/*
* 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 */
/* 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 <sys/param.h>
const mp_no mpone = {1, {1.0, 1.0}};
const mp_no mptwo = {1, {1.0, 2.0}};
/* 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)
{
long i;
long p2 = p;
for (i = 1; i <= p2; 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)
if (X[0] == ZERO)
if (Y[0] == ZERO)
i = 0;
i = -1;
else if (Y[0] == ZERO)
i = 1;
if (EX > EY)
else if (EX < EY)
i = mcr (x, y, p);
return i;
/* 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 (i = 0; i <= p; i++)
Y[i] = X[i];
/* 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
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]);
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)
z[3] += ONE;
break;
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
number *Y, Denormal case (|x| < 2**(-1022))). */
denorm (const mp_no *x, double *y, int p)
long i, k;
double c, u, z[5];
if (EX < -44 || (EX == -44 && X[1] < TWO5))
*y = ZERO;
return;
if (p2 == 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];
k = 2;
z[3] = X[1];
k = 1;
else if (p2 == 2)
z[2] = X[2];
z[3] = X[2];
z[3] = X[k];
u = (z[3] + TWO57) - TWO57;
u -= TWO5;
if (u == z[3])
for (i = k + 1; i <= p2; i++)
c = X[0] * ((z[1] + R * (z[2] + R * z[3])) - TWO10);
*y = c * TWOM1032;
/* Convert multiple precision number *X into double precision number *Y. The
result is correctly rounded to the nearest/even. */
__mp_dbl (const mp_no *x, double *y, int p)
if (EX > -42)
norm (x, y, p);
else if (EX == -42 && X[1] >= TWO10)
denorm (x, y, p);
/* Get the multiple precision equivalent of X into *Y. If the precision is too
small, the result is truncated. */
__dbl_mp (double x, mp_no *y, int p)
long i, n;
double u;
/* Sign. */
if (x == ZERO)
Y[0] = ZERO;
else if (x > ZERO)
Y[0] = ONE;
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 (p2, 4);
for (i = 1; i <= n; i++)
u = (x + TWO52) - TWO52;
if (u > x)
u -= ONE;
Y[i] = u;
x -= u;
for (; i <= p2; 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. */
add_magnitudes (const mp_no *x, const mp_no *y, mp_no *z, int p)
long i, j, k;
EZ = EX;
i = p2;
j = p2 + EY - EX;
k = p2 + 1;
if (j < 1)
__cpy (x, z, p);
Z[k] = ZERO;
for (; j > 0; i--, j--)
Z[k] += X[i] + Y[j];
if (Z[k] >= RADIX)
Z[k] -= RADIX;
Z[--k] = ONE;
Z[--k] = ZERO;
for (; i > 0; i--)
Z[k] += X[i];
if (Z[1] == ZERO)
Z[i] = Z[i + 1];
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. */
sub_magnitudes (const mp_no *x, const mp_no *y, mp_no *z, int p)
if (EX == EY)
i = j = k = p2;
Z[k] = Z[k + 1] = ZERO;
j = EX - EY;
if (j > p2)
j = p2 + 1 - j;
k = p2;
if (Y[j] > ZERO)
Z[k + 1] = RADIX - Y[j--];
Z[k] = MONE;
Z[k + 1] = ZERO;
j--;
Z[k] += (X[i] - Y[j]);
if (Z[k] < ZERO)
Z[k] += RADIX;
Z[--k] = MONE;
for (i = 1; Z[i] == ZERO; i++);
EZ = EZ - i + 1;
for (k = 1; i <= p2 + 1;)
Z[k++] = Z[i++];
for (; k <= p2;)
Z[k++] = ZERO;
/* Add *X and *Y and store the result in *Z. X and Y may overlap, but not X
__add (const mp_no *x, const mp_no *y, mp_no *z, int p)
int n;
__cpy (y, z, p);
if (X[0] == Y[0])
if (__acr (x, y, p) > 0)
add_magnitudes (x, y, z, p);
Z[0] = X[0];
add_magnitudes (y, x, z, p);
Z[0] = Y[0];
