/* mpfr_sin_cos -- sine and cosine of a floating-point number
Copyright 2002-2017 Free Software Foundation, Inc.
Contributed by the AriC and Caramba projects, INRIA.
This file is part of the GNU MPFR Library.
The GNU MPFR Library is free software; you can redistribute it and/or modify
it under the terms of the GNU Lesser General Public License as published by
the Free Software Foundation; either version 3 of the License, or (at your
option) any later version.
The GNU MPFR Library is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
License for more details.
You should have received a copy of the GNU Lesser General Public License
along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see
http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */
#define MPFR_NEED_LONGLONG_H
#include "mpfr-impl.h"
/* (y, z) <- (sin(x), cos(x)), return value is 0 iff both results are exact
ie, iff x = 0 */
int
mpfr_sin_cos (mpfr_ptr y, mpfr_ptr z, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
mpfr_prec_t prec, m;
int neg, reduce;
mpfr_t c, xr;
mpfr_srcptr xx;
mpfr_exp_t err, expx;
int inexy, inexz;
MPFR_ZIV_DECL (loop);
MPFR_SAVE_EXPO_DECL (expo);
MPFR_ASSERTN (y != z);
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
{
if (MPFR_IS_NAN(x) || MPFR_IS_INF(x))
{
MPFR_SET_NAN (y);
MPFR_SET_NAN (z);
MPFR_RET_NAN;
}
else /* x is zero */
{
MPFR_ASSERTD (MPFR_IS_ZERO (x));
MPFR_SET_ZERO (y);
MPFR_SET_SAME_SIGN (y, x);
/* y = 0, thus exact, but z is inexact in case of underflow
or overflow */
inexy = 0; /* y is exact */
inexz = mpfr_set_ui (z, 1, rnd_mode);
return INEX(inexy,inexz);
}
}
MPFR_LOG_FUNC
(("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd_mode),
("sin[%Pu]=%.*Rg cos[%Pu]=%.*Rg", mpfr_get_prec(y), mpfr_log_prec, y,
mpfr_get_prec (z), mpfr_log_prec, z));
MPFR_SAVE_EXPO_MARK (expo);
prec = MAX (MPFR_PREC (y), MPFR_PREC (z));
m = prec + MPFR_INT_CEIL_LOG2 (prec) + 13;
expx = MPFR_GET_EXP (x);
/* When x is close to 0, say 2^(-k), then there is a cancellation of about
2k bits in 1-cos(x)^2. FIXME: in that case, it would be more efficient
to compute sin(x) directly. VL: This is partly done by using
MPFR_FAST_COMPUTE_IF_SMALL_INPUT from the mpfr_sin and mpfr_cos
functions. Moreover, any overflow on m is avoided. */
if (expx < 0)
{
/* Warning: in case y = x, and the first call to
MPFR_FAST_COMPUTE_IF_SMALL_INPUT succeeds but the second fails,
we will have clobbered the original value of x.
The workaround is to first compute z = cos(x) in that case, since
y and z are different. */
if (y != x)
/* y and x differ, thus we can safely try to compute y first */
{
MPFR_FAST_COMPUTE_IF_SMALL_INPUT (
y, x, -2 * expx, 2, 0, rnd_mode,
{ inexy = _inexact;
goto small_input; });
if (0)
{
small_input:
/* we can go here only if we can round sin(x) */
MPFR_FAST_COMPUTE_IF_SMALL_INPUT (
z, __gmpfr_one, -2 * expx, 1, 0, rnd_mode,
{ inexz = _inexact;
MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags);
goto end; });
}
/* if we go here, one of the two MPFR_FAST_COMPUTE_IF_SMALL_INPUT
calls failed */
}
else /* y and x are the same variable: try to compute z first, which
necessarily differs */
{
MPFR_FAST_COMPUTE_IF_SMALL_INPUT (
z, __gmpfr_one, -2 * expx, 1, 0, rnd_mode,
