int
mpfr_add_ui (mpfr_ptr y, mpfr_srcptr x, unsigned long int u, mpfr_rnd_t rnd_mode)
{
MPFR_LOG_FUNC
(("x[%Pu]=%.*Rg u=%lu rnd=%d",
mpfr_get_prec(x), mpfr_log_prec, x, u, rnd_mode),
("y[%Pu]=%.*Rg", mpfr_get_prec (y), mpfr_log_prec, y));
if (MPFR_LIKELY(u != 0) ) /* if u=0, do nothing */
{
mpfr_t uu;
mp_limb_t up[1];
unsigned long cnt;
int inex;
MPFR_SAVE_EXPO_DECL (expo);
MPFR_TMP_INIT1 (up, uu, GMP_NUMB_BITS);
MPFR_ASSERTD (u == (mp_limb_t) u);
count_leading_zeros(cnt, (mp_limb_t) u);
up[0] = (mp_limb_t) u << cnt;
/* Optimization note: Exponent save/restore operations may be
removed if mpfr_add works even when uu is out-of-range. */
MPFR_SAVE_EXPO_MARK (expo);
MPFR_SET_EXP (uu, GMP_NUMB_BITS - cnt);
inex = mpfr_add(y, x, uu, rnd_mode);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range(y, inex, rnd_mode);
}
else
/* (unsigned long) 0 is assumed to be a real 0 (unsigned) */
return mpfr_set (y, x, rnd_mode);
}
/* This function does not change the flags. */
static void
mpfr_get_zexp (mpz_ptr ez, mpfr_srcptr x)
{
mpz_init (ez);
if (MPFR_IS_UBF (x))
mpz_set (ez, MPFR_ZEXP (x));
else
{
mp_limb_t e_limb[MPFR_EXP_LIMB_SIZE];
mpfr_t e;
int inex;
MPFR_SAVE_EXPO_DECL (expo);
/* TODO: Once this has been tested, optimize based on whether
_MPFR_EXP_FORMAT <= 3. */
MPFR_TMP_INIT1 (e_limb, e, sizeof (mpfr_exp_t) * CHAR_BIT);
MPFR_SAVE_EXPO_MARK (expo);
MPFR_DBGRES (inex = mpfr_set_exp_t (e, MPFR_GET_EXP (x), MPFR_RNDN));
MPFR_ASSERTD (inex == 0);
MPFR_DBGRES (inex = mpfr_get_z (ez, e, MPFR_RNDN));
MPFR_ASSERTD (inex == 0);
MPFR_SAVE_EXPO_FREE (expo);
}
}
int
mpfr_sqrt_ui (mpfr_ptr r, unsigned long u, mpfr_rnd_t rnd_mode)
{
if (u)
{
mpfr_t uu;
mp_limb_t up[1];
unsigned long cnt;
int inex;
MPFR_SAVE_EXPO_DECL (expo);
MPFR_TMP_INIT1 (up, uu, GMP_NUMB_BITS);
MPFR_ASSERTN (u == (mp_limb_t) u);
count_leading_zeros (cnt, (mp_limb_t) u);
*up = (mp_limb_t) u << cnt;
MPFR_SAVE_EXPO_MARK (expo);
MPFR_SET_EXP (uu, GMP_NUMB_BITS - cnt);
inex = mpfr_sqrt(r, uu, rnd_mode);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range(r, inex, rnd_mode);
}
else /* sqrt(0) = 0 */
{
MPFR_SET_ZERO(r);
MPFR_SET_POS(r);
MPFR_RET(0);
}
}
int
mpfr_sub_ui (mpfr_ptr y, mpfr_srcptr x, unsigned long int u, mp_rnd_t rnd_mode)
{
if (MPFR_LIKELY (u != 0)) /* if u=0, do nothing */
{
mpfr_t uu;
mp_limb_t up[1];
unsigned long cnt;
int inex;
MPFR_SAVE_EXPO_DECL (expo);
MPFR_TMP_INIT1 (up, uu, BITS_PER_MP_LIMB);
MPFR_ASSERTN (u == (mp_limb_t) u);
count_leading_zeros (cnt, (mp_limb_t) u);
*up = (mp_limb_t) u << cnt;
/* Optimization note: Exponent save/restore operations may be
removed if mpfr_sub works even when uu is out-of-range. */
MPFR_SAVE_EXPO_MARK (expo);
MPFR_SET_EXP (uu, BITS_PER_MP_LIMB - cnt);
inex = mpfr_sub (y, x, uu, rnd_mode);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inex, rnd_mode);
}
else
return mpfr_set (y, x, rnd_mode);
}
int
mpfr_ui_div (mpfr_ptr y, unsigned long int u, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
mpfr_t uu;
mp_limb_t up[1];
unsigned long cnt;
MPFR_LOG_FUNC
(("u=%lu x[%Pu]=%.*Rg rnd=%d",
u, mpfr_get_prec(x), mpfr_log_prec, x, rnd_mode),
("y[%Pu]=%.*Rg", mpfr_get_prec(y), mpfr_log_prec, y));
if (MPFR_UNLIKELY(MPFR_IS_SINGULAR(x)))
{
if (MPFR_IS_NAN(x))
{
MPFR_SET_NAN(y);
MPFR_RET_NAN;
}
