/** Binomial Coefficient --
* all initialization and cleanup is called in the caller
* @result R = choose(X, n)
*/
int my_mpfr_choose (mpfr_t R, long n, mpfr_t X, mpfr_rnd_t RND)
{
int ans;
long i;
mpfr_t r, x;
mpfr_prec_t p_X = mpfr_get_prec(X);
mpfr_init2(x, p_X); mpfr_set(x, X, RND);
mpfr_init2(r, p_X);
if(n > 0) {
mpfr_set(r, X, RND);
for(i=1; i < n; ) {
mpfr_sub_si(x, x, 1L, RND); // x = X - i
mpfr_mul (r, r, x, RND); // r := r * x = X(X-1)..(X-i)
mpfr_div_si(r, r, ++i, RND);
// r := r / (i+1) = X(X-1)..(X-i) / (1*2..*(i+1))
#ifdef DEBUG_Rmpfr
Rprintf("my_mpfr_choose(): X (= X_0 - %d)= ", i); R_PRT(x);
Rprintf("\n --> r ="); R_PRT(r); Rprintf("\n");
#endif
}
}
else // n = 0
mpfr_set_si(r, (long) 1, RND);
ans = mpfr_set(R, r, RND);
mpfr_clear (x);
mpfr_clear (r);
return ans;
}
int
main (int argc, char *argv[])
{
mpfr_t x, z;
int y;
int i;
tests_start_mpfr ();
mpfr_inits2 (53, x, z, (mpfr_ptr) 0);
for(i = 0 ; i < numberof (tab) ; i++)
{
mpfr_set_str (x, tab[i].op1, 16, MPFR_RNDN);
y = tab[i].op2;
mpfr_add_si (z, x, y, MPFR_RNDZ);
if (mpfr_cmp_str (z, tab[i].res_add, 16, MPFR_RNDN))
ERROR1("add_si", i, z, tab[i].res_add);
mpfr_sub_si (z, x, y, MPFR_RNDZ);
if (mpfr_cmp_str (z, tab[i].res_sub, 16, MPFR_RNDN))
ERROR1("sub_si", i, z, tab[i].res_sub);
mpfr_si_sub (z, y, x, MPFR_RNDZ);
mpfr_neg (z, z, MPFR_RNDZ);
if (mpfr_cmp_str (z, tab[i].res_sub, 16, MPFR_RNDN))
ERROR1("si_sub", i, z, tab[i].res_sub);
mpfr_mul_si (z, x, y, MPFR_RNDZ);
if (mpfr_cmp_str (z, tab[i].res_mul, 16, MPFR_RNDN))
ERROR1("mul_si", i, z, tab[i].res_mul);
mpfr_div_si (z, x, y, MPFR_RNDZ);
if (mpfr_cmp_str (z, tab[i].res_div, 16, MPFR_RNDN))
ERROR1("div_si", i, z, tab[i].res_div);
}
mpfr_set_str1 (x, "1");
mpfr_si_div (z, 1024, x, MPFR_RNDN);
if (mpfr_cmp_str1 (z, "1024"))
ERROR1("si_div", i, z, "1024");
mpfr_si_div (z, -1024, x, MPFR_RNDN);
if (mpfr_cmp_str1 (z, "-1024"))
ERROR1("si_div", i, z, "-1024");
mpfr_clears (x, z, (mpfr_ptr) 0);
check_invert ();
test_generic_add_si (2, 200, 17);
test_generic_sub_si (2, 200, 17);
test_generic_mul_si (2, 200, 17);
test_generic_div_si (2, 200, 17);
tests_end_mpfr ();
return 0;
}
static int synge_factorial(synge_t to, synge_t num, mpfr_rnd_t round) {
/* round input */
synge_t number;
mpfr_init2(number, SYNGE_PRECISION);
mpfr_abs(number, num, round);
mpfr_floor(number, number);
/* multilply original number by reverse iterator */
mpfr_set_si(to, 1, round);
while(!iszero(number)) {
mpfr_mul(to, to, number, round);
mpfr_sub_si(number, number, 1, round);
}
mpfr_copysign(to, to, num, round);
