/*
* Copyright (C) 1987-2008 Sun Microsystems, Inc. All Rights Reserved.
* Copyright (C) 2008-2011 Robert Ancell
*
* This program is free software: you can redistribute it and/or modify it under
* the terms of the GNU General Public License as published by the Free Software
* Foundation, either version 2 of the License, or (at your option) any later
* version. See http://www.gnu.org/copyleft/gpl.html the full text of the
* license.
*/
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include <errno.h>
#include "mp.h"
#include "mp-private.h"
static MPNumber eulers_number;
static gboolean have_eulers_number = FALSE;
// FIXME: Re-add overflow and underflow detection
char *mp_error = NULL;
/* THIS ROUTINE IS CALLED WHEN AN ERROR CONDITION IS ENCOUNTERED, AND
* AFTER A MESSAGE HAS BEEN WRITTEN TO STDERR.
*/
void
mperr(const char *format, ...)
{
char text[1024];
va_list args;
va_start(args, format);
vsnprintf(text, 1024, format, args);
va_end(args);
if (mp_error)
free(mp_error);
mp_error = strdup(text);
}
const char *
mp_get_error()
{
return mp_error;
}
void mp_clear_error()
{
if (mp_error)
free(mp_error);
mp_error = NULL;
}
/* ROUTINE CALLED BY MP_DIVIDE AND MP_SQRT TO ENSURE THAT
* RESULTS ARE REPRESENTED EXACTLY IN T-2 DIGITS IF THEY
* CAN BE. X IS AN MP NUMBER, I AND J ARE INTEGERS.
*/
static void
mp_ext(int i, int j, MPNumber *x)
{
int q, s;
if (mp_is_zero(x) || MP_T <= 2 || i == 0)
return;
/* COMPUTE MAXIMUM POSSIBLE ERROR IN THE LAST PLACE */
q = (j + 1) / i + 1;
s = MP_BASE * x->fraction[MP_T - 2] + x->fraction[MP_T - 1];
/* SET LAST TWO DIGITS TO ZERO */
if (s <= q) {
x->fraction[MP_T - 2] = 0;
x->fraction[MP_T - 1] = 0;
return;
}
if (s + q < MP_BASE * MP_BASE)
return;
/* ROUND UP HERE */
x->fraction[MP_T - 2] = MP_BASE - 1;
x->fraction[MP_T - 1] = MP_BASE;
/* NORMALIZE X (LAST DIGIT B IS OK IN MP_MULTIPLY_INTEGER) */
mp_multiply_integer(x, 1, x);
}
void
mp_get_eulers(MPNumber *z)
{
if (!have_eulers_number) {
MPNumber t;
mp_set_from_integer(1, &t);
mp_epowy(&t, &eulers_number);
have_eulers_number = TRUE;
}
mp_set_from_mp(&eulers_number, z);
}
void
mp_get_i(MPNumber *z)
{
mp_set_from_integer(0, z);
z->im_sign = 1;
z->im_exponent = 1;
z->im_fraction[0] = 1;
}
void
mp_abs(const MPNumber *x, MPNumber *z)
{
if (mp_is_complex(x)){
MPNumber x_real, x_im;
mp_real_component(x, &x_real);
mp_imaginary_component(x, &x_im);
mp_multiply(&x_real, &x_real, &x_real);
mp_multiply(&x_im, &x_im, &x_im);
mp_add(&x_real, &x_im, z);
mp_root(z, 2, z);
}
else {
mp_set_from_mp(x, z);
if (z->sign < 0)
z->sign = -z->sign;
}
}
void
mp_arg(const MPNumber *x, MPAngleUnit unit, MPNumber *z)
{
MPNumber x_real, x_im, pi;
if (mp_is_zero(x)) {
/* Translators: Error display when attempting to take argument of zero */
mperr(_("Argument not defined for zero"));
mp_set_from_integer(0, z);
return;
}
mp_real_component(x, &x_real);
mp_imaginary_component(x, &x_im);
mp_get_pi(&pi);
if (mp_is_zero(&x_im)) {
if (mp_is_negative(&x_real))
convert_from_radians(&pi, MP_RADIANS, z);
else
mp_set_from_integer(0, z);
}
else if (mp_is_zero(&x_real)) {
mp_set_from_mp(&pi, z);
if (mp_is_negative(&x_im))
mp_divide_integer(z, -2, z);
else
mp_divide_integer(z, 2, z);
}
else if (mp_is_negative(&x_real)) {
mp_divide(&x_im, &x_real, z);
mp_atan(z, MP_RADIANS, z);
if (mp_is_negative(&x_im))
mp_subtract(z, &pi, z);
else
mp_add(z, &pi, z);
}
else {
mp_divide(&x_im, &x_real, z);
mp_atan(z, MP_RADIANS, z);
}
convert_from_radians(z, unit, z);
}
void
mp_conjugate(const MPNumber *x, MPNumber *z)
{
mp_set_from_mp(x, z);
z->im_sign = -z->im_sign;
}
void
mp_real_component(const MPNumber *x, MPNumber *z)
{
mp_set_from_mp(x, z);
/* Clear imaginary component */
z->im_sign = 0;
z->im_exponent = 0;
memset(z->im_fraction, 0, sizeof(int) * MP_SIZE);
}
void
mp_imaginary_component(const MPNumber *x, MPNumber *z)
{
/* Copy imaginary component to real component */
z->sign = x->im_sign;
z->exponent = x->im_exponent;
memcpy(z->fraction, x->im_fraction, sizeof(int) * MP_SIZE);
/* Clear (old) imaginary component */
z->im_sign = 0;
z->im_exponent = 0;
memset(z->im_fraction, 0, sizeof(int) * MP_SIZE);
}
static void
mp_add_real(const MPNumber *x, int y_sign, const MPNumber *y, MPNumber *z)
{
int sign_prod, i, c;
int exp_diff, med;
bool x_largest = false;
const int *big_fraction, *small_fraction;
MPNumber x_copy, y_copy;
/* 0 + y = y */
if (mp_is_zero(x)) {
mp_set_from_mp(y, z);
z->sign = y_sign;
return;
}
/* x + 0 = x */
else if (mp_is_zero(y)) {
mp_set_from_mp(x, z);
return;
}
sign_prod = y_sign * x->sign;
exp_diff = x->exponent - y->exponent;
med = abs(exp_diff);
if (exp_diff < 0) {
x_largest = false;
} else if (exp_diff > 0) {
x_largest = true;
} else {
/* EXPONENTS EQUAL SO COMPARE SIGNS, THEN FRACTIONS IF NEC. */
if (sign_prod < 0) {
/* Signs are not equal. find out which mantissa is larger. */
int j;
for (j = 0; j < MP_T; j++) {
int i = x->fraction[j] - y->fraction[j];
if (i == 0)
continue;
if (i < 0)
x_largest = false;
else if (i > 0)
x_largest = true;
break;
}
/* Both mantissas equal, so result is zero. */
if (j >= MP_T) {
mp_set_from_integer(0, z);
return;
}
}
}
mp_set_from_mp(x, &x_copy);
mp_set_from_mp(y, &y_copy);
mp_set_from_integer(0, z);
if (x_largest) {
z->sign = x_copy.sign;
z->exponent = x_copy.exponent;
big_fraction = x_copy.fraction;
small_fraction = y_copy.fraction;
} else {
z->sign = y_sign;
z->exponent = y_copy.exponent;
big_fraction = y_copy.fraction;
small_fraction = x_copy.fraction;
}
/* CLEAR GUARD DIGITS TO RIGHT OF X DIGITS */
for(i = 3; i >= med; i--)
z->fraction[MP_T + i] = 0;
if (sign_prod >= 0) {
/* This is probably insufficient overflow detection, but it makes us
* not crash at least.