else if (MPFR_IS_INF(x)) /* u/Inf = 0 */
{
MPFR_SET_ZERO(y);
MPFR_SET_SAME_SIGN(y,x);
MPFR_RET(0);
}
else /* u / 0 */
{
MPFR_ASSERTD(MPFR_IS_ZERO(x));
if (u)
{
/* u > 0, so y = sign(x) * Inf */
MPFR_SET_SAME_SIGN(y, x);
MPFR_SET_INF(y);
mpfr_set_divby0 ();
MPFR_RET(0);
}
else
{
/* 0 / 0 */
MPFR_SET_NAN(y);
MPFR_RET_NAN;
}
}
}
else if (MPFR_LIKELY(u != 0))
{
MPFR_TMP_INIT1(up, uu, GMP_NUMB_BITS);
MPFR_ASSERTN(u == (mp_limb_t) u);
count_leading_zeros(cnt, (mp_limb_t) u);
up[0] = (mp_limb_t) u << cnt;
MPFR_SET_EXP (uu, GMP_NUMB_BITS - cnt);
return mpfr_div (y, uu, x, rnd_mode);
}
else /* u = 0, and x != 0 */
{
MPFR_SET_ZERO(y); /* if u=0, then set y to 0 */
MPFR_SET_SAME_SIGN(y, x); /* u considered as +0: sign(+0/x) = sign(x) */
MPFR_RET(0);
}
}
int
mpfr_mul_d (mpfr_ptr a, mpfr_srcptr b, double c, mpfr_rnd_t rnd_mode)
{
int inexact;
mpfr_t d;
mp_limb_t tmp_man[MPFR_LIMBS_PER_DOUBLE];
MPFR_SAVE_EXPO_DECL (expo);
MPFR_LOG_FUNC
(("b[%Pu]=%.*Rg c=%.20g rnd=%d",
mpfr_get_prec(b), mpfr_log_prec, b, c, rnd_mode),
("a[%Pu]=%.*Rg inexact=%d",
mpfr_get_prec (a), mpfr_log_prec, a, inexact));
MPFR_SAVE_EXPO_MARK (expo);
MPFR_TMP_INIT1(tmp_man, d, IEEE_DBL_MANT_DIG);
inexact = mpfr_set_d (d, c, rnd_mode);
MPFR_ASSERTD (inexact == 0);
MPFR_CLEAR_FLAGS ();
inexact = mpfr_mul (a, b, d, rnd_mode);
MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (a, inexact, rnd_mode);
}
int
mpfr_ui_div (mpfr_ptr y, unsigned long int u, mpfr_srcptr x, mp_rnd_t rnd_mode)
{
mpfr_t uu;
mp_limb_t up[1];
unsigned long cnt;
if (MPFR_UNLIKELY(MPFR_IS_SINGULAR(x)))
{
if (MPFR_IS_NAN(x))
{
MPFR_SET_NAN(y);
MPFR_RET_NAN;
}
else if (MPFR_IS_INF(x)) /* u/Inf = 0 */
{
MPFR_SET_ZERO(y);
MPFR_SET_SAME_SIGN(y,x);
MPFR_RET(0);
}
else /* u / 0 */
{
MPFR_ASSERTD(MPFR_IS_ZERO(x));
if (u)
{
/* u > 0, so y = sign(x) * Inf */
MPFR_SET_SAME_SIGN(y, x);
MPFR_SET_INF(y);
MPFR_RET(0);
}
else
{
/* 0 / 0 */
MPFR_SET_NAN(y);
MPFR_RET_NAN;
}
}
}
else if (MPFR_LIKELY(u != 0))
{
MPFR_TMP_INIT1(up, uu, BITS_PER_MP_LIMB);
MPFR_ASSERTN(u == (mp_limb_t) u);
count_leading_zeros(cnt, (mp_limb_t) u);
up[0] = (mp_limb_t) u << cnt;
MPFR_SET_EXP (uu, BITS_PER_MP_LIMB - cnt);
return mpfr_div (y, uu, x, rnd_mode);
}
else /* u = 0, and x != 0 */
{
MPFR_SET_ZERO(y); /* if u=0, then set y to 0 */
MPFR_SET_SAME_SIGN(y, x); /* u considered as +0: sign(+0/x) = sign(x) */
MPFR_RET(0);
}
}
double
virtual_timing_ai2 (struct speed_params *s)
{
double t;
unsigned i;
mpfr_t w, x;
mp_size_t size;
mpfr_t temp1, temp2;
SPEED_RESTRICT_COND (s->size >= MPFR_PREC_MIN);
SPEED_RESTRICT_COND (s->size <= MPFR_PREC_MAX);
size = (s->size-1)/GMP_NUMB_BITS+1;
s->xp[size-1] |= MPFR_LIMB_HIGHBIT;
MPFR_TMP_INIT1 (s->xp, x, s->size);
MPFR_SET_EXP (x, (mpfr_exp_t) s->r);
if (s->align_xp == 2) MPFR_SET_NEG (x);
mpfr_init2 (w, s->size);
speed_starttime ();
i = s->reps;
mpfr_init2 (temp1, MPFR_SMALL_PRECISION);
mpfr_init2 (temp2, MPFR_SMALL_PRECISION);
mpfr_set (temp1, x, MPFR_SMALL_PRECISION);
mpfr_set_si (temp2, MPFR_AI_THRESHOLD2, MPFR_RNDN);
mpfr_mul_ui (temp2, temp2, (unsigned int)MPFR_PREC (w), MPFR_RNDN);
if (MPFR_IS_NEG (x))
mpfr_mul_si (temp1, temp1, MPFR_AI_THRESHOLD1, MPFR_RNDN);
else
mpfr_mul_si (temp1, temp1, MPFR_AI_THRESHOLD3, MPFR_RNDN);
mpfr_add (temp1, temp1, temp2, MPFR_RNDN);
if (mpfr_cmp_si (temp1, MPFR_AI_SCALE) > 0)
