decimal r_log(const decimal& a,bool round)
{
#ifdef USE_CGAL
CGAL::Gmpfr m;
CGAL::Gmpfr n=to_gmpfr(a);
mpfr_log(m.fr(),n.fr(),MPFR_RNDN);
return r_round_preference(decimal(m),round);
#else
return r_round_preference(log(a),round);
#endif
}
/**
* Wrapper function to find the square of the log of a number of type mpz_t.
*/
void compute_logn2(mpz_t rop, mpz_t n) {
mpfr_t tmp;
mpfr_init(tmp);
mpfr_set_z(tmp, n, MPFR_RNDN);
mpfr_log(tmp, tmp, MPFR_RNDA);
mpfr_pow_ui(tmp, tmp, 2, MPFR_RNDA);
mpfr_ceil(tmp, tmp);
mpfr_get_z(rop, tmp, MPFR_RNDA);
mpfr_clear(tmp);
}
int main()
{
long iter;
flint_rand_t state;
printf("const_log10....");
fflush(stdout);
flint_randinit(state);
for (iter = 0; iter < 250; iter++)
{
fmprb_t r;
mpfr_t s;
long accuracy, prec;
prec = 2 + n_randint(state, 1 << n_randint(state, 16));
fmprb_init(r);
mpfr_init2(s, prec + 1000);
fmprb_const_log10(r, prec);
mpfr_set_ui(s, 10, MPFR_RNDN);
mpfr_log(s, s, MPFR_RNDN);
if (!fmprb_contains_mpfr(r, s))
{
printf("FAIL: containment\n\n");
printf("prec = %ld\n", prec);
printf("r = "); fmprb_printd(r, prec / 3.33); printf("\n\n");
abort();
}
accuracy = fmprb_rel_accuracy_bits(r);
if (accuracy < prec - 4)
{
printf("FAIL: poor accuracy\n\n");
printf("prec = %ld\n", prec);
printf("r = "); fmprb_printd(r, prec / 3.33); printf("\n\n");
abort();
}
fmprb_clear(r);
mpfr_clear(s);
}
flint_randclear(state);
flint_cleanup();
printf("PASS\n");
return EXIT_SUCCESS;
}
void MathUtils::GetSmoothnessBase(mpz_class& ret_base, mpz_class& N)
{
mpfr_t f_N, log_N, log_log_N;
mpz_t base_mpz;
mpz_init(base_mpz);
mpfr_init(f_N); mpfr_init(log_N); mpfr_init(log_log_N);
mpfr_set_z(f_N, N.get_mpz_t(), MPFR_RNDU); //f_N = N
mpfr_log(log_N, f_N, MPFR_RNDU); //log_N = log(N)
mpfr_log(log_log_N, log_N, MPFR_RNDU); //log_log_N = log(log(N))
mpfr_mul(f_N, log_N, log_log_N, MPFR_RNDU); //f_N = log(N) * log(log(N))
mpfr_sqrt(f_N, f_N, MPFR_RNDU); //f_N = sqrt(f_N)
mpfr_div_ui(f_N, f_N, 2, MPFR_RNDU); //f_N = f_N/2
mpfr_exp(f_N, f_N, MPFR_RNDU); //f_N = e^f_N
mpfr_get_z(base_mpz, f_N, MPFR_RNDU);
ret_base = mpz_class(base_mpz);
mpfr_clears(f_N, log_N, log_log_N, NULL);
}
// Evaluate the sign of the rejection condition v^2 + 4*u^2*log(u)
int Reject(mpfr_t u, mpfr_t v, mpfr_prec_t prec, mpfr_rnd_t round) const {
// Use x1, x2 as scratch
mpfr_set_prec(_x1, prec);
mpfr_log(_x1, u, round);
mpfr_mul(_x1, _x1, u, round); // Important to do the multiplications in
mpfr_mul(_x1, _x1, u, round); // this order so that rounding works right.
mpfr_mul_2ui(_x1, _x1, 2u, round); // 4*u^2*log(u)
mpfr_set_prec(_x2, prec);
mpfr_mul(_x2, v, v, round); // v^2
mpfr_add(_x1, _x1, _x2, round); // v^2 + 4*u^2*log(u)
return mpfr_sgn(_x1);
}
void fmpq_poly_sample_D1(fmpq_poly_t f, int n, mpfr_prec_t prec, gmp_randstate_t state) {
mpfr_t u1; mpfr_init2(u1, prec);
mpfr_t u2; mpfr_init2(u2, prec);
mpfr_t z1; mpfr_init2(z1, prec);
mpfr_t z2; mpfr_init2(z2, prec);
mpfr_t pi2; mpfr_init2(pi2, prec);
mpfr_const_pi(pi2, MPFR_RNDN);
mpfr_mul_si(pi2, pi2, 2, MPFR_RNDN);
mpf_t tmp_f;
mpq_t tmp_q;
mpf_init(tmp_f);
mpq_init(tmp_q);
assert(n%2==0);
for(long i=0; i<n; i+=2) {
mpfr_urandomb(u1, state);
mpfr_urandomb(u2, state);
mpfr_log(u1, u1, MPFR_RNDN);
mpfr_mul_si(u1, u1, -2, MPFR_RNDN);
mpfr_sqrt(u1, u1, MPFR_RNDN);
mpfr_mul(u2, pi2, u2, MPFR_RNDN);
mpfr_cos(z1, u2, MPFR_RNDN);
mpfr_mul(z1, z1, u1, MPFR_RNDN); //z1 = sqrt(-2*log(u1)) * cos(2*pi*u2)
mpfr_sin(z2, u2, MPFR_RNDN);
mpfr_mul(z2, z2, u1, MPFR_RNDN); //z1 = sqrt(-2*log(u1)) * sin(2*pi*U2)
mpfr_get_f(tmp_f, z1, MPFR_RNDN);
mpq_set_f(tmp_q, tmp_f);
fmpq_poly_set_coeff_mpq(f, i, tmp_q);
mpfr_get_f(tmp_f, z2, MPFR_RNDN);
mpq_set_f(tmp_q, tmp_f);
fmpq_poly_set_coeff_mpq(f, i+1, tmp_q);
}
mpf_clear(tmp_f);
mpq_clear(tmp_q);
mpfr_clear(pi2);
mpfr_clear(u1);
mpfr_clear(u2);
mpfr_clear(z1);
mpfr_clear(z2);
}
//------------------------------------------------------------------------------
// Name:
//------------------------------------------------------------------------------
knumber_base *knumber_float::ln() {
#ifdef KNUMBER_USE_MPFR
mpfr_t mpfr;
mpfr_init_set_f(mpfr, mpf_, rounding_mode);
mpfr_log(mpfr, mpfr, rounding_mode);
mpfr_get_f(mpf_, mpfr, rounding_mode);
mpfr_clear(mpfr);
return this;
#else
const double x = mpf_get_d(mpf_);
if(isinf(x)) {
delete this;
return new knumber_error(knumber_error::ERROR_POS_INFINITY);
} else {
return execute_libc_func< ::log>(x);
}
#endif
}
REAL _ln(REAL a, REAL, QByteArray &error)
{
if (a <= ZERO)
{
error = QByteArray("ERROR: ln(a <= 0!!!)!");
print(error.constData());
return ZERO;
}
mpfr_t tmp1; mpfr_init2(tmp1, NUMBITS);
mpfr_t result; mpfr_init2(result, NUMBITS);
// mpfr_init_set_f(tmp1, a.get_mpf_t(), MPFR_RNDN);
mpfr_set_str(tmp1, getString(a).data(), 10, MPFR_RNDN);
mpfr_log(result, tmp1, MPFR_RNDN);
mpfr_get_f(a.get_mpf_t(), result, MPFR_RNDN);
mpfr_clear(tmp1);
mpfr_clear(result);
return a;
}
static int
test_log (mpfr_ptr a, mpfr_srcptr b, mp_rnd_t rnd_mode)
{
int res;
int ok = rnd_mode == GMP_RNDN && mpfr_number_p (b) && mpfr_get_prec (a)>=53;
if (ok)
{
mpfr_print_raw (b);
}
res = mpfr_log (a, b, rnd_mode);
if (ok)
{
printf (" ");
mpfr_print_raw (a);
printf ("\n");
}
return res;
}
/* requires x != 1 */
static void
arf_log_via_mpfr(arf_t z, const arf_t x, slong prec, arf_rnd_t rnd)
{
mpfr_t xf, zf;
mp_ptr zptr, tmp;
mp_srcptr xptr;
mp_size_t xn, zn, val;
TMP_INIT;
TMP_START;
zn = (prec + FLINT_BITS - 1) / FLINT_BITS;
tmp = TMP_ALLOC(zn * sizeof(mp_limb_t));
ARF_GET_MPN_READONLY(xptr, xn, x);
xf->_mpfr_d = (mp_ptr) xptr;
xf->_mpfr_prec = xn * FLINT_BITS;
xf->_mpfr_sign = ARF_SGNBIT(x) ? -1 : 1;
xf->_mpfr_exp = ARF_EXP(x);
zf->_mpfr_d = tmp;
zf->_mpfr_prec = prec;
zf->_mpfr_sign = 1;
zf->_mpfr_exp = 0;
mpfr_set_emin(MPFR_EMIN_MIN);
mpfr_set_emax(MPFR_EMAX_MAX);
mpfr_log(zf, xf, arf_rnd_to_mpfr(rnd));
val = 0;
while (tmp[val] == 0)
val++;
ARF_GET_MPN_WRITE(zptr, zn - val, z);
flint_mpn_copyi(zptr, tmp + val, zn - val);
if (zf->_mpfr_sign < 0)
ARF_NEG(z);
fmpz_set_si(ARF_EXPREF(z), zf->_mpfr_exp);
TMP_END;
}
mpz_class FirstLogTable::function(int x)
{
mpz_class result;
double apprinv;
mpfr_t i,l;
mpz_t r;
mpfr_init(i);
mpfr_init2(l,wOut);
mpz_init2(r,400);
apprinv = fit->output2double(fit->function(x));;
// result = double2output(log(apprinv));
mpfr_set_d(i, apprinv, GMP_RNDN);
mpfr_log(l, i, GMP_RNDN);
mpfr_neg(l, l, GMP_RNDN);
// Remove the sum of small offsets that are added to the other log tables
for(int j=1; j<=op->stages; j++){
mpfr_set_d(i, 1.0, GMP_RNDN);
int pi=op->p[j];
mpfr_mul_2si(i, i, -2*pi, GMP_RNDN);
mpfr_sub(l, l, i, GMP_RNDN);
}
// code the log in 2's compliment
mpfr_mul_2si(l, l, wOut, GMP_RNDN);
mpfr_get_z(r, l, GMP_RNDN);
result = mpz_class(r); // signed
// This is a very inefficient way of converting
mpz_class t = mpz_class(1) << wOut;
result = t+result;
if(result>t) result-=t;
// cout << "x="<<x<<" apprinv="<<apprinv<<" logapprinv="<<log(apprinv)<<" result="<<result<<endl;
mpfr_clear(i);
mpfr_clear(l);
mpz_clear(r);
return result;
}
static void
x_near_one (void)
{
mpfr_t x, y;
int inex;
mpfr_init2 (x, 32);
mpfr_init2 (y, 16);
mpfr_set_ui (x, 1, GMP_RNDN);
mpfr_nextbelow (x);
inex = mpfr_log (y, x, GMP_RNDD);
if (mpfr_cmp_str (y, "-0.1000000000000001E-31", 2, GMP_RNDN)
|| inex >= 0)
{
printf ("Failure in x_near_one, got inex = %d and\ny = ", inex);
mpfr_dump (y);
}
mpfr_clears (x, y, (mpfr_ptr) 0);
}
static void
bug_ddefour(void)
{
mpfr_t ex, ex1, ex2, ex3, tot, tot1;
mpfr_init2(ex, 53);
mpfr_init2(ex1, 53);
mpfr_init2(ex2, 53);
mpfr_init2(ex3, 53);
mpfr_init2(tot, 150);
mpfr_init2(tot1, 150);
mpfr_set_ui( ex, 1, MPFR_RNDN);
mpfr_mul_2exp( ex, ex, 906, MPFR_RNDN);
mpfr_log( tot, ex, MPFR_RNDN);
mpfr_set( ex1, tot, MPFR_RNDN); /* ex1 = high(tot) */
test_sub( ex2, tot, ex1, MPFR_RNDN); /* ex2 = high(tot - ex1) */
test_sub( tot1, tot, ex1, MPFR_RNDN); /* tot1 = tot - ex1 */
mpfr_set( ex3, tot1, MPFR_RNDN); /* ex3 = high(tot - ex1) */
if (mpfr_cmp(ex2, ex3))
{
printf ("Error in ddefour test.\n");
printf ("ex2=");
mpfr_print_binary (ex2);
puts ("");
printf ("ex3=");
mpfr_print_binary (ex3);
puts ("");
exit (1);
}
mpfr_clear (ex);
mpfr_clear (ex1);
mpfr_clear (ex2);
mpfr_clear (ex3);
mpfr_clear (tot);
mpfr_clear (tot1);
}
/* try asymptotic expansion when x is large and positive:
Li2(x) = -log(x)^2/2 + Pi^2/3 - 1/x + O(1/x^2).
More precisely for x >= 2 we have for g(x) = -log(x)^2/2 + Pi^2/3:
-2 <= x * (Li2(x) - g(x)) <= -1
thus |Li2(x) - g(x)| <= 2/x.
Assumes x >= 38, which ensures log(x)^2/2 >= 2*Pi^2/3, and g(x) <= -3.3.
Return 0 if asymptotic expansion failed (unable to round), otherwise
returns correct ternary value.
*/
static int
mpfr_li2_asympt_pos (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
mpfr_t g, h;
mp_prec_t w = MPFR_PREC (y) + 20;
int inex = 0;
MPFR_ASSERTN (mpfr_cmp_ui (x, 38) >= 0);
mpfr_init2 (g, w);
mpfr_init2 (h, w);
mpfr_log (g, x, GMP_RNDN); /* rel. error <= |(1 + theta) - 1| */
mpfr_sqr (g, g, GMP_RNDN); /* rel. error <= |(1 + theta)^3 - 1| <= 2^(2-w) */
mpfr_div_2ui (g, g, 1, GMP_RNDN); /* rel. error <= 2^(2-w) */
mpfr_const_pi (h, GMP_RNDN); /* error <= 2^(1-w) */
mpfr_sqr (h, h, GMP_RNDN); /* rel. error <= 2^(2-w) */
mpfr_div_ui (h, h, 3, GMP_RNDN); /* rel. error <= |(1 + theta)^4 - 1|
<= 5 * 2^(-w) */
/* since x is chosen such that log(x)^2/2 >= 2 * (Pi^2/3), we should have
g >= 2*h, thus |g-h| >= |h|, and the relative error on g is at most
multiplied by 2 in the difference, and that by h is unchanged. */
MPFR_ASSERTN (MPFR_EXP (g) > MPFR_EXP (h));
mpfr_sub (g, h, g, GMP_RNDN); /* err <= ulp(g)/2 + g*2^(3-w) + g*5*2^(-w)
<= ulp(g) * (1/2 + 8 + 5) < 14 ulp(g).
