TWO; }

    for (i=2; i<5; i++) {
      z[i] = X[i]*a;
      u = (z[i] + CUTTER)-CUTTER;
      if  (u > z[i])  u -= RADIX;
      z[i] -= u;
      z[i-1] += u*RADIXI;
    }

    u = (z[3] + TWO71) - TWO71;
    if (u > z[3])   u -= TWO19;
    v = z[3]-u;

    if (v == TWO18) {
      if (z[4] == ZERO) {
	for (i=5; i <= p; i++) {
	  if (X[i] == ZERO)   continue;
	  else                {z[3] += ONE;   break; }
	}
      }
      else              z[3] += ONE;
    }

    c = (z[1] + R *(z[2] + R * z[3]))/a;
  }

  c *= X[0];

  for (i=1; i<EX; i++)   c *= RADIX;
  for (i=1; i>EX; i--)   c *= RADIXI;

  *y = c;
#undef R
}

/* Convert a multiple precision number *X into a double precision
   number *Y, Denormal case  (|x| < 2**(-1022))).  */
static void denorm(const mp_no *x, double *y, int p)
{
  int i,k;
  double c,u,z[5];

#define R  RADIXI
  if (EX<-44 || (EX==-44 && X[1]<TWO5))
     { *y=ZERO; return; }

  if      (p==1) {
    if      (EX==-42) {z[1]=X[1]+TWO10;  z[2]=ZERO;  z[3]=ZERO;  k=3;}
    else if (EX==-43) {z[1]=     TWO10;  z[2]=X[1];  z[3]=ZERO;  k=2;}
    else              {z[1]=     TWO10;  z[2]=ZERO;  z[3]=X[1];  k=1;}
  }
  else if (p==2) {
    if      (EX==-42) {z[1]=X[1]+TWO10;  z[2]=X[2];  z[3]=ZERO;  k=3;}
    else if (EX==-43) {z[1]=     TWO10;  z[2]=X[1];  z[3]=X[2];  k=2;}
    else              {z[1]=     TWO10;  z[2]=ZERO;  z[3]=X[1];  k=1;}
  }
  else {
    if      (EX==-42) {z[1]=X[1]+TWO10;  z[2]=X[2];  k=3;}
    else if (EX==-43) {z[1]=     TWO10;  z[2]=X[1];  k=2;}
    else              {z[1]=     TWO10;  z[2]=ZERO;  k=1;}
    z[3] = X[k];
  }

  u = (z[3] + TWO57) - TWO57;
  if  (u > z[3])   u -= TWO5;

  if (u==z[3]) {
    for (i=k+1; i <= p; i++) {
      if (X[i] == ZERO)   continue;
      else {z[3] += ONE;   break; }
    }
  }

  c = X[0]*((z[1] + R*(z[2] + R*z[3])) - TWO10);

  *y = c*TWOM1032;
#undef R
}

/* Convert multiple precision number *X into double precision number *Y.  The
   result is correctly rounded to the nearest/even.  */
void __mp_dbl(const mp_no *x, double *y, int p) {

  if (X[0] == ZERO)  {*y = ZERO;  return; }

  if (__glibc_likely (EX > -42 || (EX == -42 && X[1] >= TWO10)))
    norm(x,y,p);
  else
    denorm(x,y,p);
}
#endif

/* Get the multiple precision equivalent of X into *Y.  If the precision is too
   small, the result is truncated.  */
void
SECTION
__dbl_mp(double x, mp_no *y, int p) {

  int i,n;
  double u;

  /* Sign.  */
  if      (x == ZERO)  {Y[0] = ZERO;  return; }
  else if (x >  ZERO)   Y[0] = ONE;
  else                 {Y[0] = MONE;  x=-x;   }

  /* Exponent.  */
  for (EY=ONE; x >= RADIX; EY += ONE)   x *= RADIXI;
  for (      ; x <  ONE;   EY -= ONE)   x *= RADIX;

  /* Digits.  */
  n=MIN(p,4);
  for (i=1; i<=n; i++) {
    u = (x + TWO52) - TWO52;
    if (u>x)   u -= ONE;
    Y[i] = u;     x -= u;    x *= RADIX; }
  for (   ; i<=p; i++)     Y[i] = ZERO;
}

