static void
set_special (mpfr_ptr x, unsigned int select)
{
MPFR_ASSERTN (select < SPECIAL_MAX);
switch (select)
{
case 0:
MPFR_SET_NAN (x);
break;
case 1:
MPFR_SET_INF (x);
MPFR_SET_POS (x);
break;
case 2:
MPFR_SET_INF (x);
MPFR_SET_NEG (x);
break;
case 3:
MPFR_SET_ZERO (x);
MPFR_SET_POS (x);
break;
case 4:
MPFR_SET_ZERO (x);
MPFR_SET_NEG (x);
break;
case 5:
mpfr_set_str_binary (x, "1");
break;
case 6:
mpfr_set_str_binary (x, "-1");
break;
case 7:
mpfr_set_str_binary (x, "1e-1");
break;
case 8:
mpfr_set_str_binary (x, "1e+1");
break;
case 9:
mpfr_const_pi (x, MPFR_RNDN);
break;
case 10:
mpfr_const_pi (x, MPFR_RNDN);
MPFR_SET_EXP (x, MPFR_GET_EXP (x)-1);
break;
default:
mpfr_urandomb (x, RANDS);
if (randlimb () & 1)
mpfr_neg (x, x, MPFR_RNDN);
break;
}
}
static void
check_singular (void)
{
mpfr_t x, got;
mpfr_init2 (x, 100L);
mpfr_init2 (got, 100L);
/* sqrt(NaN) == NaN */
MPFR_SET_NAN (x);
MPFR_ASSERTN (test_sqrt (got, x, MPFR_RNDZ) == 0); /* exact */
MPFR_ASSERTN (mpfr_nan_p (got));
/* sqrt(-1) == NaN */
mpfr_set_si (x, -1L, MPFR_RNDZ);
MPFR_ASSERTN (test_sqrt (got, x, MPFR_RNDZ) == 0); /* exact */
MPFR_ASSERTN (mpfr_nan_p (got));
/* sqrt(+inf) == +inf */
MPFR_SET_INF (x);
MPFR_SET_POS (x);
MPFR_ASSERTN (test_sqrt (got, x, MPFR_RNDZ) == 0); /* exact */
MPFR_ASSERTN (mpfr_inf_p (got));
/* sqrt(-inf) == NaN */
MPFR_SET_INF (x);
MPFR_SET_NEG (x);
MPFR_ASSERTN (test_sqrt (got, x, MPFR_RNDZ) == 0); /* exact */
MPFR_ASSERTN (mpfr_nan_p (got));
/* sqrt(+0) == +0 */
mpfr_set_si (x, 0L, MPFR_RNDZ);
MPFR_ASSERTN (test_sqrt (got, x, MPFR_RNDZ) == 0); /* exact */
MPFR_ASSERTN (mpfr_number_p (got));
MPFR_ASSERTN (mpfr_cmp_ui (got, 0L) == 0);
MPFR_ASSERTN (MPFR_IS_POS (got));
/* sqrt(-0) == -0 */
mpfr_set_si (x, 0L, MPFR_RNDZ);
MPFR_SET_NEG (x);
MPFR_ASSERTN (test_sqrt (got, x, MPFR_RNDZ) == 0); /* exact */
MPFR_ASSERTN (mpfr_number_p (got));
MPFR_ASSERTN (mpfr_cmp_ui (got, 0L) == 0);
MPFR_ASSERTN (MPFR_IS_NEG (got));
mpfr_clear (x);
mpfr_clear (got);
}
void
mpfr_set_str_binary (mpfr_ptr x, const char *str)
{
int has_sign;
int res;
if (*str == 'N')
{
MPFR_SET_NAN(x);
__gmpfr_flags |= MPFR_FLAGS_NAN;
return;
}
has_sign = *str == '-' || *str == '+';
if (str[has_sign] == 'I')
{
MPFR_SET_INF(x);
if (*str == '-')
MPFR_SET_NEG(x);
else
MPFR_SET_POS(x);
return;
}
res = mpfr_strtofr (x, str, 0, 2, MPFR_RNDZ);
MPFR_ASSERTN (res == 0);
}
void
mpfr_set_zero (mpfr_ptr x, int sign)
{
mpfr_set_ui (x, 0, MPFR_RNDN);
if (sign < 0)
MPFR_SET_NEG(x);
}
static void
check_nan (void)
{
mpfr_t d, q;
mpfr_init2 (d, 100L);
mpfr_init2 (q, 100L);
/* 1/+inf == 0 */
MPFR_CLEAR_FLAGS (d);
MPFR_SET_INF (d);
MPFR_SET_POS (d);
MPFR_ASSERTN (mpfr_ui_div (q, 1L, d, GMP_RNDZ) == 0); /* exact */
MPFR_ASSERTN (mpfr_number_p (q));
MPFR_ASSERTN (mpfr_sgn (q) == 0);
/* 1/-inf == -0 */
MPFR_CLEAR_FLAGS (d);
MPFR_SET_INF (d);
MPFR_SET_NEG (d);
MPFR_ASSERTN (mpfr_ui_div (q, 1L, d, GMP_RNDZ) == 0); /* exact */
MPFR_ASSERTN (mpfr_number_p (q));
MPFR_ASSERTN (mpfr_sgn (q) == 0);
/* 1/nan == nan */
MPFR_SET_NAN (d);
MPFR_ASSERTN (mpfr_ui_div (q, 1L, d, GMP_RNDZ) == 0); /* exact */
MPFR_ASSERTN (mpfr_nan_p (q));
/* 0/0 == nan */
mpfr_set_ui (d, 0L, GMP_RNDN);
MPFR_ASSERTN (mpfr_ui_div (q, 0L, d, GMP_RNDZ) == 0); /* exact */
MPFR_ASSERTN (mpfr_nan_p (q));
/* 1/+0 = +inf */
mpfr_set_ui (d, 0L, GMP_RNDN);
MPFR_ASSERTN (mpfr_ui_div (q, 1L, d, GMP_RNDZ) == 0); /* exact */
MPFR_ASSERTN (mpfr_inf_p (q) && mpfr_sgn (q) > 0);
/* 1/-0 = -inf */
mpfr_set_ui (d, 0L, GMP_RNDN);
mpfr_neg (d, d, GMP_RNDN);
MPFR_ASSERTN (mpfr_ui_div (q, 1L, d, GMP_RNDZ) == 0); /* exact */
MPFR_ASSERTN (mpfr_inf_p (q) && mpfr_sgn (q) < 0);
/* 0/1 = +0 */
mpfr_set_ui (d, 1L, GMP_RNDN);
MPFR_ASSERTN (mpfr_ui_div (q, 0L, d, GMP_RNDZ) == 0); /* exact */
MPFR_ASSERTN (mpfr_cmp_ui (q, 0) == 0 && MPFR_IS_POS (q));
/* 0/-1 = -0 */
mpfr_set_si (d, -1, GMP_RNDN);
MPFR_ASSERTN (mpfr_ui_div (q, 0L, d, GMP_RNDZ) == 0); /* exact */
MPFR_ASSERTN (mpfr_cmp_ui (q, 0) == 0 && MPFR_IS_NEG (q));
mpfr_clear (d);
mpfr_clear (q);
}
void
mpfr_set_inf (mpfr_ptr x, int sign)
{
MPFR_SET_INF(x);
if (sign >= 0)
MPFR_SET_POS(x);
else
MPFR_SET_NEG(x);
}
static void
check_sgn(void)
{
mpfr_t x;
int i, s1, s2;
mpfr_init(x);
for(i = 0 ; i < 100 ; i++)
{
mpfr_urandomb (x, RANDS);
if (i&1)
{
MPFR_SET_POS(x);
s2 = 1;
}
else
{
MPFR_SET_NEG(x);
s2 = -1;
}
s1 = mpfr_sgn(x);
if (s1 < -1 || s1 > 1)
{
printf("Error for sgn: out of range.\n");
goto lexit;
}
else if (MPFR_IS_NAN(x) || MPFR_IS_ZERO(x))
{
if (s1 != 0)
{
printf("Error for sgn: Nan or Zero should return 0.\n");
goto lexit;
}
}
else if (s1 != s2)
{
printf("Error for sgn. Return %d instead of %d.\n", s1, s2);
goto lexit;
}
}
mpfr_clear(x);
return;
lexit:
mpfr_clear(x);
exit(1);
}
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;
}
static void
check_special (void)
{
mpfr_t x;
int ret = 0;
mpfr_init (x);
MPFR_SET_ZERO (x);
if ((mpfr_sgn) (x) != 0 || mpfr_sgn (x) != 0)
{
printf("Sgn error for 0.\n");
ret = 1;
}
MPFR_SET_INF (x);
MPFR_SET_POS (x);
if ((mpfr_sgn) (x) != 1 || mpfr_sgn (x) != 1)
{
printf("Sgn error for +Inf.\n");
ret = 1;
}
MPFR_SET_INF (x);
MPFR_SET_NEG (x);
if ((mpfr_sgn) (x) != -1 || mpfr_sgn (x) != -1)
{
printf("Sgn error for -Inf.\n");
ret = 1;
}
MPFR_SET_NAN (x);
mpfr_clear_flags ();
if ((mpfr_sgn) (x) != 0 || !mpfr_erangeflag_p ())
{
printf("Sgn error for NaN.\n");
ret = 1;
}
mpfr_clear_flags ();
if (mpfr_sgn (x) != 0 || !mpfr_erangeflag_p ())
{
printf("Sgn error for NaN.\n");
ret = 1;
}
mpfr_clear (x);
if (ret)
exit (ret);
}
int
mpfr_log1p (mpfr_ptr y, mpfr_srcptr x, mp_rnd_t rnd_mode)
{
int comp, inexact;
mp_exp_t ex;
MPFR_SAVE_EXPO_DECL (expo);
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
{
if (MPFR_IS_NAN (x))
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
/* check for inf or -inf (result is not defined) */
else if (MPFR_IS_INF (x))
{
if (MPFR_IS_POS (x))
{
MPFR_SET_INF (y);
MPFR_SET_POS (y);
MPFR_RET (0);
}
else
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
}
else /* x is zero */
{
MPFR_ASSERTD (MPFR_IS_ZERO (x));
MPFR_SET_ZERO (y); /* log1p(+/- 0) = +/- 0 */
MPFR_SET_SAME_SIGN (y, x);
MPFR_RET (0);
}
}
ex = MPFR_GET_EXP (x);
if (ex < 0) /* -0.5 < x < 0.5 */
{
/* For x > 0, abs(log(1+x)-x) < x^2/2.
For x > -0.5, abs(log(1+x)-x) < x^2. */
if (MPFR_IS_POS (x))
MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, x, - ex - 1, 0, 0, rnd_mode, {});
else
MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, x, - ex, 0, 1, rnd_mode, {});
}
comp = mpfr_cmp_si (x, -1);
/* log1p(x) is undefined for x < -1 */
if (MPFR_UNLIKELY(comp <= 0))
{
if (comp == 0)
/* x=0: log1p(-1)=-inf (division by zero) */
{
MPFR_SET_INF (y);
MPFR_SET_NEG (y);
MPFR_RET (0);
}
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
MPFR_SAVE_EXPO_MARK (expo);
/* General case */
{
/* Declaration of the intermediary variable */
mpfr_t t;
/* Declaration of the size variable */
mp_prec_t Ny = MPFR_PREC(y); /* target precision */
mp_prec_t Nt; /* working precision */
mp_exp_t err; /* error */
MPFR_ZIV_DECL (loop);
/* compute the precision of intermediary variable */
/* the optimal number of bits : see algorithms.tex */
Nt = Ny + MPFR_INT_CEIL_LOG2 (Ny) + 6;
/* if |x| is smaller than 2^(-e), we will loose about e bits
in log(1+x) */
if (MPFR_EXP(x) < 0)
Nt += -MPFR_EXP(x);
/* initialise of intermediary variable */
mpfr_init2 (t, Nt);
/* First computation of log1p */
MPFR_ZIV_INIT (loop, Nt);
for (;;)
{
/* compute log1p */
inexact = mpfr_add_ui (t, x, 1, GMP_RNDN); /* 1+x */
/* if inexact = 0, then t = x+1, and the result is simply log(t) */
if (inexact == 0)
{
inexact = mpfr_log (y, t, rnd_mode);
goto end;
}
mpfr_log (t, t, GMP_RNDN); /* log(1+x) */
/* the error is bounded by (1/2+2^(1-EXP(t))*ulp(t) (cf algorithms.tex)
if EXP(t)>=2, then error <= ulp(t)
if EXP(t)<=1, then error <= 2^(2-EXP(t))*ulp(t) */
err = Nt - MAX (0, 2 - MPFR_GET_EXP (t));
if (MPFR_LIKELY (MPFR_CAN_ROUND (t, err, Ny, rnd_mode)))
break;
/* increase the precision */
MPFR_ZIV_NEXT (loop, Nt);
mpfr_set_prec (t, Nt);
}
inexact = mpfr_set (y, t, rnd_mode);
end:
MPFR_ZIV_FREE (loop);
mpfr_clear (t);
}
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inexact, rnd_mode);
}
int
mpfr_eint (mpfr_ptr y, mpfr_srcptr x, mp_rnd_t rnd)
{
int inex;
mpfr_t tmp, ump;
mp_exp_t err, te;
mp_prec_t prec;
MPFR_SAVE_EXPO_DECL (expo);
MPFR_ZIV_DECL (loop);
MPFR_LOG_FUNC (("x[%#R]=%R rnd=%d", x, x, rnd),
("y[%#R]=%R inexact=%d", y, y, inex));
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
{
/* exp(NaN) = exp(-Inf) = NaN */
if (MPFR_IS_NAN (x) || (MPFR_IS_INF (x) && MPFR_IS_NEG(x)))
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
/* eint(+inf) = +inf */
else if (MPFR_IS_INF (x))
{
MPFR_SET_INF(y);
MPFR_SET_POS(y);
MPFR_RET(0);
}
else /* eint(+/-0) = -Inf */
{
MPFR_SET_INF(y);
MPFR_SET_NEG(y);
MPFR_RET(0);
}
}
/* eint(x) = NaN for x < 0 */
if (MPFR_IS_NEG(x))
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
MPFR_SAVE_EXPO_MARK (expo);
/* Since eint(x) >= exp(x)/x, we have log2(eint(x)) >= (x-log(x))/log(2).