t = 1.;
else
t = 1000.;
mpfr_clear (temp1);
mpfr_clear (temp2);
return t;
}
int
mpfr_ui_pow (mpfr_ptr y, unsigned long int n, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
mpfr_t t;
int inexact;
mp_limb_t tmp_mant[(sizeof (n) - 1) / sizeof (mp_limb_t) + 1];
MPFR_SAVE_EXPO_DECL (expo);
MPFR_SAVE_EXPO_MARK (expo);
MPFR_TMP_INIT1(tmp_mant, t, sizeof(n) * CHAR_BIT);
inexact = mpfr_set_ui (t, n, MPFR_RNDN);
MPFR_ASSERTD (inexact == 0);
inexact = mpfr_pow (y, t, x, rnd_mode);
MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inexact, rnd_mode);
}
int
mpfr_ui_sub (mpfr_ptr y, unsigned long int u, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
mpfr_t uu;
mp_limb_t up[1];
unsigned long cnt;
MPFR_LOG_FUNC
(("u=%lu x[%Pu]=%.*Rg rnd=%d",
u, mpfr_get_prec(x), mpfr_log_prec, x, rnd_mode),
("y[%Pu]=%.*Rg", mpfr_get_prec(y), mpfr_log_prec, y));
if (MPFR_UNLIKELY (u == 0))
return mpfr_neg (y, x, rnd_mode);
if (MPFR_UNLIKELY(MPFR_IS_SINGULAR(x)))
{
if (MPFR_IS_NAN(x))
{
MPFR_SET_NAN(y);
MPFR_RET_NAN;
}
else if (MPFR_IS_INF(x))
{
/* u - Inf = -Inf and u - -Inf = +Inf */
MPFR_SET_INF(y);
MPFR_SET_OPPOSITE_SIGN(y,x);
MPFR_RET(0); /* +/-infinity is exact */
}
else /* x is zero */
/* u - 0 = u */
return mpfr_set_ui(y, u, rnd_mode);
}
else
{
MPFR_TMP_INIT1 (up, uu, GMP_NUMB_BITS);
MPFR_ASSERTN(u == (mp_limb_t) u);
count_leading_zeros (cnt, (mp_limb_t) u);
*up = (mp_limb_t) u << cnt;
MPFR_SET_EXP (uu, GMP_NUMB_BITS - cnt);
return mpfr_sub (y, uu, x, rnd_mode);
}
}
int
mpfr_cmp_d (mpfr_srcptr b, double d)
{
mpfr_t tmp;
int res;
mp_limb_t tmp_man[MPFR_LIMBS_PER_DOUBLE];
MPFR_SAVE_EXPO_DECL (expo);
MPFR_SAVE_EXPO_MARK (expo);
MPFR_TMP_INIT1(tmp_man, tmp, IEEE_DBL_MANT_DIG);
res = mpfr_set_d (tmp, d, MPFR_RNDN);
MPFR_ASSERTD (res == 0);
MPFR_CLEAR_FLAGS ();
res = mpfr_cmp (b, tmp);
MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags);
MPFR_SAVE_EXPO_FREE (expo);
return res;
}
int
mpfr_ui_sub (mpfr_ptr y, unsigned long int u, mpfr_srcptr x, mp_rnd_t rnd_mode)
{
mpfr_t uu;
mp_limb_t up[1];
unsigned long cnt;
if (MPFR_UNLIKELY (u == 0))
return mpfr_neg (y, x, rnd_mode);
if (MPFR_UNLIKELY(MPFR_IS_SINGULAR(x)))
{
if (MPFR_IS_NAN(x))
{
MPFR_SET_NAN(y);
MPFR_RET_NAN;
}
else if (MPFR_IS_INF(x))
{
/* u - Inf = -Inf and u - -Inf = +Inf */
MPFR_SET_INF(y);
MPFR_SET_OPPOSITE_SIGN(y,x);
MPFR_RET(0); /* +/-infinity is exact */
}
else /* x is zero */
/* u - 0 = u */
return mpfr_set_ui(y, u, rnd_mode);
}
else
{
MPFR_TMP_INIT1 (up, uu, BITS_PER_MP_LIMB);
MPFR_ASSERTN(u == (mp_limb_t) u);
count_leading_zeros (cnt, (mp_limb_t) u);
*up = (mp_limb_t) u << cnt;
MPFR_SET_EXP (uu, BITS_PER_MP_LIMB - cnt);
return mpfr_sub (y, uu, x, rnd_mode);
}
}
int
mpfr_set_uj_2exp (mpfr_t x, uintmax_t j, intmax_t e, mp_rnd_t rnd)
{
unsigned int cnt, i;
mp_size_t k, len;
mp_limb_t limb;
mp_limb_t yp[sizeof(uintmax_t) / sizeof(mp_limb_t)];
mpfr_t y;
unsigned long uintmax_bit_size = sizeof(uintmax_t) * CHAR_BIT;
unsigned long bpml = BITS_PER_MP_LIMB % uintmax_bit_size;
/* Special case */
if (j == 0)
{
MPFR_SET_POS(x);
MPFR_SET_ZERO(x);
MPFR_RET(0);
}
MPFR_ASSERTN (sizeof(uintmax_t) % sizeof(mp_limb_t) == 0);
/* Create an auxillary var */
MPFR_TMP_INIT1 (yp, y, uintmax_bit_size);
k = numberof (yp);
if (k == 1)
limb = yp[0] = j;
else
{
