arb/fmprb/sin_cos_pi_fmpq.c

/*=============================================================================

    This file is part of ARB.

    ARB 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.

    ARB 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 General Public License for more details.

    You should have received a copy of the GNU General Public License
    along with ARB; if not, write to the Free Software
    Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA

=============================================================================*/
/******************************************************************************

    Copyright (C) 2012 Fredrik Johansson

******************************************************************************/

#include "fmprb.h"

/* evaluates sin(pi v/w), cos(pi v/w) assuming
   (for best performance and accuracy)
   v, w reduced and 0 <= v/w <= 1/4. */

void
_fmprb_sin_cos_pi_fmpq_oct(fmprb_t s, fmprb_t c,
            const fmpz_t v, const fmpz_t w, long prec)
{
    if (*v == 0)
    {
        fmprb_zero(s);
        fmprb_one(c);
    }
    else if (*v == 1 && *w == 4)
    {
        fmprb_sqrt_ui(s, 2, prec);
        fmprb_mul_2exp_si(s, s, -1);
        fmprb_set(c, s);
    }
    else if (*v == 1 && *w == 6)
    {
        fmprb_one(s);
        fmprb_mul_2exp_si(s, s, -1);
        fmprb_sqrt_ui(c, 3, prec);
        fmprb_mul_2exp_si(c, c, -1);
    }
    else if (*v == 1 && *w == 8)
    {
        fmprb_sqrt_ui(s, 2, prec);
        fmprb_add_ui(c, s, 2, prec);
        fmprb_sqrt(c, c, prec);
        fmprb_mul_2exp_si(c, c, -1);
        fmprb_sub_ui(s, s, 2, prec);
        fmprb_neg(s, s);
        fmprb_sqrt(s, s, prec);
        fmprb_mul_2exp_si(s, s, -1);
    }
    else if (*v == 1 && *w == 5)
    {
        fmprb_sqrt_ui(s, 5, prec);
        fmprb_add_ui(c, s, 1, prec);
        fmprb_mul_2exp_si(c, c, -2);
        fmprb_sub_ui(s, s, 5, prec);
        fmprb_neg(s, s);
        fmprb_mul_2exp_si(s, s, -3);
        fmprb_sqrt(s, s, prec);
    }
    else if (*v == 1 && *w == 10)
    {
        fmprb_sqrt_ui(c, 5, prec);
        fmprb_sub_ui(s, c, 1, prec);
        fmprb_mul_2exp_si(s, s, -2);
        fmprb_add_ui(c, c, 5, prec);
        fmprb_mul_2exp_si(c, c, -3);
        fmprb_sqrt(c, c, prec);
    }
    else
    {
        fmprb_const_pi(s, prec);
        fmprb_mul_fmpz(s, s, v, prec);
        fmprb_div_fmpz(s, s, w, prec);
        fmprb_sin_cos(s, c, s, prec);
    }
}

void
_fmprb_sin_pi_fmpq_oct(fmprb_t s, const fmpz_t v, const fmpz_t w, long prec)
{
    if (*v == 0)
    {
        fmprb_zero(s);
    }
    else if (*v == 1 && *w == 4)
    {
        fmprb_sqrt_ui(s, 2, prec);
        fmprb_mul_2exp_si(s, s, -1);
    }
    else if (*v == 1 && *w == 6)
    {
        fmprb_one(s);
        fmprb_mul_2exp_si(s, s, -1);
    }
    else if (*v == 1 && *w == 8)
    {
        fmprb_sqrt_ui(s, 2, prec);
        fmprb_sub_ui(s, s, 2, prec);
        fmprb_neg(s, s);
        fmprb_sqrt(s, s, prec);
        fmprb_mul_2exp_si(s, s, -1);
    }
    else if (*v == 1 && *w == 5)
    {
        fmprb_sqrt_ui(s, 5, prec);
        fmprb_sub_ui(s, s, 5, prec);
        fmprb_neg(s, s);
        fmprb_mul_2exp_si(s, s, -3);
        fmprb_sqrt(s, s, prec);
    }
    else if (*v == 1 && *w == 10)
    {
        fmprb_sqrt_ui(s, 5, prec);
        fmprb_sub_ui(s, s, 1, prec);
        fmprb_mul_2exp_si(s, s, -2);
    }
    else
    {
        fmprb_const_pi(s, prec);
        fmprb_mul_fmpz(s, s, v, prec);
        fmprb_div_fmpz(s, s, w, prec);
        fmprb_sin(s, s, prec);
    }
}