Let's compute k <= (x-log(x))/log(2) in a low precision. If k >= emax,
then log2(eint(x)) >= emax, and eint(x) >= 2^emax, i.e. it overflows. */
mpfr_init2 (tmp, 64);
mpfr_init2 (ump, 64);
mpfr_log (tmp, x, GMP_RNDU);
mpfr_sub (ump, x, tmp, GMP_RNDD);
mpfr_const_log2 (tmp, GMP_RNDU);
mpfr_div (ump, ump, tmp, GMP_RNDD);
/* FIXME: We really need mpfr_set_exp_t and mpfr_cmp_exp_t functions. */
MPFR_ASSERTN (MPFR_EMAX_MAX <= LONG_MAX);
if (mpfr_cmp_ui (ump, __gmpfr_emax) >= 0)
{
mpfr_clear (tmp);
mpfr_clear (ump);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_overflow (y, rnd, 1);
}
/* Init stuff */
prec = MPFR_PREC (y) + 2 * MPFR_INT_CEIL_LOG2 (MPFR_PREC (y)) + 6;
/* eint() has a root 0.37250741078136663446..., so if x is near,
already take more bits */
if (MPFR_GET_EXP(x) == -1) /* 1/4 <= x < 1/2 */
{
double d;
d = mpfr_get_d (x, GMP_RNDN) - 0.37250741078136663;
d = (d == 0.0) ? -53 : __gmpfr_ceil_log2 (d);
prec += -d;
}
mpfr_set_prec (tmp, prec);
mpfr_set_prec (ump, prec);
MPFR_ZIV_INIT (loop, prec); /* Initialize the ZivLoop controler */
for (;;) /* Infinite loop */
{
/* We need that the smallest value of k!/x^k is smaller than 2^(-p).
The minimum is obtained for x=k, and it is smaller than e*sqrt(x)/e^x
for x>=1. */
if (MPFR_GET_EXP (x) > 0 && mpfr_cmp_d (x, ((double) prec +
0.5 * (double) MPFR_GET_EXP (x)) * LOG2 + 1.0) > 0)
err = mpfr_eint_asympt (tmp, x);
else
{
err = mpfr_eint_aux (tmp, x); /* error <= 2^err ulp(tmp) */
te = MPFR_GET_EXP(tmp);
mpfr_const_euler (ump, GMP_RNDN); /* 0.577 -> EXP(ump)=0 */
mpfr_add (tmp, tmp, ump, GMP_RNDN);
/* error <= 1/2 + 1/2*2^(EXP(ump)-EXP(tmp)) + 2^(te-EXP(tmp)+err)
<= 1/2 + 2^(MAX(EXP(ump), te+err+1) - EXP(tmp))
<= 2^(MAX(0, 1 + MAX(EXP(ump), te+err+1) - EXP(tmp))) */
err = MAX(1, te + err + 2) - MPFR_GET_EXP(tmp);
err = MAX(0, err);
te = MPFR_GET_EXP(tmp);
mpfr_log (ump, x, GMP_RNDN);
mpfr_add (tmp, tmp, ump, GMP_RNDN);
/* same formula as above, except now EXP(ump) is not 0 */
err += te + 1;
if (MPFR_LIKELY (!MPFR_IS_ZERO (ump)))
err = MAX (MPFR_GET_EXP (ump), err);
err = MAX(0, err - MPFR_GET_EXP (tmp));
}
if (MPFR_LIKELY (MPFR_CAN_ROUND (tmp, prec - err, MPFR_PREC (y), rnd)))
break;
MPFR_ZIV_NEXT (loop, prec); /* Increase used precision */
mpfr_set_prec (tmp, prec);
mpfr_set_prec (ump, prec);
}
MPFR_ZIV_FREE (loop); /* Free the ZivLoop Controler */
inex = mpfr_set (y, tmp, rnd); /* Set y to the computed value */
mpfr_clear (tmp);
mpfr_clear (ump);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inex, rnd);
}
int
mpfr_pow_si (mpfr_ptr y, mpfr_srcptr x, long int n, mp_rnd_t rnd)
{
if (n >= 0)
return mpfr_pow_ui (y, x, n, rnd);
else
{
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))
{
MPFR_SET_ZERO (y);
if (MPFR_IS_POS (x) || ((unsigned) n & 1) == 0)
MPFR_SET_POS (y);
else
MPFR_SET_NEG (y);
MPFR_RET (0);
}
else /* x is zero */
{
MPFR_ASSERTD (MPFR_IS_ZERO (x));
MPFR_SET_INF(y);
if (MPFR_IS_POS (x) || ((unsigned) n & 1) == 0)
MPFR_SET_POS (y);
else
MPFR_SET_NEG (y);
MPFR_RET(0);
}
}
MPFR_CLEAR_FLAGS (y);
/* detect exact powers: x^(-n) is exact iff x is a power of 2 */
if (mpfr_cmp_si_2exp (x, MPFR_SIGN(x), MPFR_EXP(x) - 1) == 0)
{
mp_exp_t expx = MPFR_EXP (x) - 1, expy;
MPFR_ASSERTD (n < 0);
/* Warning: n * expx may overflow!
* Some systems (apparently alpha-freebsd) abort with
* LONG_MIN / 1, and LONG_MIN / -1 is undefined.
* Proof of the overflow checking. The expressions below are
* assumed to be on the rational numbers, but the word "overflow"
* still has its own meaning in the C context. / still denotes
* the integer (truncated) division, and // denotes the exact
* division.
* - First, (__gmpfr_emin - 1) / n and (__gmpfr_emax - 1) / n
* cannot overflow due to the constraints on the exponents of
* MPFR numbers.
* - If n = -1, then n * expx = - expx, which is representable
* because of the constraints on the exponents of MPFR numbers.
* - If expx = 0, then n * expx = 0, which is representable.
* - If n < -1 and expx > 0:
* + If expx > (__gmpfr_emin - 1) / n, then
* expx >= (__gmpfr_emin - 1) / n + 1
* > (__gmpfr_emin - 1) // n,
* and
* n * expx < __gmpfr_emin - 1,
* i.e.
* n * expx <= __gmpfr_emin - 2.
* This corresponds to an underflow, with a null result in
* the rounding-to-nearest mode.
* + If expx <= (__gmpfr_emin - 1) / n, then n * expx cannot
* overflow since 0 < expx <= (__gmpfr_emin - 1) / n and
* 0 > n * expx >= n * ((__gmpfr_emin - 1) / n)
* >= __gmpfr_emin - 1.
* - If n < -1 and expx < 0:
* + If expx < (__gmpfr_emax - 1) / n, then
* expx <= (__gmpfr_emax - 1) / n - 1
* < (__gmpfr_emax - 1) // n,
* and
* n * expx > __gmpfr_emax - 1,
* i.e.
* n * expx >= __gmpfr_emax.
* This corresponds to an overflow (2^(n * expx) has an
* exponent > __gmpfr_emax).
* + If expx >= (__gmpfr_emax - 1) / n, then n * expx cannot
* overflow since 0 > expx >= (__gmpfr_emax - 1) / n and
* 0 < n * expx <= n * ((__gmpfr_emax - 1) / n)
* <= __gmpfr_emax - 1.
* Note: one could use expx bounds based on MPFR_EXP_MIN and
* MPFR_EXP_MAX instead of __gmpfr_emin and __gmpfr_emax. The
* current bounds do not lead to noticeably slower code and
* allow us to avoid a bug in Sun's compiler for Solaris/x86
* (when optimizations are enabled).
*/
expy =
n != -1 && expx > 0 && expx > (__gmpfr_emin - 1) / n ?
MPFR_EMIN_MIN - 2 /* Underflow */ :
n != -1 && expx < 0 && expx < (__gmpfr_emax - 1) / n ?
MPFR_EMAX_MAX /* Overflow */ : n * expx;
return mpfr_set_si_2exp (y, n % 2 ? MPFR_INT_SIGN (x) : 1,
expy, rnd);
}
/* General case */
{
/* Declaration of the intermediary variable */
mpfr_t t;
/* Declaration of the size variable */
mp_prec_t Ny = MPFR_PREC (y); /* target precision */
mp_prec_t Nt; /* working precision */
mp_exp_t err; /* error */
int inexact;
unsigned long abs_n;
MPFR_SAVE_EXPO_DECL (expo);
MPFR_ZIV_DECL (loop);
abs_n = - (unsigned long) n;
/* compute the precision of intermediary variable */
/* the optimal number of bits : see algorithms.tex */
Nt = Ny + 3 + MPFR_INT_CEIL_LOG2 (Ny);
MPFR_SAVE_EXPO_MARK (expo);
/* initialise of intermediary variable */
mpfr_init2 (t, Nt);
MPFR_ZIV_INIT (loop, Nt);
for (;;)
{
/* compute 1/(x^n), with n > 0 */
mpfr_pow_ui (t, x, abs_n, GMP_RNDN);
mpfr_ui_div (t, 1, t, GMP_RNDN);
/* FIXME: old code improved, but I think this is still incorrect. */
if (MPFR_UNLIKELY (MPFR_IS_ZERO (t)))
{
MPFR_ZIV_FREE (loop);
mpfr_clear (t);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_underflow (y, rnd == GMP_RNDN ? GMP_RNDZ : rnd,
abs_n & 1 ? MPFR_SIGN (x) :
MPFR_SIGN_POS);
}
if (MPFR_UNLIKELY (MPFR_IS_INF (t)))
{
MPFR_ZIV_FREE (loop);
mpfr_clear (t);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_overflow (y, rnd, abs_n & 1 ? MPFR_SIGN (x) :
MPFR_SIGN_POS);
}
/* error estimate -- see pow function in algorithms.ps */
err = Nt - 3;
if (MPFR_LIKELY (MPFR_CAN_ROUND (t, err, Ny, rnd)))
break;
/* actualisation of the precision */
Nt += BITS_PER_MP_LIMB;
mpfr_set_prec (t, Nt);
}
MPFR_ZIV_FREE (loop);
inexact = mpfr_set (y, t, rnd);
mpfr_clear (t);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inexact, rnd);
}
}
}
/* 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) && is_odd (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) && is_odd (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) && is_odd (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.
*/
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_SIGN (x) > 0)
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_SIGN(x) < 0 && is_odd (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))
{
mpfr_t tmp;
mpfr_eexp_t ebound;
int inex2;
/* We must restore the flags. */
MPFR_SAVE_EXPO_MARK (expo);
mpfr_init2 (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_clear (tmp);
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_SIGN (x) < 0 && is_odd (y) ? -1 : 1);
}
}
/* 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. */
{
mpfr_exp_t b = MPFR_GET_EXP (x) - 1;
MPFR_ASSERTN (b >= LONG_MIN && b <= LONG_MAX); /* FIXME... */
if (mpfr_cmp_si_2exp (x, MPFR_SIGN(x), b) == 0)
{
mpfr_t tmp;
int sgnx = MPFR_SIGN (x);
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 && is_odd (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_t t;
mpfr_exp_t err;
/* We need an upper bound on the exponent of y * log(x). */
mpfr_init2 (t, 16);
if (MPFR_IS_POS(x))
mpfr_log (t, x, cmp_x_1 < 0 ? MPFR_RNDD : MPFR_RNDU); /* away from 0 */
else
{
/* if x < -1, round to +Inf, else round to zero */
mpfr_neg (t, x, (mpfr_cmp_si (x, -1) < 0) ? MPFR_RNDU : MPFR_RNDD);
mpfr_log (t, t, (mpfr_cmp_ui (t, 1) < 0) ? MPFR_RNDD : MPFR_RNDU);
}
MPFR_ASSERTN (MPFR_IS_PURE_FP (t));
err = MPFR_GET_EXP (y) + MPFR_GET_EXP (t);
mpfr_clear (t);
MPFR_CLEAR_FLAGS ();
MPFR_SMALL_INPUT_AFTER_SAVE_EXPO (z, __gmpfr_one, - err, 0,
(MPFR_SIGN (y) > 0) ^ (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);
}
/* Assumes that the exponent range has already been extended and if y is
an integer, then the result is not exact in unbounded exponent range. */
int
mpfr_pow_general (mpfr_ptr z, mpfr_srcptr x, mpfr_srcptr y,
mpfr_rnd_t rnd_mode, int y_is_integer, mpfr_save_expo_t *expo)
{
mpfr_t t, u, k, absx;
int neg_result = 0;
int k_non_zero = 0;
int check_exact_case = 0;
int inexact;
/* Declaration of the size variable */
mpfr_prec_t Nz = MPFR_PREC(z); /* target precision */
mpfr_prec_t Nt; /* working precision */
mpfr_exp_t err; /* error */
MPFR_ZIV_DECL (ziv_loop);
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));
/* We put the absolute value of x in absx, pointing to the significand
of x to avoid allocating memory for the significand of absx. */
MPFR_ALIAS(absx, x, /*sign=*/ 1, /*EXP=*/ MPFR_EXP(x));
/* We will compute the absolute value of the result. So, let's
invert the rounding mode if the result is negative. */
if (MPFR_IS_NEG (x) && is_odd (y))
{
neg_result = 1;
rnd_mode = MPFR_INVERT_RND (rnd_mode);
}
/* compute the precision of intermediary variable */
/* the optimal number of bits : see algorithms.tex */
Nt = Nz + 5 + MPFR_INT_CEIL_LOG2 (Nz);
/* initialise of intermediary variable */
mpfr_init2 (t, Nt);
MPFR_ZIV_INIT (ziv_loop, Nt);
for (;;)
{
MPFR_BLOCK_DECL (flags1);
/* compute exp(y*ln|x|), using MPFR_RNDU to get an upper bound, so
that we can detect underflows. */
mpfr_log (t, absx, MPFR_IS_NEG (y) ? MPFR_RNDD : MPFR_RNDU); /* ln|x| */
mpfr_mul (t, y, t, MPFR_RNDU); /* y*ln|x| */
if (k_non_zero)
{
MPFR_LOG_MSG (("subtract k * ln(2)\n", 0));
mpfr_const_log2 (u, MPFR_RNDD);
mpfr_mul (u, u, k, MPFR_RNDD);
/* Error on u = k * log(2): < k * 2^(-Nt) < 1. */
mpfr_sub (t, t, u, MPFR_RNDU);
MPFR_LOG_MSG (("t = y * ln|x| - k * ln(2)\n", 0));
MPFR_LOG_VAR (t);
}
/* estimate of the error -- see pow function in algorithms.tex.
The error on t is at most 1/2 + 3*2^(EXP(t)+1) ulps, which is
<= 2^(EXP(t)+3) for EXP(t) >= -1, and <= 2 ulps for EXP(t) <= -2.
