MpfrFloat MpfrFloat::atan2(const MpfrFloat& value1, const MpfrFloat& value2)
{
MpfrFloat retval(MpfrFloat::kNoInitialization);
mpfr_atan2(retval.mData->mFloat,
value1.mData->mFloat, value2.mData->mFloat, GMP_RNDN);
return retval;
}
real atan2(const real & b, const real & a)
{
real x;
mpfr_atan2(x.r, b.r, a.r, MPFR_RNDN);
return x;
}
/* Bug found by Robert Bajema (regression in MPFR 2.3.0).
The cause is the underflow flag set before the mpfr_atan2 call. */
static void
atan2_bug_20071003 (void)
{
mpfr_t a, x, y, z;
mpfr_inits (a, x, y, z, (mpfr_ptr) 0);
mpfr_set_underflow ();
mpfr_set_str_binary (y,
"-0.10100110110100110111010110111111100110100010001110110E2");
mpfr_set_str_binary (x,
"0.10100101010110010100010010111000110110011110001011110E3");
mpfr_set_str_binary (z,
"-0.11101111001101101100111011001101000010010111101110110E-1");
mpfr_atan2 (a, y, x, MPFR_RNDN);
if (! mpfr_equal_p (a, z))
{
printf ("mpfr_atan2 fails on:\n");
printf (" y = ");
mpfr_dump (y);
printf (" x = ");
mpfr_dump (x);
printf ("Expected ");
mpfr_dump (z);
printf ("Got ");
mpfr_dump (a);
exit (1);
}
mpfr_clears (a, x, y, z, (mpfr_ptr) 0);
}
static void
atan2_pow_of_2 (void)
{
mpfr_t x, y, r, g;
int i;
int d[] = { 0, -1, 1 };
int ntests = sizeof (d) / sizeof (int);
mpfr_init2 (x, 53);
mpfr_init2 (y, 53);
mpfr_init2 (r, 53);
mpfr_init2 (g, 53);
/* atan(42) */
mpfr_set_str_binary (g, "1100011000000011110011111001100110101000011010010011E-51");
for (i = 0; i < ntests; ++i)
{
mpfr_set_ui (y, 42, MPFR_RNDN);
mpfr_mul_2si (y, y, d[i], MPFR_RNDN);
mpfr_set_ui_2exp (x, 1, d[i], MPFR_RNDN);
mpfr_atan2 (r, y, x, MPFR_RNDN);
if (mpfr_equal_p (r, g) == 0)
{
printf ("Error in mpfr_atan2 (5)\n");
printf ("Expected "); mpfr_print_binary (g); printf ("\n");
printf ("Got "); mpfr_print_binary (r); printf ("\n");
exit (1);
}
}
mpfr_clear (x);
mpfr_clear (y);
mpfr_clear (r);
mpfr_clear (g);
}
/* from Christopher Creutzig, 18 Jul 2007 */
static void
smallvals_atan2 (void)
{
mpfr_t a, x, y;
mpfr_exp_t old_emin;
mpfr_inits (a, x, y, (mpfr_ptr) 0);
mpfr_set_ui (y, 0, MPFR_RNDN);
mpfr_nextbelow (y);
mpfr_set_ui (x, 1, MPFR_RNDN);
/* y=-2^(-emin-1), x=1 */
mpfr_atan2 (a, y, x, MPFR_RNDD);
MPFR_ASSERTN (mpfr_equal_p (a, y));
mpfr_atan2 (a, y, x, MPFR_RNDU);
MPFR_ASSERTN (mpfr_zero_p (a) && MPFR_IS_NEG(a));
mpfr_set_prec (x, 8);
mpfr_set_prec (y, 8);
mpfr_set_prec (a, 8);
old_emin = mpfr_get_emin ();
mpfr_set_emin (MPFR_EMIN_MIN);
mpfr_set_si (y, 3, MPFR_RNDN);
mpfr_set_exp (y, mpfr_get_emin ());
mpfr_set_str_binary (x, "1.1");
mpfr_atan2 (a, y, x, MPFR_RNDU);
mpfr_set_si (y, 1, MPFR_RNDN);
mpfr_set_exp (y, mpfr_get_emin ());
MPFR_ASSERTN (mpfr_equal_p (a, y));
/* From a bug reported by Christopher Creutzig on 2007-08-28.
