real atan(const real & a)
{
real x;
mpfr_atan(x.r, a.r, MPFR_RNDN);
return x;
}
mpcomplex mpcomplex::PI(const mp_prec_t &p, const mp_rnd_t &r) {
mpfr_t pi;
mpfr_t one;
mpfr_init2(one, p);
mpfr_set_ui(one , 1, r);
mpfr_init2(pi, p);
mpfr_atan(pi, one , r );
mpfr_mul_ui( pi , pi , 4 , r);
return mpcomplex( pi );
}
decimal r_atan(const decimal& a,bool round)
{
#ifdef USE_CGAL
CGAL::Gmpfr m;
CGAL::Gmpfr n=to_gmpfr(a);
mpfr_atan(m.fr(),n.fr(),MPFR_RNDN);
return r_round_preference(decimal(m),round);
#else
return r_round_preference(atan(a),round);
#endif
}
//------------------------------------------------------------------------------
// Name:
//------------------------------------------------------------------------------
knumber_base *knumber_float::atan() {
#ifdef KNUMBER_USE_MPFR
mpfr_t mpfr;
mpfr_init_set_f(mpfr, mpf_, rounding_mode);
mpfr_atan(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< ::atan>(x);
}
#endif
}
/* https://sympa.inria.fr/sympa/arc/mpfr/2011-05/msg00008.html
* Incorrect flags (in debug mode on a 32-bit machine, assertion failure).
*/
static void
reduced_expo_range (void)
{
mpfr_exp_t emin, emax;
mpfr_t x, y, ex_y;
int inex, ex_inex;
unsigned int flags, ex_flags;
emin = mpfr_get_emin ();
emax = mpfr_get_emax ();
mpfr_inits2 (12, x, y, ex_y, (mpfr_ptr) 0);
mpfr_set_str (x, "0.1e-5", 2, MPFR_RNDN);
set_emin (-5);
set_emax (-5);
mpfr_clear_flags ();
inex = mpfr_atan (y, x, MPFR_RNDN);
flags = __gmpfr_flags;
set_emin (emin);
set_emax (emax);
mpfr_set_str (ex_y, "0.1e-5", 2, MPFR_RNDN);
ex_inex = 1;
ex_flags = MPFR_FLAGS_INEXACT;
if (SIGN (inex) != ex_inex || flags != ex_flags ||
! mpfr_equal_p (y, ex_y))
{
printf ("Error in reduced_expo_range\non x = ");
mpfr_dump (x);
printf ("Expected y = ");
mpfr_out_str (stdout, 2, 0, ex_y, MPFR_RNDN);
printf ("\n inex = %d, flags = %u\n", ex_inex, ex_flags);
printf ("Got y = ");
mpfr_out_str (stdout, 2, 0, y, MPFR_RNDN);
printf ("\n inex = %d, flags = %u\n", SIGN (inex), flags);
exit (1);
}
mpfr_clears (x, y, ex_y, (mpfr_ptr) 0);
}
REAL _atan(REAL a, REAL, QByteArray &)
{
mpfr_t tmp1; mpfr_init2(tmp1, NUMBITS);
mpfr_t result; mpfr_init2(result, NUMBITS);
try
{
// mpfr_init_set_f(tmp1, a.get_mpf_t(), MPFR_RNDN);
mpfr_set_str(tmp1, getString(a).data(), 10, MPFR_RNDN);
mpfr_atan(result, tmp1, MPFR_RNDN);
mpfr_get_f(a.get_mpf_t(), result, MPFR_RNDN);
}
catch(...)
{
mpfr_clear(tmp1);
mpfr_clear(result);
return ZERO;
}
mpfr_clear(tmp1);
mpfr_clear(result);
return a;
}
void
arb_atan_arf_via_mpfr(arb_t z, const arf_t x, slong prec)
{
mpfr_t t, u;
int exact;
mpfr_init2(t, 2 + arf_bits(x));
mpfr_init2(u, prec);
mpfr_set_emin(MPFR_EMIN_MIN);
mpfr_set_emax(MPFR_EMAX_MAX);
arf_get_mpfr(t, x, MPFR_RNDD);
exact = (mpfr_atan(u, t, MPFR_RNDD) == 0);
arf_set_mpfr(arb_midref(z), u);
if (!exact)
arf_mag_set_ulp(arb_radref(z), arb_midref(z), prec);
mpfr_clear(t);
mpfr_clear(u);
}
/// @brief atan keyword implementation
///
void program::rpn_atan(void) {
MIN_ARGUMENTS(1);
if (_stack->get_type(0) == cmd_number) {
floating_t* left = &((number*)_stack->get_obj(0))->_value;
CHECK_MPFR(mpfr_atan(left->mpfr, left->mpfr, floating_t::s_mpfr_rnd));
} else if (_stack->get_type(0) == cmd_complex) {
number* num;
complex* i;
// atan(z)=0.5i(ln((1-iz)/(1+iz))
stack::copy_and_push_back(*_stack, _stack->size() - 1, _calc_stack);
i = (complex*)_calc_stack.get_obj(0);
CHECK_MPFR(mpfr_set_d(i->re()->mpfr, 0.0, floating_t::s_mpfr_rnd));
CHECK_MPFR(mpfr_set_d(i->im()->mpfr, 1.0, floating_t::s_mpfr_rnd));
stack::copy_and_push_back(_calc_stack, 0, *_stack);
rpn_mul();
num = (number*)_stack->allocate_back(number::calc_size(), cmd_number);
CHECK_MPFR(mpfr_set_d(num->_value.mpfr, 1.0, floating_t::s_mpfr_rnd));
rpn_minus(); // iz-1
rpn_neg(); // 1-iz
rpn_dup();
rpn_neg(); // iz-1
num = (number*)_stack->allocate_back(number::calc_size(), cmd_number);
CHECK_MPFR(mpfr_set_d(num->_value.mpfr, 2.0, floating_t::s_mpfr_rnd));
rpn_plus(); // iz+1
rpn_div();
rpn_ln();
CHECK_MPFR(mpfr_set_d(i->im()->mpfr, 0.5, floating_t::s_mpfr_rnd));
stack::copy_and_push_back(_calc_stack, 0, *_stack);
rpn_mul();
_calc_stack.pop_back();
} else
ERR_CONTEXT(ret_bad_operand_type);
}
MpfrFloat MpfrFloat::atan(const MpfrFloat& value)
{
MpfrFloat retval(MpfrFloat::kNoInitialization);
mpfr_atan(retval.mData->mFloat, value.mData->mFloat, GMP_RNDN);
return retval;
}
static void
special (void)
{
mpfr_t x, y, z;
int r;
int i;
mpfr_init2 (x, 53);
mpfr_init2 (y, 53);
mpfr_init2 (z, 53);
mpfr_set_str_binary (x, "1.0000100110000001100111100011001110101110100111011101");
mpfr_set_str_binary (y, "1.1001101101110100101100110011011101101000011010111110e-1");
mpfr_atan (z, x, MPFR_RNDN);
if (mpfr_cmp (y, z))
{
printf ("Error in mpfr_atan for prec=53, rnd=MPFR_RNDN\n");
printf ("x=");
mpfr_out_str (stdout, 2, 0, x, MPFR_RNDN);
printf ("\nexpected ");
mpfr_out_str (stdout, 2, 0, y, MPFR_RNDN);
printf ("\ngot ");
mpfr_out_str (stdout, 2, 0, z, MPFR_RNDN);
printf ("\n");
exit (1);
}
/* atan(+Inf) = Pi/2 */
for (r = 0; r < MPFR_RND_MAX ; r++)
{
mpfr_set_inf (x, 1);
mpfr_atan (y, x, (mpfr_rnd_t) r);
mpfr_const_pi (x, (mpfr_rnd_t) r);
mpfr_div_2exp (x, x, 1, (mpfr_rnd_t) r);
if (mpfr_cmp (x, y))
{
printf ("Error: mpfr_atan(+Inf), rnd=%s\n",
mpfr_print_rnd_mode ((mpfr_rnd_t) r));
exit (1);
}
}
/* atan(-Inf) = - Pi/2 */
for (r = 0; r < MPFR_RND_MAX ; r++)
{
mpfr_set_inf (x, -1);
mpfr_atan (y, x, (mpfr_rnd_t) r);
mpfr_const_pi (x, MPFR_INVERT_RND((mpfr_rnd_t) r));
mpfr_neg (x, x, (mpfr_rnd_t) r);