If in addition 2/x <= 2 ulp(g), i.e.,
1/x <= ulp(g), then the total error is
bounded by 16 ulp(g). */
if ((MPFR_EXP (x) >= (mp_exp_t) w - MPFR_EXP (g)) &&
MPFR_CAN_ROUND (g, w - 4, MPFR_PREC (y), rnd_mode))
inex = mpfr_set (y, g, rnd_mode);
mpfr_clear (g);
mpfr_clear (h);
return inex;
}
int
mpfr_grandom (mpfr_ptr rop1, mpfr_ptr rop2, gmp_randstate_t rstate,
mpfr_rnd_t rnd)
{
int inex1, inex2, s1, s2;
mpz_t x, y, xp, yp, t, a, b, s;
mpfr_t sfr, l, r1, r2;
mpfr_prec_t tprec, tprec0;
inex2 = inex1 = 0;
if (rop2 == NULL) /* only one output requested. */
{
tprec0 = MPFR_PREC (rop1);
}
else
{
tprec0 = MAX (MPFR_PREC (rop1), MPFR_PREC (rop2));
}
tprec0 += 11;
/* We use "Marsaglia polar method" here (cf.
George Marsaglia, Normal (Gaussian) random variables for supercomputers
The Journal of Supercomputing, Volume 5, Number 1, 49–55
DOI: 10.1007/BF00155857).
First we draw uniform x and y in [0,1] using mpz_urandomb (in
fixed precision), and scale them to [-1, 1].
*/
mpz_init (xp);
mpz_init (yp);
mpz_init (x);
mpz_init (y);
mpz_init (t);
mpz_init (s);
mpz_init (a);
mpz_init (b);
mpfr_init2 (sfr, MPFR_PREC_MIN);
mpfr_init2 (l, MPFR_PREC_MIN);
mpfr_init2 (r1, MPFR_PREC_MIN);
if (rop2 != NULL)
mpfr_init2 (r2, MPFR_PREC_MIN);
mpz_set_ui (xp, 0);
mpz_set_ui (yp, 0);
for (;;)
{
tprec = tprec0;
do
{
mpz_urandomb (xp, rstate, tprec);
mpz_urandomb (yp, rstate, tprec);
mpz_mul (a, xp, xp);
mpz_mul (b, yp, yp);
mpz_add (s, a, b);
}
while (mpz_sizeinbase (s, 2) > tprec * 2); /* x^2 + y^2 <= 2^{2tprec} */
for (;;)
{
/* FIXME: compute s as s += 2x + 2y + 2 */
mpz_add_ui (a, xp, 1);
mpz_add_ui (b, yp, 1);
mpz_mul (a, a, a);
mpz_mul (b, b, b);
mpz_add (s, a, b);
if ((mpz_sizeinbase (s, 2) <= 2 * tprec) ||
((mpz_sizeinbase (s, 2) == 2 * tprec + 1) &&
(mpz_scan1 (s, 0) == 2 * tprec)))
goto yeepee;
/* Extend by 32 bits */
mpz_mul_2exp (xp, xp, 32);
mpz_mul_2exp (yp, yp, 32);
mpz_urandomb (x, rstate, 32);
mpz_urandomb (y, rstate, 32);
mpz_add (xp, xp, x);
mpz_add (yp, yp, y);
tprec += 32;
mpz_mul (a, xp, xp);
mpz_mul (b, yp, yp);
mpz_add (s, a, b);
if (mpz_sizeinbase (s, 2) > tprec * 2)
break;
}
}
yeepee:
/* FIXME: compute s with s -= 2x + 2y + 2 */
mpz_mul (a, xp, xp);
mpz_mul (b, yp, yp);
mpz_add (s, a, b);
/* Compute the signs of the output */
mpz_urandomb (x, rstate, 2);
s1 = mpz_tstbit (x, 0);
s2 = mpz_tstbit (x, 1);
for (;;)
{
/* s = xp^2 + yp^2 (loop invariant) */
mpfr_set_prec (sfr, 2 * tprec);
mpfr_set_prec (l, tprec);
mpfr_set_z (sfr, s, MPFR_RNDN); /* exact */
mpfr_mul_2si (sfr, sfr, -2 * tprec, MPFR_RNDN); /* exact */
mpfr_log (l, sfr, MPFR_RNDN);
mpfr_neg (l, l, MPFR_RNDN);
mpfr_mul_2si (l, l, 1, MPFR_RNDN);
mpfr_div (l, l, sfr, MPFR_RNDN);
mpfr_sqrt (l, l, MPFR_RNDN);
mpfr_set_prec (r1, tprec);
mpfr_mul_z (r1, l, xp, MPFR_RNDN);
mpfr_div_2ui (r1, r1, tprec, MPFR_RNDN); /* exact */
if (s1)
mpfr_neg (r1, r1, MPFR_RNDN);
if (MPFR_CAN_ROUND (r1, tprec - 2, MPFR_PREC (rop1), rnd))
{
if (rop2 != NULL)
{
mpfr_set_prec (r2, tprec);
mpfr_mul_z (r2, l, yp, MPFR_RNDN);
mpfr_div_2ui (r2, r2, tprec, MPFR_RNDN); /* exact */
if (s2)
mpfr_neg (r2, r2, MPFR_RNDN);
if (MPFR_CAN_ROUND (r2, tprec - 2, MPFR_PREC (rop2), rnd))
break;
}
else
break;
}
/* Extend by 32 bits */
mpz_mul_2exp (xp, xp, 32);
mpz_mul_2exp (yp, yp, 32);
mpz_urandomb (x, rstate, 32);
mpz_urandomb (y, rstate, 32);
mpz_add (xp, xp, x);
mpz_add (yp, yp, y);
tprec += 32;
mpz_mul (a, xp, xp);
mpz_mul (b, yp, yp);
mpz_add (s, a, b);
}
inex1 = mpfr_set (rop1, r1, rnd);
if (rop2 != NULL)
{
inex2 = mpfr_set (rop2, r2, rnd);
inex2 = mpfr_check_range (rop2, inex2, rnd);
}
inex1 = mpfr_check_range (rop1, inex1, rnd);
if (rop2 != NULL)
mpfr_clear (r2);
mpfr_clear (r1);
mpfr_clear (l);
mpfr_clear (sfr);
mpz_clear (b);
mpz_clear (a);
mpz_clear (s);
mpz_clear (t);
mpz_clear (y);
mpz_clear (x);
mpz_clear (yp);
mpz_clear (xp);
return INEX (inex1, inex2);
}
int
mpfr_log1p (mpfr_ptr y, mpfr_srcptr x, mp_rnd_t rnd_mode)
{
int comp, inexact;
mp_exp_t ex;
MPFR_SAVE_EXPO_DECL (expo);
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
{
if (MPFR_IS_NAN (x))
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
/* check for inf or -inf (result is not defined) */
else if (MPFR_IS_INF (x))
{
if (MPFR_IS_POS (x))
{
MPFR_SET_INF (y);
MPFR_SET_POS (y);
MPFR_RET (0);
}
else
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
}
else /* x is zero */
{
MPFR_ASSERTD (MPFR_IS_ZERO (x));
MPFR_SET_ZERO (y); /* log1p(+/- 0) = +/- 0 */
MPFR_SET_SAME_SIGN (y, x);
MPFR_RET (0);
}
}
ex = MPFR_GET_EXP (x);
if (ex < 0) /* -0.5 < x < 0.5 */
{
/* For x > 0, abs(log(1+x)-x) < x^2/2.
For x > -0.5, abs(log(1+x)-x) < x^2. */
if (MPFR_IS_POS (x))
MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, x, - ex - 1, 0, 0, rnd_mode, {});
else
MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, x, - ex, 0, 1, rnd_mode, {});
}
comp = mpfr_cmp_si (x, -1);
/* log1p(x) is undefined for x < -1 */
if (MPFR_UNLIKELY(comp <= 0))
{
if (comp == 0)
/* x=0: log1p(-1)=-inf (division by zero) */
{
MPFR_SET_INF (y);
MPFR_SET_NEG (y);
MPFR_RET (0);
}
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
MPFR_SAVE_EXPO_MARK (expo);
/* General case */
{
/* Declaration of the intermediary variable */
mpfr_t t;
/* Declaration of the size variable */
mp_prec_t Ny = MPFR_PREC(y); /* target precision */
mp_prec_t Nt; /* working precision */
mp_exp_t err; /* error */
MPFR_ZIV_DECL (loop);
/* compute the precision of intermediary variable */
/* the optimal number of bits : see algorithms.tex */
Nt = Ny + MPFR_INT_CEIL_LOG2 (Ny) + 6;
/* if |x| is smaller than 2^(-e), we will loose about e bits
in log(1+x) */
if (MPFR_EXP(x) < 0)
Nt += -MPFR_EXP(x);
/* initialise of intermediary variable */
mpfr_init2 (t, Nt);
/* First computation of log1p */
MPFR_ZIV_INIT (loop, Nt);
for (;;)
{
/* compute log1p */
inexact = mpfr_add_ui (t, x, 1, GMP_RNDN); /* 1+x */
/* if inexact = 0, then t = x+1, and the result is simply log(t) */
if (inexact == 0)
{
inexact = mpfr_log (y, t, rnd_mode);
goto end;
}
mpfr_log (t, t, GMP_RNDN); /* log(1+x) */
/* the error is bounded by (1/2+2^(1-EXP(t))*ulp(t) (cf algorithms.tex)
if EXP(t)>=2, then error <= ulp(t)
if EXP(t)<=1, then error <= 2^(2-EXP(t))*ulp(t) */
err = Nt - MAX (0, 2 - MPFR_GET_EXP (t));
if (MPFR_LIKELY (MPFR_CAN_ROUND (t, err, Ny, rnd_mode)))
break;
/* increase the precision */
MPFR_ZIV_NEXT (loop, Nt);
mpfr_set_prec (t, Nt);
}
inexact = mpfr_set (y, t, rnd_mode);
end:
MPFR_ZIV_FREE (loop);
mpfr_clear (t);
}
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inexact, rnd_mode);
}
/* The computation of z = pow(x,y) is done by
z = exp(y * log(x)) = x^y
For the special cases, see Section F.9.4.4 of the C standard:
_ pow(±0, y) = ±inf for y an odd integer < 0.
_ pow(±0, y) = +inf for y < 0 and not an odd integer.
_ pow(±0, y) = ±0 for y an odd integer > 0.
_ pow(±0, y) = +0 for y > 0 and not an odd integer.
_ pow(-1, ±inf) = 1.
_ pow(+1, y) = 1 for any y, even a NaN.
_ pow(x, ±0) = 1 for any x, even a NaN.
_ pow(x, y) = NaN for finite x < 0 and finite non-integer y.
_ pow(x, -inf) = +inf for |x| < 1.
_ pow(x, -inf) = +0 for |x| > 1.
_ pow(x, +inf) = +0 for |x| < 1.
_ pow(x, +inf) = +inf for |x| > 1.
_ pow(-inf, y) = -0 for y an odd integer < 0.
_ pow(-inf, y) = +0 for y < 0 and not an odd integer.
_ pow(-inf, y) = -inf for y an odd integer > 0.
_ pow(-inf, y) = +inf for y > 0 and not an odd integer.
_ pow(+inf, y) = +0 for y < 0.
_ pow(+inf, y) = +inf for y > 0. */
int
mpfr_pow (mpfr_ptr z, mpfr_srcptr x, mpfr_srcptr y, mpfr_rnd_t rnd_mode)
{
int inexact;
int cmp_x_1;
int y_is_integer;
MPFR_SAVE_EXPO_DECL (expo);
MPFR_LOG_FUNC
(("x[%Pu]=%.*Rg y[%Pu]=%.*Rg rnd=%d",
mpfr_get_prec (x), mpfr_log_prec, x,
mpfr_get_prec (y), mpfr_log_prec, y, rnd_mode),
("z[%Pu]=%.*Rg inexact=%d",
mpfr_get_prec (z), mpfr_log_prec, z, inexact));
if (MPFR_ARE_SINGULAR (x, y))
{
/* pow(x, 0) returns 1 for any x, even a NaN. */
if (MPFR_UNLIKELY (MPFR_IS_ZERO (y)))
return mpfr_set_ui (z, 1, rnd_mode);
else if (MPFR_IS_NAN (x))
{
MPFR_SET_NAN (z);
MPFR_RET_NAN;
}
else if (MPFR_IS_NAN (y))
{
/* pow(+1, NaN) returns 1. */
if (mpfr_cmp_ui (x, 1) == 0)
return mpfr_set_ui (z, 1, rnd_mode);
MPFR_SET_NAN (z);
MPFR_RET_NAN;
}
else if (MPFR_IS_INF (y))
{
if (MPFR_IS_INF (x))
{
if (MPFR_IS_POS (y))
MPFR_SET_INF (z);
else
MPFR_SET_ZERO (z);
MPFR_SET_POS (z);
MPFR_RET (0);
}
else
{
int cmp;
cmp = mpfr_cmpabs (x, __gmpfr_one) * MPFR_INT_SIGN (y);
MPFR_SET_POS (z);
if (cmp > 0)
{
/* Return +inf. */
MPFR_SET_INF (z);
MPFR_RET (0);
}
else if (cmp < 0)
{
/* Return +0. */
MPFR_SET_ZERO (z);
MPFR_RET (0);
}
else
{
/* Return 1. */
return mpfr_set_ui (z, 1, rnd_mode);
}
}
}
else if (MPFR_IS_INF (x))
{
int negative;
/* Determine the sign now, in case y and z are the same object */
negative = MPFR_IS_NEG (x) && is_odd (y);
if (MPFR_IS_POS (y))
MPFR_SET_INF (z);
else
MPFR_SET_ZERO (z);
if (negative)
MPFR_SET_NEG (z);
else
MPFR_SET_POS (z);
MPFR_RET (0);
}
else
{
int negative;
MPFR_ASSERTD (MPFR_IS_ZERO (x));
/* Determine the sign now, in case y and z are the same object */
negative = MPFR_IS_NEG(x) && is_odd (y);
if (MPFR_IS_NEG (y))
{
MPFR_ASSERTD (! MPFR_IS_INF (y));
MPFR_SET_INF (z);
mpfr_set_divby0 ();
}
else
MPFR_SET_ZERO (z);
if (negative)
MPFR_SET_NEG (z);
else
MPFR_SET_POS (z);
MPFR_RET (0);
}
}
/* x^y for x < 0 and y not an integer is not defined */
y_is_integer = mpfr_integer_p (y);
if (MPFR_IS_NEG (x) && ! y_is_integer)
{
MPFR_SET_NAN (z);
MPFR_RET_NAN;
}
/* now the result cannot be NaN:
(1) either x > 0
(2) or x < 0 and y is an integer */
cmp_x_1 = mpfr_cmpabs (x, __gmpfr_one);
if (cmp_x_1 == 0)
return mpfr_set_si (z, MPFR_IS_NEG (x) && is_odd (y) ? -1 : 1, rnd_mode);
/* now we have:
(1) either x > 0
(2) or x < 0 and y is an integer
and in addition |x| <> 1.