Added test in each rounding mode.
Segmentation fault or assertion failure due to an infinite Ziv loop. */
mpfr_set_si (y, 1, MPFR_RNDN);
mpfr_set_exp (y, mpfr_get_emin ());
mpfr_set_str_binary (x, "1.01");
mpfr_atan2 (a, y, x, MPFR_RNDZ);
MPFR_ASSERTN (mpfr_zero_p (a));
mpfr_atan2 (a, y, x, MPFR_RNDD);
MPFR_ASSERTN (mpfr_zero_p (a));
mpfr_atan2 (a, y, x, MPFR_RNDU);
MPFR_ASSERTN (mpfr_equal_p (a, y));
mpfr_atan2 (a, y, x, MPFR_RNDN);
MPFR_ASSERTN (mpfr_equal_p (a, y));
/* trigger underflow with rounding to nearest */
mpfr_set_ui (x, 4, MPFR_RNDN);
mpfr_atan2 (a, y, x, MPFR_RNDN);
MPFR_ASSERTN (mpfr_zero_p (a));
mpfr_set_emin (old_emin);
mpfr_clears (a, x, y, (mpfr_ptr) 0);
}
decimal r_atan2(const decimal& a,const decimal& b,bool round)
{
#ifdef USE_CGAL
CGAL::Gmpfr m;
CGAL::Gmpfr n=to_gmpfr(a);
CGAL::Gmpfr o=to_gmpfr(b);
mpfr_atan2(m.fr(),n.fr(),o.fr(),MPFR_RNDN);
return r_round_preference(decimal(m),round);
#else
return r_round_preference(atan2(a,b),round);
#endif
}
/* Bug found on 2009-04-29 by Christopher Creutzig.
* With r6179: atan.c:62: MPFR assertion failed: r > n
*/
static void
atan2_different_prec (void)
{
mpfr_t a, x, y;
mpfr_init2 (a, 59);
mpfr_init2 (x, 59);
mpfr_init2 (y, 86);
mpfr_set_ui (x, 1, MPFR_RNDN);
mpfr_set_ui (y, 1, MPFR_RNDN);
mpfr_nextbelow (y);
mpfr_atan2 (a, y, x, MPFR_RNDN);
mpfr_clears (a, x, y, (mpfr_ptr) 0);
}
static void
special_atan2 (void)
{
mpfr_t x, y, z;
mpfr_inits2 (4, x, y, z, (mpfr_ptr) 0);
/* Anything with NAN should be set to NAN */
mpfr_set_ui (y, 0, MPFR_RNDN);
mpfr_set_nan (x);
mpfr_atan2 (z, y, x, MPFR_RNDN);
MPFR_ASSERTN (MPFR_IS_NAN (z));
mpfr_swap (x, y);
mpfr_atan2 (z, y, x, MPFR_RNDN);
MPFR_ASSERTN (MPFR_IS_NAN (z));
/* 0+ 0+ --> 0+ */
mpfr_set_ui (y, 0, MPFR_RNDN);
mpfr_atan2 (z, y, x, MPFR_RNDN);
MPFR_ASSERTN (MPFR_IS_ZERO (z) && MPFR_IS_POS (z));
/* 0- 0+ --> 0- */
MPFR_CHANGE_SIGN (y);
mpfr_atan2 (z, y, x, MPFR_RNDN);
MPFR_ASSERTN (MPFR_IS_ZERO (z) && MPFR_IS_NEG (z));
/* 0- 0- --> -PI */
MPFR_CHANGE_SIGN (x);
mpfr_atan2 (z, y, x, MPFR_RNDN);
MPFR_ASSERTN (mpfr_cmp_str (z, "-3.1415", 10, MPFR_RNDN) == 0);
/* 0+ 0- --> +PI */
MPFR_CHANGE_SIGN (y);
mpfr_atan2 (z, y, x, MPFR_RNDN);
MPFR_ASSERTN (mpfr_cmp_str (z, "3.1415", 10, MPFR_RNDN) == 0);
/* 0+ -1 --> PI */
mpfr_set_si (x, -1, MPFR_RNDN);
mpfr_atan2 (z, y, x, MPFR_RNDN);
MPFR_ASSERTN (mpfr_cmp_str (z, "3.1415", 10, MPFR_RNDN) == 0);