mpfr_clears(number, NULL);
return 0;
} /* synge_factorial() */
int
mpfr_exp2 (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
int inexact;
long xint;
mpfr_t xfrac;
MPFR_SAVE_EXPO_DECL (expo);
MPFR_LOG_FUNC
(("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec(x), mpfr_log_prec, x, rnd_mode),
("y[%Pu]=%.*Rg inexact=%d", mpfr_get_prec(y), mpfr_log_prec, y,
inexact));
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
{
if (MPFR_IS_NAN (x))
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
else if (MPFR_IS_INF (x))
{
if (MPFR_IS_POS (x))
MPFR_SET_INF (y);
else
MPFR_SET_ZERO (y);
MPFR_SET_POS (y);
MPFR_RET (0);
}
else /* 2^0 = 1 */
{
MPFR_ASSERTD (MPFR_IS_ZERO(x));
return mpfr_set_ui (y, 1, rnd_mode);
}
}
/* since the smallest representable non-zero float is 1/2*2^__gmpfr_emin,
if x < __gmpfr_emin - 1, the result is either 1/2*2^__gmpfr_emin or 0 */
MPFR_ASSERTN (MPFR_EMIN_MIN >= LONG_MIN + 2);
if (MPFR_UNLIKELY (mpfr_cmp_si (x, __gmpfr_emin - 1) < 0))
{
mpfr_rnd_t rnd2 = rnd_mode;
/* in round to nearest mode, round to zero when x <= __gmpfr_emin-2 */
if (rnd_mode == MPFR_RNDN &&
mpfr_cmp_si_2exp (x, __gmpfr_emin - 2, 0) <= 0)
rnd2 = MPFR_RNDZ;
return mpfr_underflow (y, rnd2, 1);
}
MPFR_ASSERTN (MPFR_EMAX_MAX <= LONG_MAX);
if (MPFR_UNLIKELY (mpfr_cmp_si (x, __gmpfr_emax) >= 0))
return mpfr_overflow (y, rnd_mode, 1);
/* We now know that emin - 1 <= x < emax. */
MPFR_SAVE_EXPO_MARK (expo);
/* 2^x = 1 + x*log(2) + O(x^2) for x near zero, and for |x| <= 1 we have
|2^x - 1| <= x < 2^EXP(x). If x > 0 we must round away from 0 (dir=1);
if x < 0 we must round toward 0 (dir=0). */
MPFR_SMALL_INPUT_AFTER_SAVE_EXPO (y, __gmpfr_one, - MPFR_GET_EXP (x), 0,
MPFR_IS_POS (x), rnd_mode, expo, {});
xint = mpfr_get_si (x, MPFR_RNDZ);
mpfr_init2 (xfrac, MPFR_PREC (x));
mpfr_sub_si (xfrac, x, xint, MPFR_RNDN); /* exact */
if (MPFR_IS_ZERO (xfrac))
{
mpfr_set_ui (y, 1, MPFR_RNDN);
inexact = 0;
}
else
{
/* Declaration of the intermediary variable */
mpfr_t t;
/* Declaration of the size variable */
mpfr_prec_t Ny = MPFR_PREC(y); /* target precision */
mpfr_prec_t Nt; /* working precision */
mpfr_exp_t err; /* error */
MPFR_ZIV_DECL (loop);
/* compute the precision of intermediary variable */
/* the optimal number of bits : see algorithms.tex */
Nt = Ny + 5 + MPFR_INT_CEIL_LOG2 (Ny);
/* initialize of intermediary variable */
mpfr_init2 (t, Nt);
/* First computation */
MPFR_ZIV_INIT (loop, Nt);
for (;;)
{
/* compute exp(x*ln(2))*/
mpfr_const_log2 (t, MPFR_RNDU); /* ln(2) */
