/*
* 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 .
*/
/************************************************************************/
/* MODULE_NAME: mpa.c */
/* */
/* FUNCTIONS: */
/* mcr */
/* acr */
/* cpy */
/* 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
#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, zk;
/* 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;
zk = Z[k2] = ZERO;
for (k = k2; k > p; k--)
{
for (i = k - p, j = p; i < p + 1; i++, j--)
zk += X[i] * Y[j];
u = (zk + CUTTER) - CUTTER;
if (u > zk)
u -= RADIX;
Z[k] = zk - u;
zk = u * RADIXI;
}
while (k > 1)
{
for (i = 1, j = k - 1; i < k; i++, j--)
zk += X[i] * Y[j];
u = (zk + CUTTER) - CUTTER;
if (u > zk)
u -= RADIX;
Z[k] = zk - u;
zk = u * RADIXI;
k--;
}
Z[k] = zk;
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 void
SECTION
__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);
}
}