void
_fmprb_cos_pi_fmpq_oct(fmprb_t c, const fmpz_t v, const fmpz_t w, long prec)
{
    if (*v == 0)
    {
        fmprb_one(c);
    }
    else if (*v == 1 && *w == 4)
    {
        fmprb_sqrt_ui(c, 2, prec);
        fmprb_mul_2exp_si(c, c, -1);
    }
    else if (*v == 1 && *w == 6)
    {
        fmprb_sqrt_ui(c, 3, prec);
        fmprb_mul_2exp_si(c, c, -1);
    }
    else if (*v == 1 && *w == 8)
    {
        fmprb_sqrt_ui(c, 2, prec);
        fmprb_add_ui(c, c, 2, prec);
        fmprb_sqrt(c, c, prec);
        fmprb_mul_2exp_si(c, c, -1);
    }
    else if (*v == 1 && *w == 5)
    {
        fmprb_sqrt_ui(c, 5, prec);
        fmprb_add_ui(c, c, 1, prec);
        fmprb_mul_2exp_si(c, c, -2);
    }
    else if (*v == 1 && *w == 10)
    {
        fmprb_sqrt_ui(c, 5, prec);
        fmprb_add_ui(c, c, 5, prec);
        fmprb_mul_2exp_si(c, c, -3);
        fmprb_sqrt(c, c, prec);
    }
    else
    {
        fmprb_const_pi(c, prec);
        fmprb_mul_fmpz(c, c, v, prec);
        fmprb_div_fmpz(c, c, w, prec);
        fmprb_cos(c, c, prec);
    }
}

unsigned int
reduce_octant(fmpz_t v, fmpz_t w, const fmpq_t x)
{
    const fmpz * p = fmpq_numref(x);
    const fmpz * q = fmpq_denref(x);
    fmpz_t u;
    unsigned int octant;
    mp_bitcnt_t vval, wval;

    fmpz_init(u);

    fmpz_mul_2exp(w, p, 2);
    fmpz_fdiv_qr(w, v, w, q);
    octant = fmpz_fdiv_ui(w, 8);
    fmpz_mul_2exp(w, q, 2);

    if (octant % 2 != 0)
        fmpz_sub(v, q, v);

    vval = fmpz_val2(v);
    wval = fmpz_val2(w);
    vval = FLINT_MIN(vval, wval);

    if (vval != 0)
    {
        fmpz_tdiv_q_2exp(v, v, vval);
        fmpz_tdiv_q_2exp(w, w, vval);
    }

    fmpz_clear(u);

    return octant;
}

void
fmprb_sin_cos_pi_fmpq(fmprb_t s, fmprb_t c, const fmpq_t x, long prec)
{
    fmpz_t v, w;
    unsigned int octant;

    fmpz_init(v);
    fmpz_init(w);

    octant = reduce_octant(v, w, x);

    if ((octant + 1) % 4 < 2)
        _fmprb_sin_cos_pi_fmpq_oct(s, c, v, w, prec);
    else
        _fmprb_sin_cos_pi_fmpq_oct(c, s, v, w, prec);

    if ((octant + 6) % 8 < 4)  fmprb_neg(c, c);
    if (octant >= 4)           fmprb_neg(s, s);

    fmpz_clear(v);
    fmpz_clear(w);
}

void
fmprb_sin_pi_fmpq(fmprb_t s, const fmpq_t x, long prec)
{
    fmpz_t v, w;
    unsigned int octant;

    fmpz_init(v);
    fmpz_init(w);

    octant = reduce_octant(v, w, x);

    if ((octant + 1) % 4 < 2)
        _fmprb_sin_pi_fmpq_oct(s, v, w, prec);
    else
        _fmprb_cos_pi_fmpq_oct(s, v, w, prec);

    if (octant >= 4)
        fmprb_neg(s, s);

    fmpz_clear(v);
    fmpz_clear(w);
}

void
fmprb_cos_pi_fmpq(fmprb_t c, const fmpq_t x, long prec)
{
    fmpz_t v, w;
    unsigned int octant;

    fmpz_init(v);
    fmpz_init(w);

    octant = reduce_octant(v, w, x);

    if ((octant + 1) % 4 < 2)
        _fmprb_cos_pi_fmpq_oct(c, v, w, prec);
    else
        _fmprb_sin_pi_fmpq_oct(c, v, w, prec);

    if ((octant + 6) % 8 < 4)
        fmprb_neg(c, c);

    fmpz_clear(v);
    fmpz_clear(w);
}