mpfr_mul (t, xfrac, t, MPFR_RNDU); /* xfrac * ln(2) */
err = Nt - (MPFR_GET_EXP (t) + 2); /* Estimate of the error */
mpfr_exp (t, t, MPFR_RNDN); /* exp(xfrac * ln(2)) */
if (MPFR_LIKELY (MPFR_CAN_ROUND (t, err, Ny, rnd_mode)))
break;
/* Actualisation of the precision */
MPFR_ZIV_NEXT (loop, Nt);
mpfr_set_prec (t, Nt);
}
MPFR_ZIV_FREE (loop);
inexact = mpfr_set (y, t, rnd_mode);
mpfr_clear (t);
}
mpfr_clear (xfrac);
MPFR_CLEAR_FLAGS ();
mpfr_mul_2si (y, y, xint, MPFR_RNDN); /* exact or overflow */
/* Note: We can have an overflow only when t was rounded up to 2. */
MPFR_ASSERTD (MPFR_IS_PURE_FP (y) || inexact > 0);
MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inexact, rnd_mode);
}
static PyObject *
GMPy_Real_Sub(PyObject *x, PyObject *y, CTXT_Object *context)
{
MPFR_Object *result;
CHECK_CONTEXT(context);
if (!(result = GMPy_MPFR_New(0, context)))
return NULL;
/* This only processes mpfr if the exponent is still in-bounds. Need
* to handle the rare case at the end. */
if (MPFR_Check(x) && MPFR_Check(y)) {
mpfr_clear_flags();
result->rc = mpfr_sub(result->f, MPFR(x), MPFR(y), GET_MPFR_ROUND(context));
goto done;
}
if (MPFR_Check(x)) {
if (PyIntOrLong_Check(y)) {
int error;
long temp = GMPy_Integer_AsLongAndError(y, &error);
if (!error) {
mpfr_clear_flags();
result->rc = mpfr_sub_si(result->f, MPFR(x), temp, GET_MPFR_ROUND(context));
goto done;
}
else {
mpz_t tempz;
mpz_inoc(tempz);
mpz_set_PyIntOrLong(tempz, y);
mpfr_clear_flags();
result->rc = mpfr_sub_z(result->f, MPFR(x), tempz, GET_MPFR_ROUND(context));
mpz_cloc(tempz);
goto done;
}
}
if (CHECK_MPZANY(y)) {
mpfr_clear_flags();
result->rc = mpfr_sub_z(result->f, MPFR(x), MPZ(y), GET_MPFR_ROUND(context));
goto done;
}
if (IS_RATIONAL(y)) {
MPQ_Object *tempy;
if (!(tempy = GMPy_MPQ_From_Number(y, context))) {
Py_DECREF((PyObject*)result);
return NULL;
}
mpfr_clear_flags();
result->rc = mpfr_sub_q(result->f, MPFR(x), tempy->q, GET_MPFR_ROUND(context));
Py_DECREF((PyObject*)tempy);
goto done;
}
if (PyFloat_Check(y)) {
mpfr_clear_flags();
result->rc = mpfr_sub_d(result->f, MPFR(x), PyFloat_AS_DOUBLE(y), GET_MPFR_ROUND(context));
goto done;
}
}
if (MPFR_Check(y)) {
if (PyIntOrLong_Check(x)) {
int error;
long temp = GMPy_Integer_AsLongAndError(x, &error);
if (!error) {
mpfr_clear_flags();
result->rc = mpfr_sub_si(result->f, MPFR(y), temp, GET_MPFR_ROUND(context));
mpfr_neg(result->f, result->f, GET_MPFR_ROUND(context));
goto done;
}
else {
mpz_t tempz;
mpz_inoc(tempz);
mpz_set_PyIntOrLong(tempz, x);
mpfr_clear_flags();