/* Note: either BITS_PER_MP_LIMB = uintmax_bit_size, then k = 1 the
shift j >>= bpml is never done, or BITS_PER_MP_LIMB < uintmax_bit_size
and bpml = BITS_PER_MP_LIMB. */
for (i = 0; i < k; i++, j >>= bpml)
yp[i] = j; /* Only the low bits are copied */
/* Find the first limb not equal to zero. */
do
{
MPFR_ASSERTD (k > 0);
limb = yp[--k];
}
while (limb == 0);
k++;
}
count_leading_zeros(cnt, limb);
len = numberof (yp) - k;
/* Normalize it: len = number of last 0 limb, k number of non-zero limbs */
if (MPFR_LIKELY(cnt))
mpn_lshift (yp+len, yp, k, cnt); /* Normalize the High Limb*/
else if (len != 0)
MPN_COPY_DECR (yp+len, yp, k); /* Must use DECR */
if (len != 0)
/* Note: when numberof(yp)==1, len is constant and null, so the compiler
can optimize out this code. */
{
if (len == 1)
yp[0] = (mp_limb_t) 0;
else
MPN_ZERO (yp, len); /* Zeroing the last limbs */
}
e += k * BITS_PER_MP_LIMB - cnt; /* Update Expo */
MPFR_ASSERTD (MPFR_LIMB_MSB(yp[numberof (yp) - 1]) != 0);
/* Check expo underflow / overflow (can't use mpfr_check_range) */
if (MPFR_UNLIKELY(e < __gmpfr_emin))
{
/* The following test is necessary because in the rounding to the
* nearest mode, mpfr_underflow always rounds away from 0. In
* this rounding mode, we need to round to 0 if:
* _ |x| < 2^(emin-2), or
* _ |x| = 2^(emin-2) and the absolute value of the exact
* result is <= 2^(emin-2). */
if (rnd == GMP_RNDN && (e+1 < __gmpfr_emin || mpfr_powerof2_raw(y)))
rnd = GMP_RNDZ;
return mpfr_underflow (x, rnd, MPFR_SIGN_POS);
}
if (MPFR_UNLIKELY(e > __gmpfr_emax))
return mpfr_overflow (x, rnd, MPFR_SIGN_POS);
MPFR_SET_EXP (y, e);
/* Final: set x to y (rounding if necessary) */
return mpfr_set (x, y, rnd);
}
int
mpfr_ui_div (mpfr_ptr y, unsigned long int u, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
MPFR_LOG_FUNC
(("u=%lu x[%Pu]=%.*Rg rnd=%d",
u, mpfr_get_prec(x), mpfr_log_prec, x, rnd_mode),
("y[%Pu]=%.*Rg", mpfr_get_prec(y), mpfr_log_prec, y));
if (MPFR_UNLIKELY(MPFR_IS_SINGULAR(x)))
{
if (MPFR_IS_NAN(x))
{
MPFR_SET_NAN(y);
MPFR_RET_NAN;
}
else if (MPFR_IS_INF(x)) /* u/Inf = 0 */
{
MPFR_SET_ZERO(y);
MPFR_SET_SAME_SIGN(y,x);
MPFR_RET(0);
}
else /* u / 0 */
{
MPFR_ASSERTD(MPFR_IS_ZERO(x));
if (u)
{
/* u > 0, so y = sign(x) * Inf */
MPFR_SET_SAME_SIGN(y, x);
MPFR_SET_INF(y);
MPFR_SET_DIVBY0 ();
MPFR_RET(0);
}
else
{
/* 0 / 0 */
MPFR_SET_NAN(y);
MPFR_RET_NAN;
}
}
}
else if (MPFR_LIKELY(u != 0))
{
mpfr_t uu;
mp_limb_t up[1];
int cnt;
int inex;
MPFR_SAVE_EXPO_DECL (expo);
MPFR_TMP_INIT1(up, uu, GMP_NUMB_BITS);
MPFR_ASSERTN(u == (mp_limb_t) u);
count_leading_zeros(cnt, (mp_limb_t) u);
up[0] = (mp_limb_t) u << cnt;
/* Optimization note: Exponent save/restore operations may be
removed if mpfr_div works even when uu is out-of-range. */
MPFR_SAVE_EXPO_MARK (expo);
MPFR_SET_EXP (uu, GMP_NUMB_BITS - cnt);
inex = mpfr_div (y, uu, x, rnd_mode);
MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inex, rnd_mode);
}
else /* u = 0, and x != 0 */
{
MPFR_SET_ZERO(y); /* if u=0, then set y to 0 */
MPFR_SET_SAME_SIGN(y, x); /* u considered as +0: sign(+0/x) = sign(x) */
MPFR_RET(0);
}
}
/* The computation of z = pow(x,y) is done by
z = exp(y * log(x)) = x^y
For the special cases, see Section F.9.4.4 of the C standard:
_ pow(±0, y) = ±inf for y an odd integer < 0.