/* Add magnitudes of *X and *Y assuming that abs (*X) >= abs (*Y) > 0.  The
   sign of the sum *Z is not changed.  X and Y may overlap but not X and Z or
   Y and Z.  No guard digit is used.  The result equals the exact sum,
   truncated.  */
static void
SECTION
add_magnitudes(const mp_no *x, const mp_no *y, mp_no *z, int p) {

  int i,j,k;

  EZ = EX;

  i=p;    j=p+ EY - EX;    k=p+1;

  if (j<1)
     {__cpy(x,z,p);  return; }
  else   Z[k] = ZERO;

  for (; j>0; i--,j--) {
    Z[k] += X[i] + Y[j];
    if (Z[k] >= RADIX) {
      Z[k]  -= RADIX;
      Z[--k] = ONE; }
    else
      Z[--k] = ZERO;
  }

  for (; i>0; i--) {
    Z[k] += X[i];
    if (Z[k] >= RADIX) {
      Z[k]  -= RADIX;
      Z[--k] = ONE; }
    else
      Z[--k] = ZERO;
  }

  if (Z[1] == ZERO) {
    for (i=1; i<=p; i++)    Z[i] = Z[i+1]; }
  else   EZ += ONE;
}

/* Subtract the magnitudes of *X and *Y assuming that abs (*x) > abs (*y) > 0.
   The sign of the difference *Z is not changed.  X and Y may overlap but not X
   and Z or Y and Z.  One guard digit is used.  The error is less than one
   ULP.  */
static void
SECTION
sub_magnitudes(const mp_no *x, const mp_no *y, mp_no *z, int p) {

  int i,j,k;

  EZ = EX;

  if (EX == EY) {
    i=j=k=p;
    Z[k] = Z[k+1] = ZERO; }
  else {
    j= EX - EY;
    if (j > p)  {__cpy(x,z,p);  return; }
    else {
      i=p;   j=p+1-j;   k=p;
      if (Y[j] > ZERO) {
	Z[k+1] = RADIX - Y[j--];
	Z[k]   = MONE; }
      else {
	Z[k+1] = ZERO;
	Z[k]   = ZERO;   j--;}
    }
  }

  for (; j>0; i--,j--) {
    Z[k] += (X[i] - Y[j]);
    if (Z[k] < ZERO) {
      Z[k]  += RADIX;
      Z[--k] = MONE; }
    else
      Z[--k] = ZERO;
  }

  for (; i>0; i--) {
    Z[k] += X[i];
    if (Z[k] < ZERO) {
      Z[k]  += RADIX;
      Z[--k] = MONE; }
    else
      Z[--k] = ZERO;
  }

  for (i=1; Z[i] == ZERO; i++) ;
  EZ = EZ - i + 1;
  for (k=1; i <= p+1; )
    Z[k++] = Z[i++];
  for (; k <= p; )
    Z[k++] = ZERO;
}

/* Add *X and *Y and store the result in *Z.  X and Y may overlap, but not X
   and Z or Y and Z.  One guard digit is used.  The error is less than one
   ULP.  */
void
SECTION
__add(const mp_no *x, const mp_no *y, mp_no *z, int p) {

  int n;

  if      (X[0] == ZERO)     {__cpy(y,z,p);  return; }
  else if (Y[0] == ZERO)     {__cpy(x,z,p);  return; }

  if (X[0] == Y[0])   {
    if (__acr(x,y,p) > 0)      {add_magnitudes(x,y,z,p);  Z[0] = X[0]; }
    else                     {add_magnitudes(y,x,z,p);  Z[0] = Y[0]; }
  }
  else                       {
    if ((n=__acr(x,y,p)) == 1) {sub_magnitudes(x,y,z,p);  Z[0] = X[0]; }
    else if (n == -1)        {sub_magnitudes(y,x,z,p);  Z[0] = Y[0]; }
    else                      Z[0] = ZERO;
  }
}