Additional error if k_no_zero: treal = t * errk, with
1 - |k| * 2^(-Nt) <= exp(-|k| * 2^(-Nt)) <= errk <= 1,
i.e., additional absolute error <= 2^(EXP(k)+EXP(t)-Nt).
Total error <= 2^err1 + 2^err2 <= 2^(max(err1,err2)+1). */
err = MPFR_NOTZERO (t) && MPFR_GET_EXP (t) >= -1 ?
MPFR_GET_EXP (t) + 3 : 1;
if (k_non_zero)
{
if (MPFR_GET_EXP (k) > err)
err = MPFR_GET_EXP (k);
err++;
}
MPFR_BLOCK (flags1, mpfr_exp (t, t, MPFR_RNDN)); /* exp(y*ln|x|)*/
/* We need to test */
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (t) || MPFR_UNDERFLOW (flags1)))
{
mpfr_prec_t Ntmin;
MPFR_BLOCK_DECL (flags2);
MPFR_ASSERTN (!k_non_zero);
MPFR_ASSERTN (!MPFR_IS_NAN (t));
/* Real underflow? */
if (MPFR_IS_ZERO (t))
{
/* Underflow. We computed rndn(exp(t)), where t >= y*ln|x|.
Therefore rndn(|x|^y) = 0, and we have a real underflow on
|x|^y. */
inexact = mpfr_underflow (z, rnd_mode == MPFR_RNDN ? MPFR_RNDZ
: rnd_mode, MPFR_SIGN_POS);
if (expo != NULL)
MPFR_SAVE_EXPO_UPDATE_FLAGS (*expo, MPFR_FLAGS_INEXACT
| MPFR_FLAGS_UNDERFLOW);
break;
}
/* Real overflow? */
if (MPFR_IS_INF (t))
{
/* Note: we can probably use a low precision for this test. */
mpfr_log (t, absx, MPFR_IS_NEG (y) ? MPFR_RNDU : MPFR_RNDD);
mpfr_mul (t, y, t, MPFR_RNDD); /* y * ln|x| */
MPFR_BLOCK (flags2, mpfr_exp (t, t, MPFR_RNDD));
/* t = lower bound on exp(y * ln|x|) */
if (MPFR_OVERFLOW (flags2))
{
/* We have computed a lower bound on |x|^y, and it
overflowed. Therefore we have a real overflow
on |x|^y. */
inexact = mpfr_overflow (z, rnd_mode, MPFR_SIGN_POS);
if (expo != NULL)
MPFR_SAVE_EXPO_UPDATE_FLAGS (*expo, MPFR_FLAGS_INEXACT
| MPFR_FLAGS_OVERFLOW);
break;
}
}
k_non_zero = 1;
Ntmin = sizeof(mpfr_exp_t) * CHAR_BIT;
if (Ntmin > Nt)
{
Nt = Ntmin;
mpfr_set_prec (t, Nt);
}
mpfr_init2 (u, Nt);
mpfr_init2 (k, Ntmin);
mpfr_log2 (k, absx, MPFR_RNDN);
mpfr_mul (k, y, k, MPFR_RNDN);
mpfr_round (k, k);
MPFR_LOG_VAR (k);
/* |y| < 2^Ntmin, therefore |k| < 2^Nt. */
continue;
}
if (MPFR_LIKELY (MPFR_CAN_ROUND (t, Nt - err, Nz, rnd_mode)))
{
inexact = mpfr_set (z, t, rnd_mode);
break;
}
/* check exact power, except when y is an integer (since the
exact cases for y integer have already been filtered out) */
if (check_exact_case == 0 && ! y_is_integer)
{
if (mpfr_pow_is_exact (z, absx, y, rnd_mode, &inexact))
break;
check_exact_case = 1;
}
/* reactualisation of the precision */
MPFR_ZIV_NEXT (ziv_loop, Nt);
mpfr_set_prec (t, Nt);
if (k_non_zero)
mpfr_set_prec (u, Nt);
}
MPFR_ZIV_FREE (ziv_loop);
if (k_non_zero)
{
int inex2;
long lk;
/* The rounded result in an unbounded exponent range is z * 2^k. As
* MPFR chooses underflow after rounding, the mpfr_mul_2si below will
* correctly detect underflows and overflows. However, in rounding to
* nearest, if z * 2^k = 2^(emin - 2), then the double rounding may
* affect the result. We need to cope with that before overwriting z.
* This can occur only if k < 0 (this test is necessary to avoid a
* potential integer overflow).
* If inexact >= 0, then the real result is <= 2^(emin - 2), so that
* o(2^(emin - 2)) = +0 is correct. If inexact < 0, then the real
* result is > 2^(emin - 2) and we need to round to 2^(emin - 1).
*/
MPFR_ASSERTN (MPFR_EXP_MAX <= LONG_MAX);
lk = mpfr_get_si (k, MPFR_RNDN);
/* Due to early overflow detection, |k| should not be much larger than
* MPFR_EMAX_MAX, and as MPFR_EMAX_MAX <= MPFR_EXP_MAX/2 <= LONG_MAX/2,
* an overflow should not be possible in mpfr_get_si (and lk is exact).
* And one even has the following assertion. TODO: complete proof.
*/
MPFR_ASSERTD (lk > LONG_MIN && lk < LONG_MAX);
/* Note: even in case of overflow (lk inexact), the code is correct.
* Indeed, for the 3 occurrences of lk:
* - The test lk < 0 is correct as sign(lk) = sign(k).
* - In the test MPFR_GET_EXP (z) == __gmpfr_emin - 1 - lk,
* if lk is inexact, then lk = LONG_MIN <= MPFR_EXP_MIN
* (the minimum value of the mpfr_exp_t type), and
* __gmpfr_emin - 1 - lk >= MPFR_EMIN_MIN - 1 - 2 * MPFR_EMIN_MIN
* >= - MPFR_EMIN_MIN - 1 = MPFR_EMAX_MAX - 1. However, from the
* choice of k, z has been chosen to be around 1, so that the
* result of the test is false, as if lk were exact.
* - In the mpfr_mul_2si (z, z, lk, rnd_mode), if lk is inexact,
* then |lk| >= LONG_MAX >= MPFR_EXP_MAX, and as z is around 1,
* mpfr_mul_2si underflows or overflows in the same way as if
* lk were exact.
* TODO: give a bound on |t|, then on |EXP(z)|.
*/
if (rnd_mode == MPFR_RNDN && inexact < 0 && lk < 0 &&
MPFR_GET_EXP (z) == __gmpfr_emin - 1 - lk && mpfr_powerof2_raw (z))
{
/* Rounding to nearest, real result > z * 2^k = 2^(emin - 2),
* underflow case: as the minimum precision is > 1, we will
* obtain the correct result and exceptions by replacing z by
* nextabove(z).
*/
MPFR_ASSERTN (MPFR_PREC_MIN > 1);
mpfr_nextabove (z);
}
MPFR_CLEAR_FLAGS ();
inex2 = mpfr_mul_2si (z, z, lk, rnd_mode);
if (inex2) /* underflow or overflow */
{
inexact = inex2;
if (expo != NULL)
MPFR_SAVE_EXPO_UPDATE_FLAGS (*expo, __gmpfr_flags);
}
mpfr_clears (u, k, (mpfr_ptr) 0);
}
mpfr_clear (t);
/* update the sign of the result if x was negative */
if (neg_result)
{
MPFR_SET_NEG(z);
inexact = -inexact;
}
return inexact;
}
int
mpfr_zeta (mpfr_t z, mpfr_srcptr s, mp_rnd_t rnd_mode)
{
mpfr_t z_pre, s1, y, p;
double sd, eps, m1, c;
long add;
mp_prec_t precz, prec1, precs, precs1;
int inex;
MPFR_GROUP_DECL (group);
MPFR_ZIV_DECL (loop);
MPFR_SAVE_EXPO_DECL (expo);
MPFR_LOG_FUNC (("s[%#R]=%R rnd=%d", s, s, rnd_mode),
("z[%#R]=%R inexact=%d", z, z, inex));
/* Zero, Nan or Inf ? */
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (s)))
{
if (MPFR_IS_NAN (s))
{
MPFR_SET_NAN (z);
MPFR_RET_NAN;
}
else if (MPFR_IS_INF (s))
{
if (MPFR_IS_POS (s))
return mpfr_set_ui (z, 1, GMP_RNDN); /* Zeta(+Inf) = 1 */
MPFR_SET_NAN (z); /* Zeta(-Inf) = NaN */
MPFR_RET_NAN;
}
else /* s iz zero */
{
MPFR_ASSERTD (MPFR_IS_ZERO (s));
mpfr_set_ui (z, 1, rnd_mode);
mpfr_div_2ui (z, z, 1, rnd_mode);
MPFR_CHANGE_SIGN (z);
MPFR_RET (0);
}
}
/* s is neither Nan, nor Inf, nor Zero */
/* check tiny s: we have zeta(s) = -1/2 - 1/2 log(2 Pi) s + ... around s=0,
and for |s| <= 0.074, we have |zeta(s) + 1/2| <= |s|.
Thus if |s| <= 1/4*ulp(1/2), we can deduce the correct rounding
(the 1/4 covers the case where |zeta(s)| < 1/2 and rounding to nearest).
A sufficient condition is that EXP(s) + 1 < -PREC(z). */
if (MPFR_EXP(s) + 1 < - (mp_exp_t) MPFR_PREC(z))
{
int signs = MPFR_SIGN(s);
mpfr_set_si_2exp (z, -1, -1, rnd_mode); /* -1/2 */
if ((rnd_mode == GMP_RNDU || rnd_mode == GMP_RNDZ) && signs < 0)
{
mpfr_nextabove (z); /* z = -1/2 + epsilon */
inex = 1;
}
else if (rnd_mode == GMP_RNDD && signs > 0)
{
mpfr_nextbelow (z); /* z = -1/2 - epsilon */
inex = -1;
}
else
{
if (rnd_mode == GMP_RNDU) /* s > 0: z = -1/2 */
inex = 1;
else if (rnd_mode == GMP_RNDD)
inex = -1; /* s < 0: z = -1/2 */
else /* (GMP_RNDZ and s > 0) or GMP_RNDN: z = -1/2 */
inex = (signs > 0) ? 1 : -1;
}
return mpfr_check_range (z, inex, rnd_mode);
}
/* Check for case s= -2n */
if (MPFR_IS_NEG (s))
{
mpfr_t tmp;
tmp[0] = *s;
MPFR_EXP (tmp) = MPFR_EXP (s) - 1;
if (mpfr_integer_p (tmp))
{
MPFR_SET_ZERO (z);
MPFR_SET_POS (z);
MPFR_RET (0);
}
}
MPFR_SAVE_EXPO_MARK (expo);
/* Compute Zeta */
if (MPFR_IS_POS (s) && MPFR_GET_EXP (s) >= 0) /* Case s >= 1/2 */
inex = mpfr_zeta_pos (z, s, rnd_mode);
else /* use reflection formula
zeta(s) = 2^s*Pi^(s-1)*sin(Pi*s/2)*gamma(1-s)*zeta(1-s) */
{
precz = MPFR_PREC (z);
precs = MPFR_PREC (s);
/* Precision precs1 needed to represent 1 - s, and s + 2,
without any truncation */
precs1 = precs + 2 + MAX (0, - MPFR_GET_EXP (s));
sd = mpfr_get_d (s, GMP_RNDN) - 1.0;
if (sd < 0.0)
sd = -sd; /* now sd = abs(s-1.0) */
/* Precision prec1 is the precision on elementary computations;
it ensures a final precision prec1 - add for zeta(s) */
/* eps = pow (2.0, - (double) precz - 14.0); */
eps = __gmpfr_ceil_exp2 (- (double) precz - 14.0);
m1 = 1.0 + MAX(1.0 / eps, 2.0 * sd) * (1.0 + eps);
c = (1.0 + eps) * (1.0 + eps * MAX(8.0, m1));
/* add = 1 + floor(log(c*c*c*(13 + m1))/log(2)); */
add = __gmpfr_ceil_log2 (c * c * c * (13.0 + m1));
prec1 = precz + add;
prec1 = MAX (prec1, precs1) + 10;
MPFR_GROUP_INIT_4 (group, prec1, z_pre, s1, y, p);
MPFR_ZIV_INIT (loop, prec1);
for (;;)
{
mpfr_sub (s1, __gmpfr_one, s, GMP_RNDN);/* s1 = 1-s */
mpfr_zeta_pos (z_pre, s1, GMP_RNDN); /* zeta(1-s) */
mpfr_gamma (y, s1, GMP_RNDN); /* gamma(1-s) */
if (MPFR_IS_INF (y)) /* Zeta(s) < 0 for -4k-2 < s < -4k,
Zeta(s) > 0 for -4k < s < -4k+2 */
{
MPFR_SET_INF (z_pre);
mpfr_div_2ui (s1, s, 2, GMP_RNDN); /* s/4, exact */
mpfr_frac (s1, s1, GMP_RNDN); /* exact, -1 < s1 < 0 */
if (mpfr_cmp_si_2exp (s1, -1, -1) > 0)
MPFR_SET_NEG (z_pre);
else
MPFR_SET_POS (z_pre);
break;
}
mpfr_mul (z_pre, z_pre, y, GMP_RNDN); /* gamma(1-s)*zeta(1-s) */
mpfr_const_pi (p, GMP_RNDD);
mpfr_mul (y, s, p, GMP_RNDN);
mpfr_div_2ui (y, y, 1, GMP_RNDN); /* s*Pi/2 */
mpfr_sin (y, y, GMP_RNDN); /* sin(Pi*s/2) */
mpfr_mul (z_pre, z_pre, y, GMP_RNDN);
mpfr_mul_2ui (y, p, 1, GMP_RNDN); /* 2*Pi */
mpfr_neg (s1, s1, GMP_RNDN); /* s-1 */
mpfr_pow (y, y, s1, GMP_RNDN); /* (2*Pi)^(s-1) */
mpfr_mul (z_pre, z_pre, y, GMP_RNDN);
mpfr_mul_2ui (z_pre, z_pre, 1, GMP_RNDN);
if (MPFR_LIKELY (MPFR_CAN_ROUND (z_pre, prec1 - add, precz,
rnd_mode)))
break;
MPFR_ZIV_NEXT (loop, prec1);
MPFR_GROUP_REPREC_4 (group, prec1, z_pre, s1, y, p);
}
MPFR_ZIV_FREE (loop);
inex = mpfr_set (z, z_pre, rnd_mode);
MPFR_GROUP_CLEAR (group);
}
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (z, inex, rnd_mode);
}
static void
check_pow_ui (void)
{
mpfr_t a, b;
int res;
mpfr_init2 (a, 53);
mpfr_init2 (b, 53);
/* check in-place operations */
mpfr_set_str (b, "0.6926773", 10, GMP_RNDN);
mpfr_pow_ui (a, b, 10, GMP_RNDN);
mpfr_pow_ui (b, b, 10, GMP_RNDN);
if (mpfr_cmp (a, b))
{
printf ("Error for mpfr_pow_ui (b, b, ...)\n");
exit (1);
}
/* check large exponents */
mpfr_set_ui (b, 1, GMP_RNDN);
mpfr_pow_ui (a, b, 4294967295UL, GMP_RNDN);
mpfr_set_inf (a, -1);
mpfr_pow_ui (a, a, 4049053855UL, GMP_RNDN);
if (!mpfr_inf_p (a) || (mpfr_sgn (a) >= 0))