mpfr_div_2exp (x, x, 1, (mpfr_rnd_t) r);
if (mpfr_cmp (x, y))
{
printf ("Error: mpfr_atan(-Inf), rnd=%s\n",
mpfr_print_rnd_mode ((mpfr_rnd_t) r));
exit (1);
}
}
/* atan(NaN) = NaN */
mpfr_set_nan (x);
mpfr_atan (y, x, MPFR_RNDN);
if (!mpfr_nan_p (y))
{
printf ("Error: mpfr_atan(NaN) <> NaN\n");
exit (1);
}
/* atan(+/-0) = +/-0 */
mpfr_set_ui (x, 0, MPFR_RNDN);
MPFR_SET_NEG (y);
mpfr_atan (y, x, MPFR_RNDN);
if (mpfr_cmp_ui (y, 0) || MPFR_IS_NEG (y))
{
printf ("Error: mpfr_atan (+0) <> +0\n");
exit (1);
}
mpfr_atan (x, x, MPFR_RNDN);
if (mpfr_cmp_ui (x, 0) || MPFR_IS_NEG (x))
{
printf ("Error: mpfr_atan (+0) <> +0 (in place)\n");
exit (1);
}
mpfr_neg (x, x, MPFR_RNDN);
MPFR_SET_POS (y);
mpfr_atan (y, x, MPFR_RNDN);
if (mpfr_cmp_ui (y, 0) || MPFR_IS_POS (y))
{
printf ("Error: mpfr_atan (-0) <> -0\n");
exit (1);
}
mpfr_atan (x, x, MPFR_RNDN);
if (mpfr_cmp_ui (x, 0) || MPFR_IS_POS (x))
{
printf ("Error: mpfr_atan (-0) <> -0 (in place)\n");
exit (1);
}
mpfr_set_prec (x, 32);
mpfr_set_prec (y, 32);
/* test one random positive argument */
mpfr_set_str_binary (x, "0.10000100001100101001001001011001");
mpfr_atan (x, x, MPFR_RNDN);
mpfr_set_str_binary (y, "0.1111010000001111001111000000011E-1");
if (mpfr_cmp (x, y))
{
printf ("Error in mpfr_atan (1)\n");
exit (1);
}
/* test one random negative argument */
mpfr_set_str_binary (x, "-0.1100001110110000010101011001011");
mpfr_atan (x, x, MPFR_RNDN);
mpfr_set_str_binary (y, "-0.101001110001010010110001110001");
if (mpfr_cmp (x, y))
{
printf ("Error in mpfr_atan (2)\n");
mpfr_print_binary (x);
printf ("\n");
mpfr_print_binary (y);
printf ("\n");
exit (1);
}
mpfr_set_prec (x, 3);
mpfr_set_prec (y, 192);
mpfr_set_prec (z, 192);
mpfr_set_str_binary (x, "-0.100e1");
mpfr_atan (z, x, MPFR_RNDD);
mpfr_set_str_binary (y, "-0.110010010000111111011010101000100010000101101000110000100011010011000100110001100110001010001011100000001101110000011100110100010010100100000010010011100000100010001010011001111100110001110101");
if (mpfr_cmp (z, y))
{
printf ("Error in mpfr_atan (3)\n");
printf ("Expected ");
mpfr_print_binary (y);
printf ("\n");
printf ("Got ");
mpfr_print_binary (z);
printf ("\n");
exit (1);
}
/* Test regression */
mpfr_set_prec (x, 51);
mpfr_set_prec (y, 51);
mpfr_set_str_binary (x,
"0.101100100000101111111010001111111000001000000000000E-11");
i = mpfr_atan (y, x, MPFR_RNDN);
if (mpfr_cmp_str (y,
"1.01100100000101111111001110011001010110100100000000e-12", 2, MPFR_RNDN)
|| i >= 0)
{
printf ("Wrong Regression test (%d)\n", i);
mpfr_dump (y);
exit (1);
}
mpfr_set_si (x, -1, MPFR_RNDN);
mpfr_atan (x, x, MPFR_RNDN);
MPFR_ASSERTN (MPFR_IS_NEG (x));
/* Test regression */
mpfr_set_prec (x, 48);
mpfr_set_prec (y, 48);
mpfr_set_str_binary (x, "1.11001110010000011111100000010000000000000000000e-19");
mpfr_atan (y, x, MPFR_RNDD);
if (mpfr_cmp_str (y, "0.111001110010000011111100000001111111110000010011E-18", 2, MPFR_RNDN))
{
printf ("Error in mpfr_atan (4)\n");
printf ("Input 1.11001110010000011111100000010000000000000000000e-19 [prec=48]\n");
printf ("Expected 0.111001110010000011111100000001111111110000010011E-18\n");
printf ("Got ");
mpfr_dump (y);
exit (1);
}
mpfr_clear (x);
mpfr_clear (y);
mpfr_clear (z);
}
static void
special_overflow (void)
{
mpfr_t x, y;
mpfr_exp_t emin, emax;
emin = mpfr_get_emin ();
emax = mpfr_get_emax ();
set_emin (-125);
set_emax (128);
mpfr_init2 (x, 24);
mpfr_init2 (y, 48);
mpfr_set_str_binary (x, "0.101101010001001101111010E0");
mpfr_atan (y, x, MPFR_RNDN);
if (mpfr_cmp_str (y, "0.100111011001100111000010111101000111010101011110E0",
2, MPFR_RNDN))
{
printf("Special Overflow error.\n");
mpfr_dump (y);
exit (1);
}
/* intermediate Pi overflows while atan(+Inf) = Pi/2 is representable */
set_emax (1);
mpfr_set_inf (x, +1);
mpfr_clear_flags ();
mpfr_atan (y, x, MPFR_RNDN);
if (mpfr_cmp_str (y, "C90FDAA22169p-47", 16, MPFR_RNDN)
|| mpfr_overflow_p ())
{
printf("atan(+Inf) = Pi/2 should not overflow when emax = %ld\n",
(long int) mpfr_get_emax ());
mpfr_dump (y);
exit (1);
}
/* atan(+Inf) = Pi/2 underflows */
set_emax (128);
set_emin (3);
mpfr_clear_flags ();
mpfr_atan (y, x, MPFR_RNDN);
if (mpfr_cmp_ui (y, 0) || !mpfr_underflow_p ())
{
printf("atan(+Inf) = Pi/2 should underflow when emin = %ld\n",
(long int) mpfr_get_emin ());
mpfr_dump (y);
exit (1);
}
/* intermediate Pi overflows while atan(+1) = Pi/4 is representable */
set_emax (1);
set_emin (-128);
mpfr_set_ui (x, 1, MPFR_RNDN);
mpfr_clear_flags ();
mpfr_atan (y, x, MPFR_RNDN);
if (mpfr_cmp_str (y, "C90FDAA22169p-48", 16, MPFR_RNDN)
|| mpfr_overflow_p ())
{
printf("atan(+1) = Pi/4 should not overflow when emax = %ld\n",
(long int) mpfr_get_emax ());
mpfr_dump (y);
exit (1);
}
/* atan(+1) = Pi/4 underflows and is rounded up to 1 */
set_emax (128);
set_emin (1);
mpfr_set_prec (y, 2);
mpfr_clear_flags ();
mpfr_atan (y, x, MPFR_RNDN);
if (mpfr_cmp_ui (y, 1) || !mpfr_underflow_p ())
{
printf("atan(+1) = Pi/4 should underflow when emin = %+ld\n",
(long int) mpfr_get_emin ());
mpfr_dump (y);
exit (1);
}
/* atan(+1) = Pi/4 underflows and is rounded down to 0 */
mpfr_clear_flags ();
mpfr_atan (y, x, MPFR_RNDD);
if (mpfr_cmp_ui (y, 0) || !mpfr_underflow_p ())
{
printf("atan(+1) = Pi/4 should underflow when emin = %+ld\n",
(long int) mpfr_get_emin ());
mpfr_dump (y);
exit (1);
}
mpfr_clear (y);
mpfr_clear (x);
set_emin (emin);
set_emax (emax);
}
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;
}
int
mpfr_asin (mpfr_ptr asin, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