_ pow(±0, y) = +inf for y < 0 and not an odd integer.
_ pow(±0, y) = ±0 for y an odd integer > 0.
_ pow(±0, y) = +0 for y > 0 and not an odd integer.
_ pow(-1, ±inf) = 1.
_ pow(+1, y) = 1 for any y, even a NaN.
_ pow(x, ±0) = 1 for any x, even a NaN.
_ pow(x, y) = NaN for finite x < 0 and finite non-integer y.
_ pow(x, -inf) = +inf for |x| < 1.
_ pow(x, -inf) = +0 for |x| > 1.
_ pow(x, +inf) = +0 for |x| < 1.
_ pow(x, +inf) = +inf for |x| > 1.
_ pow(-inf, y) = -0 for y an odd integer < 0.
_ pow(-inf, y) = +0 for y < 0 and not an odd integer.
_ pow(-inf, y) = -inf for y an odd integer > 0.
_ pow(-inf, y) = +inf for y > 0 and not an odd integer.
_ pow(+inf, y) = +0 for y < 0.
_ pow(+inf, y) = +inf for y > 0. */
int
mpfr_pow (mpfr_ptr z, mpfr_srcptr x, mpfr_srcptr y, mpfr_rnd_t rnd_mode)
{
int inexact;
int cmp_x_1;
int y_is_integer;
MPFR_SAVE_EXPO_DECL (expo);
MPFR_LOG_FUNC
(("x[%Pu]=%.*Rg y[%Pu]=%.*Rg rnd=%d",
mpfr_get_prec (x), mpfr_log_prec, x,
mpfr_get_prec (y), mpfr_log_prec, y, rnd_mode),
("z[%Pu]=%.*Rg inexact=%d",
mpfr_get_prec (z), mpfr_log_prec, z, inexact));
if (MPFR_ARE_SINGULAR (x, y))
{
/* pow(x, 0) returns 1 for any x, even a NaN. */
if (MPFR_UNLIKELY (MPFR_IS_ZERO (y)))
return mpfr_set_ui (z, 1, rnd_mode);
else if (MPFR_IS_NAN (x))
{
MPFR_SET_NAN (z);
MPFR_RET_NAN;
}
else if (MPFR_IS_NAN (y))
{
/* pow(+1, NaN) returns 1. */
if (mpfr_cmp_ui (x, 1) == 0)
return mpfr_set_ui (z, 1, rnd_mode);
MPFR_SET_NAN (z);
MPFR_RET_NAN;
}
else if (MPFR_IS_INF (y))
{
if (MPFR_IS_INF (x))
{
if (MPFR_IS_POS (y))
MPFR_SET_INF (z);
else
MPFR_SET_ZERO (z);
MPFR_SET_POS (z);
MPFR_RET (0);
}
else
{
int cmp;
cmp = mpfr_cmpabs (x, __gmpfr_one) * MPFR_INT_SIGN (y);
MPFR_SET_POS (z);
if (cmp > 0)
{
/* Return +inf. */
MPFR_SET_INF (z);
MPFR_RET (0);
}
else if (cmp < 0)
{
/* Return +0. */
MPFR_SET_ZERO (z);
MPFR_RET (0);
}
else
{
/* Return 1. */
return mpfr_set_ui (z, 1, rnd_mode);
}
}
}
else if (MPFR_IS_INF (x))
{
int negative;
/* Determine the sign now, in case y and z are the same object */
negative = MPFR_IS_NEG (x) && mpfr_odd_p (y);
if (MPFR_IS_POS (y))
MPFR_SET_INF (z);
else
MPFR_SET_ZERO (z);
if (negative)
MPFR_SET_NEG (z);
else
MPFR_SET_POS (z);
MPFR_RET (0);
}
else
{
int negative;
MPFR_ASSERTD (MPFR_IS_ZERO (x));
/* Determine the sign now, in case y and z are the same object */
negative = MPFR_IS_NEG(x) && mpfr_odd_p (y);
if (MPFR_IS_NEG (y))
{
MPFR_ASSERTD (! MPFR_IS_INF (y));
MPFR_SET_INF (z);
MPFR_SET_DIVBY0 ();
}
else
MPFR_SET_ZERO (z);
if (negative)
MPFR_SET_NEG (z);
else
MPFR_SET_POS (z);
MPFR_RET (0);
}
}
/* x^y for x < 0 and y not an integer is not defined */
y_is_integer = mpfr_integer_p (y);
if (MPFR_IS_NEG (x) && ! y_is_integer)
{
MPFR_SET_NAN (z);
MPFR_RET_NAN;
}
/* now the result cannot be NaN:
(1) either x > 0
(2) or x < 0 and y is an integer */
cmp_x_1 = mpfr_cmpabs (x, __gmpfr_one);
if (cmp_x_1 == 0)
return mpfr_set_si (z, MPFR_IS_NEG (x) && mpfr_odd_p (y) ? -1 : 1, rnd_mode);
/* now we have:
(1) either x > 0
(2) or x < 0 and y is an integer
and in addition |x| <> 1.