*/
/* detect overflow: an overflow is possible if
(a) |x| > 1 and y > 0
(b) |x| < 1 and y < 0.
FIXME: this assumes 1 is always representable.
FIXME2: maybe we can test overflow and underflow simultaneously.
The idea is the following: first compute an approximation to
y * log2|x|, using rounding to nearest. If |x| is not too near from 1,
this approximation should be accurate enough, and in most cases enable
one to prove that there is no underflow nor overflow.
Otherwise, it should enable one to check only underflow or overflow,
instead of both cases as in the present case.
*/
if (cmp_x_1 * MPFR_SIGN (y) > 0)
{
mpfr_t t;
int negative, overflow;
MPFR_SAVE_EXPO_MARK (expo);
mpfr_init2 (t, 53);
/* we want a lower bound on y*log2|x|:
(i) if x > 0, it suffices to round log2(x) toward zero, and
to round y*o(log2(x)) toward zero too;
(ii) if x < 0, we first compute t = o(-x), with rounding toward 1,
and then follow as in case (1). */
if (MPFR_SIGN (x) > 0)
mpfr_log2 (t, x, MPFR_RNDZ);
else
{
mpfr_neg (t, x, (cmp_x_1 > 0) ? MPFR_RNDZ : MPFR_RNDU);
mpfr_log2 (t, t, MPFR_RNDZ);
}
mpfr_mul (t, t, y, MPFR_RNDZ);
overflow = mpfr_cmp_si (t, __gmpfr_emax) > 0;
mpfr_clear (t);
MPFR_SAVE_EXPO_FREE (expo);
if (overflow)
{
MPFR_LOG_MSG (("early overflow detection\n", 0));
negative = MPFR_SIGN(x) < 0 && is_odd (y);
return mpfr_overflow (z, rnd_mode, negative ? -1 : 1);
}
}
/* Basic underflow checking. One has:
* - if y > 0, |x^y| < 2^(EXP(x) * y);
* - if y < 0, |x^y| <= 2^((EXP(x) - 1) * y);
* so that one can compute a value ebound such that |x^y| < 2^ebound.
* If we have ebound <= emin - 2 (emin - 1 in directed rounding modes),
* then there is an underflow and we can decide the return value.
*/
if (MPFR_IS_NEG (y) ? (MPFR_GET_EXP (x) > 1) : (MPFR_GET_EXP (x) < 0))
{
mpfr_t tmp;
mpfr_eexp_t ebound;
int inex2;
/* We must restore the flags. */
MPFR_SAVE_EXPO_MARK (expo);
mpfr_init2 (tmp, sizeof (mpfr_exp_t) * CHAR_BIT);
inex2 = mpfr_set_exp_t (tmp, MPFR_GET_EXP (x), MPFR_RNDN);
MPFR_ASSERTN (inex2 == 0);
if (MPFR_IS_NEG (y))
{
inex2 = mpfr_sub_ui (tmp, tmp, 1, MPFR_RNDN);
MPFR_ASSERTN (inex2 == 0);
}
mpfr_mul (tmp, tmp, y, MPFR_RNDU);
if (MPFR_IS_NEG (y))
mpfr_nextabove (tmp);
/* tmp doesn't necessarily fit in ebound, but that doesn't matter
since we get the minimum value in such a case. */
ebound = mpfr_get_exp_t (tmp, MPFR_RNDU);
mpfr_clear (tmp);
MPFR_SAVE_EXPO_FREE (expo);
if (MPFR_UNLIKELY (ebound <=
__gmpfr_emin - (rnd_mode == MPFR_RNDN ? 2 : 1)))
{
/* warning: mpfr_underflow rounds away from 0 for MPFR_RNDN */
MPFR_LOG_MSG (("early underflow detection\n", 0));
return mpfr_underflow (z,
rnd_mode == MPFR_RNDN ? MPFR_RNDZ : rnd_mode,
MPFR_SIGN (x) < 0 && is_odd (y) ? -1 : 1);
}
}
/* If y is an integer, we can use mpfr_pow_z (based on multiplications),
but if y is very large (I'm not sure about the best threshold -- VL),
we shouldn't use it, as it can be very slow and take a lot of memory
(and even crash or make other programs crash, as several hundred of
MBs may be necessary). Note that in such a case, either x = +/-2^b
(this case is handled below) or x^y cannot be represented exactly in
any precision supported by MPFR (the general case uses this property).
*/
if (y_is_integer && (MPFR_GET_EXP (y) <= 256))
{
mpz_t zi;
MPFR_LOG_MSG (("special code for y not too large integer\n", 0));
mpz_init (zi);
mpfr_get_z (zi, y, MPFR_RNDN);
inexact = mpfr_pow_z (z, x, zi, rnd_mode);
mpz_clear (zi);
return inexact;
}
/* Special case (+/-2^b)^Y which could be exact. If x is negative, then
necessarily y is a large integer. */
{
mpfr_exp_t b = MPFR_GET_EXP (x) - 1;
MPFR_ASSERTN (b >= LONG_MIN && b <= LONG_MAX); /* FIXME... */
if (mpfr_cmp_si_2exp (x, MPFR_SIGN(x), b) == 0)
{
mpfr_t tmp;
int sgnx = MPFR_SIGN (x);
MPFR_LOG_MSG (("special case (+/-2^b)^Y\n", 0));
/* now x = +/-2^b, so x^y = (+/-1)^y*2^(b*y) is exact whenever b*y is
an integer */
MPFR_SAVE_EXPO_MARK (expo);
mpfr_init2 (tmp, MPFR_PREC (y) + sizeof (long) * CHAR_BIT);
inexact = mpfr_mul_si (tmp, y, b, MPFR_RNDN); /* exact */
MPFR_ASSERTN (inexact == 0);
/* Note: as the exponent range has been extended, an overflow is not
possible (due to basic overflow and underflow checking above, as
the result is ~ 2^tmp), and an underflow is not possible either
because b is an integer (thus either 0 or >= 1). */
MPFR_CLEAR_FLAGS ();
inexact = mpfr_exp2 (z, tmp, rnd_mode);
mpfr_clear (tmp);
if (sgnx < 0 && is_odd (y))
{
mpfr_neg (z, z, rnd_mode);
inexact = -inexact;
}
/* Without the following, the overflows3 test in tpow.c fails. */
MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (z, inexact, rnd_mode);
}
}
MPFR_SAVE_EXPO_MARK (expo);
/* Case where |y * log(x)| is very small. Warning: x can be negative, in
that case y is a large integer. */
{
mpfr_t t;
mpfr_exp_t err;
/* We need an upper bound on the exponent of y * log(x). */
mpfr_init2 (t, 16);
if (MPFR_IS_POS(x))
mpfr_log (t, x, cmp_x_1 < 0 ? MPFR_RNDD : MPFR_RNDU); /* away from 0 */
else
{
/* if x < -1, round to +Inf, else round to zero */
mpfr_neg (t, x, (mpfr_cmp_si (x, -1) < 0) ? MPFR_RNDU : MPFR_RNDD);
mpfr_log (t, t, (mpfr_cmp_ui (t, 1) < 0) ? MPFR_RNDD : MPFR_RNDU);
}
MPFR_ASSERTN (MPFR_IS_PURE_FP (t));
err = MPFR_GET_EXP (y) + MPFR_GET_EXP (t);
mpfr_clear (t);
MPFR_CLEAR_FLAGS ();
MPFR_SMALL_INPUT_AFTER_SAVE_EXPO (z, __gmpfr_one, - err, 0,
(MPFR_SIGN (y) > 0) ^ (cmp_x_1 < 0),
rnd_mode, expo, {});
}
/* General case */
inexact = mpfr_pow_general (z, x, y, rnd_mode, y_is_integer, &expo);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (z, inexact, rnd_mode);
}
/* Assumes that the exponent range has already been extended and if y is
an integer, then the result is not exact in unbounded exponent range. */
int
mpfr_pow_general (mpfr_ptr z, mpfr_srcptr x, mpfr_srcptr y,
mpfr_rnd_t rnd_mode, int y_is_integer, mpfr_save_expo_t *expo)
{
mpfr_t t, u, k, absx;
int neg_result = 0;
int k_non_zero = 0;
int check_exact_case = 0;
int inexact;
/* Declaration of the size variable */
mpfr_prec_t Nz = MPFR_PREC(z); /* target precision */
mpfr_prec_t Nt; /* working precision */
mpfr_exp_t err; /* error */
MPFR_ZIV_DECL (ziv_loop);
MPFR_LOG_FUNC
(("x[%Pu]=%.*Rg y[%Pu]=%.*Rg rnd=%d",
mpfr_get_prec (x), mpfr_log_prec, x,
mpfr_get_prec (y), mpfr_log_prec, y, rnd_mode),
("z[%Pu]=%.*Rg inexact=%d",
mpfr_get_prec (z), mpfr_log_prec, z, inexact));
/* We put the absolute value of x in absx, pointing to the significand
of x to avoid allocating memory for the significand of absx. */
MPFR_ALIAS(absx, x, /*sign=*/ 1, /*EXP=*/ MPFR_EXP(x));
/* We will compute the absolute value of the result. So, let's
invert the rounding mode if the result is negative. */
if (MPFR_IS_NEG (x) && is_odd (y))
{
neg_result = 1;
rnd_mode = MPFR_INVERT_RND (rnd_mode);
}
/* compute the precision of intermediary variable */
/* the optimal number of bits : see algorithms.tex */
Nt = Nz + 5 + MPFR_INT_CEIL_LOG2 (Nz);
/* initialise of intermediary variable */
mpfr_init2 (t, Nt);
MPFR_ZIV_INIT (ziv_loop, Nt);
for (;;)
{
MPFR_BLOCK_DECL (flags1);
/* compute exp(y*ln|x|), using MPFR_RNDU to get an upper bound, so
that we can detect underflows. */
mpfr_log (t, absx, MPFR_IS_NEG (y) ? MPFR_RNDD : MPFR_RNDU); /* ln|x| */
mpfr_mul (t, y, t, MPFR_RNDU); /* y*ln|x| */
if (k_non_zero)
{
MPFR_LOG_MSG (("subtract k * ln(2)\n", 0));
mpfr_const_log2 (u, MPFR_RNDD);
mpfr_mul (u, u, k, MPFR_RNDD);
/* Error on u = k * log(2): < k * 2^(-Nt) < 1. */
mpfr_sub (t, t, u, MPFR_RNDU);
MPFR_LOG_MSG (("t = y * ln|x| - k * ln(2)\n", 0));
MPFR_LOG_VAR (t);
}
/* estimate of the error -- see pow function in algorithms.tex.
The error on t is at most 1/2 + 3*2^(EXP(t)+1) ulps, which is
<= 2^(EXP(t)+3) for EXP(t) >= -1, and <= 2 ulps for EXP(t) <= -2.
Additional error if k_no_zero: treal = t * errk, with
1 - |k| * 2^(-Nt) <= exp(-|k| * 2^(-Nt)) <= errk <= 1,
i.e., additional absolute error <= 2^(EXP(k)+EXP(t)-Nt).
Total error <= 2^err1 + 2^err2 <= 2^(max(err1,err2)+1). */
err = MPFR_NOTZERO (t) && MPFR_GET_EXP (t) >= -1 ?
MPFR_GET_EXP (t) + 3 : 1;
if (k_non_zero)
{
if (MPFR_GET_EXP (k) > err)
err = MPFR_GET_EXP (k);
err++;
}
MPFR_BLOCK (flags1, mpfr_exp (t, t, MPFR_RNDN)); /* exp(y*ln|x|)*/
/* We need to test */
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (t) || MPFR_UNDERFLOW (flags1)))
{
mpfr_prec_t Ntmin;
MPFR_BLOCK_DECL (flags2);
MPFR_ASSERTN (!k_non_zero);
MPFR_ASSERTN (!MPFR_IS_NAN (t));
/* Real underflow? */
if (MPFR_IS_ZERO (t))
{
/* Underflow. We computed rndn(exp(t)), where t >= y*ln|x|.
Therefore rndn(|x|^y) = 0, and we have a real underflow on
|x|^y. */
inexact = mpfr_underflow (z, rnd_mode == MPFR_RNDN ? MPFR_RNDZ
: rnd_mode, MPFR_SIGN_POS);
if (expo != NULL)
MPFR_SAVE_EXPO_UPDATE_FLAGS (*expo, MPFR_FLAGS_INEXACT
| MPFR_FLAGS_UNDERFLOW);
break;
}
/* Real overflow? */
if (MPFR_IS_INF (t))
{
/* Note: we can probably use a low precision for this test. */
mpfr_log (t, absx, MPFR_IS_NEG (y) ? MPFR_RNDU : MPFR_RNDD);
mpfr_mul (t, y, t, MPFR_RNDD); /* y * ln|x| */
MPFR_BLOCK (flags2, mpfr_exp (t, t, MPFR_RNDD));
/* t = lower bound on exp(y * ln|x|) */
if (MPFR_OVERFLOW (flags2))
{
/* We have computed a lower bound on |x|^y, and it
overflowed. Therefore we have a real overflow
on |x|^y. */
inexact = mpfr_overflow (z, rnd_mode, MPFR_SIGN_POS);
if (expo != NULL)
MPFR_SAVE_EXPO_UPDATE_FLAGS (*expo, MPFR_FLAGS_INEXACT
| MPFR_FLAGS_OVERFLOW);
break;
}
}
k_non_zero = 1;
Ntmin = sizeof(mpfr_exp_t) * CHAR_BIT;
if (Ntmin > Nt)
{
Nt = Ntmin;
mpfr_set_prec (t, Nt);
}
mpfr_init2 (u, Nt);
mpfr_init2 (k, Ntmin);
mpfr_log2 (k, absx, MPFR_RNDN);
mpfr_mul (k, y, k, MPFR_RNDN);
mpfr_round (k, k);
MPFR_LOG_VAR (k);
/* |y| < 2^Ntmin, therefore |k| < 2^Nt. */
continue;
}
if (MPFR_LIKELY (MPFR_CAN_ROUND (t, Nt - err, Nz, rnd_mode)))
{
inexact = mpfr_set (z, t, rnd_mode);
break;
}
/* check exact power, except when y is an integer (since the
exact cases for y integer have already been filtered out) */
if (check_exact_case == 0 && ! y_is_integer)
{
if (mpfr_pow_is_exact (z, absx, y, rnd_mode, &inexact))
break;
check_exact_case = 1;
}
/* reactualisation of the precision */
MPFR_ZIV_NEXT (ziv_loop, Nt);
mpfr_set_prec (t, Nt);
if (k_non_zero)
mpfr_set_prec (u, Nt);
}
MPFR_ZIV_FREE (ziv_loop);
if (k_non_zero)
{
int inex2;
long lk;
/* The rounded result in an unbounded exponent range is z * 2^k. As
* MPFR chooses underflow after rounding, the mpfr_mul_2si below will
* correctly detect underflows and overflows. However, in rounding to
* nearest, if z * 2^k = 2^(emin - 2), then the double rounding may
* affect the result. We need to cope with that before overwriting z.