mpfr_t xp;
int compared, inexact;
mpfr_prec_t prec;
mpfr_exp_t xp_exp;
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),
("asin[%Pu]=%.*Rg inexact=%d", mpfr_get_prec (asin), mpfr_log_prec, asin,
inexact));
/* Special cases */
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
{
if (MPFR_IS_NAN (x) || MPFR_IS_INF (x))
{
MPFR_SET_NAN (asin);
MPFR_RET_NAN;
}
else /* x = 0 */
{
MPFR_ASSERTD (MPFR_IS_ZERO (x));
MPFR_SET_ZERO (asin);
MPFR_SET_SAME_SIGN (asin, x);
MPFR_RET (0); /* exact result */
}
}
/* asin(x) = x + x^3/6 + ... so the error is < 2^(3*EXP(x)-2) */
MPFR_FAST_COMPUTE_IF_SMALL_INPUT (asin, x, -2 * MPFR_GET_EXP (x), 2, 1,
rnd_mode, {});
/* Set x_p=|x| (x is a normal number) */
mpfr_init2 (xp, MPFR_PREC (x));
inexact = mpfr_abs (xp, x, MPFR_RNDN);
MPFR_ASSERTD (inexact == 0);
compared = mpfr_cmp_ui (xp, 1);
MPFR_SAVE_EXPO_MARK (expo);
if (MPFR_UNLIKELY (compared >= 0))
{
mpfr_clear (xp);
if (compared > 0) /* asin(x) = NaN for |x| > 1 */
{
MPFR_SAVE_EXPO_FREE (expo);
MPFR_SET_NAN (asin);
MPFR_RET_NAN;
}
else /* x = 1 or x = -1 */
{
if (MPFR_IS_POS (x)) /* asin(+1) = Pi/2 */
inexact = mpfr_const_pi (asin, rnd_mode);
else /* asin(-1) = -Pi/2 */
{
inexact = -mpfr_const_pi (asin, MPFR_INVERT_RND(rnd_mode));
MPFR_CHANGE_SIGN (asin);
}
mpfr_div_2ui (asin, asin, 1, rnd_mode);
}
}
else
{
/* Compute exponent of 1 - ABS(x) */
mpfr_ui_sub (xp, 1, xp, MPFR_RNDD);
MPFR_ASSERTD (MPFR_GET_EXP (xp) <= 0);
MPFR_ASSERTD (MPFR_GET_EXP (x) <= 0);
xp_exp = 2 - MPFR_GET_EXP (xp);
/* Set up initial prec */
prec = MPFR_PREC (asin) + 10 + xp_exp;
/* use asin(x) = atan(x/sqrt(1-x^2)) */
MPFR_ZIV_INIT (loop, prec);
for (;;)
{
mpfr_set_prec (xp, prec);
mpfr_sqr (xp, x, MPFR_RNDN);
mpfr_ui_sub (xp, 1, xp, MPFR_RNDN);
mpfr_sqrt (xp, xp, MPFR_RNDN);
mpfr_div (xp, x, xp, MPFR_RNDN);
mpfr_atan (xp, xp, MPFR_RNDN);
if (MPFR_LIKELY (MPFR_CAN_ROUND (xp, prec - xp_exp,
MPFR_PREC (asin), rnd_mode)))
break;
MPFR_ZIV_NEXT (loop, prec);
}
MPFR_ZIV_FREE (loop);
inexact = mpfr_set (asin, xp, rnd_mode);
mpfr_clear (xp);
}
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (asin, inexact, rnd_mode);
}
int main()
{
slong iter;
flint_rand_t state;
flint_printf("atan....");
fflush(stdout);
flint_randinit(state);
/* Compare with MPFR */
for (iter = 0; iter < 10000 * 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);
arb_randtest(a, state, 1 + n_randint(state, 200), 3);
arb_randtest(b, state, 1 + n_randint(state, 200), 3);
arb_get_rand_fmpq(q, state, a, 1 + n_randint(state, 200));
fmpq_get_mpfr(t, q, MPFR_RNDN);
mpfr_atan(t, t, MPFR_RNDN);
arb_atan(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");
flint_abort();
}
arb_atan(a, a, prec);
if (!arb_equal(a, b))
{
flint_printf("FAIL: aliasing\n\n");
flint_abort();
}
arb_clear(a);
arb_clear(b);
fmpq_clear(q);
mpfr_clear(t);
}
/* Check large arguments. */
for (iter = 0; iter < 10000 * arb_test_multiplier(); iter++)
{
arb_t a, b, c, d;
slong prec1, prec2;
prec1 = 2 + n_randint(state, 1000);
prec2 = prec1 + 30;
arb_init(a);
arb_init(b);
arb_init(c);
arb_init(d);
arb_randtest_precise(a, state, 1 + n_randint(state, 1000), 100);
arb_atan(b, a, prec1);
arb_atan(c, a, prec2);
if (!arb_overlaps(b, c))
{
flint_printf("FAIL: overlap\n\n");
flint_printf("a = "); arb_print(a); flint_printf("\n\n");
flint_printf("b = "); arb_print(b); flint_printf("\n\n");
flint_printf("c = "); arb_print(c); flint_printf("\n\n");
flint_abort();
}
/* check tan(atan(x)) = x */
arb_sin_cos(c, d, b, prec1);
arb_div(c, c, d, prec1);
if (!arb_contains(c, a))
{
flint_printf("FAIL: functional equation\n\n");
flint_printf("a = "); arb_print(a); flint_printf("\n\n");
flint_printf("b = "); arb_print(b); flint_printf("\n\n");
flint_printf("c = "); arb_print(c); flint_printf("\n\n");
flint_printf("d = "); arb_print(d); flint_printf("\n\n");
flint_abort();
}
arb_clear(a);
arb_clear(b);
arb_clear(c);
arb_clear(d);
}
/* Compare with MPFR, higher precision. */
for (iter = 0; iter < 200 * arb_test_multiplier(); iter++)
{
arb_t a, b;
fmpq_t q;
mpfr_t t;
slong prec = 2 + n_randint(state, 5000);
arb_init(a);
arb_init(b);
fmpq_init(q);
mpfr_init2(t, prec + 100);
arb_randtest(a, state, 1 + n_randint(state, 5000), 8);
arb_randtest(b, state, 1 + n_randint(state, 5000), 8);
arb_get_rand_fmpq(q, state, a, 1 + n_randint(state, 200));
fmpq_get_mpfr(t, q, MPFR_RNDN);
mpfr_atan(t, t, MPFR_RNDN);
arb_atan(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("a = "); arb_printd(a, 50); flint_printf("\n\n");
flint_printf("a = "); arb_print(a); flint_printf("\n\n");
flint_printf("b = "); arb_printd(b, 50); flint_printf("\n\n");
flint_abort();
}
arb_atan(a, a, prec);
if (!arb_equal(a, b))
{
flint_printf("FAIL: aliasing\n\n");
flint_abort();
}
arb_clear(a);
arb_clear(b);
fmpq_clear(q);
mpfr_clear(t);
}
/* Higher precision + large arguments. */
for (iter = 0; iter < 2000 * arb_test_multiplier(); iter++)
{
arb_t a, b, c, d;
slong prec1, prec2;
prec1 = 2 + n_randint(state, 5000);
prec2 = prec1 + 30;
arb_init(a);
arb_init(b);
arb_init(c);
arb_init(d);
arb_randtest_precise(a, state, 1 + n_randint(state, 5000), 100);
arb_atan(b, a, prec1);
arb_atan(c, a, prec2);
if (!arb_overlaps(b, c))
{
flint_printf("FAIL: overlap\n\n");
flint_printf("a = "); arb_print(a); flint_printf("\n\n");
flint_printf("b = "); arb_print(b); flint_printf("\n\n");
flint_printf("c = "); arb_print(c); flint_printf("\n\n");
flint_abort();
}
/* check tan(atan(x)) = x */
arb_sin_cos(c, d, b, prec1);
arb_div(c, c, d, prec1);