*/
/* detect overflow: an overflow is possible if
(a) |x| > 1 and y > 0
(b) |x| < 1 and y < 0.
FIXME: this assumes 1 is always representable.
FIXME2: maybe we can test overflow and underflow simultaneously.
The idea is the following: first compute an approximation to
y * log2|x|, using rounding to nearest. If |x| is not too near from 1,
this approximation should be accurate enough, and in most cases enable
one to prove that there is no underflow nor overflow.
Otherwise, it should enable one to check only underflow or overflow,
instead of both cases as in the present case.
*/
/* fast check for cases where no overflow nor underflow is possible:
if |y| <= 2^15, and -32767 < EXP(x) <= 32767, then
|y*log2(x)| <= 2^15*32767 < 1073741823, thus for the default
emax=1073741823 and emin=-emax there can be no overflow nor underflow */
if (__gmpfr_emax >= 1073741823 && __gmpfr_emin <= -1073741823 &&
MPFR_EXP(y) <= 15 && -32767 < MPFR_EXP(x) && MPFR_EXP(x) <= 32767)
goto no_overflow_nor_underflow;
if (cmp_x_1 * MPFR_SIGN (y) > 0)
{
mpfr_t t;
int negative, overflow;
MPFR_SAVE_EXPO_MARK (expo);
mpfr_init2 (t, 53);
/* we want a lower bound on y*log2|x|:
(i) if x > 0, it suffices to round log2(x) toward zero, and
to round y*o(log2(x)) toward zero too;
(ii) if x < 0, we first compute t = o(-x), with rounding toward 1,
and then follow as in case (1). */
if (MPFR_IS_POS (x))
mpfr_log2 (t, x, MPFR_RNDZ);
else
{
mpfr_neg (t, x, (cmp_x_1 > 0) ? MPFR_RNDZ : MPFR_RNDU);
mpfr_log2 (t, t, MPFR_RNDZ);
}
mpfr_mul (t, t, y, MPFR_RNDZ);
overflow = mpfr_cmp_si (t, __gmpfr_emax) > 0;
mpfr_clear (t);
MPFR_SAVE_EXPO_FREE (expo);
if (overflow)
{
MPFR_LOG_MSG (("early overflow detection\n", 0));
negative = MPFR_IS_NEG (x) && mpfr_odd_p (y);
return mpfr_overflow (z, rnd_mode, negative ? -1 : 1);
}
}
/* Basic underflow checking. One has:
* - if y > 0, |x^y| < 2^(EXP(x) * y);
* - if y < 0, |x^y| <= 2^((EXP(x) - 1) * y);
* so that one can compute a value ebound such that |x^y| < 2^ebound.
* If we have ebound <= emin - 2 (emin - 1 in directed rounding modes),
* then there is an underflow and we can decide the return value.
*/
if (MPFR_IS_NEG (y) ? (MPFR_GET_EXP (x) > 1) : (MPFR_GET_EXP (x) < 0))
{
mp_limb_t tmp_limb[MPFR_EXP_LIMB_SIZE];
mpfr_t tmp;
mpfr_eexp_t ebound;
int inex2;
/* We must restore the flags. */
MPFR_SAVE_EXPO_MARK (expo);
MPFR_TMP_INIT1 (tmp_limb, tmp, sizeof (mpfr_exp_t) * CHAR_BIT);
inex2 = mpfr_set_exp_t (tmp, MPFR_GET_EXP (x), MPFR_RNDN);
MPFR_ASSERTN (inex2 == 0);
if (MPFR_IS_NEG (y))
{
inex2 = mpfr_sub_ui (tmp, tmp, 1, MPFR_RNDN);
MPFR_ASSERTN (inex2 == 0);
}
mpfr_mul (tmp, tmp, y, MPFR_RNDU);
if (MPFR_IS_NEG (y))
mpfr_nextabove (tmp);
/* tmp doesn't necessarily fit in ebound, but that doesn't matter
since we get the minimum value in such a case. */
ebound = mpfr_get_exp_t (tmp, MPFR_RNDU);
MPFR_SAVE_EXPO_FREE (expo);
if (MPFR_UNLIKELY (ebound <=
__gmpfr_emin - (rnd_mode == MPFR_RNDN ? 2 : 1)))
{
/* warning: mpfr_underflow rounds away from 0 for MPFR_RNDN */
MPFR_LOG_MSG (("early underflow detection\n", 0));
return mpfr_underflow (z,
rnd_mode == MPFR_RNDN ? MPFR_RNDZ : rnd_mode,
MPFR_IS_NEG (x) && mpfr_odd_p (y) ? -1 : 1);
}
}
no_overflow_nor_underflow:
/* If y is an integer, we can use mpfr_pow_z (based on multiplications),
but if y is very large (I'm not sure about the best threshold -- VL),
we shouldn't use it, as it can be very slow and take a lot of memory
(and even crash or make other programs crash, as several hundred of
MBs may be necessary). Note that in such a case, either x = +/-2^b
(this case is handled below) or x^y cannot be represented exactly in
any precision supported by MPFR (the general case uses this property).