result->rc = mpfr_sub_z(result->f, MPFR(y), tempz, GET_MPFR_ROUND(context));
mpfr_neg(result->f, result->f, GET_MPFR_ROUND(context));
mpz_cloc(tempz);
goto done;
}
}
if (CHECK_MPZANY(x)) {
mpfr_clear_flags();
result->rc = mpfr_sub_z(result->f, MPFR(y), MPZ(x), GET_MPFR_ROUND(context));
mpfr_neg(result->f, result->f, GET_MPFR_ROUND(context));
goto done;
}
if (IS_RATIONAL(x)) {
MPQ_Object *tempx;
if (!(tempx = GMPy_MPQ_From_Number(x, context))) {
Py_DECREF((PyObject*)result);
return NULL;
}
mpfr_clear_flags();
result->rc = mpfr_sub_q(result->f, MPFR(y), tempx->q, GET_MPFR_ROUND(context));
mpfr_neg(result->f, result->f, GET_MPFR_ROUND(context));
Py_DECREF((PyObject*)tempx);
goto done;
}
if (PyFloat_Check(x)) {
mpfr_clear_flags();
result->rc = mpfr_sub_d(result->f, MPFR(y), PyFloat_AS_DOUBLE(x), GET_MPFR_ROUND(context));
mpfr_neg(result->f, result->f, GET_MPFR_ROUND(context));
goto done;
}
}
if (IS_REAL(x) && IS_REAL(y)) {
MPFR_Object *tempx, *tempy;
tempx = GMPy_MPFR_From_Real(x, 1, context);
tempy = GMPy_MPFR_From_Real(y, 1, context);
if (!tempx || !tempy) {
Py_XDECREF((PyObject*)tempx);
Py_XDECREF((PyObject*)tempy);
Py_DECREF((PyObject*)result);
return NULL;
}
mpfr_clear_flags();
result->rc = mpfr_sub(result->f, MPFR(tempx), MPFR(tempy), GET_MPFR_ROUND(context));
Py_DECREF((PyObject*)tempx);
Py_DECREF((PyObject*)tempy);
goto done;
}
Py_DECREF((PyObject*)result);
Py_RETURN_NOTIMPLEMENTED;
done:
GMPY_MPFR_CLEANUP(result, context, "subtraction");
return (PyObject*)result;
}
int fmpq_poly_oz_sqrt_approx_pade(fmpq_poly_t f_sqrt, const fmpq_poly_t f, const long n, const int p, const mpfr_prec_t prec, const mpfr_prec_t bound, oz_flag_t flags, const fmpq_poly_t init) {
fmpq_poly_t y; fmpq_poly_init(y);
fmpq_poly_t y_next; fmpq_poly_init(y_next);
fmpq_poly_t z; fmpq_poly_init(z);
fmpq_poly_t z_next; fmpq_poly_init(z_next);
mpfr_t norm; mpfr_init2(norm, prec);
mpfr_t prev_norm; mpfr_init2(prev_norm, prec);
mpfr_t log_f; mpfr_init2(log_f, prec);
if (init) {
// z = y/x
fmpq_poly_set(y, init);
_fmpq_poly_oz_invert_approx(z, f, n, prec);
fmpq_poly_oz_mul(z, z, y, n);
} else {
fmpq_poly_set(y, f);
fmpq_poly_set_coeff_si(z, 0, 1);
}
fmpq_t *xi = (fmpq_t*)calloc(p, sizeof(fmpq_t));
fmpq_t *a2 = (fmpq_t*)calloc(p, sizeof(fmpq_t));
fmpq_t *c = (fmpq_t*)calloc(p, sizeof(fmpq_t));
fmpq_poly_t *t_ = (fmpq_poly_t*)calloc(p, sizeof(fmpq_poly_t));
fmpq_poly_t *s_ = (fmpq_poly_t*)calloc(p, sizeof(fmpq_poly_t));
mpfr_t pi; mpfr_init2(pi, 4*prec);
mpfr_const_pi(pi, MPFR_RNDN);
#pragma omp parallel for
for(int i=0; i<p; i++) {
mpfr_t xi_r; mpfr_init2(xi_r, 4*prec);