/* 0- -1 --> -PI */
MPFR_CHANGE_SIGN (y);
mpfr_atan2 (z, y, x, MPFR_RNDN);
MPFR_ASSERTN (mpfr_cmp_str (z, "-3.1415", 10, MPFR_RNDN) == 0);
/* 0- +1 --> 0- */
mpfr_set_ui (x, 1, MPFR_RNDN);
mpfr_atan2 (z, y, x, MPFR_RNDN);
MPFR_ASSERTN (MPFR_IS_ZERO (z) && MPFR_IS_NEG (z));
/* 0+ +1 --> 0+ */
MPFR_CHANGE_SIGN (y);
mpfr_atan2 (z, y, x, MPFR_RNDN);
MPFR_ASSERTN (MPFR_IS_ZERO (z) && MPFR_IS_POS (z));
/* +1 0+ --> PI/2 */
mpfr_swap (x, y);
mpfr_atan2 (z, y, x, MPFR_RNDN);
MPFR_ASSERTN (mpfr_cmp_str (z, "1.57075", 10, MPFR_RNDN) == 0);
/* +1 0- --> PI/2 */
MPFR_CHANGE_SIGN (x);
mpfr_atan2 (z, y, x, MPFR_RNDN);
MPFR_ASSERTN (mpfr_cmp_str (z, "1.57075", 10, MPFR_RNDN) == 0);
/* -1 0- --> -PI/2 */
MPFR_CHANGE_SIGN (y);
mpfr_atan2 (z, y, x, MPFR_RNDN);
MPFR_ASSERTN (mpfr_cmp_str (z, "-1.57075", 10, MPFR_RNDN) == 0);
/* -1 0+ --> -PI/2 */
MPFR_CHANGE_SIGN (x);
mpfr_atan2 (z, y, x, MPFR_RNDN);
MPFR_ASSERTN (mpfr_cmp_str (z, "-1.57075", 10, MPFR_RNDN) == 0);
/* -1 +INF --> -0 */
MPFR_SET_INF (x);
mpfr_atan2 (z, y, x, MPFR_RNDN);
MPFR_ASSERTN (MPFR_IS_ZERO (z) && MPFR_IS_NEG (z));
/* +1 +INF --> +0 */
MPFR_CHANGE_SIGN (y);
mpfr_atan2 (z, y, x, MPFR_RNDN);
MPFR_ASSERTN (MPFR_IS_ZERO (z) && MPFR_IS_POS (z));
/* +1 -INF --> +PI */
MPFR_CHANGE_SIGN (x);
mpfr_atan2 (z, y, x, MPFR_RNDN);
MPFR_ASSERTN (mpfr_cmp_str (z, "3.1415", 10, MPFR_RNDN) == 0);
/* -1 -INF --> -PI */
MPFR_CHANGE_SIGN (y);
mpfr_atan2 (z, y, x, MPFR_RNDN);
MPFR_ASSERTN (mpfr_cmp_str (z, "-3.1415", 10, MPFR_RNDN) == 0);
/* -INF -1 --> -PI/2 */
mpfr_swap (x, y);
mpfr_atan2 (z, y, x, MPFR_RNDN);
MPFR_ASSERTN (mpfr_cmp_str (z, "-1.57075", 10, MPFR_RNDN) == 0);
/* +INF -1 --> PI/2 */
MPFR_CHANGE_SIGN (y);
mpfr_atan2 (z, y, x, MPFR_RNDN);
MPFR_ASSERTN (mpfr_cmp_str (z, "1.57075", 10, MPFR_RNDN) == 0);
/* +INF -INF --> 3*PI/4 */
MPFR_SET_INF (x);
mpfr_atan2 (z, y, x, MPFR_RNDN);
MPFR_ASSERTN (mpfr_cmp_str (z, "2.356194490192344928", 10, MPFR_RNDN) == 0);
/* +INF +INF --> PI/4 */
MPFR_CHANGE_SIGN (x);
mpfr_atan2 (z, y, x, MPFR_RNDN);
MPFR_ASSERTN (mpfr_cmp_str (z, "0.785375", 10, MPFR_RNDN) == 0);
/* -INF +INF --> -PI/4 */
MPFR_CHANGE_SIGN (y);
mpfr_atan2 (z, y, x, MPFR_RNDN);
MPFR_ASSERTN (mpfr_cmp_str (z, "-0.785375", 10, MPFR_RNDN) == 0);
/* -INF -INF --> -3*PI/4 */
MPFR_CHANGE_SIGN (x);
mpfr_atan2 (z, y, x, MPFR_RNDN);
MPFR_ASSERTN (mpfr_cmp_str (z, "-2.356194490192344928", 10, MPFR_RNDN) == 0);