{
printf ("Error for (-Inf)^4049053855\n");
exit (1);
}
mpfr_set_inf (a, -1);
mpfr_pow_ui (a, a, (unsigned long) 30002752, GMP_RNDN);
if (!mpfr_inf_p (a) || (mpfr_sgn (a) <= 0))
{
printf ("Error for (-Inf)^30002752\n");
exit (1);
}
/* Check underflow */
mpfr_set_str_binary (a, "1E-1");
res = mpfr_pow_ui (a, a, -mpfr_get_emin (), GMP_RNDN);
if (MPFR_GET_EXP (a) != mpfr_get_emin () + 1)
{
printf ("Error for (1e-1)^MPFR_EMAX_MAX\n");
mpfr_dump (a);
exit (1);
}
mpfr_set_str_binary (a, "1E-10");
res = mpfr_pow_ui (a, a, -mpfr_get_emin (), GMP_RNDZ);
if (!MPFR_IS_ZERO (a))
{
printf ("Error for (1e-10)^MPFR_EMAX_MAX\n");
mpfr_dump (a);
exit (1);
}
/* Check overflow */
mpfr_set_str_binary (a, "1E10");
res = mpfr_pow_ui (a, a, ULONG_MAX, GMP_RNDN);
if (!MPFR_IS_INF (a) || MPFR_SIGN (a) < 0)
{
printf ("Error for (1e10)^ULONG_MAX\n");
exit (1);
}
/* Check 0 */
MPFR_SET_ZERO (a);
MPFR_SET_POS (a);
mpfr_set_si (b, -1, GMP_RNDN);
res = mpfr_pow_ui (b, a, 1, GMP_RNDN);
if (res != 0 || MPFR_IS_NEG (b))
{
printf ("Error for (0+)^1\n");
exit (1);
}
MPFR_SET_ZERO (a);
MPFR_SET_NEG (a);
mpfr_set_ui (b, 1, GMP_RNDN);
res = mpfr_pow_ui (b, a, 5, GMP_RNDN);
if (res != 0 || MPFR_IS_POS (b))
{
printf ("Error for (0-)^5\n");
exit (1);
}
MPFR_SET_ZERO (a);
MPFR_SET_NEG (a);
mpfr_set_si (b, -1, GMP_RNDN);
res = mpfr_pow_ui (b, a, 6, GMP_RNDN);
if (res != 0 || MPFR_IS_NEG (b))
{
printf ("Error for (0-)^6\n");
exit (1);
}
mpfr_set_prec (a, 122);
mpfr_set_prec (b, 122);
mpfr_set_str_binary (a, "0.10000010010000111101001110100101101010011110011100001111000001001101000110011001001001001011001011010110110110101000111011E1");
mpfr_set_str_binary (b, "0.11111111100101001001000001000001100011100000001110111111100011111000111011100111111111110100011000111011000100100011001011E51290375");
mpfr_pow_ui (a, a, 2026876995UL, GMP_RNDU);
if (mpfr_cmp (a, b) != 0)
{
printf ("Error for x^2026876995\n");
exit (1);
}
mpfr_set_prec (a, 29);
mpfr_set_prec (b, 29);
mpfr_set_str_binary (a, "1.0000000000000000000000001111");
mpfr_set_str_binary (b, "1.1001101111001100111001010111e165");
mpfr_pow_ui (a, a, 2055225053, GMP_RNDZ);
if (mpfr_cmp (a, b) != 0)
{
printf ("Error for x^2055225053\n");
printf ("Expected ");
mpfr_out_str (stdout, 2, 0, b, GMP_RNDN);
printf ("\nGot ");
mpfr_out_str (stdout, 2, 0, a, GMP_RNDN);
printf ("\n");
exit (1);
}
/* worst case found by Vincent Lefevre, 25 Nov 2006 */
mpfr_set_prec (a, 53);
mpfr_set_prec (b, 53);
mpfr_set_str_binary (a, "1.0000010110000100001000101101101001011101101011010111");
mpfr_set_str_binary (b, "1.0000110111101111011010110100001100010000001010110100E1");
mpfr_pow_ui (a, a, 35, GMP_RNDN);
if (mpfr_cmp (a, b) != 0)
{
printf ("Error in mpfr_pow_ui for worst case (1)\n");
printf ("Expected ");
mpfr_out_str (stdout, 2, 0, b, GMP_RNDN);
printf ("\nGot ");
mpfr_out_str (stdout, 2, 0, a, GMP_RNDN);
printf ("\n");
exit (1);
}
/* worst cases found on 2006-11-26 */
mpfr_set_str_binary (a, "1.0110100111010001101001010111001110010100111111000011");
mpfr_set_str_binary (b, "1.1111010011101110001111010110000101110000110110101100E17");
mpfr_pow_ui (a, a, 36, GMP_RNDD);
if (mpfr_cmp (a, b) != 0)
{
printf ("Error in mpfr_pow_ui for worst case (2)\n");
printf ("Expected ");
mpfr_out_str (stdout, 2, 0, b, GMP_RNDN);
printf ("\nGot ");
mpfr_out_str (stdout, 2, 0, a, GMP_RNDN);
printf ("\n");
exit (1);
}
mpfr_set_str_binary (a, "1.1001010100001110000110111111100011011101110011000100");
mpfr_set_str_binary (b, "1.1100011101101101100010110001000001110001111110010001E23");
mpfr_pow_ui (a, a, 36, GMP_RNDU);
if (mpfr_cmp (a, b) != 0)
{
printf ("Error in mpfr_pow_ui for worst case (3)\n");
printf ("Expected ");
mpfr_out_str (stdout, 2, 0, b, GMP_RNDN);
printf ("\nGot ");
mpfr_out_str (stdout, 2, 0, a, GMP_RNDN);
printf ("\n");
exit (1);
}
mpfr_clear (a);
mpfr_clear (b);
}
int
mpfr_log2 (mpfr_ptr r, mpfr_srcptr a, mpfr_rnd_t rnd_mode)
{
int inexact;
MPFR_SAVE_EXPO_DECL (expo);
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (a)))
{
/* If a is NaN, the result is NaN */
if (MPFR_IS_NAN (a))
{
MPFR_SET_NAN (r);
MPFR_RET_NAN;
}
/* check for infinity before zero */
else if (MPFR_IS_INF (a))
{
if (MPFR_IS_NEG (a))
/* log(-Inf) = NaN */
{
MPFR_SET_NAN (r);
MPFR_RET_NAN;
}
else /* log(+Inf) = +Inf */
{
MPFR_SET_INF (r);
MPFR_SET_POS (r);
MPFR_RET (0);
}
}
else /* a is zero */
{
MPFR_ASSERTD (MPFR_IS_ZERO (a));
MPFR_SET_INF (r);
MPFR_SET_NEG (r);
MPFR_RET (0); /* log2(0) is an exact -infinity */
}
}
/* If a is negative, the result is NaN */
if (MPFR_UNLIKELY (MPFR_IS_NEG (a)))
{
MPFR_SET_NAN (r);
MPFR_RET_NAN;
}
/* If a is 1, the result is 0 */
if (MPFR_UNLIKELY (mpfr_cmp_ui (a, 1) == 0))
{
MPFR_SET_ZERO (r);
MPFR_SET_POS (r);
MPFR_RET (0); /* only "normal" case where the result is exact */
}
/* If a is 2^N, log2(a) is exact*/
if (MPFR_UNLIKELY (mpfr_cmp_ui_2exp (a, 1, MPFR_GET_EXP (a) - 1) == 0))
return mpfr_set_si(r, MPFR_GET_EXP (a) - 1, rnd_mode);
MPFR_SAVE_EXPO_MARK (expo);
/* General case */
{
/* Declaration of the intermediary variable */
mpfr_t t, tt;
/* Declaration of the size variable */
mpfr_prec_t Ny = MPFR_PREC(r); /* target precision */
mpfr_prec_t Nt; /* working precision */
mpfr_exp_t err; /* error */
MPFR_ZIV_DECL (loop);
/* compute the precision of intermediary variable */
/* the optimal number of bits : see algorithms.tex */
Nt = Ny + 3 + MPFR_INT_CEIL_LOG2 (Ny);
/* initialise of intermediary variable */
mpfr_init2 (t, Nt);
mpfr_init2 (tt, Nt);
/* First computation of log2 */
MPFR_ZIV_INIT (loop, Nt);
for (;;)
{
/* compute log2 */
mpfr_const_log2(t,MPFR_RNDD); /* log(2) */
mpfr_log(tt,a,MPFR_RNDN); /* log(a) */
mpfr_div(t,tt,t,MPFR_RNDN); /* log(a)/log(2) */
/* estimation of the error */
err = Nt-3;
if (MPFR_LIKELY (MPFR_CAN_ROUND (t, err, Ny, rnd_mode)))
break;
/* actualisation of the precision */
MPFR_ZIV_NEXT (loop, Nt);
mpfr_set_prec (t, Nt);
mpfr_set_prec (tt, Nt);
}
MPFR_ZIV_FREE (loop);
inexact = mpfr_set (r, t, rnd_mode);
mpfr_clear (t);
mpfr_clear (tt);
}
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (r, inexact, rnd_mode);
}
int
mpfr_log10 (mpfr_ptr r, mpfr_srcptr a, mpfr_rnd_t rnd_mode)
{
int inexact;
MPFR_SAVE_EXPO_DECL (expo);
/* If a is NaN, the result is NaN */
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (a)))
{
if (MPFR_IS_NAN (a))
{
MPFR_SET_NAN (r);
MPFR_RET_NAN;
}
/* check for infinity before zero */
else if (MPFR_IS_INF (a))
{
if (MPFR_IS_NEG (a))
/* log10(-Inf) = NaN */
{
MPFR_SET_NAN (r);
MPFR_RET_NAN;
}
else /* log10(+Inf) = +Inf */
{
MPFR_SET_INF (r);
MPFR_SET_POS (r);
MPFR_RET (0); /* exact */
}
}
else /* a = 0 */
{
MPFR_ASSERTD (MPFR_IS_ZERO (a));
MPFR_SET_INF (r);
MPFR_SET_NEG (r);
MPFR_RET (0); /* log10(0) is an exact -infinity */
}
}
/* If a is negative, the result is NaN */
if (MPFR_UNLIKELY (MPFR_IS_NEG (a)))
{
MPFR_SET_NAN (r);
MPFR_RET_NAN;
}
/* If a is 1, the result is 0 */
if (mpfr_cmp_ui (a, 1) == 0)
{
MPFR_SET_ZERO (r);
MPFR_SET_POS (r);
MPFR_RET (0); /* result is exact */
}
MPFR_SAVE_EXPO_MARK (expo);
/* General case */
{
/* Declaration of the intermediary variable */
mpfr_t t, tt;
MPFR_ZIV_DECL (loop);
/* Declaration of the size variable */
mpfr_prec_t Ny = MPFR_PREC(r); /* Precision of output variable */
mpfr_prec_t Nt; /* Precision of the intermediary variable */
mpfr_exp_t err; /* Precision of error */
/* compute the precision of intermediary variable */
/* the optimal number of bits : see algorithms.tex */
Nt = Ny + 4 + MPFR_INT_CEIL_LOG2 (Ny);
/* initialise of intermediary variables */
mpfr_init2 (t, Nt);
mpfr_init2 (tt, Nt);
/* First computation of log10 */
MPFR_ZIV_INIT (loop, Nt);
for (;;)
{
/* compute log10 */
mpfr_set_ui (t, 10, MPFR_RNDN); /* 10 */
mpfr_log (t, t, MPFR_RNDD); /* log(10) */
mpfr_log (tt, a, MPFR_RNDN); /* log(a) */
mpfr_div (t, tt, t, MPFR_RNDN); /* log(a)/log(10) */
/* estimation of the error */
err = Nt - 4;
if (MPFR_LIKELY (MPFR_CAN_ROUND (t, err, Ny, rnd_mode)))
break;
/* log10(10^n) is exact:
FIXME: Can we have 10^n exactly representable as a mpfr_t
but n can't fit an unsigned long? */
if (MPFR_IS_POS (t)
&& mpfr_integer_p (t) && mpfr_fits_ulong_p (t, MPFR_RNDN)
&& !mpfr_ui_pow_ui (tt, 10, mpfr_get_ui (t, MPFR_RNDN), MPFR_RNDN)
&& mpfr_cmp (a, tt) == 0)
break;
/* actualisation of the precision */
MPFR_ZIV_NEXT (loop, Nt);
mpfr_set_prec (t, Nt);
mpfr_set_prec (tt, Nt);
}
MPFR_ZIV_FREE (loop);
inexact = mpfr_set (r, t, rnd_mode);
mpfr_clear (t);
mpfr_clear (tt);
}
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (r, inexact, rnd_mode);
}
int
main (void)
{
mpfr_t x, y, z;
int i, j, k;
tests_start_mpfr ();
mpfr_init (x);
mpfr_init (y);
mpfr_init (z);
for (i = 0; i <= 1; i++)
for (j = 0; j <= 1; j++)
for (k = 0; k <= 5; k++)
{
mpfr_set_nan (x);
i ? MPFR_SET_NEG (x) : MPFR_SET_POS (x);
mpfr_set_nan (y);
j ? MPFR_SET_NEG (y) : MPFR_SET_POS (y);
copysign_variant (z, x, y, MPFR_RNDN, k);
if (MPFR_SIGN (z) != MPFR_SIGN (y) || !mpfr_nanflag_p ())
{
printf ("Error in mpfr_copysign (%cNaN, %cNaN)\n",
i ? '-' : '+', j ? '-' : '+');
exit (1);
}
mpfr_set_si (x, i ? -1250 : 1250, MPFR_RNDN);
mpfr_set_nan (y);
j ? MPFR_SET_NEG (y) : MPFR_SET_POS (y);
copysign_variant (z, x, y, MPFR_RNDN, k);
if (i != j)
mpfr_neg (x, x, MPFR_RNDN);
if (! mpfr_equal_p (z, x) || mpfr_nanflag_p ())
{
printf ("Error in mpfr_copysign (%c1250, %cNaN)\n",
i ? '-' : '+', j ? '-' : '+');
exit (1);
}
mpfr_set_si (x, i ? -1250 : 1250, MPFR_RNDN);
mpfr_set_si (y, j ? -1717 : 1717, MPFR_RNDN);
copysign_variant (z, x, y, MPFR_RNDN, k);
if (i != j)
mpfr_neg (x, x, MPFR_RNDN);
if (! mpfr_equal_p (z, x) || mpfr_nanflag_p ())
{
printf ("Error in mpfr_copysign (%c1250, %c1717)\n",
i ? '-' : '+', j ? '-' : '+');
exit (1);
}
}
mpfr_clear (x);
mpfr_clear (y);
mpfr_clear (z);
tests_end_mpfr ();
return 0;
}
static void
special (void)
{
mpfr_t x, y, z;
int r;
int i;
mpfr_init2 (x, 53);
mpfr_init2 (y, 53);
mpfr_init2 (z, 53);
mpfr_set_str_binary (x, "1.0000100110000001100111100011001110101110100111011101");
mpfr_set_str_binary (y, "1.1001101101110100101100110011011101101000011010111110e-1");