*/
if (y_is_integer && (MPFR_GET_EXP (y) <= 256))
{
mpz_t zi;
MPFR_LOG_MSG (("special code for y not too large integer\n", 0));
mpz_init (zi);
mpfr_get_z (zi, y, MPFR_RNDN);
inexact = mpfr_pow_z (z, x, zi, rnd_mode);
mpz_clear (zi);
return inexact;
}
/* Special case (+/-2^b)^Y which could be exact. If x is negative, then
necessarily y is a large integer. */
if (mpfr_powerof2_raw (x))
{
mpfr_exp_t b = MPFR_GET_EXP (x) - 1;
mpfr_t tmp;
int sgnx = MPFR_SIGN (x);
MPFR_ASSERTN (b >= LONG_MIN && b <= LONG_MAX); /* FIXME... */
MPFR_LOG_MSG (("special case (+/-2^b)^Y\n", 0));
/* now x = +/-2^b, so x^y = (+/-1)^y*2^(b*y) is exact whenever b*y is
an integer */
MPFR_SAVE_EXPO_MARK (expo);
mpfr_init2 (tmp, MPFR_PREC (y) + sizeof (long) * CHAR_BIT);
inexact = mpfr_mul_si (tmp, y, b, MPFR_RNDN); /* exact */
MPFR_ASSERTN (inexact == 0);
/* Note: as the exponent range has been extended, an overflow is not
possible (due to basic overflow and underflow checking above, as
the result is ~ 2^tmp), and an underflow is not possible either
because b is an integer (thus either 0 or >= 1). */
MPFR_CLEAR_FLAGS ();
inexact = mpfr_exp2 (z, tmp, rnd_mode);
mpfr_clear (tmp);
if (sgnx < 0 && mpfr_odd_p (y))
{
mpfr_neg (z, z, rnd_mode);
inexact = -inexact;
}
/* Without the following, the overflows3 test in tpow.c fails. */
MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (z, inexact, rnd_mode);
}
MPFR_SAVE_EXPO_MARK (expo);
/* Case where y * log(x) is very small. Warning: x can be negative, in
that case y is a large integer. */
{
mpfr_exp_t err, expx, logt;
/* We need an upper bound on the exponent of y * log(x). */
if (MPFR_IS_POS(x))
expx = cmp_x_1 > 0 ? MPFR_EXP(x) : 1 - MPFR_EXP(x);
else
expx = mpfr_cmp_si (x, -1) > 0 ? 1 - MPFR_EXP(x) : MPFR_EXP(x);
MPFR_ASSERTD(expx >= 0);
/* now |log(x)| < expx */
logt = MPFR_INT_CEIL_LOG2 (expx);
/* now expx <= 2^logt */
err = MPFR_GET_EXP (y) + logt;
MPFR_CLEAR_FLAGS ();
MPFR_SMALL_INPUT_AFTER_SAVE_EXPO (z, __gmpfr_one, - err, 0,
(MPFR_IS_POS (y)) ^ (cmp_x_1 < 0),
rnd_mode, expo, {});
}
/* General case */
inexact = mpfr_pow_general (z, x, y, rnd_mode, y_is_integer, &expo);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (z, inexact, rnd_mode);
}
int
mpfr_erf (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
mpfr_t xf;
mp_limb_t xf_limb[(53 - 1) / GMP_NUMB_BITS + 1];
int inex, large;
MPFR_SAVE_EXPO_DECL (expo);
MPFR_LOG_FUNC
(("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd_mode),
("y[%Pu]=%.*Rg inexact=%d", mpfr_get_prec (y), mpfr_log_prec, y, inex));
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
{
if (MPFR_IS_NAN (x))
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
else if (MPFR_IS_INF (x)) /* erf(+inf) = +1, erf(-inf) = -1 */
return mpfr_set_si (y, MPFR_INT_SIGN (x), MPFR_RNDN);
else /* erf(+0) = +0, erf(-0) = -0 */
{
MPFR_ASSERTD (MPFR_IS_ZERO (x));
return mpfr_set (y, x, MPFR_RNDN); /* should keep the sign of x */
}
}
/* now x is neither NaN, Inf nor 0 */
/* first try expansion at x=0 when x is small, or asymptotic expansion
where x is large */
MPFR_SAVE_EXPO_MARK (expo);
/* around x=0, we have erf(x) = 2x/sqrt(Pi) (1 - x^2/3 + ...),
with 1 - x^2/3 <= sqrt(Pi)*erf(x)/2/x <= 1 for x >= 0. This means that
if x^2/3 < 2^(-PREC(y)-1) we can decide of the correct rounding,
unless we have a worst-case for 2x/sqrt(Pi). */
if (MPFR_EXP(x) < - (mpfr_exp_t) (MPFR_PREC(y) / 2))
{
/* we use 2x/sqrt(Pi) (1 - x^2/3) <= erf(x) <= 2x/sqrt(Pi) for x > 0
and 2x/sqrt(Pi) <= erf(x) <= 2x/sqrt(Pi) (1 - x^2/3) for x < 0.