{ inexz = _inexact;
goto small_input2; });
if (0)
{
small_input2:
/* we can go here only if we can round cos(x) */
MPFR_FAST_COMPUTE_IF_SMALL_INPUT (
y, x, -2 * expx, 2, 0, rnd_mode,
{ inexy = _inexact;
MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags);
goto end; });
}
}
m += 2 * (-expx);
}
if (prec >= MPFR_SINCOS_THRESHOLD)
{
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_sincos_fast (y, z, x, rnd_mode);
}
mpfr_init (c);
mpfr_init (xr);
MPFR_ZIV_INIT (loop, m);
for (;;)
{
/* the following is copied from sin.c */
if (expx >= 2) /* reduce the argument */
{
reduce = 1;
mpfr_set_prec (c, expx + m - 1);
mpfr_set_prec (xr, m);
mpfr_const_pi (c, MPFR_RNDN);
mpfr_mul_2ui (c, c, 1, MPFR_RNDN);
mpfr_remainder (xr, x, c, MPFR_RNDN);
mpfr_div_2ui (c, c, 1, MPFR_RNDN);
if (MPFR_SIGN (xr) > 0)
mpfr_sub (c, c, xr, MPFR_RNDZ);
else
mpfr_add (c, c, xr, MPFR_RNDZ);
if (MPFR_IS_ZERO(xr)
|| MPFR_EXP(xr) < (mpfr_exp_t) 3 - (mpfr_exp_t) m
|| MPFR_EXP(c) < (mpfr_exp_t) 3 - (mpfr_exp_t) m)
goto next_step;
xx = xr;
}
else /* the input argument is already reduced */
{
reduce = 0;
xx = x;
}
neg = MPFR_IS_NEG (xx); /* gives sign of sin(x) */
mpfr_set_prec (c, m);
mpfr_cos (c, xx, MPFR_RNDZ);
/* If no argument reduction was performed, the error is at most ulp(c),
otherwise it is at most ulp(c) + 2^(2-m). Since |c| < 1, we have
ulp(c) <= 2^(-m), thus the error is bounded by 2^(3-m) in that later
case. */
if (reduce == 0)
err = m;
else
err = MPFR_GET_EXP (c) + (mpfr_exp_t) (m - 3);
if (!mpfr_can_round (c, err, MPFR_RNDN, MPFR_RNDZ,
MPFR_PREC (z) + (rnd_mode == MPFR_RNDN)))
goto next_step;
/* we can't set z now, because in case z = x, and the mpfr_can_round()
call below fails, we will have clobbered the input */
mpfr_set_prec (xr, MPFR_PREC(c));
mpfr_swap (xr, c); /* save the approximation of the cosine in xr */
mpfr_sqr (c, xr, MPFR_RNDU); /* the absolute error is bounded by
2^(5-m) if reduce=1, and by 2^(2-m)
otherwise */
mpfr_ui_sub (c, 1, c, MPFR_RNDN); /* error bounded by 2^(6-m) if reduce
is 1, and 2^(3-m) otherwise */
mpfr_sqrt (c, c, MPFR_RNDN); /* the absolute error is bounded by
2^(6-m-Exp(c)) if reduce=1, and
2^(3-m-Exp(c)) otherwise */
err = 3 + 3 * reduce - MPFR_GET_EXP (c);
if (neg)
MPFR_CHANGE_SIGN (c);
/* the absolute error on c is at most 2^(err-m), which we must put
in the form 2^(EXP(c)-err). */
err = MPFR_GET_EXP (c) + (mpfr_exp_t) m - err;
if (mpfr_can_round (c, err, MPFR_RNDN, MPFR_RNDZ,
MPFR_PREC (y) + (rnd_mode == MPFR_RNDN)))
break;
/* check for huge cancellation */
if (err < (mpfr_exp_t) MPFR_PREC (y))
m += MPFR_PREC (y) - err;
/* Check if near 1 */
if (MPFR_GET_EXP (c) == 1
&& MPFR_MANT (c)[MPFR_LIMB_SIZE (c)-1] == MPFR_LIMB_HIGHBIT)
m += m;
next_step:
MPFR_ZIV_NEXT (loop, m);
mpfr_set_prec (c, m);
}
MPFR_ZIV_FREE (loop);
inexy = mpfr_set (y, c, rnd_mode);
inexz = mpfr_set (z, xr, rnd_mode);
mpfr_clear (c);
mpfr_clear (xr);
end:
MPFR_SAVE_EXPO_FREE (expo);
/* FIXME: add a test for bug before revision 7355 */
inexy = mpfr_check_range (y, inexy, rnd_mode);
inexz = mpfr_check_range (z, inexz, rnd_mode);
MPFR_RET (INEX(inexy,inexz));
}
/*************** asymptotically fast implementation below ********************/
/* truncate Q from R to at most prec bits.
