//------------------------------------------------------------------------------
// Name:
//------------------------------------------------------------------------------
knumber_base *knumber_float::atanh() {
#ifdef KNUMBER_USE_MPFR
mpfr_t mpfr;
mpfr_init_set_f(mpfr, mpf_, rounding_mode);
mpfr_atanh(mpfr, mpfr, rounding_mode);
mpfr_get_f(mpf_, mpfr, rounding_mode);
mpfr_clear(mpfr);
return this;
#else
const double x = mpf_get_d(mpf_);
return execute_libc_func< ::atanh>(x);
#endif
}
static PyObject *
_GMPy_MPFR_Atanh(PyObject *x, CTXT_Object *context)
{
MPFR_Object *result;
CHECK_CONTEXT(context);
if (!mpfr_nan_p(MPFR(x)) &&
(mpfr_cmp_si(MPFR(x), 1) > 0 || mpfr_cmp_si(MPFR(x), -1) < 0) &&
context->ctx.allow_complex
) {
return GMPy_Complex_Atanh(x, context);
}
if (!(result = GMPy_MPFR_New(0, context))) {
return NULL;
}
mpfr_clear_flags();
result->rc = mpfr_atanh(result->f, MPFR(x), GET_MPFR_ROUND(context));
_GMPy_MPFR_Cleanup(&result, context);
return (PyObject*)result;
}
static int
check_NAN (void)
{
mpfr_t t, ch,sh,th,ach,ash,ath;
int tester;
int fail = 0;
mpfr_init2(t,200);
mpfr_init2(ch,200);
mpfr_init2(sh,200);
mpfr_init2(th,200);
mpfr_init2(ach,200);
mpfr_init2(ash,200);
mpfr_init2(ath,200);
MPFR_SET_NAN(t);
/******cosh********/
tester=mpfr_cosh(ch,t,MPFR_RNDD);
if (!MPFR_IS_NAN(ch) || tester!=0)
{
printf("cosh NAN \n");
fail = 1;
goto clean_up;
}
/******sinh********/
tester=mpfr_sinh(sh,t,MPFR_RNDD);
if (!MPFR_IS_NAN(sh) || tester!=0)
{
printf("sinh NAN \n");
fail = 1;
goto clean_up;
}
/******tanh********/
tester=mpfr_tanh(th,t,MPFR_RNDD);
if (!MPFR_IS_NAN(th) || tester!=0)
{
printf("tanh NAN \n");
fail = 1;
goto clean_up;
}
/******acosh********/
tester=mpfr_acosh(ach,t,MPFR_RNDD);
if (!MPFR_IS_NAN(ach) || tester!=0)
{
printf("acosh NAN \n");
fail = 1;
goto clean_up;
}
/******asinh********/
tester=mpfr_asinh(ash,t,MPFR_RNDD);
if (!MPFR_IS_NAN(ash) || tester!=0)
{
printf("asinh NAN \n");
fail = 1;
goto clean_up;
}
/******atanh********/
tester=mpfr_atanh(ath,t,MPFR_RNDD);
if (!MPFR_IS_NAN(ath) || tester!=0)
{
printf("atanh NAN \n");
fail = 1;
goto clean_up;
}
clean_up:
mpfr_clear(t);
mpfr_clear(ch);
mpfr_clear(sh);
mpfr_clear(th);
mpfr_clear(ach);
mpfr_clear(ash);
mpfr_clear(ath);
return fail;
}
static int
check_INF (void)
{
mpfr_t t, ch, sh, th, ach, ash, ath;
int tester;
int fail = 0;
mpfr_init2 (t, 200);
mpfr_init2 (ch, 200);
mpfr_init2 (sh, 200);
mpfr_init2 (th, 200);
mpfr_init2 (ach, 200);
mpfr_init2 (ash, 200);
mpfr_init2 (ath, 200);
MPFR_SET_INF(t);
if(MPFR_IS_NEG (t))
MPFR_CHANGE_SIGN(t);
/******cosh********/
tester = mpfr_cosh(ch,t,MPFR_RNDD);
if (!MPFR_IS_INF(ch) || MPFR_IS_NEG (ch) || tester!=0)
{
printf("cosh(INF) \n");
fail = 1;
goto clean_up;
}
/******sinh********/
tester=mpfr_sinh(sh,t,MPFR_RNDD);
if (!MPFR_IS_INF(sh) || MPFR_IS_NEG (sh) || tester!=0)
{
printf("sinh(INF) \n");
fail = 1;
goto clean_up;
}
/******tanh********/
tester=mpfr_tanh(th,t,MPFR_RNDD);
if (mpfr_cmp_ui(th,1) != 0 || tester!=0)
{
printf("tanh(INF) \n");
fail = 1;
goto clean_up;
}
/******acosh********/