*/
if (MP_T + 3 < med) {
mperr(_("Overflow: the result couldn't be calculated"));
mp_set_from_integer(0, z);
return;
}
/* HERE DO ADDITION, EXPONENT(Y) >= EXPONENT(X) */
for (i = MP_T + 3; i >= MP_T; i--)
z->fraction[i] = small_fraction[i - med];
c = 0;
for (; i >= med; i--) {
c = big_fraction[i] + small_fraction[i - med] + c;
if (c < MP_BASE) {
/* NO CARRY GENERATED HERE */
z->fraction[i] = c;
c = 0;
} else {
/* CARRY GENERATED HERE */
z->fraction[i] = c - MP_BASE;
c = 1;
}
}
for (; i >= 0; i--)
{
c = big_fraction[i] + c;
if (c < MP_BASE) {
z->fraction[i] = c;
i--;
/* NO CARRY POSSIBLE HERE */
for (; i >= 0; i--)
z->fraction[i] = big_fraction[i];
c = 0;
break;
}
z->fraction[i] = 0;
c = 1;
}
/* MUST SHIFT RIGHT HERE AS CARRY OFF END */
if (c != 0) {
for (i = MP_T + 3; i > 0; i--)
z->fraction[i] = z->fraction[i - 1];
z->fraction[0] = 1;
z->exponent++;
}
}
else {
c = 0;
for (i = MP_T + med - 1; i >= MP_T; i--) {
/* HERE DO SUBTRACTION, ABS(Y) > ABS(X) */
z->fraction[i] = c - small_fraction[i - med];
c = 0;
/* BORROW GENERATED HERE */
if (z->fraction[i] < 0) {
c = -1;
z->fraction[i] += MP_BASE;
}
}
for(; i >= med; i--) {
c = big_fraction[i] + c - small_fraction[i - med];
if (c >= 0) {
/* NO BORROW GENERATED HERE */
z->fraction[i] = c;
c = 0;
} else {
/* BORROW GENERATED HERE */
z->fraction[i] = c + MP_BASE;
c = -1;
}
}
for (; i >= 0; i--) {
c = big_fraction[i] + c;
if (c >= 0) {
z->fraction[i] = c;
i--;
/* NO CARRY POSSIBLE HERE */
for (; i >= 0; i--)
z->fraction[i] = big_fraction[i];
break;
}
z->fraction[i] = c + MP_BASE;
c = -1;
}
}
mp_normalize(z);
}
static void
mp_add_with_sign(const MPNumber *x, int y_sign, const MPNumber *y, MPNumber *z)
{
if (mp_is_complex(x) || mp_is_complex(y)) {
MPNumber real_x, real_y, im_x, im_y, real_z, im_z;
mp_real_component(x, &real_x);
mp_imaginary_component(x, &im_x);
mp_real_component(y, &real_y);
mp_imaginary_component(y, &im_y);
mp_add_real(&real_x, y_sign * y->sign, &real_y, &real_z);
mp_add_real(&im_x, y_sign * y->im_sign, &im_y, &im_z);
mp_set_from_complex(&real_z, &im_z, z);
}
else
mp_add_real(x, y_sign * y->sign, y, z);
}
void
mp_add(const MPNumber *x, const MPNumber *y, MPNumber *z)
{
mp_add_with_sign(x, 1, y, z);
}
void
mp_add_integer(const MPNumber *x, int64_t y, MPNumber *z)
{
MPNumber t;
mp_set_from_integer(y, &t);
mp_add(x, &t, z);
}
void
mp_add_fraction(const MPNumber *x, int64_t i, int64_t j, MPNumber *y)
{
MPNumber t;
mp_set_from_fraction(i, j, &t);
mp_add(x, &t, y);
}
void
mp_subtract(const MPNumber *x, const MPNumber *y, MPNumber *z)
{
mp_add_with_sign(x, -1, y, z);
}
void
mp_sgn(const MPNumber *x, MPNumber *z)
{
if (mp_is_zero(x))
mp_set_from_integer(0, z);
else if (mp_is_negative(x))
mp_set_from_integer(-1, z);
else
mp_set_from_integer(1, z);
}
void
mp_integer_component(const MPNumber *x, MPNumber *z)
{
int i;
/* Clear fraction */
mp_set_from_mp(x, z);
for (i = z->exponent; i < MP_SIZE; i++)
z->fraction[i] = 0;
z->im_sign = 0;
z->im_exponent = 0;
memset(z->im_fraction, 0, sizeof(int) * MP_SIZE);
}
void
mp_fractional_component(const MPNumber *x, MPNumber *z)
{
int i, shift;
/* Fractional component of zero is 0 */
if (mp_is_zero(x)) {
mp_set_from_integer(0, z);
return;
}
/* All fractional */
if (x->exponent <= 0) {
mp_set_from_mp(x, z);
return;
}
/* Shift fractional component */
shift = x->exponent;
for (i = shift; i < MP_SIZE && x->fraction[i] == 0; i++)
shift++;
z->sign = x->sign;
z->exponent = x->exponent - shift;
for (i = 0; i < MP_SIZE; i++) {
if (i + shift >= MP_SIZE)
z->fraction[i] = 0;
else
z->fraction[i] = x->fraction[i + shift];
}
if (z->fraction[0] == 0)
z->sign = 0;
z->im_sign = 0;
z->im_exponent = 0;
memset(z->im_fraction, 0, sizeof(int) * MP_SIZE);
}
void
mp_fractional_part(const MPNumber *x, MPNumber *z)
{
MPNumber f;
mp_floor(x, &f);
mp_subtract(x, &f, z);
}
void
mp_floor(const MPNumber *x, MPNumber *z)
{
int i;
bool have_fraction = false, is_negative;
/* Integer component of zero = 0 */
if (mp_is_zero(x)) {
mp_set_from_mp(x, z);
return;
}
/* If all fractional then no integer component */
if (x->exponent <= 0) {
mp_set_from_integer(0, z);
return;
}
is_negative = mp_is_negative(x);
/* Clear fraction */
mp_set_from_mp(x, z);
for (i = z->exponent; i < MP_SIZE; i++) {
if (z->fraction[i])
have_fraction = true;
z->fraction[i] = 0;
}
z->im_sign = 0;
z->im_exponent = 0;
memset(z->im_fraction, 0, sizeof(int) * MP_SIZE);
if (have_fraction && is_negative)
mp_add_integer(z, -1, z);
}
void
mp_ceiling(const MPNumber *x, MPNumber *z)
{
MPNumber f;
mp_floor(x, z);
mp_fractional_component(x, &f);
if (mp_is_zero(&f))
return;
mp_add_integer(z, 1, z);
}
void
mp_round(const MPNumber *x, MPNumber *z)
{
MPNumber f, one;
bool do_floor;