/* Subtract *Y from *X and return the result in *Z.  X and Y may overlap but
   not X and Z or Y and Z.  One guard digit is used.  The error is less than
   one ULP.  */
void
SECTION
__sub(const mp_no *x, const mp_no *y, mp_no *z, int p) {

  int n;

  if      (X[0] == ZERO)     {__cpy(y,z,p);  Z[0] = -Z[0];  return; }
  else if (Y[0] == ZERO)     {__cpy(x,z,p);                 return; }

  if (X[0] != Y[0])    {
    if (__acr(x,y,p) > 0)      {add_magnitudes(x,y,z,p);  Z[0] =  X[0]; }
    else                     {add_magnitudes(y,x,z,p);  Z[0] = -Y[0]; }
  }
  else                       {
    if ((n=__acr(x,y,p)) == 1) {sub_magnitudes(x,y,z,p);  Z[0] =  X[0]; }
    else if (n == -1)        {sub_magnitudes(y,x,z,p);  Z[0] = -Y[0]; }
    else                      Z[0] = ZERO;
  }
}

/* Multiply *X and *Y and store result in *Z.  X and Y may overlap but not X
   and Z or Y and Z.  For P in [1, 2, 3], the exact result is truncated to P
   digits.  In case P > 3 the error is bounded by 1.001 ULP.  */
void
SECTION
__mul(const mp_no *x, const mp_no *y, mp_no *z, int p) {

  int i, j, k, k2;
  double u;

  /* Is z=0?  */
  if (__glibc_unlikely (X[0] * Y[0] == ZERO))
    {
      Z[0]=ZERO;
      return;
    }

  /* Multiply, add and carry.  */
  k2 = (__glibc_unlikely (p < 3)) ? p + p : p + 3;
  Z[k2] = ZERO;

  for (k = k2; k > p; )
    {
      for (i = k - p, j = p; i < p + 1; i++, j--)
	Z[k] += X[i] * Y[j];

      u = (Z[k] + CUTTER) - CUTTER;
      if (u > Z[k])
	u -= RADIX;
      Z[k] -= u;
      Z[--k] = u * RADIXI;
    }

  while (k > 1)
    {
      for (i = 1,j = k - 1; i < k; i++, j--)
	Z[k] += X[i] * Y[j];

      u = (Z[k] + CUTTER) - CUTTER;
      if (u > Z[k])
	u -= RADIX;
      Z[k] -= u;
      Z[--k] = u * RADIXI;
    }

  EZ = EX + EY;
  /* Is there a carry beyond the most significant digit?  */
  if (__glibc_unlikely (Z[1] == ZERO))
    {
      for (i = 1; i <= p; i++)
	Z[i] = Z[i+1];
      EZ--;
    }

  Z[0] = X[0] * Y[0];
}

/* Invert *X and store in *Y.  Relative error bound:
   - For P = 2: 1.001 * R ^ (1 - P)
   - For P = 3: 1.063 * R ^ (1 - P)
   - For P > 3: 2.001 * R ^ (1 - P)

   *X = 0 is not permissible.  */
static
SECTION
void __inv(const mp_no *x, mp_no *y, int p) {
  int i;
  double t;
  mp_no z,w;
  static const int np1[] = {0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,
			    4,4,4,4,4,4,4,4,4,4,4,4,4,4,4};

  __cpy(x,&z,p);  z.e=0;  __mp_dbl(&z,&t,p);
  t=ONE/t;   __dbl_mp(t,y,p);    EY -= EX;

  for (i=0; i<np1[p]; i++) {
    __cpy(y,&w,p);
    __mul(x,&w,y,p);
    __sub(&mptwo,y,&z,p);
    __mul(&w,&z,y,p);
  }
}

/* Divide *X by *Y and store result in *Z.  X and Y may overlap but not X and Z
   or Y and Z.  Relative error bound:
   - For P = 2: 2.001 * R ^ (1 - P)
   - For P = 3: 2.063 * R ^ (1 - P)
   - For P > 3: 3.001 * R ^ (1 - P)

   *X = 0 is not permissible.  */
void
SECTION
__dvd(const mp_no *x, const mp_no *y, mp_no *z, int p) {

  mp_no w;

  if (X[0] == ZERO)    Z[0] = ZERO;
  else                {__inv(y,&w,p);   __mul(x,&w,z,p);}
}