* This can occur only if k < 0 (this test is necessary to avoid a
* potential integer overflow).
* If inexact >= 0, then the real result is <= 2^(emin - 2), so that
* o(2^(emin - 2)) = +0 is correct. If inexact < 0, then the real
* result is > 2^(emin - 2) and we need to round to 2^(emin - 1).
*/
MPFR_ASSERTN (MPFR_EXP_MAX <= LONG_MAX);
lk = mpfr_get_si (k, MPFR_RNDN);
/* Due to early overflow detection, |k| should not be much larger than
* MPFR_EMAX_MAX, and as MPFR_EMAX_MAX <= MPFR_EXP_MAX/2 <= LONG_MAX/2,
* an overflow should not be possible in mpfr_get_si (and lk is exact).
* And one even has the following assertion. TODO: complete proof.
*/
MPFR_ASSERTD (lk > LONG_MIN && lk < LONG_MAX);
/* Note: even in case of overflow (lk inexact), the code is correct.
* Indeed, for the 3 occurrences of lk:
* - The test lk < 0 is correct as sign(lk) = sign(k).
* - In the test MPFR_GET_EXP (z) == __gmpfr_emin - 1 - lk,
* if lk is inexact, then lk = LONG_MIN <= MPFR_EXP_MIN
* (the minimum value of the mpfr_exp_t type), and
* __gmpfr_emin - 1 - lk >= MPFR_EMIN_MIN - 1 - 2 * MPFR_EMIN_MIN
* >= - MPFR_EMIN_MIN - 1 = MPFR_EMAX_MAX - 1. However, from the
* choice of k, z has been chosen to be around 1, so that the
* result of the test is false, as if lk were exact.
* - In the mpfr_mul_2si (z, z, lk, rnd_mode), if lk is inexact,
* then |lk| >= LONG_MAX >= MPFR_EXP_MAX, and as z is around 1,
* mpfr_mul_2si underflows or overflows in the same way as if
* lk were exact.
* TODO: give a bound on |t|, then on |EXP(z)|.
*/
if (rnd_mode == MPFR_RNDN && inexact < 0 && lk < 0 &&
MPFR_GET_EXP (z) == __gmpfr_emin - 1 - lk && mpfr_powerof2_raw (z))
{
/* Rounding to nearest, real result > z * 2^k = 2^(emin - 2),
* underflow case: as the minimum precision is > 1, we will
* obtain the correct result and exceptions by replacing z by
* nextabove(z).
*/
MPFR_ASSERTN (MPFR_PREC_MIN > 1);
mpfr_nextabove (z);
}
MPFR_CLEAR_FLAGS ();
inex2 = mpfr_mul_2si (z, z, lk, rnd_mode);
if (inex2) /* underflow or overflow */
{
inexact = inex2;
if (expo != NULL)
MPFR_SAVE_EXPO_UPDATE_FLAGS (*expo, __gmpfr_flags);
}
mpfr_clears (u, k, (mpfr_ptr) 0);
}
mpfr_clear (t);
/* update the sign of the result if x was negative */
if (neg_result)
{
MPFR_SET_NEG(z);
inexact = -inexact;
}
return inexact;
}
int
mpfr_log2 (mpfr_ptr r, mpfr_srcptr a, mpfr_rnd_t rnd_mode)
{
int inexact;
MPFR_SAVE_EXPO_DECL (expo);
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (a)))
{
/* If a is NaN, the result is NaN */
if (MPFR_IS_NAN (a))
{
MPFR_SET_NAN (r);
MPFR_RET_NAN;
}
/* check for infinity before zero */
else if (MPFR_IS_INF (a))
{
if (MPFR_IS_NEG (a))
/* log(-Inf) = NaN */
{
MPFR_SET_NAN (r);
MPFR_RET_NAN;
}
else /* log(+Inf) = +Inf */
{
MPFR_SET_INF (r);
MPFR_SET_POS (r);
MPFR_RET (0);
}
}
else /* a is zero */
{
MPFR_ASSERTD (MPFR_IS_ZERO (a));
MPFR_SET_INF (r);
MPFR_SET_NEG (r);
MPFR_RET (0); /* log2(0) is an exact -infinity */
}
}
/* If a is negative, the result is NaN */
if (MPFR_UNLIKELY (MPFR_IS_NEG (a)))
{
MPFR_SET_NAN (r);
MPFR_RET_NAN;
}
/* If a is 1, the result is 0 */
if (MPFR_UNLIKELY (mpfr_cmp_ui (a, 1) == 0))
{
MPFR_SET_ZERO (r);
MPFR_SET_POS (r);
MPFR_RET (0); /* only "normal" case where the result is exact */
}
/* If a is 2^N, log2(a) is exact*/
if (MPFR_UNLIKELY (mpfr_cmp_ui_2exp (a, 1, MPFR_GET_EXP (a) - 1) == 0))
return mpfr_set_si(r, MPFR_GET_EXP (a) - 1, rnd_mode);
MPFR_SAVE_EXPO_MARK (expo);
/* General case */
{
/* Declaration of the intermediary variable */
mpfr_t t, tt;
/* Declaration of the size variable */
mpfr_prec_t Ny = MPFR_PREC(r); /* 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 + 3 + MPFR_INT_CEIL_LOG2 (Ny);
/* initialise of intermediary variable */
mpfr_init2 (t, Nt);
mpfr_init2 (tt, Nt);
/* First computation of log2 */
MPFR_ZIV_INIT (loop, Nt);
for (;;)
{
/* compute log2 */
mpfr_const_log2(t,MPFR_RNDD); /* log(2) */
mpfr_log(tt,a,MPFR_RNDN); /* log(a) */
mpfr_div(t,tt,t,MPFR_RNDN); /* log(a)/log(2) */
/* estimation of the error */
err = Nt-3;
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_set_prec (tt, Nt);
}
MPFR_ZIV_FREE (loop);
inexact = mpfr_set (r, t, rnd_mode);
mpfr_clear (t);
mpfr_clear (tt);
}
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (r, inexact, rnd_mode);
}
int main()
{
slong iter;
flint_rand_t state;
flint_printf("d_log_upper_bound....");
fflush(stdout);
flint_randinit(state);
for (iter = 0; iter < 100000 * arb_test_multiplier(); iter++)
{
mpfr_t t;
double x, y, z;
mpfr_init2(t, 53);
x = d_randtest2(state);
x = ldexp(x, 100 - n_randint(state, 200));
switch (n_randint(state, 10))
{
case 0:
x = 1.0 + x;
break;
case 1:
x = 1.0 - x;
break;
case 2:
x = D_INF;
break;
case 3:
x = 0.0;
break;
case 4:
x = D_NAN;
break;
default:
break;
}
y = mag_d_log_upper_bound(x);
mpfr_set_d(t, x, MPFR_RNDD);
mpfr_log(t, t, MPFR_RNDU);
z = mpfr_get_d(t, MPFR_RNDD);
if (y < z || fabs(y-z) > 0.000001 * fabs(z))
{
flint_printf("FAIL\n");
flint_printf("x = %.20g\n", x);
flint_printf("y = %.20g\n", y);
flint_printf("z = %.20g\n", z);
flint_abort();
}
mpfr_clear(t);
}
flint_randclear(state);
flint_cleanup();
flint_printf("PASS\n");
return EXIT_SUCCESS;
}
int main()
{
slong iter;
flint_rand_t state;
flint_printf("log....");
fflush(stdout);
flint_randinit(state);
/* compare with mpfr */
for (iter = 0; iter < 100000 * arb_test_multiplier(); iter++)
{
arb_t a, b;
fmpq_t q;
mpfr_t t;
slong prec = 2 + n_randint(state, 200);
arb_init(a);
arb_init(b);
fmpq_init(q);
mpfr_init2(t, prec + 100);
do {
arb_randtest(a, state, 1 + n_randint(state, 200), 10);
} while (arb_contains_nonpositive(a));
arb_randtest(b, state, 1 + n_randint(state, 200), 10);
arb_get_rand_fmpq(q, state, a, 1 + n_randint(state, 200));
fmpq_get_mpfr(t, q, MPFR_RNDN);
/* todo: estimate cancellation precisely */
if (mpfr_cmp_d(t, 1 - 1e-10) > 0 && mpfr_cmp_d(t, 1 + 1e-10) < 0)
{
mpfr_set_prec(t, prec + 1000);
fmpq_get_mpfr(t, q, MPFR_RNDN);
}
mpfr_log(t, t, MPFR_RNDN);
arb_log(b, a, prec);
if (!arb_contains_mpfr(b, t))
{
flint_printf("FAIL: containment\n\n");
flint_printf("a = "); arb_print(a); flint_printf("\n\n");
flint_printf("b = "); arb_print(b); flint_printf("\n\n");
abort();
}
arb_log(a, a, prec);
if (!arb_equal(a, b))
{
flint_printf("FAIL: aliasing\n\n");
abort();
}
arb_clear(a);
arb_clear(b);
fmpq_clear(q);
mpfr_clear(t);
}
/* compare with mpfr (higher precision) */
for (iter = 0; iter < 1000 * arb_test_multiplier(); iter++)
{
arb_t a, b;
fmpq_t q;
mpfr_t t;
slong prec = 2 + n_randint(state, 6000);
arb_init(a);
arb_init(b);
fmpq_init(q);
mpfr_init2(t, prec + 100);
do {
arb_randtest(a, state, 1 + n_randint(state, 6000), 10);
} while (arb_contains_nonpositive(a));
arb_randtest(b, state, 1 + n_randint(state, 6000), 10);
arb_get_rand_fmpq(q, state, a, 1 + n_randint(state, 200));
fmpq_get_mpfr(t, q, MPFR_RNDN);
/* todo: estimate cancellation precisely */
if (mpfr_cmp_d(t, 1 - 1e-10) > 0 && mpfr_cmp_d(t, 1 + 1e-10) < 0)
{
mpfr_set_prec(t, prec + 10000);
fmpq_get_mpfr(t, q, MPFR_RNDN);
}
mpfr_log(t, t, MPFR_RNDN);
arb_log(b, a, prec);
if (!arb_contains_mpfr(b, t))
{
flint_printf("FAIL: containment\n\n");
flint_printf("a = "); arb_print(a); flint_printf("\n\n");
flint_printf("b = "); arb_print(b); flint_printf("\n\n");
abort();
}
arb_log(a, a, prec);
if (!arb_equal(a, b))
{
flint_printf("FAIL: aliasing\n\n");
abort();
}
arb_clear(a);
arb_clear(b);
fmpq_clear(q);
mpfr_clear(t);
}
/* test large numbers */
for (iter = 0; iter < 10000 * arb_test_multiplier(); iter++)
{
arb_t a, b, ab, lab, la, lb, lalb;
slong prec = 2 + n_randint(state, 6000);
arb_init(a);
arb_init(b);
arb_init(ab);
arb_init(lab);
arb_init(la);
arb_init(lb);
arb_init(lalb);
arb_randtest(a, state, 1 + n_randint(state, 400), 400);
arb_randtest(b, state, 1 + n_randint(state, 400), 400);
arb_log(la, a, prec);
arb_log(lb, b, prec);
arb_mul(ab, a, b, prec);
arb_log(lab, ab, prec);
arb_add(lalb, la, lb, prec);
if (!arb_overlaps(lab, lalb))
{
flint_printf("FAIL: containment\n\n");
flint_printf("a = "); arb_print(a); flint_printf("\n\n");
flint_printf("b = "); arb_print(b); flint_printf("\n\n");
flint_printf("la = "); arb_print(la); flint_printf("\n\n");
flint_printf("lb = "); arb_print(lb); flint_printf("\n\n");
flint_printf("ab = "); arb_print(ab); flint_printf("\n\n");
flint_printf("lab = "); arb_print(lab); flint_printf("\n\n");
flint_printf("lalb = "); arb_print(lalb); flint_printf("\n\n");
abort();
}
arb_log(a, a, prec);
if (!arb_overlaps(a, la))
{
flint_printf("FAIL: aliasing\n\n");
abort();
}
arb_clear(a);
arb_clear(b);
arb_clear(ab);
arb_clear(lab);
arb_clear(la);
arb_clear(lb);
arb_clear(lalb);
}
flint_randclear(state);
flint_cleanup();
flint_printf("PASS\n");
return EXIT_SUCCESS;
}
int
main (int argc, char *argv[])
{
int n, prec, st, st2, N, i;
mpfr_t x, y, z;
if (argc != 2 && argc != 3)
{
fprintf(stderr, "Usage: timing digits \n");
exit(1);
}
printf ("Using MPFR-%s with GMP-%s\n", mpfr_version, gmp_version);
n = atoi(argv[1]);
prec = (int) ( n * log(10.0) / log(2.0) + 1.0 );
printf("[precision is %u bits]\n", prec);
mpfr_init2(x, prec); mpfr_init2(y, prec); mpfr_init2(z, prec);
mpfr_set_d(x, 3.0, GMP_RNDN); mpfr_sqrt(x, x, GMP_RNDN); mpfr_sub_ui (x, x, 1, GMP_RNDN);
mpfr_set_d(y, 5.0, GMP_RNDN); mpfr_sqrt(y, y, GMP_RNDN);
mpfr_log (z, x, GMP_RNDN);
N=1; st = cputime();
do {
for (i=0;i<N;i++) mpfr_mul(z, x, y, GMP_RNDN);
N=2*N;
st2=cputime();
} while (st2-st<1000);
printf("x*y took %f ms (%d eval in %d ms)\n",
(double)(st2-st)/(N-1),N-1,st2-st);
N=1; st = cputime();
do {
for (i=0;i<N;i++) mpfr_mul(z, x, x, GMP_RNDN);
N=2*N;
st2=cputime();
} while (st2-st<1000);
printf("x*x took %f ms (%d eval in %d ms)\n",
(double)(st2-st)/(N-1),N-1,st2-st);
N=1; st = cputime();
do {
for (i=0;i<N;i++) mpfr_div(z, x, y, GMP_RNDN);
N=2*N;
st2=cputime();
} while (st2-st<1000);
printf("x/y took %f ms (%d eval in %d ms)\n",
(double)(st2-st)/(N-1),N-1,st2-st);
N=1; st = cputime();
do {
for (i=0;i<N;i++) mpfr_sqrt(z, x, GMP_RNDN);
N=2*N;
st2=cputime();
} while (st2-st<1000);
printf("sqrt(x) took %f ms (%d eval in %d ms)\n",
(double)(st2-st)/(N-1),N-1,st2-st);
N=1; st = cputime();
do {
for (i=0;i<N;i++) mpfr_exp(z, x, GMP_RNDN);
N=2*N;
st2=cputime();
} while (st2-st<1000);
printf("exp(x) took %f ms (%d eval in %d ms)\n",
(double)(st2-st)/(N-1),N-1,st2-st);
N=1; st = cputime();
do {
for (i=0;i<N;i++) mpfr_log(z, x, GMP_RNDN);
N=2*N;
st2=cputime();
} while (st2-st<1000);
printf("log(x) took %f ms (%d eval in %d ms)\n",
(double)(st2-st)/(N-1),N-1,st2-st);
N=1; st = cputime();
do {
for (i=0;i<N;i++) mpfr_sin(z, x, GMP_RNDN);
N=2*N;
st2=cputime();
} while (st2-st<1000);
printf("sin(x) took %f ms (%d eval in %d ms)\n",
(double)(st2-st)/(N-1),N-1,st2-st);
N=1; st = cputime();
do {
for (i=0;i<N;i++) mpfr_cos(z, x, GMP_RNDN);
N=2*N;
st2=cputime();
} while (st2-st<1000);
printf("cos(x) took %f ms (%d eval in %d ms)\n",
(double)(st2-st)/(N-1),N-1,st2-st);
N=1; st = cputime();
do {
for (i=0;i<N;i++) mpfr_acos(z, x, GMP_RNDN);
N=2*N;
st2=cputime();
} while (st2-st<1000);
printf("arccos(x) took %f ms (%d eval in %d ms)\n",
(double)(st2-st)/(N-1),N-1,st2-st);
N=1; st = cputime();
do {
for (i=0;i<N;i++) mpfr_atan(z, x, GMP_RNDN);
N=2*N;
st2=cputime();
} while (st2-st<1000);
printf("arctan(x) took %f ms (%d eval in %d ms)\n",
(double)(st2-st)/(N-1),N-1,st2-st);
mpfr_clear(x); mpfr_clear(y); mpfr_clear(z);
return 0;
}
/******************************************************************************
One forward-backward pass through a minimum-duration HMM model with a
single Gaussian in each of the states. T: totalFeatures
*******************************************************************************/
double ESHMM::mdHMMLogForwardBackward(ESHMM *mdHMM, VECTOR_OF_F_VECTORS *features, double **post, int T, mat &gamma, rowvec &gamma1,
mat &sumxi){
printf("forward backward algorithm calculation in progress...\n");
int i = 0, j = 0 , k = 0;
/* total no of states is much larger, instead of number of pdfs we have to extend
states by Min_DUR, therefore total states = Q * MD */
int Q = mdHMM->hmmStates;
int Qmd = Q * MIN_DUR;
mat logalpha(T, Qmd); // forward probability matrix
mat logbeta(T, Qmd); // backward probability matrix
mat logg(T, Qmd); // loggamma
mat m(Q, 1);
mat logA(Qmd, Qmd); /// transition matrix is already in logarithm
mat new_logp(T, Qmd); // after replication for each substates
mat logp_k(Q, T); // we have single cluster only, probability of each feature corresponding to each cluster
printf("Q: %d Qmd: %d\n", Q, Qmd);
for(i = 0; i < Qmd; i++){
for(j = 0; j < Qmd; j++){
logA(i, j) = mdHMM->trans[i]->array[j];
}
}
// minimum duration viterbi hence modify B(posterior) prob matrix
for(i = 0; i < Q; i++)
for(j = 0; j < T; j++){
logp_k(i, j) = post[i][j];
}
for(i = 0; i < Q; i++){
m(i, 0) = 1;
for(j = 0; j < T; j++)
logp_k(i, j) = 0.0; // since we have only one cluster so cluster probability and
// total probability is same. Hence subtracting cluster probability from total probability would make it zero.