In both cases |2x/sqrt(Pi) (1 - x^2/3)| <= |erf(x)| <= |2x/sqrt(Pi)|.
We will compute l and h such that l <= |2x/sqrt(Pi) (1 - x^2/3)|
and |2x/sqrt(Pi)| <= h. If l and h round to the same value to
precision PREC(y) and rounding rnd_mode, then we are done. */
mpfr_t l, h; /* lower and upper bounds for erf(x) */
int ok, inex2;
mpfr_init2 (l, MPFR_PREC(y) + 17);
mpfr_init2 (h, MPFR_PREC(y) + 17);
/* first compute l */
mpfr_mul (l, x, x, MPFR_RNDU);
mpfr_div_ui (l, l, 3, MPFR_RNDU); /* upper bound on x^2/3 */
mpfr_ui_sub (l, 1, l, MPFR_RNDZ); /* lower bound on 1 - x^2/3 */
mpfr_const_pi (h, MPFR_RNDU); /* upper bound of Pi */
mpfr_sqrt (h, h, MPFR_RNDU); /* upper bound on sqrt(Pi) */
mpfr_div (l, l, h, MPFR_RNDZ); /* lower bound on 1/sqrt(Pi) (1 - x^2/3) */
mpfr_mul_2ui (l, l, 1, MPFR_RNDZ); /* 2/sqrt(Pi) (1 - x^2/3) */
mpfr_mul (l, l, x, MPFR_RNDZ); /* |l| is a lower bound on
|2x/sqrt(Pi) (1 - x^2/3)| */
/* now compute h */
mpfr_const_pi (h, MPFR_RNDD); /* lower bound on Pi */
mpfr_sqrt (h, h, MPFR_RNDD); /* lower bound on sqrt(Pi) */
mpfr_div_2ui (h, h, 1, MPFR_RNDD); /* lower bound on sqrt(Pi)/2 */
/* since sqrt(Pi)/2 < 1, the following should not underflow */
mpfr_div (h, x, h, MPFR_IS_POS(x) ? MPFR_RNDU : MPFR_RNDD);
/* round l and h to precision PREC(y) */
inex = mpfr_prec_round (l, MPFR_PREC(y), rnd_mode);
inex2 = mpfr_prec_round (h, MPFR_PREC(y), rnd_mode);
/* Caution: we also need inex=inex2 (inex might be 0). */
ok = SAME_SIGN (inex, inex2) && mpfr_cmp (l, h) == 0;
if (ok)
mpfr_set (y, h, rnd_mode);
mpfr_clear (l);
mpfr_clear (h);
if (ok)
goto end;
/* this test can still fail for small precision, for example
for x=-0.100E-2 with a target precision of 3 bits, since
the error term x^2/3 is not that small. */
}
MPFR_TMP_INIT1(xf_limb, xf, 53);
mpfr_div (xf, x, __gmpfr_const_log2_RNDU, MPFR_RNDZ); /* round to zero
ensures we get a lower bound of |x/log(2)| */
mpfr_mul (xf, xf, x, MPFR_RNDZ);
large = mpfr_cmp_ui (xf, MPFR_PREC (y) + 1) > 0;
/* when x goes to infinity, we have erf(x) = 1 - 1/sqrt(Pi)/exp(x^2)/x + ...
and |erf(x) - 1| <= exp(-x^2) is true for any x >= 0, thus if
exp(-x^2) < 2^(-PREC(y)-1) the result is 1 or 1-epsilon.
This rewrites as x^2/log(2) > p+1. */
if (MPFR_UNLIKELY (large))
/* |erf x| = 1 or 1- */
{
mpfr_rnd_t rnd2 = MPFR_IS_POS (x) ? rnd_mode : MPFR_INVERT_RND(rnd_mode);
if (rnd2 == MPFR_RNDN || rnd2 == MPFR_RNDU || rnd2 == MPFR_RNDA)
{
inex = MPFR_INT_SIGN (x);
mpfr_set_si (y, inex, rnd2);
}
else /* round to zero */
{
inex = -MPFR_INT_SIGN (x);
mpfr_setmax (y, 0); /* warning: setmax keeps the old sign of y */
MPFR_SET_SAME_SIGN (y, x);
}
}
else /* use Taylor */
{
double xf2;
/* FIXME: get rid of doubles/mpfr_get_d here */
xf2 = mpfr_get_d (x, MPFR_RNDN);
xf2 = xf2 * xf2; /* xf2 ~ x^2 */
inex = mpfr_erf_0 (y, x, xf2, rnd_mode);
}
end:
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inex, rnd_mode);
}