if (!arb_contains(c, a))
{
flint_printf("FAIL: functional equation\n\n");
flint_printf("a = "); arb_print(a); flint_printf("\n\n");
flint_printf("b = "); arb_print(b); flint_printf("\n\n");
flint_printf("c = "); arb_print(c); flint_printf("\n\n");
flint_printf("d = "); arb_print(d); flint_printf("\n\n");
flint_abort();
}
arb_clear(a);
arb_clear(b);
arb_clear(c);
arb_clear(d);
}
/* Check wide arguments. */
for (iter = 0; iter < 100000 * arb_test_multiplier(); iter++)
{
arb_t a, b, c, d;
arb_init(a);
arb_init(b);
arb_init(c);
arb_init(d);
arb_randtest_precise(a, state, 1 + n_randint(state, 1000), 100);
arb_randtest_precise(b, state, 1 + n_randint(state, 1000), 100);
if (n_randint(state, 2))
arb_add(a, a, b, 2 + n_randint(state, 1000));
arb_union(d, a, b, 2 + n_randint(state, 1000));
arb_atan(a, a, 2 + n_randint(state, 2000));
arb_atan(b, b, 2 + n_randint(state, 2000));
arb_atan(c, d, 2 + n_randint(state, 2000));
if (!arb_overlaps(c, a) || !arb_overlaps(c, b))
{
flint_printf("FAIL: overlap\n\n");
flint_printf("d = "); arb_print(d); flint_printf("\n\n");
flint_printf("a = "); arb_print(a); flint_printf("\n\n");
flint_printf("b = "); arb_print(b); flint_printf("\n\n");
flint_printf("c = "); arb_print(c); flint_printf("\n\n");
flint_abort();
}
arb_clear(a);
arb_clear(b);
arb_clear(c);
arb_clear(d);
}
flint_randclear(state);
flint_cleanup();
flint_printf("PASS\n");
return EXIT_SUCCESS;
}
int
mpfr_acos (mpfr_ptr acos, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
mpfr_t xp, arcc, tmp;
mpfr_exp_t supplement;
mpfr_prec_t prec;
int sign, compared, inexact;
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),
("acos[%Pu]=%.*Rg inexact=%d",
mpfr_get_prec(acos), mpfr_log_prec, acos, inexact));
/* Singular cases */
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
{
if (MPFR_IS_NAN (x) || MPFR_IS_INF (x))
{
MPFR_SET_NAN (acos);
MPFR_RET_NAN;
}
else /* necessarily x=0 */
{
MPFR_ASSERTD(MPFR_IS_ZERO(x));
/* acos(0)=Pi/2 */
MPFR_SAVE_EXPO_MARK (expo);
inexact = mpfr_const_pi (acos, rnd_mode);
mpfr_div_2ui (acos, acos, 1, rnd_mode); /* exact */
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (acos, inexact, rnd_mode);
}
}
/* Set x_p=|x| */
sign = MPFR_SIGN (x);
mpfr_init2 (xp, MPFR_PREC (x));
mpfr_abs (xp, x, MPFR_RNDN); /* Exact */
compared = mpfr_cmp_ui (xp, 1);
if (MPFR_UNLIKELY (compared >= 0))
{
mpfr_clear (xp);
if (compared > 0) /* acos(x) = NaN for x > 1 */
{
MPFR_SET_NAN(acos);
MPFR_RET_NAN;
}
else
{
if (MPFR_IS_POS_SIGN (sign)) /* acos(+1) = +0 */
return mpfr_set_ui (acos, 0, rnd_mode);
else /* acos(-1) = Pi */
return mpfr_const_pi (acos, rnd_mode);
}
}
MPFR_SAVE_EXPO_MARK (expo);
/* Compute the supplement */
mpfr_ui_sub (xp, 1, xp, MPFR_RNDD);
if (MPFR_IS_POS_SIGN (sign))
supplement = 2 - 2 * MPFR_GET_EXP (xp);
else
supplement = 2 - MPFR_GET_EXP (xp);
mpfr_clear (xp);
prec = MPFR_PREC (acos);
prec += MPFR_INT_CEIL_LOG2(prec) + 10 + supplement;
/* VL: The following change concerning prec comes from r3145
"Optimize mpfr_acos by choosing a better initial precision."
but it doesn't seem to be correct and leads to problems (assertion
failure or very important inefficiency) with tiny arguments.
Therefore, I've disabled it. */
/* If x ~ 2^-N, acos(x) ~ PI/2 - x - x^3/6
If Prec < 2*N, we can't round since x^3/6 won't be counted. */
#if 0
if (MPFR_PREC (acos) >= MPFR_PREC (x) && MPFR_GET_EXP (x) < 0)
{
mpfr_uexp_t pmin = (mpfr_uexp_t) (-2 * MPFR_GET_EXP (x)) + 5;
MPFR_ASSERTN (pmin <= MPFR_PREC_MAX);
if (prec < pmin)
prec = pmin;
}
#endif
mpfr_init2 (tmp, prec);
mpfr_init2 (arcc, prec);
MPFR_ZIV_INIT (loop, prec);
for (;;)
{
/* acos(x) = Pi/2 - asin(x) = Pi/2 - atan(x/sqrt(1-x^2)) */
mpfr_sqr (tmp, x, MPFR_RNDN);
mpfr_ui_sub (tmp, 1, tmp, MPFR_RNDN);
mpfr_sqrt (tmp, tmp, MPFR_RNDN);
mpfr_div (tmp, x, tmp, MPFR_RNDN);
mpfr_atan (arcc, tmp, MPFR_RNDN);
mpfr_const_pi (tmp, MPFR_RNDN);
mpfr_div_2ui (tmp, tmp, 1, MPFR_RNDN);
mpfr_sub (arcc, tmp, arcc, MPFR_RNDN);
if (MPFR_LIKELY (MPFR_CAN_ROUND (arcc, prec - supplement,
MPFR_PREC (acos), rnd_mode)))
break;
MPFR_ZIV_NEXT (loop, prec);
mpfr_set_prec (tmp, prec);
mpfr_set_prec (arcc, prec);
}
MPFR_ZIV_FREE (loop);
inexact = mpfr_set (acos, arcc, rnd_mode);
mpfr_clear (tmp);
mpfr_clear (arcc);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (acos, inexact, rnd_mode);
}
void bvisit(const ATan &x) {
apply(result_, *(x.get_arg()));
mpfr_atan(result_, result_, rnd_);
}
int
mpfr_atan2 (mpfr_ptr dest, mpfr_srcptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
mpfr_t tmp, pi;
int inexact;
mpfr_prec_t prec;
mpfr_exp_t e;
MPFR_SAVE_EXPO_DECL (expo);
MPFR_ZIV_DECL (loop);
MPFR_LOG_FUNC
(("y[%Pu]=%.*Rg x[%Pu]=%.*Rg rnd=%d",
mpfr_get_prec (y), mpfr_log_prec, y,
mpfr_get_prec (x), mpfr_log_prec, x, rnd_mode),
("atan[%Pu]=%.*Rg inexact=%d",
mpfr_get_prec (dest), mpfr_log_prec, dest, inexact));
/* Special cases */
if (MPFR_ARE_SINGULAR (x, y))
{
/* atan2(0, 0) does not raise the "invalid" floating-point
exception, nor does atan2(y, 0) raise the "divide-by-zero"
floating-point exception.
-- atan2(±0, -0) returns ±pi.313)
-- atan2(±0, +0) returns ±0.
-- atan2(±0, x) returns ±pi, for x < 0.
-- atan2(±0, x) returns ±0, for x > 0.
-- atan2(y, ±0) returns -pi/2 for y < 0.
-- atan2(y, ±0) returns pi/2 for y > 0.
-- atan2(±oo, -oo) returns ±3pi/4.
-- atan2(±oo, +oo) returns ±pi/4.
-- atan2(±oo, x) returns ±pi/2, for finite x.
-- atan2(±y, -oo) returns ±pi, for finite y > 0.
-- atan2(±y, +oo) returns ±0, for finite y > 0.