do_floor = !mp_is_negative(x);
mp_fractional_component(x, &f);
mp_multiply_integer(&f, 2, &f);
mp_abs(&f, &f);
mp_set_from_integer(1, &one);
if (mp_is_greater_equal(&f, &one))
do_floor = !do_floor;
if (do_floor)
mp_floor(x, z);
else
mp_ceiling(x, z);
}
int
mp_compare_mp_to_mp(const MPNumber *x, const MPNumber *y)
{
int i;
if (x->sign != y->sign) {
if (x->sign > y->sign)
return 1;
else
return -1;
}
/* x = y = 0 */
if (mp_is_zero(x))
return 0;
/* See if numbers are of different magnitude */
if (x->exponent != y->exponent) {
if (x->exponent > y->exponent)
return x->sign;
else
return -x->sign;
}
/* Compare fractions */
for (i = 0; i < MP_SIZE; i++) {
if (x->fraction[i] == y->fraction[i])
continue;
if (x->fraction[i] > y->fraction[i])
return x->sign;
else
return -x->sign;
}
/* x = y */
return 0;
}
void
mp_divide(const MPNumber *x, const MPNumber *y, MPNumber *z)
{
int i, ie;
MPNumber t;
/* x/0 */
if (mp_is_zero(y)) {
/* Translators: Error displayed attempted to divide by zero */
mperr(_("Division by zero is undefined"));
mp_set_from_integer(0, z);
return;
}
/* 0/y = 0 */
if (mp_is_zero(x)) {
mp_set_from_integer(0, z);
return;
}
/* z = x × y⁻¹ */
/* FIXME: Set exponent to zero to avoid overflow in mp_multiply??? */
mp_reciprocal(y, &t);
ie = t.exponent;
t.exponent = 0;
i = t.fraction[0];
mp_multiply(x, &t, z);
mp_ext(i, z->fraction[0], z);
z->exponent += ie;
}
static void
mp_divide_integer_real(const MPNumber *x, int64_t y, MPNumber *z)
{
int c, i, k, b2, c2, j1, j2;
MPNumber x_copy;
/* x/0 */
if (y == 0) {
/* Translators: Error displayed attempted to divide by zero */
mperr(_("Division by zero is undefined"));
mp_set_from_integer(0, z);
return;
}
/* 0/y = 0 */
if (mp_is_zero(x)) {
mp_set_from_integer(0, z);
return;
}
/* Division by -1 or 1 just changes sign */
if (y == 1 || y == -1) {
if (y < 0)
mp_invert_sign(x, z);
else
mp_set_from_mp(x, z);
return;
}
/* Copy x as z may also refer to x */
mp_set_from_mp(x, &x_copy);
mp_set_from_integer(0, z);
if (y < 0) {
y = -y;
z->sign = -x_copy.sign;
}
else
z->sign = x_copy.sign;
z->exponent = x_copy.exponent;
c = 0;
i = 0;
/* IF y*B NOT REPRESENTABLE AS AN INTEGER HAVE TO SIMULATE
* LONG DIVISION. ASSUME AT LEAST 16-BIT WORD.
*/
/* Computing MAX */
b2 = max(MP_BASE << 3, 32767 / MP_BASE);
if (y < b2) {
int kh, r1;
/* LOOK FOR FIRST NONZERO DIGIT IN QUOTIENT */
do {
c = MP_BASE * c;
if (i < MP_T)
c += x_copy.fraction[i];
i++;
r1 = c / y;
if (r1 < 0)
goto L210;
} while (r1 == 0);
/* ADJUST EXPONENT AND GET T+4 DIGITS IN QUOTIENT */
z->exponent += 1 - i;
z->fraction[0] = r1;
c = MP_BASE * (c - y * r1);
kh = 1;
if (i < MP_T) {
kh = MP_T + 1 - i;
for (k = 1; k < kh; k++) {
c += x_copy.fraction[i];
z->fraction[k] = c / y;
c = MP_BASE * (c - y * z->fraction[k]);
i++;
}
if (c < 0)
goto L210;
}
for (k = kh; k < MP_T + 4; k++) {
z->fraction[k] = c / y;
c = MP_BASE * (c - y * z->fraction[k]);
}
if (c < 0)
goto L210;
mp_normalize(z);
return;
}
/* HERE NEED SIMULATED DOUBLE-PRECISION DIVISION */
j1 = y / MP_BASE;
j2 = y - j1 * MP_BASE;
/* LOOK FOR FIRST NONZERO DIGIT */
c2 = 0;
do {
c = MP_BASE * c + c2;
c2 = i < MP_T ? x_copy.fraction[i] : 0;
i++;
} while (c < j1 || (c == j1 && c2 < j2));
/* COMPUTE T+4 QUOTIENT DIGITS */
z->exponent += 1 - i;
i--;
/* MAIN LOOP FOR LARGE ABS(y) CASE */
for (k = 1; k <= MP_T + 4; k++) {
int ir, iq, iqj;
/* GET APPROXIMATE QUOTIENT FIRST */
ir = c / (j1 + 1);
/* NOW REDUCE SO OVERFLOW DOES NOT OCCUR */
iq = c - ir * j1;
if (iq >= b2) {
/* HERE IQ*B WOULD POSSIBLY OVERFLOW SO INCREASE IR */
++ir;
iq -= j1;
}
iq = iq * MP_BASE - ir * j2;
if (iq < 0) {
/* HERE IQ NEGATIVE SO IR WAS TOO LARGE */
ir--;
iq += y;
}
if (i < MP_T)
iq += x_copy.fraction[i];
i++;
iqj = iq / y;
/* R(K) = QUOTIENT, C = REMAINDER */
z->fraction[k - 1] = iqj + ir;
c = iq - y * iqj;
if (c < 0)
goto L210;
}
mp_normalize(z);
L210:
/* CARRY NEGATIVE SO OVERFLOW MUST HAVE OCCURRED */
mperr("*** INTEGER OVERFLOW IN MP_DIVIDE_INTEGER, B TOO LARGE ***");
mp_set_from_integer(0, z);
}
void
mp_divide_integer(const MPNumber *x, int64_t y, MPNumber *z)
{
if (mp_is_complex(x)) {
MPNumber re_z, im_z;
mp_real_component(x, &re_z);
mp_imaginary_component(x, &im_z);
mp_divide_integer_real(&re_z, y, &re_z);
mp_divide_integer_real(&im_z, y, &im_z);
mp_set_from_complex(&re_z, &im_z, z);
}
else
mp_divide_integer_real(x, y, z);
}
bool
mp_is_integer(const MPNumber *x)
{
MPNumber t1, t2, t3;
if (mp_is_complex(x))
return false;
/* This fix is required for 1/3 repiprocal not being detected as an integer */
/* Multiplication and division by 10000 is used to get around a
* limitation to the "fix" for Sun bugtraq bug #4006391 in the
* mp_floor() routine in mp.c, when the exponent is less than 1.