Return the number of truncated bits.
*/
static mpfr_prec_t
reduce (mpz_t Q, mpz_srcptr R, mpfr_prec_t prec)
{
mpfr_prec_t l = mpz_sizeinbase (R, 2);
l = (l > prec) ? l - prec : 0;
mpz_fdiv_q_2exp (Q, R, l);
return l;
}
/* truncate S and C so that the smaller has prec bits.
Return the number of truncated bits.
*/
static unsigned long
reduce2 (mpz_t S, mpz_t C, mpfr_prec_t prec)
{
unsigned long ls = mpz_sizeinbase (S, 2);
unsigned long lc = mpz_sizeinbase (C, 2);
unsigned long l;
l = (ls < lc) ? ls : lc; /* smaller length */
l = (l > prec) ? l - prec : 0;
mpz_fdiv_q_2exp (S, S, l);
mpz_fdiv_q_2exp (C, C, l);
return l;
}
/* return in S0/Q0 a rational approximation of sin(X) with absolute error
bounded by 9*2^(-prec), where 0 <= X=p/2^r <= 1/2,
and in C0/Q0 a rational approximation of cos(X), with relative error
bounded by 9*2^(-prec) (and also absolute error, since
|cos(X)| <= 1).
We have sin(X)/X = sum((-1)^i*(p/2^r)^i/(2i+1)!, i=0..infinity).
We use the following binary splitting formula:
P(a,b) = (-p)^(b-a)
Q(a,b) = (2a)*(2a+1)*2^r if a+1=b [except Q(0,1)=1], Q(a,c)*Q(c,b) otherwise
T(a,b) = 1 if a+1=b, Q(c,b)*T(a,c)+P(a,c)*T(c,b) otherwise.
Since we use P(a,b) for b-a=2^k only, we compute only p^(2^k).
We do not store the factor 2^r in Q().
Then sin(X)/X ~ T(0,i)/Q(0,i) for i so that (p/2^r)^i/i! is small enough.
Return l such that Q0 has to be multiplied by 2^l.
Assumes prec >= 10.
*/
static unsigned long
sin_bs_aux (mpz_t Q0, mpz_t S0, mpz_t C0, mpz_srcptr p, mpfr_prec_t r,
mpfr_prec_t prec)
{
mpz_t T[GMP_NUMB_BITS], Q[GMP_NUMB_BITS], ptoj[GMP_NUMB_BITS], pp;
mpfr_prec_t log2_nb_terms[GMP_NUMB_BITS], mult[GMP_NUMB_BITS];
mpfr_prec_t accu[GMP_NUMB_BITS], size_ptoj[GMP_NUMB_BITS];
mpfr_prec_t prec_i_have, r0 = r;
unsigned long alloc, i, j, k;
mpfr_prec_t l;
if (MPFR_UNLIKELY(mpz_cmp_ui (p, 0) == 0)) /* sin(x)/x -> 1 */
{
mpz_set_ui (Q0, 1);
mpz_set_ui (S0, 1);
mpz_set_ui (C0, 1);
return 0;
}
/* check that X=p/2^r <= 1/2 */
MPFR_ASSERTN(mpz_sizeinbase (p, 2) - (mpfr_exp_t) r <= -1);
mpz_init (pp);
/* normalize p (non-zero here) */
l = mpz_scan1 (p, 0);
mpz_fdiv_q_2exp (pp, p, l); /* p = pp * 2^l */
mpz_mul (pp, pp, pp);
r = 2 * (r - l); /* x^2 = (p/2^r0)^2 = pp / 2^r */
/* now p is odd */
alloc = 2;
mpz_init_set_ui (T[0], 6);
mpz_init_set_ui (Q[0], 6);
mpz_init_set (ptoj[0], pp); /* ptoj[i] = pp^(2^i) */
mpz_init (T[1]);
mpz_init (Q[1]);
mpz_init (ptoj[1]);
mpz_mul (ptoj[1], pp, pp); /* ptoj[1] = pp^2 */
size_ptoj[1] = mpz_sizeinbase (ptoj[1], 2);
mpz_mul_2exp (T[0], T[0], r);
