void
arb_exp_arf_bb(arb_t z, const arf_t x, slong prec, int minus_one)
{
slong k, iter, bits, r, mag, q, wp, N;
slong argred_bits, start_bits;
mp_bitcnt_t Qexp[1];
int inexact;
fmpz_t t, u, T, Q;
arb_t w;
if (arf_is_zero(x))
{
if (minus_one)
arb_zero(z);
else
arb_one(z);
return;
}
if (arf_is_special(x))
{
abort();
}
mag = arf_abs_bound_lt_2exp_si(x);
/* We assume that this function only gets called with something
reasonable as input (huge/tiny input will be handled by
the main exp wrapper). */
if (mag > 200 || mag < -2 * prec - 100)
{
flint_printf("arb_exp_arf_bb: unexpectedly large/small input\n");
abort();
}
if (prec < 100000000)
{
argred_bits = 16;
start_bits = 32;
}
else
{
argred_bits = 32;
start_bits = 64;
}
/* Argument reduction: exp(x) -> exp(x/2^q). This improves efficiency
of the first iteration in the bit-burst algorithm. */
q = FLINT_MAX(0, mag + argred_bits);
/* Determine working precision. */
wp = prec + 10 + 2 * q + 2 * FLINT_BIT_COUNT(prec);
if (minus_one && mag < 0)
wp += (-mag);
fmpz_init(t);
fmpz_init(u);
fmpz_init(Q);
fmpz_init(T);
arb_init(w);
/* Convert x/2^q to a fixed-point number. */
inexact = arf_get_fmpz_fixed_si(t, x, -wp + q);
/* Aliasing of z and x is safe now that only use t. */
/* Start with z = 1. */
arb_one(z);
/* Bit-burst loop. */
for (iter = 0, bits = start_bits; !fmpz_is_zero(t);
iter++, bits *= 2)
{
/* Extract bits. */
r = FLINT_MIN(bits, wp);
fmpz_tdiv_q_2exp(u, t, wp - r);
/* Binary splitting (+1 fixed-point ulp truncation error). */
mag = fmpz_bits(u) - r;
N = bs_num_terms(mag, wp);
_arb_exp_sum_bs_powtab(T, Q, Qexp, u, r, N);
/* T = T / Q (+1 fixed-point ulp error). */
if (*Qexp >= wp)
{
fmpz_tdiv_q_2exp(T, T, *Qexp - wp);
fmpz_tdiv_q(T, T, Q);
}
else
{
fmpz_mul_2exp(T, T, wp - *Qexp);
fmpz_tdiv_q(T, T, Q);
}
/* T = 1 + T */
fmpz_one(Q);
fmpz_mul_2exp(Q, Q, wp);
fmpz_add(T, T, Q);
/* Now T = exp(u) with at most 2 fixed-point ulp error. */
/* Set z = z * T. */
arf_set_fmpz(arb_midref(w), T);
arf_mul_2exp_si(arb_midref(w), arb_midref(w), -wp);
mag_set_ui_2exp_si(arb_radref(w), 2, -wp);
arb_mul(z, z, w, wp);
/* Remove used bits. */
fmpz_mul_2exp(u, u, wp - r);
fmpz_sub(t, t, u);
}
/* We have exp(x + eps) - exp(x) < 2*eps (by assumption that the argument
reduction is large enough). */
if (inexact)
arb_add_error_2exp_si(z, -wp + 1);
fmpz_clear(t);
fmpz_clear(u);
fmpz_clear(Q);
fmpz_clear(T);
arb_clear(w);
/* exp(x) = exp(x/2^q)^(2^q) */
for (k = 0; k < q; k++)
arb_mul(z, z, z, wp);
if (minus_one)
arb_sub_ui(z, z, 1, wp);
arb_set_round(z, z, prec);
}
slong
fmpr_div(fmpr_t z, const fmpr_t x, const fmpr_t y, slong prec, fmpr_rnd_t rnd)
{
if (fmpr_is_special(x) || fmpr_is_special(y))
{
_fmpr_div_special(z, x, y);
return FMPR_RESULT_EXACT;
}
/* division by power of two <=> shift exponents */
if (fmpz_is_pm1(fmpr_manref(y)))
{
if (fmpz_is_one(fmpr_manref(y)))
fmpz_set(fmpr_manref(z), fmpr_manref(x));
else
fmpz_neg(fmpr_manref(z), fmpr_manref(x));
fmpz_sub(fmpr_expref(z), fmpr_expref(x), fmpr_expref(y));
return _fmpr_normalise(fmpr_manref(z), fmpr_expref(z), prec, rnd);
}
else
{
slong xbits, ybits, extra, extra_pad, extra_control;
int negative;
fmpz_t t, u;
/* todo: work out exact needed shift */
xbits = fmpz_bits(fmpr_manref(x));
ybits = fmpz_bits(fmpr_manref(y));
extra = prec - xbits + ybits;
extra = FLINT_MAX(extra, 0);
extra_pad = 32;
extra_control = 24;
extra += extra_pad;
fmpz_init(t);
fmpz_init(u);
fmpz_mul_2exp(t, fmpr_manref(x), extra);
fmpz_tdiv_q(u, t, fmpr_manref(y));
if (low_bits_are_zero(u, extra_control))
{
fmpz_t v;
fmpz_init(v);
fmpz_mul(v, u, fmpr_manref(y));
negative = fmpz_sgn(fmpr_manref(x)) != fmpz_sgn(fmpr_manref(y));
if (!fmpz_equal(t, v))
{
if (negative)
fmpz_sub_ui(u, u, 1);
else
fmpz_add_ui(u, u, 1);
}
fmpz_clear(v);
}
fmpz_swap(fmpr_manref(z), u);
fmpz_clear(t);
fmpz_clear(u);
fmpz_sub(fmpr_expref(z), fmpr_expref(x), fmpr_expref(y));
fmpz_sub_ui(fmpr_expref(z), fmpr_expref(z), extra);
return _fmpr_normalise(fmpr_manref(z), fmpr_expref(z), prec, rnd);
}
}