*/
mp_set_from_integer(10000, &t3);
mp_multiply(x, &t3, &t1);
mp_divide(&t1, &t3, &t1);
mp_floor(&t1, &t2);
return mp_is_equal(&t1, &t2);
/* Correct way to check for integer */
/*int i;
// Zero is an integer
if (mp_is_zero(x))
return true;
// Fractional
if (x->exponent <= 0)
return false;
// Look for fractional components
for (i = x->exponent; i < MP_SIZE; i++) {
if (x->fraction[i] != 0)
return false;
}
return true;*/
}
bool
mp_is_positive_integer(const MPNumber *x)
{
if (mp_is_complex(x))
return false;
else
return x->sign >= 0 && mp_is_integer(x);
}
bool
mp_is_natural(const MPNumber *x)
{
if (mp_is_complex(x))
return false;
else
return x->sign > 0 && mp_is_integer(x);
}
bool
mp_is_complex(const MPNumber *x)
{
return x->im_sign != 0;
}
bool
mp_is_equal(const MPNumber *x, const MPNumber *y)
{
return mp_compare_mp_to_mp(x, y) == 0;
}
/* Return e^x for |x| < 1 USING AN O(SQRT(T).M(T)) ALGORITHM
* DESCRIBED IN - R. P. BRENT, THE COMPLEXITY OF MULTIPLE-
* PRECISION ARITHMETIC (IN COMPLEXITY OF COMPUTATIONAL PROBLEM
* SOLVING, UNIV. OF QUEENSLAND PRESS, BRISBANE, 1976, 126-165).
* ASYMPTOTICALLY FASTER METHODS EXIST, BUT ARE NOT USEFUL
* UNLESS T IS VERY LARGE. SEE COMMENTS TO MP_ATAN AND MPPIGL.
*/
static void
mp_exp(const MPNumber *x, MPNumber *z)
{
int i, q;
float rlb;
MPNumber t1, t2;
/* e^0 = 1 */
if (mp_is_zero(x)) {
mp_set_from_integer(1, z);
return;
}
/* Only defined for |x| < 1 */
if (x->exponent > 0) {
mperr("*** ABS(X) NOT LESS THAN 1 IN CALL TO MP_EXP ***");
mp_set_from_integer(0, z);
return;
}
mp_set_from_mp(x, &t1);
rlb = log((float)MP_BASE);
/* Compute approximately optimal q (and divide x by 2^q) */
q = (int)(sqrt((float)MP_T * 0.48f * rlb) + (float) x->exponent * 1.44f * rlb);
/* HALVE Q TIMES */
if (q > 0) {
int ib, ic;
ib = MP_BASE << 2;
ic = 1;
for (i = 1; i <= q; ++i) {
ic *= 2;
if (ic < ib && ic != MP_BASE && i < q)
continue;
mp_divide_integer(&t1, ic, &t1);
ic = 1;
}
}
if (mp_is_zero(&t1)) {
mp_set_from_integer(0, z);
return;
}
/* Sum series, reducing t where possible */
mp_set_from_mp(&t1, z);
mp_set_from_mp(&t1, &t2);
for (i = 2; MP_T + t2.exponent - z->exponent > 0; i++) {
mp_multiply(&t1, &t2, &t2);
mp_divide_integer(&t2, i, &t2);
mp_add(&t2, z, z);
if (mp_is_zero(&t2))
break;
}
/* Apply (x+1)^2 - 1 = x(2 + x) for q iterations */
for (i = 1; i <= q; ++i) {
mp_add_integer(z, 2, &t1);
mp_multiply(&t1, z, z);
}
mp_add_integer(z, 1, z);
}
static void
mp_epowy_real(const MPNumber *x, MPNumber *z)
{
float r__1;
int i, ix, xs, tss;
float rx, rz;
MPNumber t1, t2;
/* e^0 = 1 */
if (mp_is_zero(x)) {
mp_set_from_integer(1, z);
return;
}
/* If |x| < 1 use mp_exp */
if (x->exponent <= 0) {
mp_exp(x, z);
return;
}
/* NOW SAFE TO CONVERT X TO REAL */
rx = mp_cast_to_float(x);
/* SAVE SIGN AND WORK WITH ABS(X) */
xs = x->sign;
mp_abs(x, &t2);
/* GET FRACTIONAL AND INTEGER PARTS OF ABS(X) */
ix = mp_cast_to_int(&t2);
mp_fractional_component(&t2, &t2);
/* ATTACH SIGN TO FRACTIONAL PART AND COMPUTE EXP OF IT */
t2.sign *= xs;
mp_exp(&t2, z);
/* COMPUTE E-2 OR 1/E USING TWO EXTRA DIGITS IN CASE ABS(X) LARGE
* (BUT ONLY ONE EXTRA DIGIT IF T < 4)
*/
if (MP_T < 4)
tss = MP_T + 1;
else
tss = MP_T + 2;
/* LOOP FOR E COMPUTATION. DECREASE T IF POSSIBLE. */
/* Computing MIN */
mp_set_from_integer(xs, &t1);
t2.sign = 0;
for (i = 2 ; ; i++) {
if (min(tss, tss + 2 + t1.exponent) <= 2)
break;
mp_divide_integer(&t1, i * xs, &t1);
mp_add(&t2, &t1, &t2);
if (mp_is_zero(&t1))
break;
}
/* RAISE E OR 1/E TO POWER IX */
if (xs > 0)
mp_add_integer(&t2, 2, &t2);
mp_xpowy_integer(&t2, ix, &t2);
/* MULTIPLY EXPS OF INTEGER AND FRACTIONAL PARTS */
mp_multiply(z, &t2, z);
/* CHECK THAT RELATIVE ERROR LESS THAN 0.01 UNLESS ABS(X) LARGE
* (WHEN EXP MIGHT OVERFLOW OR UNDERFLOW)
*/
if (fabs(rx) > 10.0f)
return;
rz = mp_cast_to_float(z);
r__1 = rz - exp(rx);
if (fabs(r__1) < rz * 0.01f)
return;
/* THE FOLLOWING MESSAGE MAY INDICATE THAT
* B**(T-1) IS TOO SMALL, OR THAT M IS TOO SMALL SO THE
* RESULT UNDERFLOWED.
*/
mperr("*** ERROR OCCURRED IN MP_EPOWY, RESULT INCORRECT ***");
}
void
mp_epowy(const MPNumber *x, MPNumber *z)
{
/* e^0 = 1 */
if (mp_is_zero(x)) {
mp_set_from_integer(1, z);
return;
}
if (mp_is_complex(x)) {
MPNumber x_real, r, theta;
mp_real_component(x, &x_real);
mp_imaginary_component(x, &theta);
mp_epowy_real(&x_real, &r);
mp_set_from_polar(&r, MP_RADIANS, &theta, z);
}
else
mp_epowy_real(x, z);
}
/* RETURNS K = K/GCD AND L = L/GCD, WHERE GCD IS THE
* GREATEST COMMON DIVISOR OF K AND L.