}
// modifying logp matrix according to minimum duration
for(i = 0; i < Q; i++){
for(j = 0; j < T; j++){
for(k = i*MIN_DUR; k < (i+1)*MIN_DUR; k++){
new_logp(j, k) = post[i][j];
}
}
}
/* forward initialization */
// for summing log probabilties, first sum probs and then take logarithm
printf("forward initialization...\n\n");
for(i = 0; i < Qmd; i++){
logalpha(0, i) = mdHMM->prior->array[i] + new_logp(0, i) ;
}
///print logalpha after initialization
for(i = 0; i < Qmd; i++)
printf("%lf ", logalpha(0, i));
/* forward induction */
printf("forward induction in progress...\n");
int t = 0;
mpfr_t summation3;
mpfr_init(summation3);
mpfr_t var11, var21;
mpfr_init(var11);
mpfr_init(var21);
mpfr_set_d(var11, 0.0, MPFR_RNDN);
mpfr_set_d(var21, 0.0, MPFR_RNDN);
mpfr_set_d(summation3, 0.0, MPFR_RNDN);
for(t = 1; t < T; t++){
//printf("%d ", t);
for(j = 0; j < Qmd; j++){
vec v1(Qmd), v2(Qmd); vec v3(Qmd);
//first find logalpha vector
for(i = 0; i < Qmd; i++)
v1(i) = logalpha(t-1, i);
// if(t < 20)
// v1.print("v1:\n");
// extract transition probability vector
for(i = 0; i < Qmd; i++)
v2(i) = logA(i, j);
// if(t < 20)
// v2.print("v2:\n");
// Now sum both the vectors into one
for(i = 0; i < Qmd; i++)
v3(i) = v1(i) + v2(i);
double *temp = (double *)calloc(Qmd, sizeof(double ));
for(i = 0; i < Qmd; i++)
temp[i] = v3(i);
// if(t < 20)
// v3.print("v3:\n");
//printf("printed\n");
// now sum over whole column vector
mpfr_set_d(summation3, 0.0, MPFR_RNDN);
// take the exponentiation and summation in one loop
// getting double from mpfr variable
/// double mpfr_get_d(mpfr_t op, mpfr_rnd_t rnd);
//mpfr_set_d(var1, 0.0, MPFR_RNDD);
//mpfr_set_d(var2, 0.0, MPFR_RNDD);
// now take the exponentiation
for(i = 0; i < Qmd; i++){
double elem = temp[i];
mpfr_set_d(var21, elem, MPFR_RNDD);
//mpfr_printf("var2: %lf\n", var21);
mpfr_exp(var11, var21, MPFR_RNDD); ///take exp(v2) and store in v1
// take sum of all elements in total
mpfr_add(summation3, summation3, var11, MPFR_RNDD); // add summation and v1
}
// now take the logarithm of sum
mpfr_log(summation3, summation3, MPFR_RNDD);
// now convert this sum to double
double sum2 = mpfr_get_d(summation3, MPFR_RNDD);
// now assign this double to logalpha
// now add logp(t, j)
sum2 += new_logp(t, j);
// if(t < 20)
// printf("sum: %lf\n", sum2);
logalpha(t, j) = sum2;
/// clear mpfr variables
}
if(t < 20){
printf("logalpha:\n");
for(j = 0; j < Qmd; j++)
printf("%lf ", logalpha(t, j));
printf("\n");
}
} // close the forward induction loop
mpfr_clear(var11);
mpfr_clear(var21);
mpfr_clear(summation3);
/* forward termination */
double ll = 0; // total log likelihood of all observation given this HMM
for(i = 0; i < Qmd; i++){
ll += logalpha(T-1, i);
}
///===================================================================
// for(i = 0; i < 100; i++){
// for(j = 0; j < Qmd; j++)
// printf("%lf ", logalpha(i, j));
// printf("\n");
// }
printf("\nprinting last column of logalpha...\n");
for(i = 1; i < 6; i++){
for(j = 0; j < Qmd; j++)
printf("%lf ", logalpha(T-i, j));
printf("\n");
}
printf("total loglikelihood: %lf\n", ll);
///===================================================================
double sum = 0;
/* calculate logalpha last row sum */
for(i = 0; i < Qmd; i++)
sum += logalpha(T-1, i);
ll = sum;
printf("LL: %lf........\n", ll);
/* backward initilization */
/// intialize mpfr variables
mpfr_t summation;
mpfr_init(summation);
mpfr_t var1, var2;
mpfr_init(var1);
mpfr_init(var2);
mpfr_set_d(summation, 0.0, MPFR_RNDN);
mpfr_set_d(var1, 0.0, MPFR_RNDN);
mpfr_set_d(var2, 0.0, MPFR_RNDN);
printf("backward initialization...\n");
mpfr_set_d(summation, 0.0, MPFR_RNDN);
double *temp = (double *)calloc(Qmd, sizeof(double ));
for(i = 0; i < Qmd; i++)
temp[i] = logalpha(T-1, i);
for(i = 0; i < Qmd; i++){
//double elem = logalpha(T-1, i);
double elem = temp[i-1];
mpfr_set_d(var2, elem, MPFR_RNDN);
mpfr_exp(var1, var2, MPFR_RNDN);
mpfr_add(summation, summation, var1, MPFR_RNDN);
}
// take logarithm
mpfr_log(summation, summation, MPFR_RNDN);
double sum2 = mpfr_get_d(summation, MPFR_RNDN);
for(i = 0; i < Qmd; i++){
logg(T-1, i) = logalpha(T-1, i) - sum2 ;
}
// gamma matrix
for(j = 0; j < Q; j++){
gamma(j, T-1) = exp(logp_k(j, T-1) + logg(T-1, j));
}
mat lognewxi(Qmd, Qmd); // declare lognewxi matrix
/* backward induction */
printf("backward induction in progress...\n");
for(t = T-2; t >= 0 ; t--){
for(j = 0; j < Qmd; j++){
vec v1(Qmd);
vec v2(Qmd);
vec v3(Qmd);
sum = 0;
for(i = 0; i < Qmd; i++)
v1(i) = logA(j, i);
for(i = 0; i < Qmd; i++)
v2(i) = logbeta(t+1, i);
for(i = 0; i < Qmd; i++)
v3(i) = new_logp(t+1, i);
// add all three vectors
for(i = 0; i < Qmd; i++)
v1(i) += v2(i) + v3(i);
mpfr_set_d(summation, 0.0, MPFR_RNDN);
for(i = 0; i < Qmd; i++){
double elem = v1(i);
mpfr_set_d(var2, elem, MPFR_RNDN);
mpfr_exp(var1, var2, MPFR_RNDN);
mpfr_add(summation, summation, var1, MPFR_RNDN);
}
mpfr_log(summation, summation, MPFR_RNDN);
sum2 = mpfr_get_d(summation, MPFR_RNDN);
logbeta(t, j) = sum2;
}
// computation of log(gamma) is now possible called logg here
for(i = 0; i < Qmd; i++){
logg(t, i) = logalpha(t, i) + logbeta(t, i);
}
mpfr_set_d(summation, 0.0, MPFR_RNDN);
for(i = 0; i < Qmd; i++){
double elem = logg(t, i);
mpfr_set_d(var2, elem, MPFR_RNDN);
mpfr_exp(var1, var2, MPFR_RNDN);
mpfr_add(summation, summation, var1, MPFR_RNDN);
}
mpfr_log(summation, summation, MPFR_RNDN);
sum2 = mpfr_get_d(summation, MPFR_RNDN);
for(i = 0; i < Qmd; i++)
logg(t, i) = logg(t, i) - sum2;
// finally the gamma_k is computed (called gamma here )
mpfr_set_d(summation, 0.0, MPFR_RNDN);
for(j = 0; j < Q; j++){
// for(i = j*MIN_DUR; i < (j+1) * MIN_DUR; i++){
// sum += exp(logg(t, i));
// }
gamma(j, t) = exp( logp_k(j, t) + logg(t, j) );
}
/* for the EM algorithm we need the sum
over xi all over t */
// replicate logalpha(t, :)' matrix along columns
mat m1(Qmd, Qmd);
for(i = 0; i < Qmd; i++){
for(j = 0; j < Qmd; j++){
m1(i, j) = logalpha(t, i);
}
}
// replicate logbeta matrix
vec v1(Qmd);
for(i = 0; i < Qmd; i++)
v1(i) = logbeta(t+1, i);
vec v2(Qmd);
for(i = 0; i < Qmd; i++)
v2(i) = new_logp(t+1, i);
vec v3(Qmd);
for(i = 0; i < Qmd; i++)
v3(i) = v1(i) + v2(i);
// replicate v3 row vector along all rows of matrix m2
mat m2(Qmd, Qmd);
for(i = 0; i < Qmd; i++){
for(j = 0; j < Qmd; j++){
m2(i, j) = v3(i);
}
}
// add both matrices m1 and m2
mat m3(Qmd, Qmd);
m3 = m1 + m2; // can do direct addition
///mat lognewxi(Qmd, Qmd); // declare lognewxi matrix
lognewxi.zeros();
lognewxi = m3 + logA;
// add new sum to older sumxi
/// first subtract total sum from lognewxi
mpfr_set_d(summation, 0.0, MPFR_RNDN);
for(i = 0; i < Qmd; i++){
for(j = 0; j < Qmd; j++){
double elem = lognewxi(i, j);
mpfr_set_d(var2, elem, MPFR_RNDN);
mpfr_exp(var1, var2, MPFR_RNDN);
mpfr_add(summation, summation, var1, MPFR_RNDN);
//sum += exp(lognewxi(i, j));
}
}
// now take the logarithm of sum
mpfr_log(summation, summation, MPFR_RNDN);
sum2 = mpfr_get_d(summation, MPFR_RNDN);
// subtract sum from lognewxi
for(i = 0; i < Qmd; i++){
for(j = 0; j < Qmd; j++){
lognewxi(i, j) = lognewxi(i, j) - sum2;
}
}
mat newxi(Qmd, Qmd);
newxi = lognewxi;
// add sumxi and newlogsumxi
/// take exponential of each element
for(i = 0; i < Qmd; i++){
for(j = 0; j < Qmd; j++){
newxi(i, j) = exp(newxi(i, j));
}
}
sumxi = sumxi + newxi;
} // close the backward induction loop
/* handle annoying numerics */
/// calculate sum of lognewxi along each row (lognewxi is already modified in our case)
for(i = 0; i < Qmd; i++){
mpfr_set_d(summation, 0.0, MPFR_RNDN);
for(j = 0; j < Qmd; j++){
//sum += lognewxi(i, j);
double elem = lognewxi(i, j);
mpfr_set_d(var2, elem, MPFR_RNDN);
mpfr_exp(var1, var2, MPFR_RNDN);
mpfr_add(summation, summation, var1, MPFR_RNDN);
}
sum2 = mpfr_get_d(summation, MPFR_RNDN);
gamma1(i) = sum2;
}
// normalize gamma1 which is prior and normalize sumxi which is transition matrix
sum = 0;
for(i = 0; i < Qmd; i++)
sum += gamma1(i);
for(i = 0; i < Qmd; i++)
gamma1(i) /= sum;
// transition probability matrix will be normalized in train_hmm function
/// clear mpfr variables
mpfr_clear(summation);
mpfr_clear(var1);
mpfr_clear(var2);
printf("forward-backward algorithm calculation is done...\n");
/* finished forward-backward algorithm */
return ll;
}
int main()
{
slong i;
mpfr_t tabx, logx, y1, y2;
mpz_t tt;
flint_printf("log_tab....");
fflush(stdout);
{
slong prec, bits, num;
prec = ARB_LOG_TAB1_LIMBS * FLINT_BITS;
bits = ARB_LOG_TAB11_BITS;
num = 1 << ARB_LOG_TAB11_BITS;
mpfr_init2(tabx, prec);
mpfr_init2(logx, prec);
mpfr_init2(y1, prec);
mpfr_init2(y2, prec);
for (i = 0; i < num; i++)
{
tt->_mp_d = (mp_ptr) arb_log_tab11[i];