mpz_sub (T[0], T[0], pp); /* 6*2^r - pp = 6*2^r*(1 - x^2/6) */
log2_nb_terms[0] = 1;
/* already take into account the factor x=p/2^r in sin(x) = x * (...) */
mult[0] = r - mpz_sizeinbase (pp, 2) + r0 - mpz_sizeinbase (p, 2);
/* we have x^3 < 1/2^mult[0] */
for (i = 2, k = 0, prec_i_have = mult[0]; prec_i_have < prec; i += 2)
{
/* i is even here */
/* invariant: Q[0]*Q[1]*...*Q[k] equals (2i-1)!,
we have already summed terms of index < i
in S[0]/Q[0], ..., S[k]/Q[k] */
k ++;
if (k + 1 >= alloc) /* necessarily k + 1 = alloc */
{
alloc ++;
mpz_init (T[k+1]);
mpz_init (Q[k+1]);
mpz_init (ptoj[k+1]);
mpz_mul (ptoj[k+1], ptoj[k], ptoj[k]); /* pp^(2^(k+1)) */
size_ptoj[k+1] = mpz_sizeinbase (ptoj[k+1], 2);
}
/* for i even, we have Q[k] = (2*i)*(2*i+1), T[k] = 1,
then Q[k+1] = (2*i+2)*(2*i+3), T[k+1] = 1,
which reduces to T[k] = (2*i+2)*(2*i+3)*2^r-pp,
Q[k] = (2*i)*(2*i+1)*(2*i+2)*(2*i+3). */
log2_nb_terms[k] = 1;
mpz_set_ui (Q[k], 2 * i + 2);
mpz_mul_ui (Q[k], Q[k], 2 * i + 3);
mpz_mul_2exp (T[k], Q[k], r);
mpz_sub (T[k], T[k], pp);
mpz_mul_ui (Q[k], Q[k], 2 * i);
mpz_mul_ui (Q[k], Q[k], 2 * i + 1);
/* the next term of the series is divided by Q[k] and multiplied
by pp^2/2^(2r), thus the mult. factor < 1/2^mult[k] */
mult[k] = mpz_sizeinbase (Q[k], 2) + 2 * r - size_ptoj[1] - 1;
/* the absolute contribution of the next term is 1/2^accu[k] */
accu[k] = (k == 0) ? mult[k] : mult[k] + accu[k-1];
prec_i_have = accu[k]; /* the current term is < 1/2^accu[k] */
j = (i + 2) / 2;
l = 1;
while ((j & 1) == 0) /* combine and reduce */
{
mpz_mul (T[k], T[k], ptoj[l]);
mpz_mul (T[k-1], T[k-1], Q[k]);
mpz_mul_2exp (T[k-1], T[k-1], r << l);
mpz_add (T[k-1], T[k-1], T[k]);
mpz_mul (Q[k-1], Q[k-1], Q[k]);
log2_nb_terms[k-1] ++; /* number of terms in S[k-1]
is a power of 2 by construction */
prec_i_have = mpz_sizeinbase (Q[k], 2);
mult[k-1] += prec_i_have + (r << l) - size_ptoj[l] - 1;
accu[k-1] = (k == 1) ? mult[k-1] : mult[k-1] + accu[k-2];
prec_i_have = accu[k-1];
l ++;
j >>= 1;
k --;
}
}
/* accumulate all products in T[0] and Q[0]. Warning: contrary to above,
here we do not have log2_nb_terms[k-1] = log2_nb_terms[k]+1. */
l = 0; /* number of accumulated terms in the right part T[k]/Q[k] */
while (k > 0)
{
j = log2_nb_terms[k-1];
mpz_mul (T[k], T[k], ptoj[j]);
mpz_mul (T[k-1], T[k-1], Q[k]);
l += 1 << log2_nb_terms[k];
mpz_mul_2exp (T[k-1], T[k-1], r * l);
mpz_add (T[k-1], T[k-1], T[k]);
mpz_mul (Q[k-1], Q[k-1], Q[k]);
k--;
}
l = r0 + r * (i - 1); /* implicit multiplier 2^r for Q0 */