* SAVE INPUT PARAMETERS IN LOCAL VARIABLES
*/
void
mp_gcd(int64_t *k, int64_t *l)
{
int64_t i, j;
i = abs(*k);
j = abs(*l);
if (j == 0) {
/* IF J = 0 RETURN (1, 0) UNLESS I = 0, THEN (0, 0) */
*k = 1;
*l = 0;
if (i == 0)
*k = 0;
return;
}
/* EUCLIDEAN ALGORITHM LOOP */
do {
i %= j;
if (i == 0) {
*k = *k / j;
*l = *l / j;
return;
}
j %= i;
} while (j != 0);
/* HERE J IS THE GCD OF K AND L */
*k = *k / i;
*l = *l / i;
}
bool
mp_is_zero(const MPNumber *x)
{
return x->sign == 0 && x->im_sign == 0;
}
bool
mp_is_negative(const MPNumber *x)
{
return x->sign < 0;
}
bool
mp_is_greater_equal(const MPNumber *x, const MPNumber *y)
{
return mp_compare_mp_to_mp(x, y) >= 0;
}
bool
mp_is_greater_than(const MPNumber *x, const MPNumber *y)
{
return mp_compare_mp_to_mp(x, y) > 0;
}
bool
mp_is_less_equal(const MPNumber *x, const MPNumber *y)
{
return mp_compare_mp_to_mp(x, y) <= 0;
}
/* RETURNS MP Y = LN(1+X) IF X IS AN MP NUMBER SATISFYING THE
* CONDITION ABS(X) < 1/B, ERROR OTHERWISE.
* USES NEWTONS METHOD TO SOLVE THE EQUATION
* EXP1(-Y) = X, THEN REVERSES SIGN OF Y.
*/
static void
mp_lns(const MPNumber *x, MPNumber *z)
{
int t, it0;
MPNumber t1, t2, t3;
/* ln(1+0) = 0 */
if (mp_is_zero(x)) {
mp_set_from_integer(0, z);
return;
}
/* Get starting approximation -ln(1+x) ~= -x + x^2/2 - x^3/3 + x^4/4 */
mp_set_from_mp(x, &t2);
mp_divide_integer(x, 4, &t1);
mp_add_fraction(&t1, -1, 3, &t1);
mp_multiply(x, &t1, &t1);
mp_add_fraction(&t1, 1, 2, &t1);
mp_multiply(x, &t1, &t1);
mp_add_integer(&t1, -1, &t1);
mp_multiply(x, &t1, z);
/* Solve using Newtons method */
it0 = t = 5;
while(1)
{
int ts2, ts3;
/* t3 = (e^t3 - 1) */
/* z = z - (t2 + t3 + (t2 * t3)) */
mp_epowy(z, &t3);
mp_add_integer(&t3, -1, &t3);
mp_multiply(&t2, &t3, &t1);
mp_add(&t3, &t1, &t3);
mp_add(&t2, &t3, &t3);
mp_subtract(z, &t3, z);
if (t >= MP_T)
break;
/* FOLLOWING LOOP COMPUTES NEXT VALUE OF T TO USE.
* BECAUSE NEWTONS METHOD HAS 2ND ORDER CONVERGENCE,
* WE CAN ALMOST DOUBLE T EACH TIME.
*/
ts3 = t;
t = MP_T;
do {
ts2 = t;
t = (t + it0) / 2;
} while (t > ts3);
t = ts2;
}
/* CHECK THAT NEWTON ITERATION WAS CONVERGING AS EXPECTED */
if (t3.sign != 0 && t3.exponent << 1 > it0 - MP_T) {
mperr("*** ERROR OCCURRED IN MP_LNS, NEWTON ITERATION NOT CONVERGING PROPERLY ***");
}
z->sign = -z->sign;
}
static void
mp_ln_real(const MPNumber *x, MPNumber *z)
{
int e, k;
float rx, rlx;
MPNumber t1, t2;
/* LOOP TO GET APPROXIMATE LN(X) USING SINGLE-PRECISION */
mp_set_from_mp(x, &t1);
mp_set_from_integer(0, z);
for(k = 0; k < 10; k++)
{
/* COMPUTE FINAL CORRECTION ACCURATELY USING MP_LNS */
mp_add_integer(&t1, -1, &t2);
if (mp_is_zero(&t2) || t2.exponent + 1 <= 0) {
mp_lns(&t2, &t2);
mp_add(z, &t2, z);
return;
}
/* REMOVE EXPONENT TO AVOID FLOATING-POINT OVERFLOW */
e = t1.exponent;
t1.exponent = 0;
rx = mp_cast_to_float(&t1);
t1.exponent = e;
rlx = log(rx) + (float)e * log((float)MP_BASE);
mp_set_from_float(-(double)rlx, &t2);
/* UPDATE Z AND COMPUTE ACCURATE EXP OF APPROXIMATE LOG */
mp_subtract(z, &t2, z);
mp_epowy(&t2, &t2);
/* COMPUTE RESIDUAL WHOSE LOG IS STILL TO BE FOUND */
mp_multiply(&t1, &t2, &t1);
}
mperr("*** ERROR IN MP_LN, ITERATION NOT CONVERGING ***");
}
void
mp_ln(const MPNumber *x, MPNumber *z)
{
/* ln(0) undefined */
if (mp_is_zero(x)) {
/* Translators: Error displayed when attempting to take logarithm of zero */
mperr(_("Logarithm of zero is undefined"));
mp_set_from_integer(0, z);
return;
}
/* ln(-x) complex */
/* FIXME: Make complex numbers optional */
/*if (mp_is_negative(x)) {
// Translators: Error displayed attempted to take logarithm of negative value
mperr(_("Logarithm of negative values is undefined"));
mp_set_from_integer(0, z);
return;
}*/
if (mp_is_complex(x) || mp_is_negative(x)) {
MPNumber r, theta, z_real;
/* ln(re^iθ) = e^(ln(r)+iθ) */
mp_abs(x, &r);
mp_arg(x, MP_RADIANS, &theta);
mp_ln_real(&r, &z_real);
mp_set_from_complex(&z_real, &theta, z);
}
else
mp_ln_real(x, z);
}
void
mp_logarithm(int64_t n, const MPNumber *x, MPNumber *z)
{
MPNumber t1, t2;
/* log(0) undefined */
if (mp_is_zero(x)) {
/* Translators: Error displayed when attempting to take logarithm of zero */
mperr(_("Logarithm of zero is undefined"));
mp_set_from_integer(0, z);
return;
}
/* logn(x) = ln(x) / ln(n) */
mp_set_from_integer(n, &t1);
mp_ln(&t1, &t1);
mp_ln(x, &t2);
mp_divide(&t2, &t1, z);
}
bool
mp_is_less_than(const MPNumber *x, const MPNumber *y)
{
return mp_compare_mp_to_mp(x, y) < 0;
}
static void
mp_multiply_real(const MPNumber *x, const MPNumber *y, MPNumber *z)
{
int c, i, xi;
MPNumber r;
/* x*0 = 0*y = 0 */
if (x->sign == 0 || y->sign == 0) {
mp_set_from_integer(0, z);
return;
}
z->sign = x->sign * y->sign;
z->exponent = x->exponent + y->exponent;
memset(&r, 0, sizeof(MPNumber));
/* PERFORM MULTIPLICATION */
c = 8;
for (i = 0; i < MP_T; i++) {
int j;
xi = x->fraction[i];
/* FOR SPEED, PUT THE NUMBER WITH MANY ZEROS FIRST */
if (xi == 0)
continue;
/* Computing MIN */
for (j = 0; j < min(MP_T, MP_T + 3 - i); j++)
r.fraction[i+j+1] += xi * y->fraction[j];
c--;
if (c > 0)
continue;
/* CHECK FOR LEGAL BASE B DIGIT */
if (xi < 0 || xi >= MP_BASE) {
mperr("*** ILLEGAL BASE B DIGIT IN CALL TO MP_MULTIPLY, POSSIBLE OVERWRITING PROBLEM ***");
mp_set_from_integer(0, z);
return;
}
/* PROPAGATE CARRIES AT END AND EVERY EIGHTH TIME,
* FASTER THAN DOING IT EVERY TIME.