mpfr_atan (z, x, MPFR_RNDN);
if (mpfr_cmp (y, z))
{
printf ("Error in mpfr_atan for prec=53, rnd=MPFR_RNDN\n");
printf ("x=");
mpfr_out_str (stdout, 2, 0, x, MPFR_RNDN);
printf ("\nexpected ");
mpfr_out_str (stdout, 2, 0, y, MPFR_RNDN);
printf ("\ngot ");
mpfr_out_str (stdout, 2, 0, z, MPFR_RNDN);
printf ("\n");
exit (1);
}
/* atan(+Inf) = Pi/2 */
for (r = 0; r < MPFR_RND_MAX ; r++)
{
mpfr_set_inf (x, 1);
mpfr_atan (y, x, (mpfr_rnd_t) r);
mpfr_const_pi (x, (mpfr_rnd_t) r);
mpfr_div_2exp (x, x, 1, (mpfr_rnd_t) r);
if (mpfr_cmp (x, y))
{
printf ("Error: mpfr_atan(+Inf), rnd=%s\n",
mpfr_print_rnd_mode ((mpfr_rnd_t) r));
exit (1);
}
}
/* atan(-Inf) = - Pi/2 */
for (r = 0; r < MPFR_RND_MAX ; r++)
{
mpfr_set_inf (x, -1);
mpfr_atan (y, x, (mpfr_rnd_t) r);
mpfr_const_pi (x, MPFR_INVERT_RND((mpfr_rnd_t) r));
mpfr_neg (x, x, (mpfr_rnd_t) r);
mpfr_div_2exp (x, x, 1, (mpfr_rnd_t) r);
if (mpfr_cmp (x, y))
{
printf ("Error: mpfr_atan(-Inf), rnd=%s\n",
mpfr_print_rnd_mode ((mpfr_rnd_t) r));
exit (1);
}
}
/* atan(NaN) = NaN */
mpfr_set_nan (x);
mpfr_atan (y, x, MPFR_RNDN);
if (!mpfr_nan_p (y))
{
printf ("Error: mpfr_atan(NaN) <> NaN\n");
exit (1);
}
/* atan(+/-0) = +/-0 */
mpfr_set_ui (x, 0, MPFR_RNDN);
MPFR_SET_NEG (y);
mpfr_atan (y, x, MPFR_RNDN);
if (mpfr_cmp_ui (y, 0) || MPFR_IS_NEG (y))
{
printf ("Error: mpfr_atan (+0) <> +0\n");
exit (1);
}
mpfr_atan (x, x, MPFR_RNDN);
if (mpfr_cmp_ui (x, 0) || MPFR_IS_NEG (x))
{
printf ("Error: mpfr_atan (+0) <> +0 (in place)\n");
exit (1);
}
mpfr_neg (x, x, MPFR_RNDN);
MPFR_SET_POS (y);
mpfr_atan (y, x, MPFR_RNDN);
if (mpfr_cmp_ui (y, 0) || MPFR_IS_POS (y))
{
printf ("Error: mpfr_atan (-0) <> -0\n");
exit (1);
}
mpfr_atan (x, x, MPFR_RNDN);
if (mpfr_cmp_ui (x, 0) || MPFR_IS_POS (x))
{
printf ("Error: mpfr_atan (-0) <> -0 (in place)\n");
exit (1);
}
mpfr_set_prec (x, 32);
mpfr_set_prec (y, 32);
/* test one random positive argument */
mpfr_set_str_binary (x, "0.10000100001100101001001001011001");
mpfr_atan (x, x, MPFR_RNDN);
mpfr_set_str_binary (y, "0.1111010000001111001111000000011E-1");
if (mpfr_cmp (x, y))
{
printf ("Error in mpfr_atan (1)\n");
exit (1);
}
/* test one random negative argument */
mpfr_set_str_binary (x, "-0.1100001110110000010101011001011");
mpfr_atan (x, x, MPFR_RNDN);
mpfr_set_str_binary (y, "-0.101001110001010010110001110001");
if (mpfr_cmp (x, y))
{
printf ("Error in mpfr_atan (2)\n");
mpfr_print_binary (x);
printf ("\n");
mpfr_print_binary (y);
printf ("\n");
exit (1);
}
mpfr_set_prec (x, 3);
mpfr_set_prec (y, 192);
mpfr_set_prec (z, 192);
mpfr_set_str_binary (x, "-0.100e1");
mpfr_atan (z, x, MPFR_RNDD);
mpfr_set_str_binary (y, "-0.110010010000111111011010101000100010000101101000110000100011010011000100110001100110001010001011100000001101110000011100110100010010100100000010010011100000100010001010011001111100110001110101");
if (mpfr_cmp (z, y))
{
printf ("Error in mpfr_atan (3)\n");
printf ("Expected ");
mpfr_print_binary (y);
printf ("\n");
printf ("Got ");
mpfr_print_binary (z);
printf ("\n");
exit (1);
}
/* Test regression */
mpfr_set_prec (x, 51);
mpfr_set_prec (y, 51);
mpfr_set_str_binary (x,
"0.101100100000101111111010001111111000001000000000000E-11");
i = mpfr_atan (y, x, MPFR_RNDN);
if (mpfr_cmp_str (y,
"1.01100100000101111111001110011001010110100100000000e-12", 2, MPFR_RNDN)
|| i >= 0)
{
printf ("Wrong Regression test (%d)\n", i);
mpfr_dump (y);
exit (1);
}
mpfr_set_si (x, -1, MPFR_RNDN);
mpfr_atan (x, x, MPFR_RNDN);
MPFR_ASSERTN (MPFR_IS_NEG (x));
/* Test regression */
mpfr_set_prec (x, 48);
mpfr_set_prec (y, 48);
mpfr_set_str_binary (x, "1.11001110010000011111100000010000000000000000000e-19");
mpfr_atan (y, x, MPFR_RNDD);
if (mpfr_cmp_str (y, "0.111001110010000011111100000001111111110000010011E-18", 2, MPFR_RNDN))
{
printf ("Error in mpfr_atan (4)\n");
printf ("Input 1.11001110010000011111100000010000000000000000000e-19 [prec=48]\n");
printf ("Expected 0.111001110010000011111100000001111111110000010011E-18\n");
printf ("Got ");
mpfr_dump (y);
exit (1);
}
mpfr_clear (x);
mpfr_clear (y);
mpfr_clear (z);
}
int
mpfr_log (mpfr_ptr r, mpfr_srcptr a, mpfr_rnd_t rnd_mode)
{
int inexact;
mpfr_prec_t p, q;
mpfr_t tmp1, tmp2;
MPFR_SAVE_EXPO_DECL (expo);
MPFR_ZIV_DECL (loop);
MPFR_GROUP_DECL(group);
MPFR_LOG_FUNC
(("a[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (a), mpfr_log_prec, a, rnd_mode),
("r[%Pu]=%.*Rg inexact=%d", mpfr_get_prec (r), mpfr_log_prec, r,
inexact));
/* Special cases */
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (a)))
{
/* If a is NaN, the result is NaN */
if (MPFR_IS_NAN (a))
{
MPFR_SET_NAN (r);
MPFR_RET_NAN;
}
/* check for infinity before zero */
else if (MPFR_IS_INF (a))
{
if (MPFR_IS_NEG (a))
/* log(-Inf) = NaN */
{
MPFR_SET_NAN (r);
MPFR_RET_NAN;
}
else /* log(+Inf) = +Inf */
{
MPFR_SET_INF (r);
MPFR_SET_POS (r);
MPFR_RET (0);
}
}
else /* a is zero */
{
MPFR_ASSERTD (MPFR_IS_ZERO (a));
MPFR_SET_INF (r);
MPFR_SET_NEG (r);
mpfr_set_divby0 ();
MPFR_RET (0); /* log(0) is an exact -infinity */
}
}
/* If a is negative, the result is NaN */
else if (MPFR_UNLIKELY (MPFR_IS_NEG (a)))
{
MPFR_SET_NAN (r);
MPFR_RET_NAN;
}
/* If a is 1, the result is 0 */
else if (MPFR_UNLIKELY (MPFR_GET_EXP (a) == 1 && mpfr_cmp_ui (a, 1) == 0))
{
MPFR_SET_ZERO (r);
MPFR_SET_POS (r);
MPFR_RET (0); /* only "normal" case where the result is exact */
}
q = MPFR_PREC (r);
/* use initial precision about q+lg(q)+5 */
p = q + 5 + 2 * MPFR_INT_CEIL_LOG2 (q);
/* % ~(mpfr_prec_t)GMP_NUMB_BITS ;
m=q; while (m) { p++; m >>= 1; } */
/* if (MPFR_LIKELY(p % GMP_NUMB_BITS != 0))
p += GMP_NUMB_BITS - (p%GMP_NUMB_BITS); */
MPFR_SAVE_EXPO_MARK (expo);
MPFR_GROUP_INIT_2 (group, p, tmp1, tmp2);
MPFR_ZIV_INIT (loop, p);
for (;;)
{
long m;
mpfr_exp_t cancel;
/* Calculus of m (depends on p) */
m = (p + 1) / 2 - MPFR_GET_EXP (a) + 1;
mpfr_mul_2si (tmp2, a, m, MPFR_RNDN); /* s=a*2^m, err<=1 ulp */
mpfr_div (tmp1, __gmpfr_four, tmp2, MPFR_RNDN);/* 4/s, err<=2 ulps */
mpfr_agm (tmp2, __gmpfr_one, tmp1, MPFR_RNDN); /* AG(1,4/s),err<=3 ulps */
mpfr_mul_2ui (tmp2, tmp2, 1, MPFR_RNDN); /* 2*AG(1,4/s), err<=3 ulps */
mpfr_const_pi (tmp1, MPFR_RNDN); /* compute pi, err<=1ulp */
mpfr_div (tmp2, tmp1, tmp2, MPFR_RNDN); /* pi/2*AG(1,4/s), err<=5ulps */
mpfr_const_log2 (tmp1, MPFR_RNDN); /* compute log(2), err<=1ulp */
mpfr_mul_si (tmp1, tmp1, m, MPFR_RNDN); /* compute m*log(2),err<=2ulps */
mpfr_sub (tmp1, tmp2, tmp1, MPFR_RNDN); /* log(a), err<=7ulps+cancel */
if (MPFR_LIKELY (MPFR_IS_PURE_FP (tmp1) && MPFR_IS_PURE_FP (tmp2)))
{
cancel = MPFR_GET_EXP (tmp2) - MPFR_GET_EXP (tmp1);
MPFR_LOG_MSG (("canceled bits=%ld\n", (long) cancel));
MPFR_LOG_VAR (tmp1);
if (MPFR_UNLIKELY (cancel < 0))
cancel = 0;
/* we have 7 ulps of error from the above roundings,
4 ulps from the 4/s^2 second order term,
plus the canceled bits */
if (MPFR_LIKELY (MPFR_CAN_ROUND (tmp1, p-cancel-4, q, rnd_mode)))
break;
/* VL: I think it is better to have an increment that it isn't
too low; in particular, the increment must be positive even
if cancel = 0 (can this occur?). */
p += cancel >= 8 ? cancel : 8;
}
else
{
/* TODO: find why this case can occur and what is best to do
with it. */
p += 32;
}
MPFR_ZIV_NEXT (loop, p);
MPFR_GROUP_REPREC_2 (group, p, tmp1, tmp2);
}
MPFR_ZIV_FREE (loop);
inexact = mpfr_set (r, tmp1, rnd_mode);
/* We clean */
MPFR_GROUP_CLEAR (group);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (r, inexact, rnd_mode);
}
int
main (void)
{
mpfr_t x;
FILE *f;
int i;
tests_start_mpfr ();
/* Check RND_MODE */
if (strcmp (mpfr_print_rnd_mode(MPFR_RNDN), "MPFR_RNDN"))
{
printf ("Error for printing MPFR_RNDN\n");
exit (1);
}
if (strcmp (mpfr_print_rnd_mode(MPFR_RNDU), "MPFR_RNDU"))
{
printf ("Error for printing MPFR_RNDU\n");
exit (1);
}
if (strcmp (mpfr_print_rnd_mode(MPFR_RNDD), "MPFR_RNDD"))
{
printf ("Error for printing MPFR_RNDD\n");
exit (1);
}
if (strcmp (mpfr_print_rnd_mode(MPFR_RNDZ), "MPFR_RNDZ"))
{
printf ("Error for printing MPFR_RNDZ\n");
exit (1);
}
if (mpfr_print_rnd_mode ((mpfr_rnd_t) -1) != NULL ||
mpfr_print_rnd_mode (MPFR_RND_MAX) != NULL)
{
printf ("Error for illegal rounding mode values.\n");
exit (1);
}
/* Reopen stdout to a file. All errors will be put to stderr
Can't use tmpname since it is unsecure */
if (freopen (FILE_NAME, "w", stdout) == NULL)
{
printf ("Error can't redirect stdout\n");
exit (1);
}
mpfr_init (x);
mpfr_set_nan (x);
mpfr_dump (x);
mpfr_set_inf (x, -1);
mpfr_dump (x);
MPFR_SET_ZERO (x); MPFR_SET_NEG (x);
mpfr_dump (x);
mpfr_set_str_binary (x, "0.101010101010111110010001100011000100001E32");
mpfr_dump (x);
mpfr_print_mant_binary ("x=",MPFR_MANT(x), MPFR_PREC(x));
mpfr_clear (x);
fclose (stdout);
/* Open it and check for it */
f = fopen (FILE_NAME, "r");
if (f == NULL)
{
fprintf (stderr, "Can't reopen file!\n");
exit (1);
}
for(i = 0 ; i < sizeof(Buffer)-1 ; i++)
{
if (feof (f))
{
fprintf (stderr, "Error EOF\n");
exit (1);
}
if (Buffer[i] != fgetc (f))
{
fprintf (stderr, "Character mismatch for i=%d / %lu\n",
i, (unsigned long) sizeof(Buffer));
exit (1);
}
}
fclose (f);
remove (FILE_NAME);
tests_end_mpfr ();
return 0;
}
int
mpfr_pow_si (mpfr_ptr y, mpfr_srcptr x, long int n, mpfr_rnd_t rnd)
{
MPFR_LOG_FUNC
(("x[%Pu]=%.*Rg n=%ld rnd=%d",
mpfr_get_prec (x), mpfr_log_prec, x, n, rnd),
("y[%Pu]=%.*Rg", mpfr_get_prec (y), mpfr_log_prec, y));
if (n >= 0)
return mpfr_pow_ui (y, x, n, rnd);
else
{
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
{
if (MPFR_IS_NAN (x))
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
else
{
int positive = MPFR_IS_POS (x) || ((unsigned long) n & 1) == 0;
if (MPFR_IS_INF (x))
MPFR_SET_ZERO (y);
else /* x is zero */
{
MPFR_ASSERTD (MPFR_IS_ZERO (x));
MPFR_SET_INF (y);
mpfr_set_divby0 ();
}
if (positive)
MPFR_SET_POS (y);
else
MPFR_SET_NEG (y);
MPFR_RET (0);
}
}
/* detect exact powers: x^(-n) is exact iff x is a power of 2 */
if (mpfr_cmp_si_2exp (x, MPFR_SIGN(x), MPFR_EXP(x) - 1) == 0)
{
mpfr_exp_t expx = MPFR_EXP (x) - 1, expy;
MPFR_ASSERTD (n < 0);
/* Warning: n * expx may overflow!