/* at this point T[0]/(2^l*Q[0]) is an approximation of sin(x) where the 1st
neglected term has contribution < 1/2^prec, thus since the series has
alternate signs, the error is < 1/2^prec */
/* we truncate Q0 to prec bits: the relative error is at most 2^(1-prec),
which means that Q0 = Q[0] * (1+theta) with |theta| <= 2^(1-prec)
[up to a power of two] */
l += reduce (Q0, Q[0], prec);
l -= reduce (T[0], T[0], prec);
/* multiply by x = p/2^l */
mpz_mul (S0, T[0], p);
l -= reduce (S0, S0, prec); /* S0 = T[0] * (1 + theta)^2 up to power of 2 */
/* sin(X) ~ S0/Q0*(1 + theta)^3 + err with |theta| <= 2^(1-prec) and
|err| <= 2^(-prec), thus since |S0/Q0| <= 1:
|sin(X) - S0/Q0| <= 4*|theta*S0/Q0| + |err| <= 9*2^(-prec) */
mpz_clear (pp);
for (j = 0; j < alloc; j ++)
{
mpz_clear (T[j]);
mpz_clear (Q[j]);
mpz_clear (ptoj[j]);
}
/* compute cos(X) from sin(X): sqrt(1-(S/Q)^2) = sqrt(Q^2-S^2)/Q
= sqrt(Q0^2*2^(2l)-S0^2)/Q0.
Write S/Q = sin(X) + eps with |eps| <= 9*2^(-prec),
then sqrt(Q^2-S^2) = sqrt(Q^2-Q^2*(sin(X)+eps)^2)
= sqrt(Q^2*cos(X)^2-Q^2*(2*sin(X)*eps+eps^2))
= sqrt(Q^2*cos(X)^2-Q^2*eps1) with |eps1|<=9*2^(-prec)
[using X<=1/2 and eps<=9*2^(-prec) and prec>=10]
Since we truncate the square root, we get:
sqrt(Q^2*cos(X)^2-Q^2*eps1)+eps2 with |eps2|<1
= Q*sqrt(cos(X)^2-eps1)+eps2
= Q*cos(X)*(1+eps3)+eps2 with |eps3| <= 6*2^(-prec)
= Q*cos(X)*(1+eps3+eps2/(Q*cos(X)))
= Q*cos(X)*(1+eps4) with |eps4| <= 9*2^(-prec)
since |Q| >= 2^(prec-1) */
/* we assume that Q0*2^l >= 2^(prec-1) */
MPFR_ASSERTN(l + mpz_sizeinbase (Q0, 2) >= prec);
mpz_mul (C0, Q0, Q0);
mpz_mul_2exp (C0, C0, 2 * l);
mpz_submul (C0, S0, S0);
mpz_sqrt (C0, C0);
return l;
}
/* Put in s and c approximations of sin(x) and cos(x) respectively.
Assumes 0 < x < Pi/4 and PREC(s) = PREC(c) >= 10.
Return err such that the relative error is bounded by 2^err ulps.
*/
static int
sincos_aux (mpfr_t s, mpfr_t c, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
mpfr_prec_t prec_s, sh;
mpz_t Q, S, C, Q2, S2, C2, y;
mpfr_t x2;
unsigned long l, l2, j, err;
MPFR_ASSERTD(MPFR_PREC(s) == MPFR_PREC(c));
prec_s = MPFR_PREC(s);
mpfr_init2 (x2, MPFR_PREC(x));
mpz_init (Q);
mpz_init (S);
mpz_init (C);
mpz_init (Q2);
mpz_init (S2);
mpz_init (C2);
mpz_init (y);
mpfr_set (x2, x, MPFR_RNDN); /* exact */
mpz_set_ui (Q, 1);
l = 0;
mpz_set_ui (S, 0); /* sin(0) = S/(2^l*Q), exact */
mpz_set_ui (C, 1); /* cos(0) = C/(2^l*Q), exact */
/* Invariant: x = X + x2/2^(sh-1), where the part X was already treated,
S/(2^l*Q) ~ sin(X), C/(2^l*Q) ~ cos(X), and x2/2^(sh-1) < Pi/4.