*/
for (j = MP_T + 3; j >= 0; j--) {
int ri = r.fraction[j] + c;
if (ri < 0) {
mperr("*** INTEGER OVERFLOW IN MP_MULTIPLY, B TOO LARGE ***");
mp_set_from_integer(0, z);
return;
}
c = ri / MP_BASE;
r.fraction[j] = ri - MP_BASE * c;
}
if (c != 0) {
mperr("*** ILLEGAL BASE B DIGIT IN CALL TO MP_MULTIPLY, POSSIBLE OVERWRITING PROBLEM ***");
mp_set_from_integer(0, z);
return;
}
c = 8;
}
if (c != 8) {
if (xi < 0 || xi >= MP_BASE) {
mperr("*** ILLEGAL BASE B DIGIT IN CALL TO MP_MULTIPLY, POSSIBLE OVERWRITING PROBLEM ***");
mp_set_from_integer(0, z);
return;
}
c = 0;
for (i = MP_T + 3; i >= 0; i--) {
int ri = r.fraction[i] + c;
if (ri < 0) {
mperr("*** INTEGER OVERFLOW IN MP_MULTIPLY, B TOO LARGE ***");
mp_set_from_integer(0, z);
return;
}
c = ri / MP_BASE;
r.fraction[i] = ri - MP_BASE * c;
}
if (c != 0) {
mperr("*** ILLEGAL BASE B DIGIT IN CALL TO MP_MULTIPLY, POSSIBLE OVERWRITING PROBLEM ***");
mp_set_from_integer(0, z);
return;
}
}
/* Clear complex part */
z->im_sign = 0;
z->im_exponent = 0;
memset(z->im_fraction, 0, sizeof(int) * MP_SIZE);
/* NORMALIZE AND ROUND RESULT */
// FIXME: Use stack variable because of mp_normalize brokeness
for (i = 0; i < MP_SIZE; i++)
z->fraction[i] = r.fraction[i];
mp_normalize(z);
}
void
mp_multiply(const MPNumber *x, const MPNumber *y, MPNumber *z)
{
/* x*0 = 0*y = 0 */
if (mp_is_zero(x) || mp_is_zero(y)) {
mp_set_from_integer(0, z);
return;
}
/* (a+bi)(c+di) = (ac-bd)+(ad+bc)i */
if (mp_is_complex(x) || mp_is_complex(y)) {
MPNumber real_x, real_y, im_x, im_y, t1, t2, real_z, im_z;
mp_real_component(x, &real_x);
mp_imaginary_component(x, &im_x);
mp_real_component(y, &real_y);
mp_imaginary_component(y, &im_y);
mp_multiply_real(&real_x, &real_y, &t1);
mp_multiply_real(&im_x, &im_y, &t2);
mp_subtract(&t1, &t2, &real_z);
mp_multiply_real(&real_x, &im_y, &t1);
mp_multiply_real(&im_x, &real_y, &t2);
mp_add(&t1, &t2, &im_z);
mp_set_from_complex(&real_z, &im_z, z);
}
else {
mp_multiply_real(x, y, z);
}
}
static void
mp_multiply_integer_real(const MPNumber *x, int64_t y, MPNumber *z)
{
int c, i;
MPNumber x_copy;
/* x*0 = 0*y = 0 */
if (mp_is_zero(x) || y == 0) {
mp_set_from_integer(0, z);
return;
}
/* x*1 = x, x*-1 = -x */
// FIXME: Why is this not working? mp_ext is using this function to do a normalization
/*if (y == 1 || y == -1) {
if (y < 0)
mp_invert_sign(x, z);
else
mp_set_from_mp(x, z);
return;
}*/
/* Copy x as z may also refer to x */
mp_set_from_mp(x, &x_copy);
mp_set_from_integer(0, z);
if (y < 0) {
y = -y;
z->sign = -x_copy.sign;
}
else
z->sign = x_copy.sign;
z->exponent = x_copy.exponent + 4;
/* FORM PRODUCT IN ACCUMULATOR */
c = 0;
/* IF y*B NOT REPRESENTABLE AS AN INTEGER WE HAVE TO SIMULATE
* DOUBLE-PRECISION MULTIPLICATION.
*/
/* Computing MAX */
if (y >= max(MP_BASE << 3, 32767 / MP_BASE)) {
int64_t j1, j2;
/* HERE J IS TOO LARGE FOR SINGLE-PRECISION MULTIPLICATION */
j1 = y / MP_BASE;
j2 = y - j1 * MP_BASE;
/* FORM PRODUCT */
for (i = MP_T + 3; i >= 0; i--) {
int64_t c1, c2, is, ix, t;
c1 = c / MP_BASE;
c2 = c - MP_BASE * c1;
ix = 0;
if (i > 3)
ix = x_copy.fraction[i - 4];
t = j2 * ix + c2;
is = t / MP_BASE;
c = j1 * ix + c1 + is;
z->fraction[i] = t - MP_BASE * is;
}
}
else
{
int64_t ri = 0;
for (i = MP_T + 3; i >= 4; i--) {
ri = y * x_copy.fraction[i - 4] + c;
c = ri / MP_BASE;
z->fraction[i] = ri - MP_BASE * c;
}
/* CHECK FOR INTEGER OVERFLOW */
if (ri < 0) {
mperr("*** INTEGER OVERFLOW IN mp_multiply_integer, B TOO LARGE ***");
mp_set_from_integer(0, z);
return;
}
/* HAVE TO TREAT FIRST FOUR WORDS OF R SEPARATELY */
for (i = 3; i >= 0; i--) {
int t;
t = c;
c = t / MP_BASE;
z->fraction[i] = t - MP_BASE * c;
}
}
/* HAVE TO SHIFT RIGHT HERE AS CARRY OFF END */
while (c != 0) {
int64_t t;
for (i = MP_T + 3; i >= 1; i--)
z->fraction[i] = z->fraction[i - 1];
t = c;
c = t / MP_BASE;
z->fraction[0] = t - MP_BASE * c;
z->exponent++;
}
if (c < 0) {
mperr("*** INTEGER OVERFLOW IN mp_multiply_integer, B TOO LARGE ***");
mp_set_from_integer(0, z);
return;
}
z->im_sign = 0;
z->im_exponent = 0;
memset(z->im_fraction, 0, sizeof(int) * MP_SIZE);
mp_normalize(z);
}
void
mp_multiply_integer(const MPNumber *x, int64_t y, MPNumber *z)
{
if (mp_is_complex(x)) {
MPNumber re_z, im_z;
mp_real_component(x, &re_z);
mp_imaginary_component(x, &im_z);
mp_multiply_integer_real(&re_z, y, &re_z);
mp_multiply_integer_real(&im_z, y, &im_z);
mp_set_from_complex(&re_z, &im_z, z);
}
else
mp_multiply_integer_real(x, y, z);
}
void
mp_multiply_fraction(const MPNumber *x, int64_t numerator, int64_t denominator, MPNumber *z)
{
if (denominator == 0) {
mperr(_("Division by zero is undefined"));
mp_set_from_integer(0, z);
return;
}
if (numerator == 0) {
mp_set_from_integer(0, z);
return;
}
/* Reduce to lowest terms */
mp_gcd(&numerator, &denominator);
mp_divide_integer(x, denominator, z);