*
* Some systems (apparently alpha-freebsd) abort with
* LONG_MIN / 1, and LONG_MIN / -1 is undefined.
* http://www.freebsd.org/cgi/query-pr.cgi?pr=72024
*
* Proof of the overflow checking. The expressions below are
* assumed to be on the rational numbers, but the word "overflow"
* still has its own meaning in the C context. / still denotes
* the integer (truncated) division, and // denotes the exact
* division.
* - First, (__gmpfr_emin - 1) / n and (__gmpfr_emax - 1) / n
* cannot overflow due to the constraints on the exponents of
* MPFR numbers.
* - If n = -1, then n * expx = - expx, which is representable
* because of the constraints on the exponents of MPFR numbers.
* - If expx = 0, then n * expx = 0, which is representable.
* - If n < -1 and expx > 0:
* + If expx > (__gmpfr_emin - 1) / n, then
* expx >= (__gmpfr_emin - 1) / n + 1
* > (__gmpfr_emin - 1) // n,
* and
* n * expx < __gmpfr_emin - 1,
* i.e.
* n * expx <= __gmpfr_emin - 2.
* This corresponds to an underflow, with a null result in
* the rounding-to-nearest mode.
* + If expx <= (__gmpfr_emin - 1) / n, then n * expx cannot
* overflow since 0 < expx <= (__gmpfr_emin - 1) / n and
* 0 > n * expx >= n * ((__gmpfr_emin - 1) / n)
* >= __gmpfr_emin - 1.
* - If n < -1 and expx < 0:
* + If expx < (__gmpfr_emax - 1) / n, then
* expx <= (__gmpfr_emax - 1) / n - 1
* < (__gmpfr_emax - 1) // n,
* and
* n * expx > __gmpfr_emax - 1,
* i.e.
* n * expx >= __gmpfr_emax.
* This corresponds to an overflow (2^(n * expx) has an
* exponent > __gmpfr_emax).
* + If expx >= (__gmpfr_emax - 1) / n, then n * expx cannot
* overflow since 0 > expx >= (__gmpfr_emax - 1) / n and
* 0 < n * expx <= n * ((__gmpfr_emax - 1) / n)
* <= __gmpfr_emax - 1.
* Note: one could use expx bounds based on MPFR_EXP_MIN and
* MPFR_EXP_MAX instead of __gmpfr_emin and __gmpfr_emax. The
* current bounds do not lead to noticeably slower code and
* allow us to avoid a bug in Sun's compiler for Solaris/x86
* (when optimizations are enabled); known affected versions:
* cc: Sun C 5.8 2005/10/13
* cc: Sun C 5.8 Patch 121016-02 2006/03/31
* cc: Sun C 5.8 Patch 121016-04 2006/10/18
*/
expy =
n != -1 && expx > 0 && expx > (__gmpfr_emin - 1) / n ?
MPFR_EMIN_MIN - 2 /* Underflow */ :
n != -1 && expx < 0 && expx < (__gmpfr_emax - 1) / n ?
MPFR_EMAX_MAX /* Overflow */ : n * expx;
return mpfr_set_si_2exp (y, n % 2 ? MPFR_INT_SIGN (x) : 1,
expy, rnd);
}
/* General case */
{
/* Declaration of the intermediary variable */
mpfr_t t;
/* Declaration of the size variable */
mpfr_prec_t Ny; /* target precision */
mpfr_prec_t Nt; /* working precision */
mpfr_rnd_t rnd1;
int size_n;
int inexact;
unsigned long abs_n;
MPFR_SAVE_EXPO_DECL (expo);
MPFR_ZIV_DECL (loop);
abs_n = - (unsigned long) n;
count_leading_zeros (size_n, (mp_limb_t) abs_n);
size_n = GMP_NUMB_BITS - size_n;
/* initial working precision */
Ny = MPFR_PREC (y);
Nt = Ny + size_n + 3 + MPFR_INT_CEIL_LOG2 (Ny);
MPFR_SAVE_EXPO_MARK (expo);
/* initialise of intermediary variable */
mpfr_init2 (t, Nt);
/* We will compute rnd(rnd1(1/x) ^ |n|), where rnd1 is the rounding
toward sign(x), to avoid spurious overflow or underflow, as in
mpfr_pow_z. */
rnd1 = MPFR_EXP (x) < 1 ? MPFR_RNDZ :
(MPFR_SIGN (x) > 0 ? MPFR_RNDU : MPFR_RNDD);
MPFR_ZIV_INIT (loop, Nt);
for (;;)
{
MPFR_BLOCK_DECL (flags);
/* compute (1/x)^|n| */
MPFR_BLOCK (flags, mpfr_ui_div (t, 1, x, rnd1));
MPFR_ASSERTD (! MPFR_UNDERFLOW (flags));
/* t = (1/x)*(1+theta) where |theta| <= 2^(-Nt) */
if (MPFR_UNLIKELY (MPFR_OVERFLOW (flags)))
goto overflow;
MPFR_BLOCK (flags, mpfr_pow_ui (t, t, abs_n, rnd));
/* t = (1/x)^|n|*(1+theta')^(|n|+1) where |theta'| <= 2^(-Nt).
If (|n|+1)*2^(-Nt) <= 1/2, which is satisfied as soon as
Nt >= bits(n)+2, then we can use Lemma \ref{lemma_graillat}
from algorithms.tex, which yields x^n*(1+theta) with
|theta| <= 2(|n|+1)*2^(-Nt), thus the error is bounded by
2(|n|+1) ulps <= 2^(bits(n)+2) ulps. */
if (MPFR_UNLIKELY (MPFR_OVERFLOW (flags)))
{
overflow:
MPFR_ZIV_FREE (loop);
mpfr_clear (t);
MPFR_SAVE_EXPO_FREE (expo);
MPFR_LOG_MSG (("overflow\n", 0));
return mpfr_overflow (y, rnd, abs_n & 1 ?
MPFR_SIGN (x) : MPFR_SIGN_POS);
}
if (MPFR_UNLIKELY (MPFR_UNDERFLOW (flags)))
{
MPFR_ZIV_FREE (loop);
mpfr_clear (t);
MPFR_LOG_MSG (("underflow\n", 0));
if (rnd == MPFR_RNDN)
{
mpfr_t y2, nn;
/* We cannot decide now whether the result should be
rounded toward zero or away from zero. So, like
in mpfr_pow_pos_z, let's use the general case of
mpfr_pow in precision 2. */
MPFR_ASSERTD (mpfr_cmp_si_2exp (x, MPFR_SIGN (x),
MPFR_EXP (x) - 1) != 0);
mpfr_init2 (y2, 2);
mpfr_init2 (nn, sizeof (long) * CHAR_BIT);
inexact = mpfr_set_si (nn, n, MPFR_RNDN);
MPFR_ASSERTN (inexact == 0);
inexact = mpfr_pow_general (y2, x, nn, rnd, 1,
(mpfr_save_expo_t *) NULL);
mpfr_clear (nn);
mpfr_set (y, y2, MPFR_RNDN);
mpfr_clear (y2);
MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_UNDERFLOW);
goto end;
}
else
{
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_underflow (y, rnd, abs_n & 1 ?
MPFR_SIGN (x) : MPFR_SIGN_POS);
}
}
/* error estimate -- see pow function in algorithms.ps */
if (MPFR_LIKELY (MPFR_CAN_ROUND (t, Nt - size_n - 2, Ny, rnd)))
break;
/* actualisation of the precision */
MPFR_ZIV_NEXT (loop, Nt);
mpfr_set_prec (t, Nt);
}
MPFR_ZIV_FREE (loop);
inexact = mpfr_set (y, t, rnd);
mpfr_clear (t);
end:
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inexact, rnd);
}
}
}
int
mpfr_yn (mpfr_ptr res, long n, mpfr_srcptr z, mpfr_rnd_t r)
{
int inex;
unsigned long absn;
MPFR_SAVE_EXPO_DECL (expo);
MPFR_LOG_FUNC
(("n=%ld x[%Pu]=%.*Rg rnd=%d", n, mpfr_get_prec (z), mpfr_log_prec, z, r),
("y[%Pu]=%.*Rg inexact=%d", mpfr_get_prec (res), mpfr_log_prec, res, inex));
absn = SAFE_ABS (unsigned long, n);
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (z)))
{
if (MPFR_IS_NAN (z))
{
MPFR_SET_NAN (res); /* y(n,NaN) = NaN */
MPFR_RET_NAN;
}
/* y(n,z) tends to zero when z goes to +Inf, oscillating around
0. We choose to return +0 in that case. */
else if (MPFR_IS_INF (z))
{
if (MPFR_SIGN(z) > 0)
return mpfr_set_ui (res, 0, r);
else /* y(n,-Inf) = NaN */
{
MPFR_SET_NAN (res);
MPFR_RET_NAN;
}
}
else /* y(n,z) tends to -Inf for n >= 0 or n even, to +Inf otherwise,
when z goes to zero */
{
MPFR_SET_INF(res);
if (n >= 0 || ((unsigned long) n & 1) == 0)
MPFR_SET_NEG(res);
else
MPFR_SET_POS(res);
mpfr_set_divby0 ();
MPFR_RET(0);
}
}
/* for z < 0, y(n,z) is imaginary except when j(n,|z|) = 0, which we
assume does not happen for a rational z. */
if (MPFR_SIGN(z) < 0)
{
MPFR_SET_NAN (res);
MPFR_RET_NAN;
}
/* now z is not singular, and z > 0 */
MPFR_SAVE_EXPO_MARK (expo);
/* Deal with tiny arguments. We have:
y0(z) = 2 log(z)/Pi + 2 (euler - log(2))/Pi + O(log(z)*z^2), more
precisely for 0 <= z <= 1/2, with g(z) = 2/Pi + 2(euler-log(2))/Pi/log(z),
g(z) - 0.41*z^2 < y0(z)/log(z) < g(z)
thus since log(z) is negative:
g(z)*log(z) < y0(z) < (g(z) - z^2/2)*log(z)
and since |g(z)| >= 0.63 for 0 <= z <= 1/2, the relative error on
y0(z)/log(z) is bounded by 0.41*z^2/0.63 <= 0.66*z^2.
Note: we use both the main term in log(z) and the constant term, because
otherwise the relative error would be only in 1/log(|log(z)|).