*/
if (MPFR_IS_NAN (x) || MPFR_IS_NAN (y))
{
MPFR_SET_NAN (dest);
MPFR_RET_NAN;
}
if (MPFR_IS_ZERO (y))
{
if (MPFR_IS_NEG (x)) /* +/- PI */
{
set_pi:
if (MPFR_IS_NEG (y))
{
inexact = mpfr_const_pi (dest, MPFR_INVERT_RND (rnd_mode));
MPFR_CHANGE_SIGN (dest);
return -inexact;
}
else
return mpfr_const_pi (dest, rnd_mode);
}
else /* +/- 0 */
{
set_zero:
MPFR_SET_ZERO (dest);
MPFR_SET_SAME_SIGN (dest, y);
return 0;
}
}
if (MPFR_IS_ZERO (x))
{
return pi_div_2ui (dest, 1, MPFR_IS_NEG (y), rnd_mode);
}
if (MPFR_IS_INF (y))
{
if (!MPFR_IS_INF (x)) /* +/- PI/2 */
return pi_div_2ui (dest, 1, MPFR_IS_NEG (y), rnd_mode);
else if (MPFR_IS_POS (x)) /* +/- PI/4 */
return pi_div_2ui (dest, 2, MPFR_IS_NEG (y), rnd_mode);
else /* +/- 3*PI/4: Ugly since we have to round properly */
{
mpfr_t tmp2;
MPFR_ZIV_DECL (loop2);
mpfr_prec_t prec2 = MPFR_PREC (dest) + 10;
MPFR_SAVE_EXPO_MARK (expo);
mpfr_init2 (tmp2, prec2);
MPFR_ZIV_INIT (loop2, prec2);
for (;;)
{
mpfr_const_pi (tmp2, MPFR_RNDN);
mpfr_mul_ui (tmp2, tmp2, 3, MPFR_RNDN); /* Error <= 2 */
mpfr_div_2ui (tmp2, tmp2, 2, MPFR_RNDN);
if (mpfr_round_p (MPFR_MANT (tmp2), MPFR_LIMB_SIZE (tmp2),
MPFR_PREC (tmp2) - 2,
MPFR_PREC (dest) + (rnd_mode == MPFR_RNDN)))
break;
MPFR_ZIV_NEXT (loop2, prec2);
mpfr_set_prec (tmp2, prec2);
}
MPFR_ZIV_FREE (loop2);
if (MPFR_IS_NEG (y))
MPFR_CHANGE_SIGN (tmp2);
inexact = mpfr_set (dest, tmp2, rnd_mode);
mpfr_clear (tmp2);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (dest, inexact, rnd_mode);
}
}
MPFR_ASSERTD (MPFR_IS_INF (x));
if (MPFR_IS_NEG (x))
goto set_pi;
else
goto set_zero;
}
/* When x is a power of two, we call directly atan(y/x) since y/x is
exact. */
if (MPFR_UNLIKELY (MPFR_IS_POWER_OF_2 (x)))
{
int r;
mpfr_t yoverx;
unsigned int saved_flags = __gmpfr_flags;
mpfr_init2 (yoverx, MPFR_PREC (y));
if (MPFR_LIKELY (mpfr_div_2si (yoverx, y, MPFR_GET_EXP (x) - 1,
MPFR_RNDN) == 0))
{
/* Here the flags have not changed due to mpfr_div_2si. */
r = mpfr_atan (dest, yoverx, rnd_mode);
mpfr_clear (yoverx);
return r;
}
else
{
/* Division is inexact because of a small exponent range */
mpfr_clear (yoverx);
__gmpfr_flags = saved_flags;
}
}
MPFR_SAVE_EXPO_MARK (expo);
/* Set up initial prec */
prec = MPFR_PREC (dest) + 3 + MPFR_INT_CEIL_LOG2 (MPFR_PREC (dest));
mpfr_init2 (tmp, prec);
MPFR_ZIV_INIT (loop, prec);
if (MPFR_IS_POS (x))
/* use atan2(y,x) = atan(y/x) */
for (;;)
{
int div_inex;
MPFR_BLOCK_DECL (flags);
MPFR_BLOCK (flags, div_inex = mpfr_div (tmp, y, x, MPFR_RNDN));
if (div_inex == 0)
{
/* Result is exact. */
inexact = mpfr_atan (dest, tmp, rnd_mode);
goto end;
}
/* Error <= ulp (tmp) except in case of underflow or overflow. */
/* If the division underflowed, since |atan(z)/z| < 1, we have
an underflow. */
if (MPFR_UNDERFLOW (flags))
{
int sign;
/* In the case MPFR_RNDN with 2^(emin-2) < |y/x| < 2^(emin-1):
The smallest significand value S > 1 of |y/x| is:
* 1 / (1 - 2^(-px)) if py <= px,
* (1 - 2^(-px) + 2^(-py)) / (1 - 2^(-px)) if py >= px.
Therefore S - 1 > 2^(-pz), where pz = max(px,py). We have:
atan(|y/x|) > atan(z), where z = 2^(emin-2) * (1 + 2^(-pz)).
> z - z^3 / 3.
> 2^(emin-2) * (1 + 2^(-pz) - 2^(2 emin - 5))
Assuming pz <= -2 emin + 5, we can round away from zero
(this is what mpfr_underflow always does on MPFR_RNDN).
In the case MPFR_RNDN with |y/x| <= 2^(emin-2), we round
toward zero, as |atan(z)/z| < 1. */
MPFR_ASSERTN (MPFR_PREC_MAX <=
2 * (mpfr_uexp_t) - MPFR_EMIN_MIN + 5);
if (rnd_mode == MPFR_RNDN && MPFR_IS_ZERO (tmp))
rnd_mode = MPFR_RNDZ;
sign = MPFR_SIGN (tmp);
mpfr_clear (tmp);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_underflow (dest, rnd_mode, sign);
}
mpfr_atan (tmp, tmp, MPFR_RNDN); /* Error <= 2*ulp (tmp) since
abs(D(arctan)) <= 1 */
/* TODO: check that the error bound is correct in case of overflow. */
/* FIXME: Error <= ulp(tmp) ? */
if (MPFR_LIKELY (MPFR_CAN_ROUND (tmp, prec - 2, MPFR_PREC (dest),
rnd_mode)))
break;
MPFR_ZIV_NEXT (loop, prec);
mpfr_set_prec (tmp, prec);
}
else /* x < 0 */
/* Use sign(y)*(PI - atan (|y/x|)) */
{
mpfr_init2 (pi, prec);
for (;;)
{
mpfr_div (tmp, y, x, MPFR_RNDN); /* Error <= ulp (tmp) */
/* If tmp is 0, we have |y/x| <= 2^(-emin-2), thus
atan|y/x| < 2^(-emin-2). */
MPFR_SET_POS (tmp); /* no error */
mpfr_atan (tmp, tmp, MPFR_RNDN); /* Error <= 2*ulp (tmp) since
abs(D(arctan)) <= 1 */
mpfr_const_pi (pi, MPFR_RNDN); /* Error <= ulp(pi) /2 */
e = MPFR_NOTZERO(tmp) ? MPFR_GET_EXP (tmp) : __gmpfr_emin - 1;
mpfr_sub (tmp, pi, tmp, MPFR_RNDN); /* see above */
if (MPFR_IS_NEG (y))
MPFR_CHANGE_SIGN (tmp);
/* Error(tmp) <= (1/2+2^(EXP(pi)-EXP(tmp)-1)+2^(e-EXP(tmp)+1))*ulp
<= 2^(MAX (MAX (EXP(PI)-EXP(tmp)-1, e-EXP(tmp)+1),
-1)+2)*ulp(tmp) */
e = MAX (MAX (MPFR_GET_EXP (pi)-MPFR_GET_EXP (tmp) - 1,
e - MPFR_GET_EXP (tmp) + 1), -1) + 2;
if (MPFR_LIKELY (MPFR_CAN_ROUND (tmp, prec - e, MPFR_PREC (dest),
rnd_mode)))
break;
MPFR_ZIV_NEXT (loop, prec);
mpfr_set_prec (tmp, prec);
mpfr_set_prec (pi, prec);
}
mpfr_clear (pi);
}
inexact = mpfr_set (dest, tmp, rnd_mode);
end:
MPFR_ZIV_FREE (loop);
mpfr_clear (tmp);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (dest, inexact, rnd_mode);
}
int
mpfr_asin (mpfr_ptr asin, mpfr_srcptr x, mp_rnd_t rnd_mode)
{
mpfr_t xp;
mpfr_t arcs;
int signe, suplement;
mpfr_t tmp;
int Prec;
int prec_asin;
int good = 0;
int realprec;
int estimated_delta;
int compared;
/* Trivial cases */
if (MPFR_IS_NAN(x) || MPFR_IS_INF(x))