mpfr_t a2_r; mpfr_init2(a2_r, 4*prec);
/* ζ_i = 1/2 * (1 + cos( (2·i -1)·π/(2·p) )) */
mpfr_set_si(xi_r, 2*i+1, MPFR_RNDN);
mpfr_mul(xi_r, xi_r, pi, MPFR_RNDN);
mpfr_div_si(xi_r, xi_r, 2*p, MPFR_RNDN);
mpfr_cos(xi_r, xi_r, MPFR_RNDN);
mpfr_add_si(xi_r, xi_r, 1, MPFR_RNDN);
mpfr_div_si(xi_r, xi_r, 2, MPFR_RNDN);
/* α_i^2 = 1/ζ_i -1 */
mpfr_set_si(a2_r, 1, MPFR_RNDN);
mpfr_div(a2_r, a2_r, xi_r, MPFR_RNDN);
mpfr_sub_si(a2_r, a2_r, 1, MPFR_RNDN);
fmpq_init(xi[i]);
fmpq_init(a2[i]);
fmpq_set_mpfr(xi[i], xi_r, MPFR_RNDN);
fmpq_set_mpfr(a2[i], a2_r, MPFR_RNDN);
fmpq_init(c[i]);
fmpq_poly_init(t_[i]);
fmpq_poly_init(s_[i]);
mpfr_clear(xi_r);
mpfr_clear(a2_r);
}
mpfr_clear(pi);
uint64_t t = oz_walltime(0);
int r = 0;
int cont = 1;
for(long k=0; cont; k++) {
if (k == 0 || mpfr_cmp_ui(prev_norm, 1) > 0)
_fmpq_poly_oz_sqrt_approx_scale(y, z, n, prec);
/* T = sum([1/xi[i] * ~(Z*Y + a2[i]) for i in range(p)]) */
#pragma omp parallel for
for(int i=0; i<p; i++) {
fmpq_poly_oz_mul(t_[i], z, y, n);
fmpq_poly_get_coeff_fmpq(c[i], t_[i], 0);
fmpq_add(c[i], c[i], a2[i]);
fmpq_poly_set_coeff_fmpq(t_[i], 0, c[i]);
fmpq_poly_scalar_mul_fmpq(t_[i], t_[i], xi[i]);
_fmpq_poly_oz_invert_approx(s_[i], t_[i], n, prec);
}
for(int i=1; i<p; i++)
fmpq_poly_add(s_[0], s_[0], s_[i]);
#pragma omp parallel sections
{
#pragma omp section
{
fmpq_poly_oz_mul(y_next, y, s_[0], n);
fmpq_poly_scalar_div_si(y_next, y_next, p);
fmpq_poly_set(y, y_next);
}
#pragma omp section
{
fmpq_poly_oz_mul(z_next, z, s_[0], n);
fmpq_poly_scalar_div_si(z_next, z_next, p);
fmpq_poly_set(z, z_next);
}
}
cont = !_fmpq_poly_oz_sqrt_approx_break(norm, y, f, n, bound, prec);
if(flags & OZ_VERBOSE) {
mpfr_log2(log_f, norm, MPFR_RNDN);
mpfr_fprintf(stderr, "Computing sqrt(Σ):: k: %4d, Δ=|sqrt(Σ)^2-Σ|: %7.2Rf", k, log_f);
fprintf(stderr, " <? %4ld, ", -bound);
fprintf(stderr, "t: %8.2fs\n", oz_seconds(oz_walltime(t)));
fflush(0);
}
if (cont) {
if (k>0 && mpfr_cmp_ui_2exp(norm, 1, bound) >= 0) {
/* something went really wrong */
r = -1;
break;
}
if (k>0 && mpfr_cmp(norm, prev_norm) >= 0) {
/* we don't converge any more */
r = 1;
break;
}
mpfr_set(prev_norm, norm, MPFR_RNDN);
}
}
for(int i=0; i<p; i++) {
fmpq_clear(xi[i]);
fmpq_clear(a2[i]);
fmpq_clear(c[i]);
fmpq_poly_clear(t_[i]);
fmpq_poly_clear(s_[i]);
}
free(xi);
free(a2);
free(c);
free(t_);
free(s_);
mpfr_clear(log_f);
fmpq_poly_set(f_sqrt, y);
mpfr_clear(norm);
mpfr_clear(prev_norm);
fmpq_poly_clear(y_next);
fmpq_poly_clear(y);
fmpq_poly_clear(z_next);
fmpq_poly_clear(z);
return r;
}