tester=mpfr_acosh(ach,t,MPFR_RNDD);
if (!MPFR_IS_INF(ach) || MPFR_IS_NEG (ach) || tester!=0)
{
printf("acosh(INF) \n");
fail = 1;
goto clean_up;
}
/******asinh********/
tester=mpfr_asinh(ash,t,MPFR_RNDD);
if (!MPFR_IS_INF(ash) || MPFR_IS_NEG (ash) || tester!=0)
{
printf("asinh(INF) \n");
fail = 1;
goto clean_up;
}
/******atanh********/
tester = mpfr_atanh (ath, t, MPFR_RNDD);
if (!MPFR_IS_NAN(ath) || tester != 0)
{
printf("atanh(INF) \n");
fail = 1;
goto clean_up;
}
MPFR_CHANGE_SIGN(t);
/******cosh********/
tester=mpfr_cosh(ch,t,MPFR_RNDD);
if (!MPFR_IS_INF(ch) || MPFR_IS_NEG (ch) || tester!=0)
{
printf("cosh(-INF) \n");
fail = 1;
goto clean_up;
}
/******sinh********/
tester=mpfr_sinh(sh,t,MPFR_RNDD);
if (!MPFR_IS_INF(sh) || MPFR_IS_POS (sh) || tester!=0)
{
printf("sinh(-INF) \n");
fail = 1;
goto clean_up;
}
/******tanh********/
tester=mpfr_tanh(th,t,MPFR_RNDD);
if (!mpfr_cmp_ui(th,-1) || tester!=0)
{
printf("tanh(-INF) \n");
fail = 1;
goto clean_up;
}
/******acosh********/
tester=mpfr_acosh(ach,t,MPFR_RNDD);
if (!MPFR_IS_NAN(ach) || tester!=0)
{
printf("acosh(-INF) \n");
fail = 1;
goto clean_up;
}
/******asinh********/
tester=mpfr_asinh(ash,t,MPFR_RNDD);
if (!MPFR_IS_INF(ash) || MPFR_IS_POS (ash) || tester!=0)
{
printf("asinh(-INF) \n");
fail = 1;
goto clean_up;
}
/******atanh********/
tester = mpfr_atanh (ath, t, MPFR_RNDD);
if (!MPFR_IS_NAN(ath) || tester != 0)
{
printf("atanh(-INF) \n");
fail = 1;
goto clean_up;
}
clean_up:
mpfr_clear(t);
mpfr_clear(ch);
mpfr_clear(sh);
mpfr_clear(th);
mpfr_clear(ach);
mpfr_clear(ash);
mpfr_clear(ath);
return fail;
}
static int
check_zero (void)
{
mpfr_t t, ch,sh,th,ach,ash,ath;
int tester;
int fail = 0;
mpfr_init2(t,200);
mpfr_init2(ch,200);
mpfr_init2(sh,200);
mpfr_init2(th,200);
mpfr_init2(ach,200);
mpfr_init2(ash,200);
mpfr_init2(ath,200);
mpfr_set_ui(t,0,MPFR_RNDD);
/******cosh********/
tester = mpfr_cosh (ch, t, MPFR_RNDD);
if (mpfr_cmp_ui(ch, 1) || tester)
{
printf("cosh(0) \n");
fail = 1;
goto clean_up;
}
/******sinh********/
tester = mpfr_sinh (sh, t, MPFR_RNDD);
if (!MPFR_IS_ZERO(sh) || tester)
{
printf("sinh(0) \n");
fail = 1;
goto clean_up;
}
/******tanh********/
tester = mpfr_tanh (th, t, MPFR_RNDD);
if (!MPFR_IS_ZERO(th) || tester)
{
printf("tanh(0) \n");
fail = 1;
goto clean_up;
}
/******acosh********/
tester=mpfr_acosh(ach,t,MPFR_RNDD);
if (!MPFR_IS_NAN(ach) || tester)
{
printf("acosh(0) \n");
fail = 1;
goto clean_up;
}
/******asinh********/
tester=mpfr_asinh(ash,t,MPFR_RNDD);
if (!MPFR_IS_ZERO(ash) || tester)
{
printf("asinh(0) \n");
fail = 1;
goto clean_up;
}
/******atanh********/
tester=mpfr_atanh(ath,t,MPFR_RNDD);
if (!MPFR_IS_ZERO(ath) || tester)
{
printf("atanh(0) \n");
fail = 1;
goto clean_up;
}
clean_up:
mpfr_clear(t);
mpfr_clear(ch);
mpfr_clear(sh);
mpfr_clear(th);
mpfr_clear(ach);
mpfr_clear(ash);
mpfr_clear(ath);
return fail;
}
static void
special (void)
{
mpfr_t x, y, z;
int i;
mpfr_init (x);
mpfr_init (y);
MPFR_SET_INF(x);
mpfr_set_ui (y, 0, GMP_RNDN);
mpfr_atanh (x, y, GMP_RNDN);
if (MPFR_IS_INF(x) || MPFR_IS_NAN(x) )