{
MPFR_SET_NAN(asin);
MPFR_RET_NAN;
}
/* Set x_p=|x| */
signe = MPFR_SIGN(x);
mpfr_init2 (xp, MPFR_PREC(x));
mpfr_set (xp, x, rnd_mode);
if (signe == -1)
MPFR_CHANGE_SIGN(xp);
compared = mpfr_cmp_ui (xp, 1);
if (compared > 0) /* asin(x) = NaN for |x| > 1 */
{
MPFR_SET_NAN(asin);
mpfr_clear (xp);
MPFR_RET_NAN;
}
if (compared == 0) /* x = 1 or x = -1 */
{
if (signe > 0) /* asin(+1) = Pi/2 */
mpfr_const_pi (asin, rnd_mode);
else /* asin(-1) = -Pi/2 */
{
if (rnd_mode == GMP_RNDU)
rnd_mode = GMP_RNDD;
else if (rnd_mode == GMP_RNDD)
rnd_mode = GMP_RNDU;
mpfr_const_pi (asin, rnd_mode);
mpfr_neg (asin, asin, rnd_mode);
}
MPFR_EXP(asin)--;
mpfr_clear (xp);
return 1; /* inexact */
}
if (MPFR_IS_ZERO(x)) /* x = 0 */
{
mpfr_set_ui (asin, 0, GMP_RNDN);
mpfr_clear(xp);
return 0; /* exact result */
}
prec_asin = MPFR_PREC(asin);
mpfr_ui_sub (xp, 1, xp, GMP_RNDD);
suplement = 2 - MPFR_EXP(xp);
#ifdef DEBUG
printf("suplement=%d\n", suplement);
#endif
realprec = prec_asin + 10;
while (!good)
{
estimated_delta = 1 + suplement;
Prec = realprec+estimated_delta;
/* Initialisation */
mpfr_init2 (tmp, Prec);
mpfr_init2 (arcs, Prec);
#ifdef DEBUG
printf("Prec=%d\n", Prec);
printf(" x=");
mpfr_out_str (stdout, 2, 0, x, GMP_RNDN);
printf ("\n");
#endif
mpfr_mul (tmp, x, x, GMP_RNDN);
#ifdef DEBUG
printf(" x^2=");
mpfr_out_str (stdout, 2, 0, tmp, GMP_RNDN);
printf ("\n");
#endif
mpfr_ui_sub (tmp, 1, tmp, GMP_RNDN);
#ifdef DEBUG
printf(" 1-x^2=");
mpfr_out_str (stdout, 2, 0, tmp, GMP_RNDN);
printf ("\n");
printf("10: 1-x^2=");
mpfr_out_str (stdout, 10, 0, tmp, GMP_RNDN);
printf ("\n");
#endif
mpfr_sqrt (tmp, tmp, GMP_RNDN);
#ifdef DEBUG
printf(" sqrt(1-x^2)=");
mpfr_out_str (stdout, 2, 0, tmp, GMP_RNDN);
printf ("\n");
printf("10: sqrt(1-x^2)=");
mpfr_out_str (stdout, 10, 0, tmp, GMP_RNDN);
printf ("\n");
#endif
mpfr_div (tmp, x, tmp, GMP_RNDN);
#ifdef DEBUG
printf("x/sqrt(1-x^2)=");
mpfr_out_str (stdout, 2, 0, tmp, GMP_RNDN);
printf ("\n");
#endif
mpfr_atan (arcs, tmp, GMP_RNDN);
#ifdef DEBUG
printf("atan(x/..x^2)=");
mpfr_out_str (stdout, 2, 0, arcs, GMP_RNDN);
printf ("\n");
#endif
if (mpfr_can_round (arcs, realprec, GMP_RNDN, rnd_mode, MPFR_PREC(asin)))
{
mpfr_set (asin, arcs, rnd_mode);
#ifdef DEBUG
printf("asin =");
mpfr_out_str (stdout, 2, prec_asin, asin, GMP_RNDN);
printf ("\n");
#endif
good = 1;
}
else
{
realprec += _mpfr_ceil_log2 ((double) realprec);
#ifdef DEBUG
printf("RETRY\n");
#endif
}
mpfr_clear (tmp);
mpfr_clear (arcs);
}
mpfr_clear (xp);
return 1; /* inexact result */
}
int
mpc_atan (mpc_ptr rop, mpc_srcptr op, mpc_rnd_t rnd)
{
int s_re;
int s_im;
int inex_re;
int inex_im;
int inex;
inex_re = 0;
inex_im = 0;
s_re = mpfr_signbit (mpc_realref (op));
s_im = mpfr_signbit (mpc_imagref (op));
/* special values */
if (mpfr_nan_p (mpc_realref (op)) || mpfr_nan_p (mpc_imagref (op)))
{
if (mpfr_nan_p (mpc_realref (op)))
{
mpfr_set_nan (mpc_realref (rop));
if (mpfr_zero_p (mpc_imagref (op)) || mpfr_inf_p (mpc_imagref (op)))
{
mpfr_set_ui (mpc_imagref (rop), 0, GMP_RNDN);
if (s_im)
mpc_conj (rop, rop, MPC_RNDNN);
}
else
mpfr_set_nan (mpc_imagref (rop));
}
else
{
if (mpfr_inf_p (mpc_realref (op)))
{
inex_re = set_pi_over_2 (mpc_realref (rop), -s_re, MPC_RND_RE (rnd));
mpfr_set_ui (mpc_imagref (rop), 0, GMP_RNDN);
}
else
{
mpfr_set_nan (mpc_realref (rop));
mpfr_set_nan (mpc_imagref (rop));
}
}
return MPC_INEX (inex_re, 0);
}
if (mpfr_inf_p (mpc_realref (op)) || mpfr_inf_p (mpc_imagref (op)))
{
inex_re = set_pi_over_2 (mpc_realref (rop), -s_re, MPC_RND_RE (rnd));
mpfr_set_ui (mpc_imagref (rop), 0, GMP_RNDN);
if (s_im)
mpc_conj (rop, rop, GMP_RNDN);
return MPC_INEX (inex_re, 0);
}
/* pure real argument */
if (mpfr_zero_p (mpc_imagref (op)))
{
inex_re = mpfr_atan (mpc_realref (rop), mpc_realref (op), MPC_RND_RE (rnd));
mpfr_set_ui (mpc_imagref (rop), 0, GMP_RNDN);
if (s_im)
mpc_conj (rop, rop, GMP_RNDN);
return MPC_INEX (inex_re, 0);
}
/* pure imaginary argument */
if (mpfr_zero_p (mpc_realref (op)))
{
int cmp_1;
if (s_im)
cmp_1 = -mpfr_cmp_si (mpc_imagref (op), -1);
else
cmp_1 = mpfr_cmp_ui (mpc_imagref (op), +1);
if (cmp_1 < 0)
{
/* atan(+0+iy) = +0 +i*atanh(y), if |y| < 1
atan(-0+iy) = -0 +i*atanh(y), if |y| < 1 */
mpfr_set_ui (mpc_realref (rop), 0, GMP_RNDN);
if (s_re)
mpfr_neg (mpc_realref (rop), mpc_realref (rop), GMP_RNDN);
inex_im = mpfr_atanh (mpc_imagref (rop), mpc_imagref (op), MPC_RND_IM (rnd));
}
else if (cmp_1 == 0)
{
/* atan(+/-0+i) = NaN +i*inf
atan(+/-0-i) = NaN -i*inf */
mpfr_set_nan (mpc_realref (rop));
mpfr_set_inf (mpc_imagref (rop), s_im ? -1 : +1);
}
else
{
/* atan(+0+iy) = +pi/2 +i*atanh(1/y), if |y| > 1
atan(-0+iy) = -pi/2 +i*atanh(1/y), if |y| > 1 */
mpfr_rnd_t rnd_im, rnd_away;
mpfr_t y;
mpfr_prec_t p, p_im;
int ok;
rnd_im = MPC_RND_IM (rnd);
mpfr_init (y);
p_im = mpfr_get_prec (mpc_imagref (rop));
p = p_im;
/* a = o(1/y) with error(a) < 1 ulp(a)
b = o(atanh(a)) with error(b) < (1+2^{1+Exp(a)-Exp(b)}) ulp(b)
As |atanh (1/y)| > |1/y| we have Exp(a)-Exp(b) <=0 so, at most,
2 bits of precision are lost.
We round atanh(1/y) away from 0.
*/
do
{
p += mpc_ceil_log2 (p) + 2;
mpfr_set_prec (y, p);
rnd_away = s_im == 0 ? GMP_RNDU : GMP_RNDD;
inex_im = mpfr_ui_div (y, 1, mpc_imagref (op), rnd_away);
/* FIXME: should we consider the case with unreasonably huge
precision prec(y)>3*exp_min, where atanh(1/Im(op)) could be
representable while 1/Im(op) underflows ?