mpfr_set_prec (z, 905); /* exercises Ziv's loop */
mpfr_atan2 (z, y, x, MPFR_RNDZ);
MPFR_ASSERTN (mpfr_cmp_str (z, "-2.35619449019234492884698253745962716314787704953132936573120844423086230471465674897102611900658780098661106488496172998532038345716293667379401955609636083808771307702645389082916973346721171619778647332160823174945008459635673617534008737395340143185923642519259526145784", 10, MPFR_RNDN) == 0);
mpfr_clears (x, y, z, (mpfr_ptr) 0);
}
int main()
{
long iter;
flint_rand_t state;
printf("atan2....");
fflush(stdout);
flint_randinit(state);
for (iter = 0; iter < 100000; iter++)
{
arb_t a, b, c;
fmpq_t q, r;
mpfr_t t, u;
long prec = 2 + n_randint(state, 200);
arb_init(a);
arb_init(b);
arb_init(c);
fmpq_init(q);
fmpq_init(r);
mpfr_init2(t, prec + 100);
mpfr_init2(u, prec + 100);
arb_randtest(a, state, 1 + n_randint(state, 200), 3);
arb_randtest(b, state, 1 + n_randint(state, 200), 3);
arb_randtest(c, state, 1 + n_randint(state, 200), 3);
arb_get_rand_fmpq(q, state, a, 1 + n_randint(state, 200));
arb_get_rand_fmpq(r, state, b, 1 + n_randint(state, 200));
fmpq_get_mpfr(t, q, MPFR_RNDN);
fmpq_get_mpfr(u, r, MPFR_RNDN);
mpfr_atan2(t, u, t, MPFR_RNDN);
arb_atan2(c, b, a, prec);
if (!arb_contains_mpfr(c, t))
{
printf("FAIL: containment\n\n");
printf("a = "); arb_print(a); printf("\n\n");
printf("b = "); arb_print(b); printf("\n\n");
printf("c = "); arb_print(c); printf("\n\n");
abort();
}
arb_clear(a);
arb_clear(b);
arb_clear(c);
fmpq_clear(q);
fmpq_clear(r);
mpfr_clear(t);
mpfr_clear(u);
}
flint_randclear(state);
flint_cleanup();
printf("PASS\n");
return EXIT_SUCCESS;
}
int
mpc_log (mpc_ptr rop, mpc_srcptr op, mpc_rnd_t rnd){
int ok, underflow = 0;
mpfr_srcptr x, y;
mpfr_t v, w;
mpfr_prec_t prec;
int loops;
int re_cmp, im_cmp;
int inex_re, inex_im;
int err;
mpfr_exp_t expw;
int sgnw;
/* special values: NaN and infinities */
if (!mpc_fin_p (op)) {
if (mpfr_nan_p (mpc_realref (op))) {
if (mpfr_inf_p (mpc_imagref (op)))
mpfr_set_inf (mpc_realref (rop), +1);
else
mpfr_set_nan (mpc_realref (rop));
mpfr_set_nan (mpc_imagref (rop));
inex_im = 0; /* Inf/NaN is exact */
}
else if (mpfr_nan_p (mpc_imagref (op))) {
if (mpfr_inf_p (mpc_realref (op)))
mpfr_set_inf (mpc_realref (rop), +1);
else
mpfr_set_nan (mpc_realref (rop));
mpfr_set_nan (mpc_imagref (rop));
inex_im = 0; /* Inf/NaN is exact */
}
else /* We have an infinity in at least one part. */ {