*/
if (n == 0 && MPFR_EXP(z) < - (mpfr_exp_t) (MPFR_PREC(res) / 2))
{
mpfr_t l, h, t, logz;
mpfr_prec_t prec;
int ok, inex2;
prec = MPFR_PREC(res) + 10;
mpfr_init2 (l, prec);
mpfr_init2 (h, prec);
mpfr_init2 (t, prec);
mpfr_init2 (logz, prec);
/* first enclose log(z) + euler - log(2) = log(z/2) + euler */
mpfr_log (logz, z, MPFR_RNDD); /* lower bound of log(z) */
mpfr_set (h, logz, MPFR_RNDU); /* exact */
mpfr_nextabove (h); /* upper bound of log(z) */
mpfr_const_euler (t, MPFR_RNDD); /* lower bound of euler */
mpfr_add (l, logz, t, MPFR_RNDD); /* lower bound of log(z) + euler */
mpfr_nextabove (t); /* upper bound of euler */
mpfr_add (h, h, t, MPFR_RNDU); /* upper bound of log(z) + euler */
mpfr_const_log2 (t, MPFR_RNDU); /* upper bound of log(2) */
mpfr_sub (l, l, t, MPFR_RNDD); /* lower bound of log(z/2) + euler */
mpfr_nextbelow (t); /* lower bound of log(2) */
mpfr_sub (h, h, t, MPFR_RNDU); /* upper bound of log(z/2) + euler */
mpfr_const_pi (t, MPFR_RNDU); /* upper bound of Pi */
mpfr_div (l, l, t, MPFR_RNDD); /* lower bound of (log(z/2)+euler)/Pi */
mpfr_nextbelow (t); /* lower bound of Pi */
mpfr_div (h, h, t, MPFR_RNDD); /* upper bound of (log(z/2)+euler)/Pi */
mpfr_mul_2ui (l, l, 1, MPFR_RNDD); /* lower bound on g(z)*log(z) */
mpfr_mul_2ui (h, h, 1, MPFR_RNDU); /* upper bound on g(z)*log(z) */
/* we now have l <= g(z)*log(z) <= h, and we need to add -z^2/2*log(z)
to h */
mpfr_mul (t, z, z, MPFR_RNDU); /* upper bound on z^2 */
/* since logz is negative, a lower bound corresponds to an upper bound
for its absolute value */
mpfr_neg (t, t, MPFR_RNDD);
mpfr_div_2ui (t, t, 1, MPFR_RNDD);
mpfr_mul (t, t, logz, MPFR_RNDU); /* upper bound on z^2/2*log(z) */
mpfr_add (h, h, t, MPFR_RNDU);
inex = mpfr_prec_round (l, MPFR_PREC(res), r);
inex2 = mpfr_prec_round (h, MPFR_PREC(res), r);
/* we need h=l and inex=inex2 */
ok = (inex == inex2) && mpfr_equal_p (l, h);
if (ok)
mpfr_set (res, h, r); /* exact */
mpfr_clear (l);
mpfr_clear (h);
mpfr_clear (t);
mpfr_clear (logz);
if (ok)
goto end;
}
/* small argument check for y1(z) = -2/Pi/z + O(log(z)):
for 0 <= z <= 1, |y1(z) + 2/Pi/z| <= 0.25 */
if (n == 1 && MPFR_EXP(z) + 1 < - (mpfr_exp_t) MPFR_PREC(res))
{
mpfr_t y;
mpfr_prec_t prec;
mpfr_exp_t err1;
int ok;
MPFR_BLOCK_DECL (flags);
/* since 2/Pi > 0.5, and |y1(z)| >= |2/Pi/z|, if z <= 2^(-emax-1),
then |y1(z)| > 2^emax */
prec = MPFR_PREC(res) + 10;
mpfr_init2 (y, prec);
mpfr_const_pi (y, MPFR_RNDU); /* Pi*(1+u)^2, where here and below u
represents a quantity <= 1/2^prec */
mpfr_mul (y, y, z, MPFR_RNDU); /* Pi*z * (1+u)^4, upper bound */
MPFR_BLOCK (flags, mpfr_ui_div (y, 2, y, MPFR_RNDZ));
/* 2/Pi/z * (1+u)^6, lower bound, with possible overflow */
if (MPFR_OVERFLOW (flags))
{
mpfr_clear (y);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_overflow (res, r, -1);
}
mpfr_neg (y, y, MPFR_RNDN);
/* (1+u)^6 can be written 1+7u [for another value of u], thus the
error on 2/Pi/z is less than 7ulp(y). The truncation error is less
than 1/4, thus if ulp(y)>=1/4, the total error is less than 8ulp(y),
otherwise it is less than 1/4+7/8 <= 2. */
if (MPFR_EXP(y) + 2 >= MPFR_PREC(y)) /* ulp(y) >= 1/4 */
err1 = 3;
else /* ulp(y) <= 1/8 */
err1 = (mpfr_exp_t) MPFR_PREC(y) - MPFR_EXP(y) + 1;
ok = MPFR_CAN_ROUND (y, prec - err1, MPFR_PREC(res), r);
if (ok)
inex = mpfr_set (res, y, r);
mpfr_clear (y);
if (ok)
goto end;
}
/* we can use the asymptotic expansion as soon as z > p log(2)/2,
but to get some margin we use it for z > p/2 */
if (mpfr_cmp_ui (z, MPFR_PREC(res) / 2 + 3) > 0)
{
inex = mpfr_yn_asympt (res, n, z, r);
if (inex != 0)
goto end;
}
/* General case */
{
mpfr_prec_t prec;
mpfr_exp_t err1, err2, err3;
mpfr_t y, s1, s2, s3;
MPFR_ZIV_DECL (loop);
mpfr_init (y);
mpfr_init (s1);
mpfr_init (s2);
mpfr_init (s3);
prec = MPFR_PREC(res) + 2 * MPFR_INT_CEIL_LOG2 (MPFR_PREC (res)) + 13;
MPFR_ZIV_INIT (loop, prec);
for (;;)
{
mpfr_set_prec (y, prec);
mpfr_set_prec (s1, prec);
mpfr_set_prec (s2, prec);
mpfr_set_prec (s3, prec);
mpfr_mul (y, z, z, MPFR_RNDN);
mpfr_div_2ui (y, y, 2, MPFR_RNDN); /* z^2/4 */
/* store (z/2)^n temporarily in s2 */
mpfr_pow_ui (s2, z, absn, MPFR_RNDN);
mpfr_div_2si (s2, s2, absn, MPFR_RNDN);
/* compute S1 * (z/2)^(-n) */
if (n == 0)
{
mpfr_set_ui (s1, 0, MPFR_RNDN);
err1 = 0;
}
else
err1 = mpfr_yn_s1 (s1, y, absn - 1);
mpfr_div (s1, s1, s2, MPFR_RNDN); /* (z/2)^(-n) * S1 */
/* See algorithms.tex: the relative error on s1 is bounded by
(3n+3)*2^(e+1-prec). */
err1 = MPFR_INT_CEIL_LOG2 (3 * absn + 3) + err1 + 1;
/* rel_err(s1) <= 2^(err1-prec), thus err(s1) <= 2^err1 ulps */
/* compute (z/2)^n * S3 */
mpfr_neg (y, y, MPFR_RNDN); /* -z^2/4 */
err3 = mpfr_yn_s3 (s3, y, s2, absn); /* (z/2)^n * S3 */
/* the error on s3 is bounded by 2^err3 ulps */
/* add s1+s3 */
err1 += MPFR_EXP(s1);
mpfr_add (s1, s1, s3, MPFR_RNDN);
/* the error is bounded by 1/2 + 2^err1*2^(- EXP(s1))
+ 2^err3*2^(EXP(s3) - EXP(s1)) */
err3 += MPFR_EXP(s3);
err1 = (err3 > err1) ? err3 + 1 : err1 + 1;
err1 -= MPFR_EXP(s1);
err1 = (err1 >= 0) ? err1 + 1 : 1;
/* now the error on s1 is bounded by 2^err1*ulp(s1) */
/* compute S2 */
mpfr_div_2ui (s2, z, 1, MPFR_RNDN); /* z/2 */
mpfr_log (s2, s2, MPFR_RNDN); /* log(z/2) */
mpfr_const_euler (s3, MPFR_RNDN);
err2 = MPFR_EXP(s2) > MPFR_EXP(s3) ? MPFR_EXP(s2) : MPFR_EXP(s3);
mpfr_add (s2, s2, s3, MPFR_RNDN); /* log(z/2) + gamma */
err2 -= MPFR_EXP(s2);
mpfr_mul_2ui (s2, s2, 1, MPFR_RNDN); /* 2*(log(z/2) + gamma) */
mpfr_jn (s3, absn, z, MPFR_RNDN); /* Jn(z) */
mpfr_mul (s2, s2, s3, MPFR_RNDN); /* 2*(log(z/2) + gamma)*Jn(z) */
err2 += 4; /* the error on s2 is bounded by 2^err2 ulps, see
algorithms.tex */
/* add all three sums */
err1 += MPFR_EXP(s1); /* the error on s1 is bounded by 2^err1 */
err2 += MPFR_EXP(s2); /* the error on s2 is bounded by 2^err2 */
mpfr_sub (s2, s2, s1, MPFR_RNDN); /* s2 - (s1+s3) */
err2 = (err1 > err2) ? err1 + 1 : err2 + 1;
err2 -= MPFR_EXP(s2);
err2 = (err2 >= 0) ? err2 + 1 : 1;
/* now the error on s2 is bounded by 2^err2*ulp(s2) */
mpfr_const_pi (y, MPFR_RNDN); /* error bounded by 1 ulp */
mpfr_div (s2, s2, y, MPFR_RNDN); /* error bounded by
2^(err2+1)*ulp(s2) */
err2 ++;
if (MPFR_LIKELY (MPFR_CAN_ROUND (s2, prec - err2, MPFR_PREC(res), r)))
break;
MPFR_ZIV_NEXT (loop, prec);
}
MPFR_ZIV_FREE (loop);
/* Assume two's complement for the test n & 1 */
inex = mpfr_set4 (res, s2, r, n >= 0 || (n & 1) == 0 ?
MPFR_SIGN (s2) : - MPFR_SIGN (s2));
mpfr_clear (y);
mpfr_clear (s1);
mpfr_clear (s2);
mpfr_clear (s3);
}
end:
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (res, inex, r);
}
static
void check_special (void)
{
mpfr_t tab[3], r;
mpfr_ptr tabp[3];
int i;
mpfr_inits (tab[0], tab[1], tab[2], r, (mpfr_ptr) 0);
tabp[0] = tab[0];
tabp[1] = tab[1];
tabp[2] = tab[2];
i = mpfr_sum (r, tabp, 0, MPFR_RNDN);
if (!MPFR_IS_ZERO (r) || !MPFR_IS_POS (r) || i != 0)
{
printf ("Special case n==0 failed!\n");
exit (1);
}
mpfr_set_ui (tab[0], 42, MPFR_RNDN);
i = mpfr_sum (r, tabp, 1, MPFR_RNDN);
if (mpfr_cmp_ui (r, 42) || i != 0)
{
printf ("Special case n==1 failed!\n");
exit (1);
}
mpfr_set_ui (tab[1], 17, MPFR_RNDN);
MPFR_SET_NAN (tab[2]);
i = mpfr_sum (r, tabp, 3, MPFR_RNDN);
if (!MPFR_IS_NAN (r) || i != 0)
{
printf ("Special case NAN failed!\n");
exit (1);
}
MPFR_SET_INF (tab[2]);
MPFR_SET_POS (tab[2]);
i = mpfr_sum (r, tabp, 3, MPFR_RNDN);
if (!MPFR_IS_INF (r) || !MPFR_IS_POS (r) || i != 0)
{
printf ("Special case +INF failed!\n");
exit (1);
}
MPFR_SET_INF (tab[2]);
MPFR_SET_NEG (tab[2]);
i = mpfr_sum (r, tabp, 3, MPFR_RNDN);
if (!MPFR_IS_INF (r) || !MPFR_IS_NEG (r) || i != 0)
{
printf ("Special case -INF failed!\n");
exit (1);
}
MPFR_SET_ZERO (tab[1]);
i = mpfr_sum (r, tabp, 2, MPFR_RNDN);
if (mpfr_cmp_ui (r, 42) || i != 0)
{
printf ("Special case 42+0 failed!\n");
exit (1);
}
MPFR_SET_NAN (tab[0]);
i = mpfr_sum (r, tabp, 3, MPFR_RNDN);
if (!MPFR_IS_NAN (r) || i != 0)
{
printf ("Special case NAN+0+-INF failed!\n");
exit (1);
}
mpfr_set_inf (tab[0], 1);
mpfr_set_ui (tab[1], 59, MPFR_RNDN);
mpfr_set_inf (tab[2], -1);
i = mpfr_sum (r, tabp, 3, MPFR_RNDN);
if (!MPFR_IS_NAN (r) || i != 0)
{
printf ("Special case +INF + 59 +-INF failed!\n");
exit (1);
}
mpfr_clears (tab[0], tab[1], tab[2], r, (mpfr_ptr) 0);
}
static void
special ()
{
mpfr_t x, y, z, t;
mpfr_init2 (x, 53);
mpfr_init2 (y, 53);
mpfr_init2 (z, 53);
mpfr_init2 (t, 2);
mpfr_set_ui (x, 2, GMP_RNDN);
mpfr_pow_si (x, x, -2, GMP_RNDN);
if (mpfr_cmp_ui_2exp (x, 1, -2))
{
printf ("Error in pow_si(x,x,-2) for x=2\n");
exit (1);
}
mpfr_set_ui (x, 2, GMP_RNDN);
mpfr_set_si (y, -2, GMP_RNDN);
test_pow (x, x, y, GMP_RNDN);
if (mpfr_cmp_ui_2exp (x, 1, -2))
{
printf ("Error in pow(x,x,y) for x=2, y=-2\n");
exit (1);
}
mpfr_set_prec (x, 2);
mpfr_set_str_binary (x, "1.0e-1");
mpfr_set_prec (y, 53);
mpfr_set_str_binary (y, "0.11010110011100101010110011001010100111000001000101110E-1");
mpfr_set_prec (z, 2);
test_pow (z, x, y, GMP_RNDZ);
mpfr_set_str_binary (x, "1.0e-1");
if (mpfr_cmp (x, z))
{
printf ("Error in mpfr_pow (1)\n");
exit (1);
}
mpfr_set_prec (x, 64);
mpfr_set_prec (y, 64);
mpfr_set_prec (z, 64);
mpfr_set_prec (t, 64);
mpfr_set_str_binary (x, "0.111011000111100000111010000101010100110011010000011");