{
printf ("Inf flag not clears in atanh!\n");
exit (1);
}
MPFR_SET_NAN(x);
mpfr_atanh (x, y, GMP_RNDN);
if (MPFR_IS_NAN(x) || MPFR_IS_INF(x))
{
printf ("NAN flag not clears in atanh!\n");
exit (1);
}
/* atanh(+/-x) = NaN if x > 1 */
for (i = 3; i <= 6; i++)
{
mpfr_set_si (x, i, GMP_RNDN);
mpfr_div_2ui (x, x, 1, GMP_RNDN);
mpfr_atanh (y, x, GMP_RNDN);
if (!mpfr_nan_p (y))
{
printf ("Error: mpfr_atanh(%d/2) <> NaN\n", i);
exit (1);
}
mpfr_neg (x, x, GMP_RNDN);
mpfr_atanh (y, x, GMP_RNDN);
if (!mpfr_nan_p (y))
{
printf ("Error: mpfr_atanh(-%d/2) <> NaN\n", i);
exit (1);
}
}
/* atanh(+0) = +0, atanh(-0) = -0 */
mpfr_set_ui (x, 0, GMP_RNDN);
mpfr_atanh (y, x, GMP_RNDN);
if (mpfr_cmp_ui (y, 0) || mpfr_sgn (y) < 0)
{
printf ("Error: mpfr_atanh(+0) <> +0\n");
exit (1);
}
mpfr_neg (x, x, GMP_RNDN);
mpfr_atanh (y, x, GMP_RNDN);
if (mpfr_cmp_ui (y, 0) || mpfr_sgn (y) > 0)
{
printf ("Error: mpfr_atanh(-0) <> -0\n");
exit (1);
}
MPFR_SET_NAN(x);
mpfr_atanh (y, x, GMP_RNDN);
if (!mpfr_nan_p (y))
{
printf ("Error: mpfr_atanh(NaN) <> NaN\n");
exit (1);
}
mpfr_set_inf (x, 1);
mpfr_atanh (y, x, GMP_RNDN);
if (!mpfr_nan_p (y))
{
printf ("Error: mpfr_atanh(+Inf) <> NaN\n");
mpfr_print_binary (y); printf ("\n");
exit (1);
}
mpfr_set_inf (x, -1);
mpfr_atanh (y, x, GMP_RNDN);
if (!mpfr_nan_p (y))
{
printf ("Error: mpfr_atanh(-Inf) <> NaN\n");
exit (1);
}
mpfr_set_ui (x, 1, GMP_RNDN);
mpfr_atanh (y, x, GMP_RNDN);
if (!mpfr_inf_p (y) || mpfr_sgn (y) < 0)
{
printf ("Error: mpfr_atanh(1) <> +Inf\n");
exit (1);
}
mpfr_set_si (x, -1, GMP_RNDN);
mpfr_atanh (y, x, GMP_RNDN);
if (!mpfr_inf_p (y) || mpfr_sgn (y) > 0)
{
printf ("Error: mpfr_atanh(-1) <> -Inf\n");
exit (1);
}
mpfr_set_prec (x, 32);
mpfr_set_prec (y, 32);
mpfr_set_str_binary (x, "0.10001000001001011000100001E-6");
mpfr_atanh (x, x, GMP_RNDN);
mpfr_set_str_binary (y, "0.10001000001001100101010110100001E-6");
if (mpfr_cmp (x, y))
{
printf ("Error: mpfr_atanh (1)\n");
exit (1);
}
mpfr_set_str_binary (x, "-0.1101011110111100111010011001011E-1");
mpfr_atanh (x, x, GMP_RNDN);
mpfr_set_str_binary (y, "-0.11100110000100001111101100010111E-1");
if (mpfr_cmp (x, y))
{
printf ("Error: mpfr_atanh (2)\n");
exit (1);
}
mpfr_set_prec (x, 33);
mpfr_set_prec (y, 43);
mpfr_set_str_binary (x, "0.111001101100000110011001010000101");
mpfr_atanh (y, x, GMP_RNDZ);
mpfr_init2 (z, 43);
mpfr_set_str_binary (z, "1.01111010110001101001000000101101011110101");
if (mpfr_cmp (y, z))
{
printf ("Error: mpfr_atanh (3)\n");
mpfr_print_binary (y); printf ("\n");
exit (1);
}
mpfr_clear (x);
mpfr_clear (y);
mpfr_clear (z);
}
MpfrFloat MpfrFloat::atanh(const MpfrFloat& value)
{
MpfrFloat retval(MpfrFloat::kNoInitialization);
mpfr_atanh(retval.mData->mFloat, value.mData->mFloat, GMP_RNDN);
return retval;
}
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 ACoth &x) {
apply(result_, *(x.get_arg()));
mpfr_ui_div(result_, 1, result_, rnd_);
mpfr_atanh(result_, result_, rnd_);
}
void bvisit(const ATanh &x) {
apply(result_, *(x.get_arg()));
mpfr_atanh(result_, result_, rnd_);
}