tt->_mp_size = prec / FLINT_BITS;
tt->_mp_alloc = tt->_mp_size;
while (tt->_mp_size > 0 && tt->_mp_d[tt->_mp_size-1] == 0)
tt->_mp_size--;
mpfr_set_z(tabx, tt, MPFR_RNDD);
mpfr_div_2ui(tabx, tabx, prec, MPFR_RNDD);
mpfr_set_ui(logx, i, MPFR_RNDD);
mpfr_div_2ui(logx, logx, bits, MPFR_RNDD);
mpfr_add_ui(logx, logx, 1, MPFR_RNDD);
mpfr_log(logx, logx, MPFR_RNDD);
mpfr_mul_2ui(y1, tabx, prec, MPFR_RNDD);
mpfr_floor(y1, y1);
mpfr_div_2ui(y1, y1, prec, MPFR_RNDD);
mpfr_mul_2ui(y2, logx, prec, MPFR_RNDD);
mpfr_floor(y2, y2);
mpfr_div_2ui(y2, y2, prec, MPFR_RNDD);
if (!mpfr_equal_p(y1, y2))
{
flint_printf("FAIL: i = %wd, bits = %wd, prec = %wd\n", i, bits, prec);
mpfr_printf("y1 = %.1500Rg\n", y1);
mpfr_printf("y2 = %.1500Rg\n", y2);
flint_abort();
}
}
mpfr_clear(tabx);
mpfr_clear(logx);
mpfr_clear(y1);
mpfr_clear(y2);
}
{
slong prec, bits, num;
prec = ARB_LOG_TAB1_LIMBS * FLINT_BITS;
bits = ARB_LOG_TAB11_BITS + ARB_LOG_TAB12_BITS;
num = 1 << ARB_LOG_TAB12_BITS;
mpfr_init2(tabx, prec);
mpfr_init2(logx, prec);
mpfr_init2(y1, prec);
mpfr_init2(y2, prec);
for (i = 0; i < num; i++)
{
tt->_mp_d = (mp_ptr) arb_log_tab12[i];
tt->_mp_size = prec / FLINT_BITS;
tt->_mp_alloc = tt->_mp_size;
while (tt->_mp_size > 0 && tt->_mp_d[tt->_mp_size-1] == 0)
tt->_mp_size--;
mpfr_set_z(tabx, tt, MPFR_RNDD);
mpfr_div_2ui(tabx, tabx, prec, MPFR_RNDD);
mpfr_set_ui(logx, i, MPFR_RNDD);
mpfr_div_2ui(logx, logx, bits, MPFR_RNDD);
mpfr_add_ui(logx, logx, 1, MPFR_RNDD);
mpfr_log(logx, logx, MPFR_RNDD);
mpfr_mul_2ui(y1, tabx, prec, MPFR_RNDD);
mpfr_floor(y1, y1);
mpfr_div_2ui(y1, y1, prec, MPFR_RNDD);
mpfr_mul_2ui(y2, logx, prec, MPFR_RNDD);
mpfr_floor(y2, y2);
mpfr_div_2ui(y2, y2, prec, MPFR_RNDD);
if (!mpfr_equal_p(y1, y2))
{
flint_printf("FAIL: i = %wd, bits = %wd, prec = %wd\n", i, bits, prec);
mpfr_printf("y1 = %.1500Rg\n", y1);
mpfr_printf("y2 = %.1500Rg\n", y2);
flint_abort();
}
}
mpfr_clear(tabx);
mpfr_clear(logx);
mpfr_clear(y1);
mpfr_clear(y2);
}
{
slong prec, bits, num;
prec = ARB_LOG_TAB2_LIMBS * FLINT_BITS;
bits = ARB_LOG_TAB21_BITS;
num = 1 << ARB_LOG_TAB21_BITS;
mpfr_init2(tabx, prec);
mpfr_init2(logx, prec);
mpfr_init2(y1, prec);
mpfr_init2(y2, prec);
for (i = 0; i < num; i++)
{
tt->_mp_d = (mp_ptr) arb_log_tab21[i];
tt->_mp_size = prec / FLINT_BITS;
tt->_mp_alloc = tt->_mp_size;
while (tt->_mp_size > 0 && tt->_mp_d[tt->_mp_size-1] == 0)
tt->_mp_size--;
mpfr_set_z(tabx, tt, MPFR_RNDD);
mpfr_div_2ui(tabx, tabx, prec, MPFR_RNDD);
mpfr_set_ui(logx, i, MPFR_RNDD);
mpfr_div_2ui(logx, logx, bits, MPFR_RNDD);
mpfr_add_ui(logx, logx, 1, MPFR_RNDD);
mpfr_log(logx, logx, MPFR_RNDD);
mpfr_mul_2ui(y1, tabx, prec, MPFR_RNDD);
mpfr_floor(y1, y1);
mpfr_div_2ui(y1, y1, prec, MPFR_RNDD);
mpfr_mul_2ui(y2, logx, prec, MPFR_RNDD);
mpfr_floor(y2, y2);
mpfr_div_2ui(y2, y2, prec, MPFR_RNDD);
if (!mpfr_equal_p(y1, y2))
{
flint_printf("FAIL: i = %wd, bits = %wd, prec = %wd\n", i, bits, prec);
mpfr_printf("y1 = %.1500Rg\n", y1);
mpfr_printf("y2 = %.1500Rg\n", y2);
flint_abort();
}
}
mpfr_clear(tabx);
mpfr_clear(logx);
mpfr_clear(y1);
mpfr_clear(y2);
}
{
slong prec, bits, num;
prec = ARB_LOG_TAB2_LIMBS * FLINT_BITS;
bits = ARB_LOG_TAB21_BITS + ARB_LOG_TAB22_BITS;
num = 1 << ARB_LOG_TAB22_BITS;
mpfr_init2(tabx, prec);
mpfr_init2(logx, prec);
mpfr_init2(y1, prec);
mpfr_init2(y2, prec);
for (i = 0; i < num; i++)
{
tt->_mp_d = (mp_ptr) arb_log_tab22[i];
tt->_mp_size = prec / FLINT_BITS;
tt->_mp_alloc = tt->_mp_size;
while (tt->_mp_size > 0 && tt->_mp_d[tt->_mp_size-1] == 0)
tt->_mp_size--;
mpfr_set_z(tabx, tt, MPFR_RNDD);
mpfr_div_2ui(tabx, tabx, prec, MPFR_RNDD);
mpfr_set_ui(logx, i, MPFR_RNDD);
mpfr_div_2ui(logx, logx, bits, MPFR_RNDD);
mpfr_add_ui(logx, logx, 1, MPFR_RNDD);
mpfr_log(logx, logx, MPFR_RNDD);
mpfr_mul_2ui(y1, tabx, prec, MPFR_RNDD);
mpfr_floor(y1, y1);
mpfr_div_2ui(y1, y1, prec, MPFR_RNDD);
mpfr_mul_2ui(y2, logx, prec, MPFR_RNDD);
mpfr_floor(y2, y2);
mpfr_div_2ui(y2, y2, prec, MPFR_RNDD);
if (!mpfr_equal_p(y1, y2))
{
flint_printf("FAIL: i = %wd, bits = %wd, prec = %wd\n", i, bits, prec);
mpfr_printf("y1 = %.1500Rg\n", y1);
mpfr_printf("y2 = %.1500Rg\n", y2);
flint_abort();
}
}
mpfr_clear(tabx);
mpfr_clear(logx);
mpfr_clear(y1);
mpfr_clear(y2);
}
flint_cleanup();
flint_printf("PASS\n");
return EXIT_SUCCESS;
}
mpfr_t* double_pole_case_c(long pole_order_max, mpfr_t base, mpfr_t pole_position, mpfr_t incomplete_gamma_factor, mpfr_prec_t prec){
mpfr_t* result=malloc(sizeof(mpfr_t)*(pole_order_max+1));
mpfr_t temp1;
mpfr_init2(temp1,prec);
mpfr_t double_pole_integral;
mpfr_init2(double_pole_integral,prec);
mpfr_t temp2;
mpfr_init2(temp2,prec);
mpfr_t minus_pole_position;
mpfr_init2(minus_pole_position,prec);
mpfr_neg(minus_pole_position,pole_position,MPFR_RNDN);
// mpfr_t factorial;
// mpfr_init2(factorial,prec);
// mpfr_set_ui(factorial,1,MPFR_RNDN);
mpfr_t minus_log_base;
mpfr_init2(minus_log_base,prec);
mpfr_log(minus_log_base,base,prec);
mpfr_neg(minus_log_base,minus_log_base,MPFR_RNDN);
mpfr_ui_div(minus_log_base,1,minus_log_base,MPFR_RNDN);
mpfr_t log_base_power;
mpfr_init2(log_base_power,prec);
mpfr_set(log_base_power,minus_log_base,MPFR_RNDN);
//mpfr_set_ui(temp1,0,MPFR_RNDN);
//mpfr_mul(double_pole_integral,temp1,incomplete_gamma_factor,MPFR_RNDN);
mpfr_log(temp1,base,MPFR_RNDN);
mpfr_mul(double_pole_integral,incomplete_gamma_factor,temp1,MPFR_RNDN);
mpfr_ui_div(temp1,1,minus_pole_position,MPFR_RNDN);
mpfr_add(double_pole_integral,double_pole_integral,temp1,MPFR_RNDN);
//mpfr_printf("before loop: double_pole_integral = %.32RNf\n",double_pole_integral);
for(int i=0;i<=pole_order_max;i++){
mpfr_init2(result[i],prec);
mpfr_set(result[i],double_pole_integral,MPFR_RNDN);
if(i<pole_order_max){
mpfr_mul(double_pole_integral,double_pole_integral,pole_position,MPFR_RNDN);
}
}
#ifdef debug_mode
for(int i=0;i<=pole_order_max;i++){
mpfr_printf("result[%d] = %.16RNf\n",i,result[i],MPFR_RNDN);
}
#endif
mpfr_t * factorial_times_power_lnb;
mpfr_t * single_pole_coeffs;
/*
* x**1/(x+a)**2 case
* */
if(pole_order_max >= 1){
single_pole_coeffs=malloc(sizeof(mpfr_t)*(pole_order_max));
/* x^(j+1)=(
* single_pole_coeffs[0] x^(j-1) +
* single_pole_coeffs[1] x^(j-2) + ... +
* single_pole_coeffs[j-1] x^0
* )*(x-a)^2 +
*
* single_pole_coeffs[j](x-a) * ((x-a) + a)
*
* + a^(j+1)
*
* => single_pole_coeffs[j+1]
*
* single_pole_coeffs[j+1] = single_pole_coeffs[j]*a + a^j+1
* single_pole_coeffs[0] =
* */
if(pole_order_max>=2){
factorial_times_power_lnb=malloc(sizeof(mpfr_t)*(pole_order_max-1));
mpfr_init2(factorial_times_power_lnb[0],prec);
mpfr_set(factorial_times_power_lnb[0],minus_log_base,MPFR_RNDN);
}
mpfr_set(temp1,pole_position,MPFR_RNDN);
/* below temp1 is used as pole_position^j*/
mpfr_init2(single_pole_coeffs[0],prec);
mpfr_set_ui(single_pole_coeffs[0], 1 ,MPFR_RNDN);
mpfr_add(result[1],result[1],incomplete_gamma_factor,MPFR_RNDN);
for(int j=1;j<=pole_order_max-1;j++){
mpfr_init2(single_pole_coeffs[j],prec);
mpfr_mul(single_pole_coeffs[j],single_pole_coeffs[j-1],pole_position,MPFR_RNDN);
mpfr_add(single_pole_coeffs[j],single_pole_coeffs[j],temp1,MPFR_RNDN);
mpfr_mul(temp2,single_pole_coeffs[j],incomplete_gamma_factor,MPFR_RNDN);
mpfr_add(result[j+1],result[j+1],temp2,MPFR_RNDN);
if(j<=pole_order_max-2){
mpfr_init2(factorial_times_power_lnb[j],prec);
mpfr_mul(temp1,temp1,pole_position,MPFR_RNDN);
mpfr_mul(factorial_times_power_lnb[j],factorial_times_power_lnb[j-1],minus_log_base,MPFR_RNDN);
mpfr_mul_ui(factorial_times_power_lnb[j],factorial_times_power_lnb[j],j,MPFR_RNDN);
}
}
}
#ifdef debug_mode
for(int j=0;j<=pole_order_max-1;j++){
mpfr_printf("single_pole_coeffs[%d] = %.16RNf\n",j,single_pole_coeffs[j]);
}
#endif
#ifdef debug_mode
for(int j=0;j<=pole_order_max-2;j++){
mpfr_printf("factorial_times_power_lnb[%d] = %.16RNf\n",j,factorial_times_power_lnb[j]);
}
#endif
for(int j=0;j<=pole_order_max-2;j++){
for(int k=0;k<=j;k++){
mpfr_mul(temp2,factorial_times_power_lnb[k],single_pole_coeffs[j-k],MPFR_RNDN);
mpfr_add(result[j+2],result[j+2],temp2,MPFR_RNDN);
}
}
for(int i=0;i<=pole_order_max-1;i++){
mpfr_clear(single_pole_coeffs[i]);
}
for(int i=0;i<=pole_order_max-2;i++){
mpfr_clear(factorial_times_power_lnb[i]);
}
if(pole_order_max > 0){
free(single_pole_coeffs);
}
if(pole_order_max > 1){
free(factorial_times_power_lnb);
}
mpfr_clear(temp1);
mpfr_clear(double_pole_integral);
mpfr_clear(temp2);
mpfr_clear(minus_pole_position);
//mpfr_clear(factorial);
mpfr_clear(minus_log_base);
mpfr_clear(log_base_power);
return result;
}
/* Put in s an approximation of digamma(x).
Assumes x >= 2.
Assumes s does not overlap with x.
Returns an integer e such that the error is bounded by 2^e ulps
of the result s.