mp_multiply_integer(z, numerator, z);
}
void
mp_invert_sign(const MPNumber *x, MPNumber *z)
{
mp_set_from_mp(x, z);
z->sign = -z->sign;
z->im_sign = -z->im_sign;
}
// FIXME: Is r->fraction large enough? It seems to be in practise but it may be MP_T+4 instead of MP_T
// FIXME: There is some sort of stack corruption/use of unitialised variables here. Some functions are
// using stack variables as x otherwise there are corruption errors. e.g. "Cos(45) - 1/Sqrt(2) = -0"
// (try in scientific mode)
void
mp_normalize(MPNumber *x)
{
int start_index;
/* Find first non-zero digit */
for (start_index = 0; start_index < MP_SIZE && x->fraction[start_index] == 0; start_index++);
/* Mark as zero */
if (start_index >= MP_SIZE) {
x->sign = 0;
x->exponent = 0;
return;
}
/* Shift left so first digit is non-zero */
if (start_index > 0) {
int i;
x->exponent -= start_index;
for (i = 0; (i + start_index) < MP_SIZE; i++)
x->fraction[i] = x->fraction[i + start_index];
for (; i < MP_SIZE; i++)
x->fraction[i] = 0;
}
}
static void
mp_pwr(const MPNumber *x, const MPNumber *y, MPNumber *z)
{
MPNumber t;
/* (-x)^y imaginary */
/* FIXME: Make complex numbers optional */
/*if (x->sign < 0) {
mperr(_("The power of negative numbers is only defined for integer exponents"));
mp_set_from_integer(0, z);
return;
}*/
/* 0^y = 0, 0^-y undefined */
if (mp_is_zero(x)) {
mp_set_from_integer(0, z);
if (y->sign < 0)
mperr(_("The power of zero is undefined for a negative exponent"));
return;
}
/* x^0 = 1 */
if (mp_is_zero(y)) {
mp_set_from_integer(1, z);
return;
}
mp_ln(x, &t);
mp_multiply(y, &t, z);
mp_epowy(z, z);
}
static void
mp_reciprocal_real(const MPNumber *x, MPNumber *z)
{
MPNumber t1, t2;
int it0, t;
/* 1/0 invalid */
if (mp_is_zero(x)) {
mperr(_("Reciprocal of zero is undefined"));
mp_set_from_integer(0, z);
return;
}
/* Start by approximating value using floating point */
mp_set_from_mp(x, &t1);
t1.exponent = 0;
mp_set_from_float(1.0f / mp_cast_to_float(&t1), &t1);
t1.exponent -= x->exponent;
it0 = t = 3;
while(1) {
int ts2, ts3;
/* t1 = t1 - (t1 * ((x * t1) - 1)) (2*t1 - t1^2*x) */
mp_multiply(x, &t1, &t2);
mp_add_integer(&t2, -1, &t2);
mp_multiply(&t1, &t2, &t2);
mp_subtract(&t1, &t2, &t1);
if (t >= MP_T)
break;
/* FOLLOWING LOOP ALMOST DOUBLES T (POSSIBLE
* BECAUSE NEWTONS METHOD HAS 2ND ORDER CONVERGENCE).
*/
ts3 = t;
t = MP_T;
do {
ts2 = t;
t = (t + it0) / 2;
} while (t > ts3);
t = min(ts2, MP_T);
}
/* RETURN IF NEWTON ITERATION WAS CONVERGING */
if (t2.sign != 0 && (t1.exponent - t2.exponent) << 1 < MP_T - it0) {
/* THE FOLLOWING MESSAGE MAY INDICATE THAT B**(T-1) IS TOO SMALL,
* OR THAT THE STARTING APPROXIMATION IS NOT ACCURATE ENOUGH.
*/
mperr("*** ERROR OCCURRED IN MP_RECIPROCAL, NEWTON ITERATION NOT CONVERGING PROPERLY ***");
}
mp_set_from_mp(&t1, z);
}
void
mp_reciprocal(const MPNumber *x, MPNumber *z)
{
if (mp_is_complex(x)) {
MPNumber t1, t2;
MPNumber real_x, im_x;
mp_real_component(x, &real_x);
mp_imaginary_component(x, &im_x);
/* 1/(a+bi) = (a-bi)/(a+bi)(a-bi) = (a-bi)/(a²+b²) */
mp_multiply(&real_x, &real_x, &t1);
mp_multiply(&im_x, &im_x, &t2);
mp_add(&t1, &t2, &t1);
mp_reciprocal_real(&t1, z);
mp_conjugate(x, &t1);
mp_multiply(&t1, z, z);
}
else
mp_reciprocal_real(x, z);
}
static void
mp_root_real(const MPNumber *x, int64_t n, MPNumber *z)
{
float approximation;
int ex, np, it0, t;
MPNumber t1, t2;
/* x^(1/1) = x */
if (n == 1) {
mp_set_from_mp(x, z);
return;
}
/* x^(1/0) invalid */
if (n == 0) {
mperr(_("Root must be non-zero"));
mp_set_from_integer(0, z);
return;
}
np = abs(n);
/* LOSS OF ACCURACY IF NP LARGE, SO ONLY ALLOW NP <= MAX (B, 64) */
if (np > max(MP_BASE, 64)) {
mperr("*** ABS(N) TOO LARGE IN CALL TO MP_ROOT ***");
mp_set_from_integer(0, z);
return;
}
/* 0^(1/n) = 0 for positive n */
if (mp_is_zero(x)) {
mp_set_from_integer(0, z);
if (n <= 0)
mperr(_("Negative root of zero is undefined"));
return;
}
// FIXME: Imaginary root
if (x->sign < 0 && np % 2 == 0) {
mperr(_("nth root of negative number is undefined for even n"));
mp_set_from_integer(0, z);
return;
}
/* DIVIDE EXPONENT BY NP */
ex = x->exponent / np;
/* Get initial approximation */
mp_set_from_mp(x, &t1);
t1.exponent = 0;
approximation = exp(((float)(np * ex - x->exponent) * log((float)MP_BASE) -
log((fabs(mp_cast_to_float(&t1))))) / (float)np);
mp_set_from_float(approximation, &t1);
t1.sign = x->sign;
t1.exponent -= ex;
/* MAIN ITERATION LOOP */
it0 = t = 3;
while(1) {
int ts2, ts3;
/* t1 = t1 - ((t1 * ((x * t1^np) - 1)) / np) */
mp_xpowy_integer(&t1, np, &t2);
mp_multiply(x, &t2, &t2);
mp_add_integer(&t2, -1, &t2);
mp_multiply(&t1, &t2, &t2);
mp_divide_integer(&t2, np, &t2);
mp_subtract(&t1, &t2, &t1);
/* FOLLOWING LOOP ALMOST DOUBLES T (POSSIBLE BECAUSE
* NEWTONS METHOD HAS 2ND ORDER CONVERGENCE).