'sh-1' is the number of already shifted bits in x2.
*/
for (sh = 1, j = 0; mpfr_cmp_ui (x2, 0) != 0 && sh <= prec_s; sh <<= 1, j++)
{
if (sh > prec_s / 2) /* sin(x) = x + O(x^3), cos(x) = 1 + O(x^2) */
{
l2 = -mpfr_get_z_2exp (S2, x2); /* S2/2^l2 = x2 */
l2 += sh - 1;
mpz_set_ui (Q2, 1);
mpz_set_ui (C2, 1);
mpz_mul_2exp (C2, C2, l2);
mpfr_set_ui (x2, 0, MPFR_RNDN);
}
else
{
/* y <- trunc(x2 * 2^sh) = trunc(x * 2^(2*sh-1)) */
mpfr_mul_2exp (x2, x2, sh, MPFR_RNDN); /* exact */
mpfr_get_z (y, x2, MPFR_RNDZ); /* round towards zero: now
0 <= x2 < 2^sh, thus
0 <= x2/2^(sh-1) < 2^(1-sh) */
if (mpz_cmp_ui (y, 0) == 0)
continue;
mpfr_sub_z (x2, x2, y, MPFR_RNDN); /* should be exact */
l2 = sin_bs_aux (Q2, S2, C2, y, 2 * sh - 1, prec_s);
/* we now have |S2/Q2/2^l2 - sin(X)| <= 9*2^(prec_s)
and |C2/Q2/2^l2 - cos(X)| <= 6*2^(prec_s), with X=y/2^(2sh-1) */
}
if (sh == 1) /* S=0, C=1 */
{
l = l2;
mpz_swap (Q, Q2);
mpz_swap (S, S2);
mpz_swap (C, C2);
}
else
{
/* s <- s*c2+c*s2, c <- c*c2-s*s2, using Karatsuba:
a = s+c, b = s2+c2, t = a*b, d = s*s2, e = c*c2,
s <- t - d - e, c <- e - d */
mpz_add (y, S, C); /* a */
mpz_mul (C, C, C2); /* e */
mpz_add (C2, C2, S2); /* b */
mpz_mul (S2, S, S2); /* d */
mpz_mul (y, y, C2); /* a*b */
mpz_sub (S, y, S2); /* t - d */
mpz_sub (S, S, C); /* t - d - e */
mpz_sub (C, C, S2); /* e - d */
mpz_mul (Q, Q, Q2);
/* after j loops, the error is <= (11j-2)*2^(prec_s) */
l += l2;
/* reduce Q to prec_s bits */
l += reduce (Q, Q, prec_s);
/* reduce S,C to prec_s bits, error <= 11*j*2^(prec_s) */
l -= reduce2 (S, C, prec_s);
}
}
j = 11 * j;
for (err = 0; j > 1; j = (j + 1) / 2, err ++);
mpfr_set_z (s, S, MPFR_RNDN);
mpfr_div_z (s, s, Q, MPFR_RNDN);
mpfr_div_2exp (s, s, l, MPFR_RNDN);
mpfr_set_z (c, C, MPFR_RNDN);
mpfr_div_z (c, c, Q, MPFR_RNDN);
mpfr_div_2exp (c, c, l, MPFR_RNDN);
mpz_clear (Q);
mpz_clear (S);
mpz_clear (C);
mpz_clear (Q2);
mpz_clear (S2);
mpz_clear (C2);
mpz_clear (y);
mpfr_clear (x2);
return err;
}
/* Assumes x is neither NaN, +/-Inf, nor +/- 0.
One of s and c might be NULL, in which case the corresponding value is
not computed.
Assumes s differs from c.