*/
static mpfr_exp_t
mpfr_digamma_approx (mpfr_ptr s, mpfr_srcptr x)
{
mpfr_prec_t p = MPFR_PREC (s);
mpfr_t t, u, invxx;
mpfr_exp_t e, exps, f, expu;
mpz_t *INITIALIZED(B); /* variable B declared as initialized */
unsigned long n0, n; /* number of allocated B[] */
MPFR_ASSERTN(MPFR_IS_POS(x) && (MPFR_EXP(x) >= 2));
mpfr_init2 (t, p);
mpfr_init2 (u, p);
mpfr_init2 (invxx, p);
mpfr_log (s, x, MPFR_RNDN); /* error <= 1/2 ulp */
mpfr_ui_div (t, 1, x, MPFR_RNDN); /* error <= 1/2 ulp */
mpfr_div_2exp (t, t, 1, MPFR_RNDN); /* exact */
mpfr_sub (s, s, t, MPFR_RNDN);
/* error <= 1/2 + 1/2*2^(EXP(olds)-EXP(s)) + 1/2*2^(EXP(t)-EXP(s)).
For x >= 2, log(x) >= 2*(1/(2x)), thus olds >= 2t, and olds - t >= olds/2,
thus 0 <= EXP(olds)-EXP(s) <= 1, and EXP(t)-EXP(s) <= 0, thus
error <= 1/2 + 1/2*2 + 1/2 <= 2 ulps. */
e = 2; /* initial error */
mpfr_mul (invxx, x, x, MPFR_RNDZ); /* invxx = x^2 * (1 + theta)
for |theta| <= 2^(-p) */
mpfr_ui_div (invxx, 1, invxx, MPFR_RNDU); /* invxx = 1/x^2 * (1 + theta)^2 */
/* in the following we note err=xxx when the ratio between the approximation
and the exact result can be written (1 + theta)^xxx for |theta| <= 2^(-p),
following Higham's method */
B = mpfr_bernoulli_internal ((mpz_t *) 0, 0);
mpfr_set_ui (t, 1, MPFR_RNDN); /* err = 0 */
for (n = 1;; n++)
{
/* compute next Bernoulli number */
B = mpfr_bernoulli_internal (B, n);
/* The main term is Bernoulli[2n]/(2n)/x^(2n) = B[n]/(2n+1)!(2n)/x^(2n)
= B[n]*t[n]/(2n) where t[n]/t[n-1] = 1/(2n)/(2n+1)/x^2. */
mpfr_mul (t, t, invxx, MPFR_RNDU); /* err = err + 3 */
mpfr_div_ui (t, t, 2 * n, MPFR_RNDU); /* err = err + 1 */
mpfr_div_ui (t, t, 2 * n + 1, MPFR_RNDU); /* err = err + 1 */
/* we thus have err = 5n here */
mpfr_div_ui (u, t, 2 * n, MPFR_RNDU); /* err = 5n+1 */
mpfr_mul_z (u, u, B[n], MPFR_RNDU); /* err = 5n+2, and the
absolute error is bounded
by 10n+4 ulp(u) [Rule 11] */
/* if the terms 'u' are decreasing by a factor two at least,
then the error coming from those is bounded by
sum((10n+4)/2^n, n=1..infinity) = 24 */
exps = mpfr_get_exp (s);
expu = mpfr_get_exp (u);
if (expu < exps - (mpfr_exp_t) p)
break;
mpfr_sub (s, s, u, MPFR_RNDN); /* error <= 24 + n/2 */
if (mpfr_get_exp (s) < exps)
e <<= exps - mpfr_get_exp (s);
e ++; /* error in mpfr_sub */
f = 10 * n + 4;
while (expu < exps)
{
f = (1 + f) / 2;
expu ++;
}
e += f; /* total rouding error coming from 'u' term */
}
n0 = ++n;
while (n--)
mpz_clear (B[n]);
(*__gmp_free_func) (B, n0 * sizeof (mpz_t));
mpfr_clear (t);
mpfr_clear (u);
mpfr_clear (invxx);
f = 0;
while (e > 1)
{
f++;
e = (e + 1) / 2;
/* Invariant: 2^f * e does not decrease */
}
return f;
}
int
mpfr_atanh (mpfr_ptr y, mpfr_srcptr xt , mpfr_rnd_t rnd_mode)
{
int inexact;
mpfr_t x, t, te;
mpfr_prec_t Nx, Ny, Nt;
mpfr_exp_t err;
MPFR_ZIV_DECL (loop);
MPFR_SAVE_EXPO_DECL (expo);
MPFR_LOG_FUNC
(("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (xt), mpfr_log_prec, xt, rnd_mode),
("y[%Pu]=%.*Rg inexact=%d",
mpfr_get_prec (y), mpfr_log_prec, y, inexact));
/* Special cases */
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (xt)))
{
/* atanh(NaN) = NaN, and atanh(+/-Inf) = NaN since tanh gives a result
between -1 and 1 */
if (MPFR_IS_NAN (xt) || MPFR_IS_INF (xt))
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
else /* necessarily xt is 0 */
{
MPFR_ASSERTD (MPFR_IS_ZERO (xt));
MPFR_SET_ZERO (y); /* atanh(0) = 0 */
MPFR_SET_SAME_SIGN (y,xt);
MPFR_RET (0);
}
}
/* atanh (x) = NaN as soon as |x| > 1, and arctanh(+/-1) = +/-Inf */
if (MPFR_UNLIKELY (MPFR_GET_EXP (xt) > 0))
{
if (MPFR_GET_EXP (xt) == 1 && mpfr_powerof2_raw (xt))
{
MPFR_SET_INF (y);
MPFR_SET_SAME_SIGN (y, xt);
mpfr_set_divby0 ();
MPFR_RET (0);
}
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
/* atanh(x) = x + x^3/3 + ... so the error is < 2^(3*EXP(x)-1) */
MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, xt, -2 * MPFR_GET_EXP (xt), 1, 1,
rnd_mode, {});
MPFR_SAVE_EXPO_MARK (expo);
/* Compute initial precision */
Nx = MPFR_PREC (xt);
MPFR_TMP_INIT_ABS (x, xt);
Ny = MPFR_PREC (y);
Nt = MAX (Nx, Ny);
/* the optimal number of bits : see algorithms.ps */
Nt = Nt + MPFR_INT_CEIL_LOG2 (Nt) + 4;
/* initialise of intermediary variable */
mpfr_init2 (t, Nt);
mpfr_init2 (te, Nt);
/* First computation of cosh */
MPFR_ZIV_INIT (loop, Nt);
for (;;)
{
/* compute atanh */
mpfr_ui_sub (te, 1, x, MPFR_RNDU); /* (1-xt)*/
mpfr_add_ui (t, x, 1, MPFR_RNDD); /* (xt+1)*/
mpfr_div (t, t, te, MPFR_RNDN); /* (1+xt)/(1-xt)*/
mpfr_log (t, t, MPFR_RNDN); /* ln((1+xt)/(1-xt))*/
mpfr_div_2ui (t, t, 1, MPFR_RNDN); /* (1/2)*ln((1+xt)/(1-xt))*/
/* error estimate: see algorithms.tex */
/* FIXME: this does not correspond to the value in algorithms.tex!!! */
/* err=Nt-__gmpfr_ceil_log2(1+5*pow(2,1-MPFR_EXP(t)));*/
err = Nt - (MAX (4 - MPFR_GET_EXP (t), 0) + 1);
if (MPFR_LIKELY (MPFR_IS_ZERO (t)
|| MPFR_CAN_ROUND (t, err, Ny, rnd_mode)))
break;
/* reactualisation of the precision */
MPFR_ZIV_NEXT (loop, Nt);
mpfr_set_prec (t, Nt);
mpfr_set_prec (te, Nt);
}
MPFR_ZIV_FREE (loop);
inexact = mpfr_set4 (y, t, rnd_mode, MPFR_SIGN (xt));
mpfr_clear(t);
mpfr_clear(te);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inexact, rnd_mode);
}
int
mpfr_atanh (mpfr_ptr y, mpfr_srcptr xt , mp_rnd_t rnd_mode)
{
int inexact = 0;
mpfr_t x;
mp_prec_t Nx = MPFR_PREC(xt); /* Precision of input variable */
/* Special cases */
if (MPFR_UNLIKELY( MPFR_IS_SINGULAR(xt) ))
{
/* atanh(NaN) = NaN, and atanh(+/-Inf) = NaN since tanh gives a result
between -1 and 1 */
if (MPFR_IS_NAN(xt) || MPFR_IS_INF(xt))
{
MPFR_SET_NAN(y);
MPFR_RET_NAN;
}
else /* necessarily xt is 0 */
{
MPFR_ASSERTD(MPFR_IS_ZERO(xt));
MPFR_SET_ZERO(y); /* atanh(0) = 0 */
MPFR_SET_SAME_SIGN(y,xt);
MPFR_RET(0);
}
}
/* Useless due to final mpfr_set
MPFR_CLEAR_FLAGS(y);*/
/* atanh(x) = NaN as soon as |x| > 1, and arctanh(+/-1) = +/-Inf */
if (MPFR_EXP(xt) > 0)
{
if (MPFR_EXP(xt) == 1 && mpfr_powerof2_raw (xt))
{
MPFR_SET_INF(y);
MPFR_SET_SAME_SIGN(y, xt);
MPFR_RET(0);
}
MPFR_SET_NAN(y);
MPFR_RET_NAN;
}
mpfr_init2 (x, Nx);
mpfr_abs (x, xt, GMP_RNDN);
/* General case */
{
/* Declaration of the intermediary variable */
mpfr_t t, te,ti;
/* Declaration of the size variable */
mp_prec_t Nx = MPFR_PREC(x); /* Precision of input variable */
mp_prec_t Ny = MPFR_PREC(y); /* Precision of input variable */
mp_prec_t Nt; /* Precision of the intermediary variable */
long int err; /* Precision of error */
/* compute the precision of intermediary variable */
Nt=MAX(Nx,Ny);
/* the optimal number of bits : see algorithms.ps */
Nt=Nt+4+__gmpfr_ceil_log2(Nt);
/* initialise of intermediary variable */
mpfr_init(t);
mpfr_init(te);
mpfr_init(ti);
/* First computation of cosh */
do
{
/* reactualisation of the precision */
mpfr_set_prec(t,Nt);
mpfr_set_prec(te,Nt);
mpfr_set_prec(ti,Nt);
/* compute atanh */
mpfr_ui_sub(te,1,x,GMP_RNDU); /* (1-xt)*/
mpfr_add_ui(ti,x,1,GMP_RNDD); /* (xt+1)*/
mpfr_div(te,ti,te,GMP_RNDN); /* (1+xt)/(1-xt)*/
mpfr_log(te,te,GMP_RNDN); /* ln((1+xt)/(1-xt))*/
mpfr_div_2ui(t,te,1,GMP_RNDN); /* (1/2)*ln((1+xt)/(1-xt))*/
/* error estimate see- algorithms.ps*/
/* err=Nt-__gmpfr_ceil_log2(1+5*pow(2,1-MPFR_EXP(t)));*/
err = Nt - (MAX (4 - MPFR_GET_EXP (t), 0) + 1);
/* actualisation of the precision */
Nt += 10;
}
while ((err < 0) || (!mpfr_can_round (t, err, GMP_RNDN, GMP_RNDZ,
Ny + (rnd_mode == GMP_RNDN))
|| MPFR_IS_ZERO(t)));
if (MPFR_IS_NEG(xt))
MPFR_CHANGE_SIGN(t);
inexact = mpfr_set (y, t, rnd_mode);
mpfr_clear(t);
mpfr_clear(ti);
mpfr_clear(te);
}
mpfr_clear(x);
return inexact;
}
/* We use the reflection formula
Gamma(1+t) Gamma(1-t) = - Pi t / sin(Pi (1 + t))
in order to treat the case x <= 1,
i.e. with x = 1-t, then Gamma(x) = -Pi*(1-x)/sin(Pi*(2-x))/GAMMA(2-x)
*/
int
mpfr_gamma (mpfr_ptr gamma, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
mpfr_t xp, GammaTrial, tmp, tmp2;
mpz_t fact;
mpfr_prec_t realprec;
int compared, is_integer;
int inex = 0; /* 0 means: result gamma not set yet */
MPFR_GROUP_DECL (group);
MPFR_SAVE_EXPO_DECL (expo);
MPFR_ZIV_DECL (loop);
MPFR_LOG_FUNC
(("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd_mode),
("gamma[%Pu]=%.*Rg inexact=%d",
mpfr_get_prec (gamma), mpfr_log_prec, gamma, inex));
/* Trivial cases */
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
{
if (MPFR_IS_NAN (x))
{
MPFR_SET_NAN (gamma);
MPFR_RET_NAN;
}
else if (MPFR_IS_INF (x))
{
if (MPFR_IS_NEG (x))
{
MPFR_SET_NAN (gamma);
MPFR_RET_NAN;
}
else
{
MPFR_SET_INF (gamma);
MPFR_SET_POS (gamma);
MPFR_RET (0); /* exact */
}
}
else /* x is zero */
{
MPFR_ASSERTD(MPFR_IS_ZERO(x));
MPFR_SET_INF(gamma);
MPFR_SET_SAME_SIGN(gamma, x);
MPFR_SET_DIVBY0 ();
MPFR_RET (0); /* exact */
}
}
/* Check for tiny arguments, where gamma(x) ~ 1/x - euler + ....
We know from "Bound on Runs of Zeros and Ones for Algebraic Functions",
Proceedings of Arith15, T. Lang and J.-M. Muller, 2001, that the maximal
number of consecutive zeroes or ones after the round bit is n-1 for an
input of n bits. But we need a more precise lower bound. Assume x has
n bits, and 1/x is near a floating-point number y of n+1 bits. We can
write x = X*2^e, y = Y/2^f with X, Y integers of n and n+1 bits.
Thus X*Y^2^(e-f) is near from 1, i.e., X*Y is near from 2^(f-e).
Two cases can happen:
(i) either X*Y is exactly 2^(f-e), but this can happen only if X and Y
are themselves powers of two, i.e., x is a power of two;
(ii) or X*Y is at distance at least one from 2^(f-e), thus
|xy-1| >= 2^(e-f), or |y-1/x| >= 2^(e-f)/x = 2^(-f)/X >= 2^(-f-n).
Since ufp(y) = 2^(n-f) [ufp = unit in first place], this means
that the distance |y-1/x| >= 2^(-2n) ufp(y).
Now assuming |gamma(x)-1/x| <= 1, which is true for x <= 1,
if 2^(-2n) ufp(y) >= 2, the error is at most 2^(-2n-1) ufp(y),
and round(1/x) with precision >= 2n+2 gives the correct result.
If x < 2^E, then y > 2^(-E), thus ufp(y) > 2^(-E-1).
A sufficient condition is thus EXP(x) + 2 <= -2 MAX(PREC(x),PREC(Y)).