inex_im = mpfr_atan2 (mpc_imagref (rop), mpc_imagref (op), mpc_realref (op),
MPC_RND_IM (rnd));
mpfr_set_inf (mpc_realref (rop), +1);
}
return MPC_INEX(0, inex_im);
}
/* special cases: real and purely imaginary numbers */
re_cmp = mpfr_cmp_ui (mpc_realref (op), 0);
im_cmp = mpfr_cmp_ui (mpc_imagref (op), 0);
if (im_cmp == 0) {
if (re_cmp == 0) {
inex_im = mpfr_atan2 (mpc_imagref (rop), mpc_imagref (op), mpc_realref (op),
MPC_RND_IM (rnd));
mpfr_set_inf (mpc_realref (rop), -1);
inex_re = 0; /* -Inf is exact */
}
else if (re_cmp > 0) {
inex_re = mpfr_log (mpc_realref (rop), mpc_realref (op), MPC_RND_RE (rnd));
inex_im = mpfr_set (mpc_imagref (rop), mpc_imagref (op), MPC_RND_IM (rnd));
}
else {
/* op = x + 0*y; let w = -x = |x| */
int negative_zero;
mpfr_rnd_t rnd_im;
negative_zero = mpfr_signbit (mpc_imagref (op));
if (negative_zero)
rnd_im = INV_RND (MPC_RND_IM (rnd));
else
rnd_im = MPC_RND_IM (rnd);
w [0] = *mpc_realref (op);
MPFR_CHANGE_SIGN (w);
inex_re = mpfr_log (mpc_realref (rop), w, MPC_RND_RE (rnd));
inex_im = mpfr_const_pi (mpc_imagref (rop), rnd_im);
if (negative_zero) {
mpc_conj (rop, rop, MPC_RNDNN);
inex_im = -inex_im;
}
}
return MPC_INEX(inex_re, inex_im);
}
else if (re_cmp == 0) {
if (im_cmp > 0) {
inex_re = mpfr_log (mpc_realref (rop), mpc_imagref (op), MPC_RND_RE (rnd));
inex_im = mpfr_const_pi (mpc_imagref (rop), MPC_RND_IM (rnd));
/* division by 2 does not change the ternary flag */
mpfr_div_2ui (mpc_imagref (rop), mpc_imagref (rop), 1, GMP_RNDN);
}
else {
w [0] = *mpc_imagref (op);
MPFR_CHANGE_SIGN (w);
inex_re = mpfr_log (mpc_realref (rop), w, MPC_RND_RE (rnd));
inex_im = mpfr_const_pi (mpc_imagref (rop), INV_RND (MPC_RND_IM (rnd)));
/* division by 2 does not change the ternary flag */
mpfr_div_2ui (mpc_imagref (rop), mpc_imagref (rop), 1, GMP_RNDN);
mpfr_neg (mpc_imagref (rop), mpc_imagref (rop), GMP_RNDN);
inex_im = -inex_im; /* negate the ternary flag */
}
return MPC_INEX(inex_re, inex_im);
}
prec = MPC_PREC_RE(rop);
mpfr_init2 (w, 2);
/* let op = x + iy; log = 1/2 log (x^2 + y^2) + i atan2 (y, x) */
/* loop for the real part: 1/2 log (x^2 + y^2), fast, but unsafe */
/* implementation */
ok = 0;
for (loops = 1; !ok && loops <= 2; loops++) {
prec += mpc_ceil_log2 (prec) + 4;
mpfr_set_prec (w, prec);
mpc_abs (w, op, GMP_RNDN);
/* error 0.5 ulp */
if (mpfr_inf_p (w))
/* intermediate overflow; the logarithm may be representable.