This corresponds to |y| = 0.5*2^emin, in which case the
result may be wrong. */
/* atanh cannot underflow: |atanh(x)| > |x| for |x| < 1 */
inex_im |= mpfr_atanh (y, y, rnd_away);
ok = inex_im == 0
|| mpfr_can_round (y, p - 2, rnd_away, GMP_RNDZ,
p_im + (rnd_im == GMP_RNDN));
} while (ok == 0);
inex_re = set_pi_over_2 (mpc_realref (rop), -s_re, MPC_RND_RE (rnd));
inex_im = mpfr_set (mpc_imagref (rop), y, rnd_im);
mpfr_clear (y);
}
return MPC_INEX (inex_re, inex_im);
}
/* regular number argument */
{
mpfr_t a, b, x, y;
mpfr_prec_t prec, p;
mpfr_exp_t err, expo;
int ok = 0;
mpfr_t minus_op_re;
mpfr_exp_t op_re_exp, op_im_exp;
mpfr_rnd_t rnd1, rnd2;
mpfr_inits2 (MPFR_PREC_MIN, a, b, x, y, (mpfr_ptr) 0);
/* real part: Re(arctan(x+i*y)) = [arctan2(x,1-y) - arctan2(-x,1+y)]/2 */
minus_op_re[0] = mpc_realref (op)[0];
MPFR_CHANGE_SIGN (minus_op_re);
op_re_exp = mpfr_get_exp (mpc_realref (op));
op_im_exp = mpfr_get_exp (mpc_imagref (op));
prec = mpfr_get_prec (mpc_realref (rop)); /* result precision */
/* a = o(1-y) error(a) < 1 ulp(a)
b = o(atan2(x,a)) error(b) < [1+2^{3+Exp(x)-Exp(a)-Exp(b)}] ulp(b)
= kb ulp(b)
c = o(1+y) error(c) < 1 ulp(c)
d = o(atan2(-x,c)) error(d) < [1+2^{3+Exp(x)-Exp(c)-Exp(d)}] ulp(d)
= kd ulp(d)
e = o(b - d) error(e) < [1 + kb*2^{Exp(b}-Exp(e)}
+ kd*2^{Exp(d)-Exp(e)}] ulp(e)
error(e) < [1 + 2^{4+Exp(x)-Exp(a)-Exp(e)}
+ 2^{4+Exp(x)-Exp(c)-Exp(e)}] ulp(e)
because |atan(u)| < |u|
< [1 + 2^{5+Exp(x)-min(Exp(a),Exp(c))
-Exp(e)}] ulp(e)
f = e/2 exact
*/
/* p: working precision */
p = (op_im_exp > 0 || prec > SAFE_ABS (mpfr_prec_t, op_im_exp)) ? prec
: (prec - op_im_exp);
rnd1 = mpfr_sgn (mpc_realref (op)) > 0 ? GMP_RNDD : GMP_RNDU;
rnd2 = mpfr_sgn (mpc_realref (op)) < 0 ? GMP_RNDU : GMP_RNDD;
do
{
p += mpc_ceil_log2 (p) + 2;
mpfr_set_prec (a, p);
mpfr_set_prec (b, p);
mpfr_set_prec (x, p);
/* x = upper bound for atan (x/(1-y)). Since atan is increasing, we
need an upper bound on x/(1-y), i.e., a lower bound on 1-y for
x positive, and an upper bound on 1-y for x negative */
mpfr_ui_sub (a, 1, mpc_imagref (op), rnd1);
if (mpfr_sgn (a) == 0) /* y is near 1, thus 1+y is near 2, and
expo will be 1 or 2 below */
{
MPC_ASSERT (mpfr_cmp_ui (mpc_imagref(op), 1) == 0);
/* check for intermediate underflow */
err = 2; /* ensures err will be expo below */
}
else
err = mpfr_get_exp (a); /* err = Exp(a) with the notations above */
mpfr_atan2 (x, mpc_realref (op), a, GMP_RNDU);
/* b = lower bound for atan (-x/(1+y)): for x negative, we need a
lower bound on -x/(1+y), i.e., an upper bound on 1+y */
mpfr_add_ui (a, mpc_imagref(op), 1, rnd2);
/* if a is exactly zero, i.e., Im(op) = -1, then the error on a is 0,
and we can simply ignore the terms involving Exp(a) in the error */
if (mpfr_sgn (a) == 0)
{
MPC_ASSERT (mpfr_cmp_si (mpc_imagref(op), -1) == 0);
/* check for intermediate underflow */
expo = err; /* will leave err unchanged below */
}
else
expo = mpfr_get_exp (a); /* expo = Exp(c) with the notations above */
mpfr_atan2 (b, minus_op_re, a, GMP_RNDD);
err = err < expo ? err : expo; /* err = min(Exp(a),Exp(c)) */
mpfr_sub (x, x, b, GMP_RNDU);
err = 5 + op_re_exp - err - mpfr_get_exp (x);
/* error is bounded by [1 + 2^err] ulp(e) */
err = err < 0 ? 1 : err + 1;
mpfr_div_2ui (x, x, 1, GMP_RNDU);
/* Note: using RND2=RNDD guarantees that if x is exactly representable
on prec + ... bits, mpfr_can_round will return 0 */
ok = mpfr_can_round (x, p - err, GMP_RNDU, GMP_RNDD,
prec + (MPC_RND_RE (rnd) == GMP_RNDN));
} while (ok == 0);
/* Imaginary part
Im(atan(x+I*y)) = 1/4 * [log(x^2+(1+y)^2) - log (x^2 +(1-y)^2)] */
prec = mpfr_get_prec (mpc_imagref (rop)); /* result precision */
/* a = o(1+y) error(a) < 1 ulp(a)
b = o(a^2) error(b) < 5 ulp(b)
c = o(x^2) error(c) < 1 ulp(c)
d = o(b+c) error(d) < 7 ulp(d)
e = o(log(d)) error(e) < [1 + 7*2^{2-Exp(e)}] ulp(e) = ke ulp(e)
f = o(1-y) error(f) < 1 ulp(f)
g = o(f^2) error(g) < 5 ulp(g)
h = o(c+f) error(h) < 7 ulp(h)
i = o(log(h)) error(i) < [1 + 7*2^{2-Exp(i)}] ulp(i) = ki ulp(i)
j = o(e-i) error(j) < [1 + ke*2^{Exp(e)-Exp(j)}
+ ki*2^{Exp(i)-Exp(j)}] ulp(j)
error(j) < [1 + 2^{Exp(e)-Exp(j)} + 2^{Exp(i)-Exp(j)}
+ 7*2^{3-Exp(j)}] ulp(j)
< [1 + 2^{max(Exp(e),Exp(i))-Exp(j)+1}
+ 7*2^{3-Exp(j)}] ulp(j)
k = j/4 exact
*/
err = 2;
p = prec; /* working precision */
do
{
p += mpc_ceil_log2 (p) + err;
mpfr_set_prec (a, p);
mpfr_set_prec (b, p);
mpfr_set_prec (y, p);
/* a = upper bound for log(x^2 + (1+y)^2) */
ROUND_AWAY (mpfr_add_ui (a, mpc_imagref (op), 1, MPFR_RNDA), a);
mpfr_sqr (a, a, GMP_RNDU);
mpfr_sqr (y, mpc_realref (op), GMP_RNDU);
mpfr_add (a, a, y, GMP_RNDU);
mpfr_log (a, a, GMP_RNDU);
/* b = lower bound for log(x^2 + (1-y)^2) */
mpfr_ui_sub (b, 1, mpc_imagref (op), GMP_RNDZ); /* round to zero */
mpfr_sqr (b, b, GMP_RNDZ);
/* we could write mpfr_sqr (y, mpc_realref (op), GMP_RNDZ) but it is
more efficient to reuse the value of y (x^2) above and subtract
one ulp */
mpfr_nextbelow (y);
mpfr_add (b, b, y, GMP_RNDZ);
mpfr_log (b, b, GMP_RNDZ);
mpfr_sub (y, a, b, GMP_RNDU);
if (mpfr_zero_p (y))
/* FIXME: happens when x and y have very different magnitudes;
could be handled more efficiently */
ok = 0;
else
{
expo = MPC_MAX (mpfr_get_exp (a), mpfr_get_exp (b));
expo = expo - mpfr_get_exp (y) + 1;
err = 3 - mpfr_get_exp (y);
/* error(j) <= [1 + 2^expo + 7*2^err] ulp(j) */
if (expo <= err) /* error(j) <= [1 + 2^{err+1}] ulp(j) */
err = (err < 0) ? 1 : err + 2;
else
err = (expo < 0) ? 1 : expo + 2;
mpfr_div_2ui (y, y, 2, GMP_RNDN);
MPC_ASSERT (!mpfr_zero_p (y));
/* FIXME: underflow. Since the main term of the Taylor series
in y=0 is 1/(x^2+1) * y, this means that y is very small
and/or x very large; but then the mpfr_zero_p (y) above
should be true. This needs a proof, or better yet,
special code. */
ok = mpfr_can_round (y, p - err, GMP_RNDU, GMP_RNDD,
prec + (MPC_RND_IM (rnd) == GMP_RNDN));
}
} while (ok == 0);
inex = mpc_set_fr_fr (rop, x, y, rnd);
mpfr_clears (a, b, x, y, (mpfr_ptr) 0);
return inex;
}
}
void bvisit(const ACot &x) {
apply(result_, *(x.get_arg()));