if ((n = __acr (x, y, p)) == 1)
sub_magnitudes (x, y, z, p);
else if (n == -1)
sub_magnitudes (y, x, z, p);
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. */
__sub (const mp_no *x, const mp_no *y, mp_no *z, int p)
Z[0] = -Z[0];
if (X[0] != Y[0])
Z[0] = -Y[0];
/* 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. */
__mul (const mp_no *x, const mp_no *y, mp_no *z, int p)
long i, i1, i2, j, k, k2;
double u, zk, zk2;
/* Is z=0? */
if (X[0] * Y[0] == ZERO)
/* Multiply, add and carry */
k2 = (p2 < 3) ? p2 + p2 : p2 + 3;
zk = Z[k2] = ZERO;
for (k = k2; k > 1;)
if (k > p2)
i1 = k - p2;
i2 = p2 + 1;
i1 = 1;
i2 = k;
#if 1
/* Rearrange this inner loop to allow the fmadd instructions to be
independent and execute in parallel on processors that have
dual symmetrical FP pipelines. */
if (i1 < (i2 - 1))
/* Make sure we have at least 2 iterations. */
if (((i2 - i1) & 1L) == 1L)
/* Handle the odd iterations case. */
zk2 = x->d[i2 - 1] * y->d[i1];
zk2 = 0.0;
/* Do two multiply/adds per loop iteration, using independent
accumulators; zk and zk2. */
for (i = i1, j = i2 - 1; i < i2 - 1; i += 2, j -= 2)
zk += x->d[i] * y->d[j];
zk2 += x->d[i + 1] * y->d[j - 1];
zk += zk2; /* Final sum. */
/* Special case when iterations is 1. */
zk += x->d[i1] * y->d[i1];
#else
/* The original code. */
for (i = i1, j = i2 - 1; i < i2; i++, j--)
zk += X[i] * Y[j];
#endif
u = (zk + CUTTER) - CUTTER;
if (u > zk)
Z[k] = zk - u;
zk = u * RADIXI;
--k;
Z[k] = zk;
/* Is there a carry beyond the most significant digit? */
EZ = EX + EY - 1;
EZ = EX + EY;
Z[0] = X[0] * Y[0];
/* Square *X and store result in *Y. X and Y may not overlap. 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. This is a faster special case of
multiplication. */
__sqr (const mp_no *x, mp_no *y, int p)
long i, j, k, ip;
double u, yk;
if (__glibc_unlikely (X[0] == ZERO))
/* We need not iterate through all X's since it's pointless to
multiply zeroes. */
for (ip = p; ip > 0; ip--)
if (X[ip] != ZERO)
k = (__glibc_unlikely (p < 3)) ? p + p : p + 3;
while (k > 2 * ip + 1)
Y[k--] = ZERO;
yk = ZERO;
while (k > p)
double yk2 = 0.0;
long lim = k / 2;
if (k % 2 == 0)
yk += X[lim] * X[lim];
lim--;
/* In __mul, this loop (and the one within the next while loop) run
between a range to calculate the mantissa as follows:
Z[k] = X[k] * Y[n] + X[k+1] * Y[n-1] ... + X[n-1] * Y[k+1]
+ X[n] * Y[k]
For X == Y, we can get away with summing halfway and doubling the
result. For cases where the range size is even, the mid-point needs
to be added separately (above). */
for (i = k - p, j = p; i <= lim; i++, j--)
yk2 += X[i] * X[j];
yk += 2.0 * yk2;
u = (yk + CUTTER) - CUTTER;
if (u > yk)
Y[k--] = yk - u;
yk = u * RADIXI;
while (k > 1)
/* Likewise for this loop. */
for (i = 1, j = k - 1; i <= lim; i++, j--)
Y[k] = yk;
/* Squares are always positive. */
Y[0] = 1.0;
EY = 2 * EX;
if (__glibc_unlikely (Y[1] == ZERO))
for (i = 1; i <= p; i++)
Y[i] = Y[i + 1];
EY--;
/* 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. */
__inv (const mp_no *x, mp_no *y, int p)
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)
__dvd (const mp_no *x, const mp_no *y, mp_no *z, int p)
mp_no w;
__inv (y, &w, p);
__mul (x, &w, z, p);