Intermediate underflow is impossible. */
break;
mpfr_log (w, w, GMP_RNDN);
/* generic error of log: (2^(- exp(w)) + 0.5) ulp */
if (mpfr_zero_p (w))
/* impossible to round, switch to second algorithm */
break;
err = MPC_MAX (-mpfr_get_exp (w), 0) + 1;
/* number of lost digits */
ok = mpfr_can_round (w, prec - err, GMP_RNDN, GMP_RNDZ,
mpfr_get_prec (mpc_realref (rop)) + (MPC_RND_RE (rnd) == GMP_RNDN));
}
if (!ok) {
prec = MPC_PREC_RE(rop);
mpfr_init2 (v, 2);
/* compute 1/2 log (x^2 + y^2) = log |x| + 1/2 * log (1 + (y/x)^2)
if |x| >= |y|; otherwise, exchange x and y */
if (mpfr_cmpabs (mpc_realref (op), mpc_imagref (op)) >= 0) {
x = mpc_realref (op);
y = mpc_imagref (op);
}
else {
x = mpc_imagref (op);
y = mpc_realref (op);
}
do {
prec += mpc_ceil_log2 (prec) + 4;
mpfr_set_prec (v, prec);
mpfr_set_prec (w, prec);
mpfr_div (v, y, x, GMP_RNDD); /* error 1 ulp */
mpfr_sqr (v, v, GMP_RNDD);
/* generic error of multiplication:
1 + 2*1*(2+1*2^(1-prec)) <= 5.0625 since prec >= 6 */
mpfr_log1p (v, v, GMP_RNDD);
/* error 1 + 4*5.0625 = 21.25 , see algorithms.tex */
mpfr_div_2ui (v, v, 1, GMP_RNDD);
/* If the result is 0, then there has been an underflow somewhere. */
mpfr_abs (w, x, GMP_RNDN); /* exact */
mpfr_log (w, w, GMP_RNDN); /* error 0.5 ulp */
expw = mpfr_get_exp (w);
sgnw = mpfr_signbit (w);
mpfr_add (w, w, v, GMP_RNDN);
if (!sgnw) /* v is positive, so no cancellation;
error 22.25 ulp; error counts lost bits */
err = 5;
else
err = MPC_MAX (5 + mpfr_get_exp (v),
/* 21.25 ulp (v) rewritten in ulp (result, now in w) */
-1 + expw - mpfr_get_exp (w)
/* 0.5 ulp (previous w), rewritten in ulp (result) */
) + 2;
/* handle one special case: |x|=1, and (y/x)^2 underflows;
then 1/2*log(x^2+y^2) \approx 1/2*y^2 also underflows. */
if ( (mpfr_cmp_si (x, -1) == 0 || mpfr_cmp_ui (x, 1) == 0)
&& mpfr_zero_p (w))
underflow = 1;
} while (!underflow &&
!mpfr_can_round (w, prec - err, GMP_RNDN, GMP_RNDZ,
mpfr_get_prec (mpc_realref (rop)) + (MPC_RND_RE (rnd) == GMP_RNDN)));
mpfr_clear (v);
}
/* imaginary part */
inex_im = mpfr_atan2 (mpc_imagref (rop), mpc_imagref (op), mpc_realref (op),
MPC_RND_IM (rnd));
/* set the real part; cannot be done before if rop==op */
if (underflow)
/* create underflow in result */
inex_re = mpfr_set_ui_2exp (mpc_realref (rop), 1,
mpfr_get_emin_min () - 2, MPC_RND_RE (rnd));
else
inex_re = mpfr_set (mpc_realref (rop), w, MPC_RND_RE (rnd));
mpfr_clear (w);
return MPC_INEX(inex_re, inex_im);
}
void
ovm_q_atan2(oregister_t *l, oregister_t *r)
{
switch (r->t) {
case t_void:
if (!cfg_float_format) {
l->t = t_float;
l->v.d = atan2(mpq_get_d(oqr(l)), 0.0);
}
else {
mpfr_set_ui(orr(r), 0, thr_rnd);
goto mpr;
}
break;
case t_word:
if (!cfg_float_format) {
l->t = t_float;
l->v.d = atan2(mpq_get_d(oqr(l)), r->v.w);
}
else {