mpfr_ui_div(result_, 1, result_, rnd_);
mpfr_atan(result_, result_, rnd_);
}
FixAtan2ByCORDIC::FixAtan2ByCORDIC(Target* target_, int wIn_, int wOut_, map<string, double> inputDelays_) :
FixAtan2(target_, wIn_, wOut_, inputDelays_)
{
int stage;
srcFileName="FixAtan2ByCORDIC";
setCopyrightString ( "Matei Istoan, Florent de Dinechin (2012-...)" );
useNumericStd_Unsigned();
ostringstream name;
name << "FixAtan2ByCORDIC_" << wIn_ << "_" << wOut_ << "_uid" << getNewUId();
setNameWithFreq( name.str() );
mpfr_t zatan;
mpfr_init2(zatan, 10*wOut);
computeGuardBits();
//Defining the various parameters according to method
///////////// VHDL GENERATION
/////////////////////////////////////////////////////////////////////////////
//
// First range reduction
//
/////////////////////////////////////////////////////////////////////////////
buildQuadrantRangeReduction();
// No scaling RR: just a copy
vhdl << tab << declare("XRS", wIn-1) << " <= XR;" << endl;
vhdl << tab << declare("YRS", wIn-1) << " <= YR;" << endl;
int sizeZ=wOut-2+gA; // w-2 because two bits come from arg red
////////////////////////////////////////////////////////////////////////////
//
// CORDIC iterations
//
////////////////////////////////////////////////////////////////////////////
// Fixed-point considerations:
// Y -> 0 and X -> K.sqrt(x1^2+y1^2)
// Max value attained by X is sqrt(2)*K which is smaller than 2
int zMSB=-1; // -1 because these two bits have weight 0 and -1, but we must keep the sign
int zLSB = zMSB-sizeZ+1;
int sizeX = wIn+gXY;
int sizeY = sizeX;
vhdl << tab << declare("X1", sizeX) << " <= '0' & XRS & " << zg(sizeX-(wIn-1)-1) << ";" <<endl;
vhdl << tab << declare("Y1", sizeY) << " <= '0' & YRS & " << zg(sizeY-(wIn-1)-1) << ";" <<endl;
stage=1;
manageCriticalPath( getTarget()->adderDelay(sizeX));
vhdl << tab << "--- Iteration " << stage << " : sign is known positive ---" << endl;
vhdl << tab << declare(join("YShift", stage), sizeX) << " <= " << rangeAssign(sizeX-1, sizeX-stage, "'0'") << " & Y" << stage << range(sizeX-1, stage) << ";" << endl;
vhdl << tab << declare(join("X", stage+1), sizeX) << " <= "
<< join("X", stage) << " + " << join("YShift", stage) << " ;" << endl;
vhdl << tab << declare(join("XShift", stage), sizeY) << " <= " << zg(stage) << " & X" << stage << range(sizeY-1, stage) << ";" <<endl;
vhdl << tab << declare(join("Y", stage+1), sizeY) << " <= "
<< join("Y", stage) << " - " << join("XShift", stage) << " ;" << endl;
//create the constant signal for the arctan
mpfr_set_d(zatan, 1.0, GMP_RNDN);
mpfr_div_2si(zatan, zatan, stage, GMP_RNDN);
mpfr_atan(zatan, zatan, GMP_RNDN);
mpfr_div(zatan, zatan, constPi, GMP_RNDN);
mpfr_t roundbit;
mpfr_init2(roundbit, 30); // should be enough for anybody
mpfr_set_d(roundbit, 1.0, GMP_RNDN);
mpfr_div_2si(roundbit, roundbit, wOut, GMP_RNDN); // roundbit is in position 2^-wOut
REPORT(DEBUG, "stage=" << stage << " atancst=" << printMPFR(zatan));
mpfr_add(zatan, zatan, roundbit, GMP_RNDN);
vhdl << tab << declare("Z2", sizeZ) << " <= " << unsignedFixPointNumber(zatan, zMSB, zLSB) << "; -- initial atan, plus round bit" <<endl;
for(stage=2; stage<=maxIterations; stage++, sizeY--){
// Invariant: sizeX-sizeY = stage-2
vhdl << tab << "--- Iteration " << stage << " ---" << endl;
manageCriticalPath( getTarget()->localWireDelay(sizeX+1) + getTarget()->adderDelay(max(sizeX,sizeZ)) );
vhdl << tab << declare(join("sgnY", stage)) << " <= " << join("Y", stage) << of(sizeY-1) << ";" << endl;
if(-2*stage+1 >= -wIn+1-gXY) {
vhdl << tab << declare(join("YShift", stage), sizeX)
<< " <= " << rangeAssign(sizeX-1, sizeX -(sizeX-sizeY+stage), join("sgnY", stage))
<< " & Y" << stage << range(sizeY-1, stage) << ";" << endl;
vhdl << tab << declare(join("X", stage+1), sizeX) << " <= "
<< join("X", stage) << " - " << join("YShift", stage) << " when " << join("sgnY", stage) << "=\'1\' else "
<< join("X", stage) << " + " << join("YShift", stage) << " ;" << endl;
}
else { // autant pisser dans un violon
vhdl << tab << declare(join("X", stage+1), sizeX) << " <= " << join("X", stage) << " ;" << endl;
}
vhdl << tab << declare(join("XShift", stage), sizeY) << " <= " << zg(2) << " & X" << stage << range(sizeX-1, sizeX - sizeY + 2) << ";" <<endl;
vhdl << tab << declare(join("YY", stage+1), sizeY) << " <= "
<< join("Y", stage) << " + " << join("XShift", stage) << " when " << join("sgnY", stage) << "=\'1\' else "
<< join("Y", stage) << " - " << join("XShift", stage) << " ;" << endl;
vhdl << tab << declare(join("Y", stage+1), sizeY-1) << " <= " << join("YY", stage+1) << range(sizeY-2, 0) << ";" <<endl;
//create the constant signal for the arctan
mpfr_set_d(zatan, 1.0, GMP_RNDN);
mpfr_div_2si(zatan, zatan, stage, GMP_RNDN);
mpfr_atan(zatan, zatan, GMP_RNDN);
mpfr_div(zatan, zatan, constPi, GMP_RNDN);
REPORT(DEBUG, "stage=" << stage << " atancst=" << printMPFR(zatan));
// rounding here in unsignedFixPointNumber()
vhdl << tab << declare(join("atan2PowStage", stage), sizeZ) << " <= " << unsignedFixPointNumber(zatan, zMSB, zLSB) << ";" <<endl;
vhdl << tab << declare(join("Z", stage+1), sizeZ) << " <= "
<< join("Z", stage) << " + " << join("atan2PowStage", stage) << " when " << join("sgnY", stage) << "=\'0\' else "
<< join("Z", stage) << " - " << join("atan2PowStage", stage) << " ;" << endl;
} //end for loop
// Give the time to finish the last rotation
// manageCriticalPath( getTarget()->localWireDelay(w+1) + getTarget()->adderDelay(w+1) // actual CP delay
// - (getTarget()->localWireDelay(sizeZ+1) + getTarget()->adderDelay(sizeZ+1))); // CP delay that was already added
manageCriticalPath( getTarget()->localWireDelay(wOut+1) + getTarget()->adderDelay(2) );
vhdl << tab << declare("finalZ", wOut) << " <= Z" << stage << of(sizeZ-1) << " & Z" << stage << range(sizeZ-1, sizeZ-wOut+1) << "; -- sign-extended and rounded" << endl;
buildQuadrantReconstruction();
};
CGAL::Gmpfr atan(const CGAL::Gmpfr &x)
{
CGAL::Gmpfr result(0, gmp_result_precision(x));
mpfr_atan(result.fr(), x.fr(), gmp_rounding_mode(CGAL::Gmpfr::get_default_rndmode()));
return result;
}