*/
if (MPFR_GET_EXP (x) + 2
<= -2 * (mpfr_exp_t) MAX(MPFR_PREC(x), MPFR_PREC(gamma)))
{
int sign = MPFR_SIGN (x); /* retrieve sign before possible override */
int special;
MPFR_BLOCK_DECL (flags);
MPFR_SAVE_EXPO_MARK (expo);
/* for overflow cases, see below; this needs to be done
before x possibly gets overridden. */
special =
MPFR_GET_EXP (x) == 1 - MPFR_EMAX_MAX &&
MPFR_IS_POS_SIGN (sign) &&
MPFR_IS_LIKE_RNDD (rnd_mode, sign) &&
mpfr_powerof2_raw (x);
MPFR_BLOCK (flags, inex = mpfr_ui_div (gamma, 1, x, rnd_mode));
if (inex == 0) /* x is a power of two */
{
/* return RND(1/x - euler) = RND(+/- 2^k - eps) with eps > 0 */
if (rnd_mode == MPFR_RNDN || MPFR_IS_LIKE_RNDU (rnd_mode, sign))
inex = 1;
else
{
mpfr_nextbelow (gamma);
inex = -1;
}
}
else if (MPFR_UNLIKELY (MPFR_OVERFLOW (flags)))
{
/* Overflow in the division 1/x. This is a real overflow, except
in RNDZ or RNDD when 1/x = 2^emax, i.e. x = 2^(-emax): due to
the "- euler", the rounded value in unbounded exponent range
is 0.111...11 * 2^emax (not an overflow). */
if (!special)
MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, flags);
}
MPFR_SAVE_EXPO_FREE (expo);
/* Note: an overflow is possible with an infinite result;
in this case, the overflow flag will automatically be
restored by mpfr_check_range. */
return mpfr_check_range (gamma, inex, rnd_mode);
}
is_integer = mpfr_integer_p (x);
/* gamma(x) for x a negative integer gives NaN */
if (is_integer && MPFR_IS_NEG(x))
{
MPFR_SET_NAN (gamma);
MPFR_RET_NAN;
}
compared = mpfr_cmp_ui (x, 1);
if (compared == 0)
return mpfr_set_ui (gamma, 1, rnd_mode);
/* if x is an integer that fits into an unsigned long, use mpfr_fac_ui
if argument is not too large.
If precision is p, fac_ui costs O(u*p), whereas gamma costs O(p*M(p)),
so for u <= M(p), fac_ui should be faster.
We approximate here M(p) by p*log(p)^2, which is not a bad guess.
Warning: since the generic code does not handle exact cases,
we want all cases where gamma(x) is exact to be treated here.
*/
if (is_integer && mpfr_fits_ulong_p (x, MPFR_RNDN))
{
unsigned long int u;
mpfr_prec_t p = MPFR_PREC(gamma);
u = mpfr_get_ui (x, MPFR_RNDN);
if (u < 44787929UL && bits_fac (u - 1) <= p + (rnd_mode == MPFR_RNDN))
/* bits_fac: lower bound on the number of bits of m,
where gamma(x) = (u-1)! = m*2^e with m odd. */
return mpfr_fac_ui (gamma, u - 1, rnd_mode);
/* if bits_fac(...) > p (resp. p+1 for rounding to nearest),
then gamma(x) cannot be exact in precision p (resp. p+1).
FIXME: remove the test u < 44787929UL after changing bits_fac
to return a mpz_t or mpfr_t. */
}
MPFR_SAVE_EXPO_MARK (expo);
/* check for overflow: according to (6.1.37) in Abramowitz & Stegun,
gamma(x) >= exp(-x) * x^(x-1/2) * sqrt(2*Pi)
>= 2 * (x/e)^x / x for x >= 1 */
if (compared > 0)
{
mpfr_t yp;
mpfr_exp_t expxp;
MPFR_BLOCK_DECL (flags);
/* quick test for the default exponent range */
if (mpfr_get_emax () >= 1073741823UL && MPFR_GET_EXP(x) <= 25)
{
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_gamma_aux (gamma, x, rnd_mode);
}
/* 1/e rounded down to 53 bits */
#define EXPM1_STR "0.010111100010110101011000110110001011001110111100111"
mpfr_init2 (xp, 53);
mpfr_init2 (yp, 53);
mpfr_set_str_binary (xp, EXPM1_STR);
mpfr_mul (xp, x, xp, MPFR_RNDZ);
mpfr_sub_ui (yp, x, 2, MPFR_RNDZ);
mpfr_pow (xp, xp, yp, MPFR_RNDZ); /* (x/e)^(x-2) */
mpfr_set_str_binary (yp, EXPM1_STR);
mpfr_mul (xp, xp, yp, MPFR_RNDZ); /* x^(x-2) / e^(x-1) */
mpfr_mul (xp, xp, yp, MPFR_RNDZ); /* x^(x-2) / e^x */
mpfr_mul (xp, xp, x, MPFR_RNDZ); /* lower bound on x^(x-1) / e^x */
MPFR_BLOCK (flags, mpfr_mul_2ui (xp, xp, 1, MPFR_RNDZ));
expxp = MPFR_GET_EXP (xp);
mpfr_clear (xp);
mpfr_clear (yp);
MPFR_SAVE_EXPO_FREE (expo);
return MPFR_OVERFLOW (flags) || expxp > __gmpfr_emax ?
mpfr_overflow (gamma, rnd_mode, 1) :
mpfr_gamma_aux (gamma, x, rnd_mode);
}
/* now compared < 0 */
/* check for underflow: for x < 1,
gamma(x) = Pi*(x-1)/sin(Pi*(2-x))/gamma(2-x).
Since gamma(2-x) >= 2 * ((2-x)/e)^(2-x) / (2-x), we have
|gamma(x)| <= Pi*(1-x)*(2-x)/2/((2-x)/e)^(2-x) / |sin(Pi*(2-x))|
<= 12 * ((2-x)/e)^x / |sin(Pi*(2-x))|.
To avoid an underflow in ((2-x)/e)^x, we compute the logarithm.
*/
if (MPFR_IS_NEG(x))
{
int underflow = 0, sgn, ck;
mpfr_prec_t w;
mpfr_init2 (xp, 53);
mpfr_init2 (tmp, 53);
mpfr_init2 (tmp2, 53);
/* we want an upper bound for x * [log(2-x)-1].
since x < 0, we need a lower bound on log(2-x) */
mpfr_ui_sub (xp, 2, x, MPFR_RNDD);
mpfr_log (xp, xp, MPFR_RNDD);
mpfr_sub_ui (xp, xp, 1, MPFR_RNDD);
mpfr_mul (xp, xp, x, MPFR_RNDU);
/* we need an upper bound on 1/|sin(Pi*(2-x))|,
thus a lower bound on |sin(Pi*(2-x))|.
If 2-x is exact, then the error of Pi*(2-x) is (1+u)^2 with u = 2^(-p)
thus the error on sin(Pi*(2-x)) is less than 1/2ulp + 3Pi(2-x)u,
assuming u <= 1, thus <= u + 3Pi(2-x)u */
w = mpfr_gamma_2_minus_x_exact (x); /* 2-x is exact for prec >= w */
w += 17; /* to get tmp2 small enough */
mpfr_set_prec (tmp, w);
mpfr_set_prec (tmp2, w);
MPFR_DBGRES (ck = mpfr_ui_sub (tmp, 2, x, MPFR_RNDN));
MPFR_ASSERTD (ck == 0); /* tmp = 2-x exactly */
mpfr_const_pi (tmp2, MPFR_RNDN);
mpfr_mul (tmp2, tmp2, tmp, MPFR_RNDN); /* Pi*(2-x) */
mpfr_sin (tmp, tmp2, MPFR_RNDN); /* sin(Pi*(2-x)) */
sgn = mpfr_sgn (tmp);
mpfr_abs (tmp, tmp, MPFR_RNDN);
mpfr_mul_ui (tmp2, tmp2, 3, MPFR_RNDU); /* 3Pi(2-x) */
mpfr_add_ui (tmp2, tmp2, 1, MPFR_RNDU); /* 3Pi(2-x)+1 */
mpfr_div_2ui (tmp2, tmp2, mpfr_get_prec (tmp), MPFR_RNDU);
/* if tmp2<|tmp|, we get a lower bound */
if (mpfr_cmp (tmp2, tmp) < 0)
{
mpfr_sub (tmp, tmp, tmp2, MPFR_RNDZ); /* low bnd on |sin(Pi*(2-x))| */
mpfr_ui_div (tmp, 12, tmp, MPFR_RNDU); /* upper bound */
mpfr_log2 (tmp, tmp, MPFR_RNDU);
mpfr_add (xp, tmp, xp, MPFR_RNDU);
/* The assert below checks that expo.saved_emin - 2 always
fits in a long. FIXME if we want to allow mpfr_exp_t to
be a long long, for instance. */
MPFR_ASSERTN (MPFR_EMIN_MIN - 2 >= LONG_MIN);
underflow = mpfr_cmp_si (xp, expo.saved_emin - 2) <= 0;
}
mpfr_clear (xp);
mpfr_clear (tmp);
mpfr_clear (tmp2);
if (underflow) /* the sign is the opposite of that of sin(Pi*(2-x)) */
{
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_underflow (gamma, (rnd_mode == MPFR_RNDN) ? MPFR_RNDZ : rnd_mode, -sgn);
}
}
realprec = MPFR_PREC (gamma);
/* we want both 1-x and 2-x to be exact */
{
mpfr_prec_t w;
w = mpfr_gamma_1_minus_x_exact (x);
if (realprec < w)
realprec = w;
w = mpfr_gamma_2_minus_x_exact (x);
if (realprec < w)
realprec = w;
}
realprec = realprec + MPFR_INT_CEIL_LOG2 (realprec) + 20;
MPFR_ASSERTD(realprec >= 5);
MPFR_GROUP_INIT_4 (group, realprec + MPFR_INT_CEIL_LOG2 (realprec) + 20,
xp, tmp, tmp2, GammaTrial);
mpz_init (fact);
MPFR_ZIV_INIT (loop, realprec);
for (;;)
{
mpfr_exp_t err_g;
int ck;
MPFR_GROUP_REPREC_4 (group, realprec, xp, tmp, tmp2, GammaTrial);
/* reflection formula: gamma(x) = Pi*(x-1)/sin(Pi*(2-x))/gamma(2-x) */
ck = mpfr_ui_sub (xp, 2, x, MPFR_RNDN); /* 2-x, exact */
MPFR_ASSERTD(ck == 0); (void) ck; /* use ck to avoid a warning */
mpfr_gamma (tmp, xp, MPFR_RNDN); /* gamma(2-x), error (1+u) */
mpfr_const_pi (tmp2, MPFR_RNDN); /* Pi, error (1+u) */
mpfr_mul (GammaTrial, tmp2, xp, MPFR_RNDN); /* Pi*(2-x), error (1+u)^2 */
err_g = MPFR_GET_EXP(GammaTrial);
mpfr_sin (GammaTrial, GammaTrial, MPFR_RNDN); /* sin(Pi*(2-x)) */
/* If tmp is +Inf, we compute exp(lngamma(x)). */
if (mpfr_inf_p (tmp))
{
inex = mpfr_explgamma (gamma, x, &expo, tmp, tmp2, rnd_mode);
if (inex)
goto end;
else
goto ziv_next;
}
err_g = err_g + 1 - MPFR_GET_EXP(GammaTrial);
/* let g0 the true value of Pi*(2-x), g the computed value.
We have g = g0 + h with |h| <= |(1+u^2)-1|*g.
Thus sin(g) = sin(g0) + h' with |h'| <= |(1+u^2)-1|*g.
The relative error is thus bounded by |(1+u^2)-1|*g/sin(g)
<= |(1+u^2)-1|*2^err_g. <= 2.25*u*2^err_g for |u|<=1/4.
With the rounding error, this gives (0.5 + 2.25*2^err_g)*u. */
ck = mpfr_sub_ui (xp, x, 1, MPFR_RNDN); /* x-1, exact */
MPFR_ASSERTD(ck == 0); (void) ck; /* use ck to avoid a warning */
mpfr_mul (xp, tmp2, xp, MPFR_RNDN); /* Pi*(x-1), error (1+u)^2 */
mpfr_mul (GammaTrial, GammaTrial, tmp, MPFR_RNDN);
/* [1 + (0.5 + 2.25*2^err_g)*u]*(1+u)^2 = 1 + (2.5 + 2.25*2^err_g)*u
+ (0.5 + 2.25*2^err_g)*u*(2u+u^2) + u^2.
For err_g <= realprec-2, we have (0.5 + 2.25*2^err_g)*u <=
0.5*u + 2.25/4 <= 0.6875 and u^2 <= u/4, thus
(0.5 + 2.25*2^err_g)*u*(2u+u^2) + u^2 <= 0.6875*(2u+u/4) + u/4
<= 1.8*u, thus the rel. error is bounded by (4.5 + 2.25*2^err_g)*u. */
mpfr_div (GammaTrial, xp, GammaTrial, MPFR_RNDN);
/* the error is of the form (1+u)^3/[1 + (4.5 + 2.25*2^err_g)*u].
For realprec >= 5 and err_g <= realprec-2, [(4.5 + 2.25*2^err_g)*u]^2
<= 0.71, and for |y|<=0.71, 1/(1-y) can be written 1+a*y with a<=4.
(1+u)^3 * (1+4*(4.5 + 2.25*2^err_g)*u)
= 1 + (21 + 9*2^err_g)*u + (57+27*2^err_g)*u^2 + (55+27*2^err_g)*u^3
+ (18+9*2^err_g)*u^4
<= 1 + (21 + 9*2^err_g)*u + (57+27*2^err_g)*u^2 + (56+28*2^err_g)*u^3
<= 1 + (21 + 9*2^err_g)*u + (59+28*2^err_g)*u^2
<= 1 + (23 + 10*2^err_g)*u.
The final error is thus bounded by (23 + 10*2^err_g) ulps,
which is <= 2^6 for err_g<=2, and <= 2^(err_g+4) for err_g >= 2. */
err_g = (err_g <= 2) ? 6 : err_g + 4;
if (MPFR_LIKELY (MPFR_CAN_ROUND (GammaTrial, realprec - err_g,
MPFR_PREC(gamma), rnd_mode)))
break;
ziv_next:
MPFR_ZIV_NEXT (loop, realprec);
}
end:
MPFR_ZIV_FREE (loop);
if (inex == 0)
inex = mpfr_set (gamma, GammaTrial, rnd_mode);
MPFR_GROUP_CLEAR (group);
mpz_clear (fact);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (gamma, inex, rnd_mode);
}
static void
mpfr_const_log10 (mpfr_ptr log10)
{
mpfr_set_ui (log10, 10, MPFR_RNDN); /* exact since prec >= 4 */
mpfr_log (log10, log10, MPFR_RNDN); /* error <= 1/2 ulp */
}