mpfr_set_si(orr(r), r->v.w, thr_rnd);
goto mpr;
}
break;
case t_float:
l->t = t_float;
l->v.d = atan2(mpq_get_d(oqr(l)), r->v.d);
break;
case t_mpz:
if (!cfg_float_format) {
l->t = t_float;
l->v.d = atan2(mpq_get_d(oqr(l)), mpz_get_d(ozr(r)));
}
else {
mpfr_set_z(orr(r), ozr(r), thr_rnd);
goto mpr;
}
break;
case t_rat:
if (!cfg_float_format) {
l->t = t_float;
l->v.d = atan2(mpq_get_d(oqr(l)), rat_get_d(r->v.r));
}
else {
mpq_set_si(oqr(r), rat_num(r->v.r), rat_den(r->v.r));
mpfr_set_q(orr(r), oqr(r), thr_rnd);
goto mpr;
}
break;
case t_mpq:
if (!cfg_float_format) {
l->t = t_float;
l->v.d = atan2(mpq_get_d(oqr(l)), mpq_get_d(oqr(r)));
}
else {
mpfr_set_q(orr(r), oqr(r), thr_rnd);
goto mpr;
}
break;
case t_mpr:
mpr:
mpfr_set_q(orr(l), oqr(l), thr_rnd);
l->t = t_mpr;
mpfr_atan2(orr(l), orr(l), orr(r), thr_rnd);
break;
case t_cdd:
cdd:
l->t = t_cdd;
real(l->v.dd) = mpq_get_d(oqr(l));
imag(l->v.dd) = 0.0;
l->v.dd = catan(l->v.dd / r->v.dd);
check_cdd(l);
break;
case t_cqq:
if (!cfg_float_format) {
real(r->v.dd) = mpq_get_d(oqr(r));
imag(r->v.dd) = mpq_get_d(oqi(r));
goto cdd;
}
mpc_set_q_q(occ(r), oqr(r), oqi(r), thr_rndc);
case t_mpc:
l->t = t_mpc;
mpc_set_q(occ(l), oqr(l), thr_rndc);
mpc_div(occ(l), occ(l), occ(r), thr_rndc);
mpc_atan(occ(l), occ(l), thr_rndc);
check_mpc(l);
break;
default:
ovm_raise(except_not_a_number);
}
}
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 ATan2 &x) {
mpfr_class t(mpfr_get_prec(result_));
apply(t.get_mpfr_t(), *(x.get_num()));
apply(result_, *(x.get_den()));
mpfr_atan2(result_, t.get_mpfr_t(), result_, rnd_);
}
void mexFunction( int nlhs, mxArray *plhs[],
int nrhs, const mxArray *prhs[] )
{
double *prec,*eoutr,*eouti;
int mrows,ncols;
char *input_buf;
char *w1,*w2;
int buflen,status;
mpfr_t xr,xi,yr,yi,zr,zi,temp,temp1,temp2,temp3,temp4;
mp_exp_t expptr;
/* Check for proper number of arguments. */
if(nrhs!=5) {
mexErrMsgTxt("5 inputs required.");
} else if(nlhs>4) {
mexErrMsgTxt("Too many output arguments");
}
/* The input must be a noncomplex scalar double.*/
mrows = mxGetM(prhs[0]);
ncols = mxGetN(prhs[0]);
if( !mxIsDouble(prhs[0]) || mxIsComplex(prhs[0]) ||
!(mrows==1 && ncols==1) ) {
mexErrMsgTxt("Input must be a noncomplex scalar double.");
}
/* Set precision and initialize mpfr variables */
prec = mxGetPr(prhs[0]);
mpfr_set_default_prec(*prec);
mpfr_init(xr); mpfr_init(xi);
mpfr_init(yr); mpfr_init(yi);
mpfr_init(zr); mpfr_init(zi);
mpfr_init(temp); mpfr_init(temp1);
mpfr_init(temp2); mpfr_init(temp3);
mpfr_init(temp4);
/* Read the input strings into mpfr x real */
buflen = (mxGetM(prhs[1]) * mxGetN(prhs[1])) + 1;
input_buf=mxCalloc(buflen, sizeof(char));
status = mxGetString(prhs[1], input_buf, buflen);
mpfr_set_str(xr,input_buf,10,GMP_RNDN);
/* Read the input strings into mpfr x imag */