*/
int
mpfr_sincos_fast (mpfr_t s, mpfr_t c, mpfr_srcptr x, mpfr_rnd_t rnd)
{
int inexs, inexc;
mpfr_t x_red, ts, tc;
mpfr_prec_t w;
mpfr_exp_t err, errs, errc;
MPFR_ZIV_DECL (loop);
MPFR_ASSERTN(s != c);
if (s == NULL)
w = MPFR_PREC(c);
else if (c == NULL)
w = MPFR_PREC(s);
else
w = MPFR_PREC(s) >= MPFR_PREC(c) ? MPFR_PREC(s) : MPFR_PREC(c);
w += MPFR_INT_CEIL_LOG2(w) + 9; /* ensures w >= 10 (needed by sincos_aux) */
mpfr_init2 (ts, w);
mpfr_init2 (tc, w);
MPFR_ZIV_INIT (loop, w);
for (;;)
{
/* if 0 < x <= Pi/4, we can call sincos_aux directly */
if (MPFR_IS_POS(x) && mpfr_cmp_ui_2exp (x, 1686629713, -31) <= 0)
{
err = sincos_aux (ts, tc, x, MPFR_RNDN);
}
/* if -Pi/4 <= x < 0, use sin(-x)=-sin(x) */
else if (MPFR_IS_NEG(x) && mpfr_cmp_si_2exp (x, -1686629713, -31) >= 0)
{
mpfr_init2 (x_red, MPFR_PREC(x));
mpfr_neg (x_red, x, rnd); /* exact */
err = sincos_aux (ts, tc, x_red, MPFR_RNDN);
mpfr_neg (ts, ts, MPFR_RNDN);
mpfr_clear (x_red);
}
else /* argument reduction is needed */
{
long q;
mpfr_t pi;
int neg = 0;
mpfr_init2 (x_red, w);
mpfr_init2 (pi, (MPFR_EXP(x) > 0) ? w + MPFR_EXP(x) : w);
mpfr_const_pi (pi, MPFR_RNDN);
mpfr_div_2exp (pi, pi, 1, MPFR_RNDN); /* Pi/2 */
mpfr_remquo (x_red, &q, x, pi, MPFR_RNDN);
/* x = q * (Pi/2 + eps1) + x_red + eps2,
where |eps1| <= 1/2*ulp(Pi/2) = 2^(-w-MAX(0,EXP(x))),
and eps2 <= 1/2*ulp(x_red) <= 1/2*ulp(Pi/2) = 2^(-w)
Since |q| <= x/(Pi/2) <= |x|, we have
q*|eps1| <= 2^(-w), thus
|x - q * Pi/2 - x_red| <= 2^(1-w) */
/* now -Pi/4 <= x_red <= Pi/4: if x_red < 0, consider -x_red */
if (MPFR_IS_NEG(x_red))
{
mpfr_neg (x_red, x_red, MPFR_RNDN);
neg = 1;
}
err = sincos_aux (ts, tc, x_red, MPFR_RNDN);
err ++; /* to take into account the argument reduction */
if (neg) /* sin(-x) = -sin(x), cos(-x) = cos(x) */
mpfr_neg (ts, ts, MPFR_RNDN);
if (q & 2) /* sin(x+Pi) = -sin(x), cos(x+Pi) = -cos(x) */
{
mpfr_neg (ts, ts, MPFR_RNDN);
mpfr_neg (tc, tc, MPFR_RNDN);
}
if (q & 1) /* sin(x+Pi/2) = cos(x), cos(x+Pi/2) = -sin(x) */
{
mpfr_neg (ts, ts, MPFR_RNDN);
mpfr_swap (ts, tc);
}
mpfr_clear (x_red);
mpfr_clear (pi);
}
/* adjust errors with respect to absolute values */
errs = err - MPFR_EXP(ts);
errc = err - MPFR_EXP(tc);
if ((s == NULL || MPFR_CAN_ROUND (ts, w - errs, MPFR_PREC(s), rnd)) &&
(c == NULL || MPFR_CAN_ROUND (tc, w - errc, MPFR_PREC(c), rnd)))
break;
MPFR_ZIV_NEXT (loop, w);
mpfr_set_prec (ts, w);
mpfr_set_prec (tc, w);
}
MPFR_ZIV_FREE (loop);
inexs = (s == NULL) ? 0 : mpfr_set (s, ts, rnd);
inexc = (c == NULL) ? 0 : mpfr_set (c, tc, rnd);
mpfr_clear (ts);
mpfr_clear (tc);
return INEX(inexs,inexc);
}