*/
if (t >= MP_T)
break;
ts3 = t;
t = MP_T;
do {
ts2 = t;
t = (t + it0) / 2;
} while (t > ts3);
t = min(ts2, MP_T);
}
/* NOW R(I2) IS X**(-1/NP)
* CHECK THAT NEWTON ITERATION WAS CONVERGING
*/
if (t2.sign != 0 && (t1.exponent - t2.exponent) << 1 < MP_T - it0) {
/* THE FOLLOWING MESSAGE MAY INDICATE THAT B**(T-1) IS TOO SMALL,
* OR THAT THE INITIAL APPROXIMATION OBTAINED USING ALOG AND EXP
* IS NOT ACCURATE ENOUGH.
*/
mperr("*** ERROR OCCURRED IN MP_ROOT, NEWTON ITERATION NOT CONVERGING PROPERLY ***");
}
if (n >= 0) {
mp_xpowy_integer(&t1, n - 1, &t1);
mp_multiply(x, &t1, z);
return;
}
mp_set_from_mp(&t1, z);
}
void
mp_root(const MPNumber *x, int64_t n, MPNumber *z)
{
if (!mp_is_complex(x) && mp_is_negative(x) && n % 2 == 1) {
mp_abs(x, z);
mp_root_real(z, n, z);
mp_invert_sign(z, z);
}
else if (mp_is_complex(x) || mp_is_negative(x)) {
MPNumber r, theta;
mp_abs(x, &r);
mp_arg(x, MP_RADIANS, &theta);
mp_root_real(&r, n, &r);
mp_divide_integer(&theta, n, &theta);
mp_set_from_polar(&r, MP_RADIANS, &theta, z);
}
else
mp_root_real(x, n, z);
}
void
mp_sqrt(const MPNumber *x, MPNumber *z)
{
if (mp_is_zero(x))
mp_set_from_integer(0, z);
/* FIXME: Make complex numbers optional */
/*else if (x->sign < 0) {
mperr(_("Square root is undefined for negative values"));
mp_set_from_integer(0, z);
}*/
else {
MPNumber t;
mp_root(x, -2, &t);
mp_multiply(x, &t, z);
mp_ext(t.fraction[0], z->fraction[0], z);
}
}
void
mp_factorial(const MPNumber *x, MPNumber *z)
{
int i, value;
/* 0! == 1 */
if (mp_is_zero(x)) {
mp_set_from_integer(1, z);
return;
}
if (!mp_is_natural(x)) {
/* Translators: Error displayed when attempted take the factorial of a fractional number */
mperr(_("Factorial is only defined for natural numbers"));
mp_set_from_integer(0, z);
return;
}
/* Convert to integer - if couldn't be converted then the factorial would be too big anyway */
value = mp_cast_to_int(x);
mp_set_from_mp(x, z);
for (i = 2; i < value; i++)
mp_multiply_integer(z, i, z);
}
void
mp_modulus_divide(const MPNumber *x, const MPNumber *y, MPNumber *z)
{
MPNumber t1, t2;
if (!mp_is_integer(x) || !mp_is_integer(y)) {
/* Translators: Error displayed when attemping to do a modulus division on non-integer numbers */
mperr(_("Modulus division is only defined for integers"));
mp_set_from_integer(0, z);
}
mp_divide(x, y, &t1);
mp_floor(&t1, &t1);
mp_multiply(&t1, y, &t2);
mp_subtract(x, &t2, z);
mp_set_from_integer(0, &t1);
if ((mp_is_less_than(y, &t1) && mp_is_greater_than(z, &t1)) || mp_is_less_than(z, &t1))
mp_add(z, y, z);
}
void
mp_xpowy(const MPNumber *x, const MPNumber *y, MPNumber *z)
{
if (mp_is_integer(y)) {
mp_xpowy_integer(x, mp_cast_to_int(y), z);
} else {
MPNumber reciprocal;
mp_reciprocal(y, &reciprocal);
if (mp_is_integer(&reciprocal))
mp_root(x, mp_cast_to_int(&reciprocal), z);
else
mp_pwr(x, y, z);
}
}
void
mp_xpowy_integer(const MPNumber *x, int64_t n, MPNumber *z)
{
int i;
MPNumber t;
/* 0^-n invalid */
if (mp_is_zero(x) && n < 0) {
/* Translators: Error displayed when attempted to raise 0 to a negative exponent */
mperr(_("The power of zero is undefined for a negative exponent"));
mp_set_from_integer(0, z);
return;
}
/* x^0 = 1 */
if (n == 0) {
mp_set_from_integer(1, z);
return;
}
/* 0^n = 0 */
if (mp_is_zero(x)) {
mp_set_from_integer(0, z);
return;
}
/* x^1 = x */
if (n == 1) {
mp_set_from_mp(x, z);
return;
}
if (n < 0) {
mp_reciprocal(x, &t);
n = -n;
}
else
mp_set_from_mp(x, &t);
/* Multply x n times */
// FIXME: Can do mp_multiply(z, z, z) until close to answer (each call doubles number of multiples) */
mp_set_from_integer(1, z);
for (i = 0; i < n; i++)
mp_multiply(z, &t, z);
}
GList*
mp_factorize(const MPNumber *x)
{
GList *list = NULL;
MPNumber *factor = g_slice_alloc0(sizeof(MPNumber));
MPNumber value, tmp, divisor, root;
mp_abs(x, &value);
if (mp_is_zero(&value)) {
mp_set_from_mp(&value, factor);
list = g_list_append(list, factor);
return list;
}
mp_set_from_integer(1, &tmp);
if (mp_is_equal(&value, &tmp)) {
mp_set_from_mp(x, factor);
list = g_list_append(list, factor);
return list;
}
mp_set_from_integer(2, &divisor);
while (TRUE) {
mp_divide(&value, &divisor, &tmp);
if (mp_is_integer(&tmp)) {
value = tmp;
mp_set_from_mp(&divisor, factor);
list = g_list_append(list, factor);
factor = g_slice_alloc0(sizeof(MPNumber));
} else {
break;
}
}
mp_set_from_integer(3, &divisor);
mp_sqrt(&value, &root);
while (mp_is_less_equal(&divisor, &root)) {
mp_divide(&value, &divisor, &tmp);
if (mp_is_integer(&tmp)) {
value = tmp;
mp_sqrt(&value, &root);
mp_set_from_mp(&divisor, factor);
list = g_list_append(list, factor);
factor = g_slice_alloc0(sizeof(MPNumber));
} else {
mp_add_integer(&divisor, 2, &tmp);
divisor = tmp;
}
}
mp_set_from_integer(1, &tmp);
if (mp_is_greater_than(&value, &tmp)) {
mp_set_from_mp(&value, factor);
list = g_list_append(list, factor);
} else {
g_slice_free(MPNumber, factor);
}
if (mp_is_negative(x)) {
mp_invert_sign(list->data, list->data);
}
return list;
}