buflen = (mxGetM(prhs[2]) * mxGetN(prhs[2])) + 1;
input_buf=mxCalloc(buflen, sizeof(char));
status = mxGetString(prhs[2], input_buf, buflen);
mpfr_set_str(xi,input_buf,10,GMP_RNDN);
/* Read the input strings into mpfr y real */
buflen = (mxGetM(prhs[3]) * mxGetN(prhs[3])) + 1;
input_buf=mxCalloc(buflen, sizeof(char));
status = mxGetString(prhs[3], input_buf, buflen);
mpfr_set_str(yr,input_buf,10,GMP_RNDN);
/* Read the input strings into mpfr y imag */
buflen = (mxGetM(prhs[4]) * mxGetN(prhs[4])) + 1;
input_buf=mxCalloc(buflen, sizeof(char));
status = mxGetString(prhs[4], input_buf, buflen);
mpfr_set_str(yi,input_buf,10,GMP_RNDN);
/* Mathematical operation */
/* ln(magnitude) */
mpfr_mul(temp,xr,xr,GMP_RNDN);
mpfr_mul(temp1,xi,xi,GMP_RNDN);
mpfr_add(temp,temp,temp1,GMP_RNDN);
mpfr_sqrt(temp,temp,GMP_RNDN);
mpfr_log(temp,temp,GMP_RNDN);
/* angle */
mpfr_atan2(temp1,xi,xr,GMP_RNDN);
/* real exp */
mpfr_mul(temp3,temp,yr,GMP_RNDN);
mpfr_mul(temp2,temp1,yi,GMP_RNDN);
mpfr_sub(temp3,temp3,temp2,GMP_RNDN);
mpfr_exp(temp3,temp3,GMP_RNDN);
/* cos sin argument */
mpfr_mul(temp2,temp1,yr,GMP_RNDN);
mpfr_mul(temp4,temp,yi,GMP_RNDN);
mpfr_add(temp2,temp2,temp4,GMP_RNDN);
mpfr_cos(zr,temp2,GMP_RNDN);
mpfr_mul(zr,zr,temp3,GMP_RNDN);
mpfr_sin(zi,temp2,GMP_RNDN);
mpfr_mul(zi,zi,temp3,GMP_RNDN);
/* Retrieve results */
mxFree(input_buf);
input_buf=mpfr_get_str (NULL, &expptr, 10, 0, zr, GMP_RNDN);
w1=malloc(strlen(input_buf)+20);
w2=malloc(strlen(input_buf)+20);
if (strncmp(input_buf, "-", 1)==0){
strcpy(w2,&input_buf[1]);
sprintf(w1,"-.%se%i",w2,expptr);
} else {
strcpy(w2,&input_buf[0]);
sprintf(w1,"+.%se%i",w2,expptr);
}
plhs[0] = mxCreateString(w1);
/* plhs[1] = mxCreateDoubleMatrix(mrows,ncols, mxREAL); */
/* eoutr=mxGetPr(plhs[1]); */
/* *eoutr=expptr; */
mpfr_free_str(input_buf);
input_buf=mpfr_get_str (NULL, &expptr, 10, 0, zi, GMP_RNDN);
free(w1);
free(w2);
w1=malloc(strlen(input_buf)+20);
w2=malloc(strlen(input_buf)+20);
if (strncmp(input_buf, "-", 1)==0){
strcpy(w2,&input_buf[1]);
sprintf(w1,"-.%se%i",w2,expptr);
} else {
strcpy(w2,&input_buf[0]);
sprintf(w1,"+.%se%i",w2,expptr);
}
plhs[1] = mxCreateString(w1);
/* plhs[3] = mxCreateDoubleMatrix(mrows,ncols, mxREAL); */
/* eouti=mxGetPr(plhs[3]); */
/* *eouti=expptr; */
mpfr_clear(xr); mpfr_clear(xi);
mpfr_clear(yr); mpfr_clear(yi);
mpfr_clear(zr); mpfr_clear(zi);
mpfr_clear(temp); mpfr_clear(temp1);
mpfr_clear(temp2); mpfr_clear(temp3);
mpfr_clear(temp4);
mpfr_free_str(input_buf);
free(w1);
free(w2);
}
CGAL::Gmpfr atan2(const CGAL::Gmpfr &y, const CGAL::Gmpfr &x)
{
CGAL::Gmpfr result(0, std::max(gmp_result_precision(y), gmp_result_precision(x)));
mpfr_atan2(result.fr(), y.fr(), x.fr(), gmp_rounding_mode(CGAL::Gmpfr::get_default_rndmode()));
return result;
}