mpfr_set_str_binary (y, "0.111110010100110000011101100011010111000010000100101");
mpfr_set_str_binary (t, "0.1110110011110110001000110100100001001111010011111000010000011001");
test_pow (z, x, y, GMP_RNDN);
if (mpfr_cmp (z, t))
{
printf ("Error in mpfr_pow for prec=64, rnd=GMP_RNDN\n");
exit (1);
}
mpfr_set_prec (x, 53);
mpfr_set_prec (y, 53);
mpfr_set_prec (z, 53);
mpfr_set_str (x, "5.68824667828621954868e-01", 10, GMP_RNDN);
mpfr_set_str (y, "9.03327850535952658895e-01", 10, GMP_RNDN);
test_pow (z, x, y, GMP_RNDZ);
if (mpfr_cmp_str1 (z, "0.60071044650456473235"))
{
printf ("Error in mpfr_pow for prec=53, rnd=GMP_RNDZ\n");
exit (1);
}
mpfr_set_prec (t, 2);
mpfr_set_prec (x, 30);
mpfr_set_prec (y, 30);
mpfr_set_prec (z, 30);
mpfr_set_str (x, "1.00000000001010111110001111011e1", 2, GMP_RNDN);
mpfr_set_str (t, "-0.5", 10, GMP_RNDN);
test_pow (z, x, t, GMP_RNDN);
mpfr_set_str (y, "1.01101001111010101110000101111e-1", 2, GMP_RNDN);
if (mpfr_cmp (z, y))
{
printf ("Error in mpfr_pow for prec=30, rnd=GMP_RNDN\n");
exit (1);
}
mpfr_set_prec (x, 21);
mpfr_set_prec (y, 21);
mpfr_set_prec (z, 21);
mpfr_set_str (x, "1.11111100100001100101", 2, GMP_RNDN);
test_pow (z, x, t, GMP_RNDZ);
mpfr_set_str (y, "1.01101011010001100000e-1", 2, GMP_RNDN);
if (mpfr_cmp (z, y))
{
printf ("Error in mpfr_pow for prec=21, rnd=GMP_RNDZ\n");
printf ("Expected ");
mpfr_out_str (stdout, 2, 0, y, GMP_RNDN);
printf ("\nGot ");
mpfr_out_str (stdout, 2, 0, z, GMP_RNDN);
printf ("\n");
exit (1);
}
/* From http://www.terra.es/personal9/ismaeljc/hall.htm */
mpfr_set_prec (x, 113);
mpfr_set_prec (y, 2);
mpfr_set_prec (z, 169);
mpfr_set_str1 (x, "6078673043126084065007902175846955");
mpfr_set_ui_2exp (y, 3, -1, GMP_RNDN);
test_pow (z, x, y, GMP_RNDZ);
if (mpfr_cmp_str1 (z, "473928882491000966028828671876527456070714790264144"))
{
printf ("Error in mpfr_pow for 6078673043126084065007902175846955");
printf ("^(3/2), GMP_RNDZ\nExpected ");
printf ("4.73928882491000966028828671876527456070714790264144e50");
printf ("\nGot ");
mpfr_out_str (stdout, 10, 0, z, GMP_RNDN);
printf ("\n");
exit (1);
}
test_pow (z, x, y, GMP_RNDU);
if (mpfr_cmp_str1 (z, "473928882491000966028828671876527456070714790264145"))
{
printf ("Error in mpfr_pow for 6078673043126084065007902175846955");
printf ("^(3/2), GMP_RNDU\nExpected ");
printf ("4.73928882491000966028828671876527456070714790264145e50");
printf ("\nGot ");
mpfr_out_str (stdout, 10, 0, z, GMP_RNDN);
printf ("\n");
exit (1);
}
mpfr_set_inf (x, 1);
mpfr_set_prec (y, 2);
mpfr_set_str_binary (y, "1E10");
test_pow (z, x, y, GMP_RNDN);
MPFR_ASSERTN(mpfr_inf_p (z) && MPFR_IS_POS(z));
mpfr_set_inf (x, -1);
test_pow (z, x, y, GMP_RNDN);
MPFR_ASSERTN(mpfr_inf_p (z) && MPFR_IS_POS(z));
mpfr_set_prec (y, 10);
mpfr_set_str_binary (y, "1.000000001E9");
test_pow (z, x, y, GMP_RNDN);
MPFR_ASSERTN(mpfr_inf_p (z) && MPFR_IS_NEG(z));
mpfr_set_str_binary (y, "1.000000001E8");
test_pow (z, x, y, GMP_RNDN);
MPFR_ASSERTN(mpfr_inf_p (z) && MPFR_IS_POS(z));
mpfr_set_inf (x, -1);
mpfr_set_prec (y, 2 * mp_bits_per_limb);
mpfr_set_ui (y, 1, GMP_RNDN);
mpfr_mul_2exp (y, y, mp_bits_per_limb - 1, GMP_RNDN);
/* y = 2^(mp_bits_per_limb - 1) */
test_pow (z, x, y, GMP_RNDN);
MPFR_ASSERTN(mpfr_inf_p (z) && MPFR_IS_POS(z));
mpfr_nextabove (y);
test_pow (z, x, y, GMP_RNDN);
/* y = 2^(mp_bits_per_limb - 1) + epsilon */
MPFR_ASSERTN(mpfr_inf_p (z) && MPFR_IS_POS(z));
mpfr_nextbelow (y);
mpfr_div_2exp (y, y, 1, GMP_RNDN);
mpfr_nextabove (y);
test_pow (z, x, y, GMP_RNDN);
/* y = 2^(mp_bits_per_limb - 2) + epsilon */
MPFR_ASSERTN(mpfr_inf_p (z) && MPFR_IS_POS(z));
mpfr_set_si (x, -1, GMP_RNDN);
mpfr_set_prec (y, 2);
mpfr_set_str_binary (y, "1E10");
test_pow (z, x, y, GMP_RNDN);
MPFR_ASSERTN(mpfr_cmp_ui (z, 1) == 0);
/* Check (-0)^(17.0001) */
mpfr_set_prec (x, 6);
mpfr_set_prec (y, 640);
MPFR_SET_ZERO (x);
MPFR_SET_NEG (x);
mpfr_set_ui (y, 17, GMP_RNDN);
mpfr_nextabove (y);
test_pow (z, x, y, GMP_RNDN);
MPFR_ASSERTN (MPFR_IS_ZERO (z) && MPFR_IS_POS (z));
mpfr_clear (x);
mpfr_clear (y);
mpfr_clear (z);
mpfr_clear (t);
}
static void
check_special (void)
{
mpfr_t a, d, q;
mpfr_exp_t emax, emin;
int i;
mpfr_init2 (a, 100L);
mpfr_init2 (d, 100L);
mpfr_init2 (q, 100L);
/* 1/nan == nan */
mpfr_set_ui (a, 1L, MPFR_RNDN);
MPFR_SET_NAN (d);
mpfr_clear_flags ();
MPFR_ASSERTN (test_div (q, a, d, MPFR_RNDZ) == 0); /* exact */
MPFR_ASSERTN (__gmpfr_flags == MPFR_FLAGS_NAN);
/* nan/1 == nan */
MPFR_SET_NAN (a);
mpfr_set_ui (d, 1L, MPFR_RNDN);
mpfr_clear_flags ();
MPFR_ASSERTN (test_div (q, a, d, MPFR_RNDZ) == 0); /* exact */
MPFR_ASSERTN (__gmpfr_flags == MPFR_FLAGS_NAN);
/* +inf/1 == +inf */
MPFR_SET_INF (a);
MPFR_SET_POS (a);
mpfr_set_ui (d, 1L, MPFR_RNDN);
mpfr_clear_flags ();
MPFR_ASSERTN (test_div (q, a, d, MPFR_RNDZ) == 0); /* exact */
MPFR_ASSERTN (mpfr_inf_p (q));
MPFR_ASSERTN (mpfr_sgn (q) > 0);
MPFR_ASSERTN (__gmpfr_flags == 0);
/* +inf/-1 == -inf */
MPFR_SET_INF (a);
MPFR_SET_POS (a);
mpfr_set_si (d, -1, MPFR_RNDN);
mpfr_clear_flags ();
MPFR_ASSERTN (test_div (q, a, d, MPFR_RNDZ) == 0); /* exact */
MPFR_ASSERTN (mpfr_inf_p (q));
MPFR_ASSERTN (mpfr_sgn (q) < 0);
MPFR_ASSERTN (__gmpfr_flags == 0);
/* -inf/1 == -inf */
MPFR_SET_INF (a);
MPFR_SET_NEG (a);
mpfr_set_ui (d, 1L, MPFR_RNDN);
mpfr_clear_flags ();
MPFR_ASSERTN (test_div (q, a, d, MPFR_RNDZ) == 0); /* exact */
MPFR_ASSERTN (mpfr_inf_p (q));
MPFR_ASSERTN (mpfr_sgn (q) < 0);
MPFR_ASSERTN (__gmpfr_flags == 0);
/* -inf/-1 == +inf */
MPFR_SET_INF (a);
MPFR_SET_NEG (a);
mpfr_set_si (d, -1, MPFR_RNDN);
mpfr_clear_flags ();
MPFR_ASSERTN (test_div (q, a, d, MPFR_RNDZ) == 0); /* exact */
MPFR_ASSERTN (mpfr_inf_p (q));
MPFR_ASSERTN (mpfr_sgn (q) > 0);
MPFR_ASSERTN (__gmpfr_flags == 0);
/* 1/+inf == +0 */
mpfr_set_ui (a, 1L, MPFR_RNDN);
MPFR_SET_INF (d);
MPFR_SET_POS (d);
mpfr_clear_flags ();
MPFR_ASSERTN (test_div (q, a, d, MPFR_RNDZ) == 0); /* exact */
MPFR_ASSERTN (mpfr_number_p (q));
MPFR_ASSERTN (mpfr_sgn (q) == 0);
MPFR_ASSERTN (MPFR_IS_POS (q));
MPFR_ASSERTN (__gmpfr_flags == 0);
/* 1/-inf == -0 */
mpfr_set_ui (a, 1L, MPFR_RNDN);
MPFR_SET_INF (d);
MPFR_SET_NEG (d);
mpfr_clear_flags ();
MPFR_ASSERTN (test_div (q, a, d, MPFR_RNDZ) == 0); /* exact */
MPFR_ASSERTN (mpfr_number_p (q));
MPFR_ASSERTN (mpfr_sgn (q) == 0);
MPFR_ASSERTN (MPFR_IS_NEG (q));
MPFR_ASSERTN (__gmpfr_flags == 0);
/* -1/+inf == -0 */
mpfr_set_si (a, -1, MPFR_RNDN);
MPFR_SET_INF (d);
MPFR_SET_POS (d);
mpfr_clear_flags ();
MPFR_ASSERTN (test_div (q, a, d, MPFR_RNDZ) == 0); /* exact */
MPFR_ASSERTN (mpfr_number_p (q));
MPFR_ASSERTN (mpfr_sgn (q) == 0);
MPFR_ASSERTN (MPFR_IS_NEG (q));
MPFR_ASSERTN (__gmpfr_flags == 0);
/* -1/-inf == +0 */
mpfr_set_si (a, -1, MPFR_RNDN);
MPFR_SET_INF (d);
MPFR_SET_NEG (d);
mpfr_clear_flags ();
MPFR_ASSERTN (test_div (q, a, d, MPFR_RNDZ) == 0); /* exact */
MPFR_ASSERTN (mpfr_number_p (q));
MPFR_ASSERTN (mpfr_sgn (q) == 0);
MPFR_ASSERTN (MPFR_IS_POS (q));
MPFR_ASSERTN (__gmpfr_flags == 0);
/* 0/0 == nan */
mpfr_set_ui (a, 0L, MPFR_RNDN);
mpfr_set_ui (d, 0L, MPFR_RNDN);
mpfr_clear_flags ();
MPFR_ASSERTN (test_div (q, a, d, MPFR_RNDZ) == 0); /* exact */
MPFR_ASSERTN (__gmpfr_flags == MPFR_FLAGS_NAN);
/* +inf/+inf == nan */
MPFR_SET_INF (a);
MPFR_SET_POS (a);
MPFR_SET_INF (d);
MPFR_SET_POS (d);
mpfr_clear_flags ();
MPFR_ASSERTN (test_div (q, a, d, MPFR_RNDZ) == 0); /* exact */
MPFR_ASSERTN (__gmpfr_flags == MPFR_FLAGS_NAN);
/* 1/+0 = +inf */
mpfr_set_ui (a, 1, MPFR_RNDZ);
mpfr_set_ui (d, 0, MPFR_RNDZ);
mpfr_clear_flags ();
MPFR_ASSERTN (test_div (q, a, d, MPFR_RNDZ) == 0); /* exact */
MPFR_ASSERTN (mpfr_inf_p (q) && mpfr_sgn (q) > 0);
MPFR_ASSERTN (__gmpfr_flags == MPFR_FLAGS_DIVBY0);
/* 1/-0 = -inf */
mpfr_set_ui (a, 1, MPFR_RNDZ);
mpfr_set_ui (d, 0, MPFR_RNDZ);
mpfr_neg (d, d, MPFR_RNDZ);
mpfr_clear_flags ();
MPFR_ASSERTN (test_div (q, a, d, MPFR_RNDZ) == 0); /* exact */
MPFR_ASSERTN (mpfr_inf_p (q) && mpfr_sgn (q) < 0);
MPFR_ASSERTN (__gmpfr_flags == MPFR_FLAGS_DIVBY0);
/* -1/+0 = -inf */
mpfr_set_si (a, -1, MPFR_RNDZ);
mpfr_set_ui (d, 0, MPFR_RNDZ);
mpfr_clear_flags ();
MPFR_ASSERTN (test_div (q, a, d, MPFR_RNDZ) == 0); /* exact */
MPFR_ASSERTN (mpfr_inf_p (q) && mpfr_sgn (q) < 0);
MPFR_ASSERTN (__gmpfr_flags == MPFR_FLAGS_DIVBY0);
/* -1/-0 = +inf */
mpfr_set_si (a, -1, MPFR_RNDZ);
mpfr_set_ui (d, 0, MPFR_RNDZ);
mpfr_neg (d, d, MPFR_RNDZ);
mpfr_clear_flags ();
MPFR_ASSERTN (test_div (q, a, d, MPFR_RNDZ) == 0); /* exact */
MPFR_ASSERTN (mpfr_inf_p (q) && mpfr_sgn (q) > 0);
MPFR_ASSERTN (__gmpfr_flags == MPFR_FLAGS_DIVBY0);
/* +inf/+0 = +inf */
MPFR_SET_INF (a);
MPFR_SET_POS (a);
mpfr_set_ui (d, 0, MPFR_RNDZ);
mpfr_clear_flags ();
MPFR_ASSERTN (test_div (q, a, d, MPFR_RNDZ) == 0); /* exact */
MPFR_ASSERTN (mpfr_inf_p (q) && mpfr_sgn (q) > 0);
MPFR_ASSERTN (__gmpfr_flags == 0);
/* +inf/-0 = -inf */
MPFR_SET_INF (a);
MPFR_SET_POS (a);
mpfr_set_ui (d, 0, MPFR_RNDZ);
mpfr_neg (d, d, MPFR_RNDZ);
mpfr_clear_flags ();
MPFR_ASSERTN (test_div (q, a, d, MPFR_RNDZ) == 0); /* exact */
MPFR_ASSERTN (mpfr_inf_p (q) && mpfr_sgn (q) < 0);
MPFR_ASSERTN (__gmpfr_flags == 0);
/* -inf/+0 = -inf */
MPFR_SET_INF (a);
MPFR_SET_NEG (a);
mpfr_set_ui (d, 0, MPFR_RNDZ);
mpfr_clear_flags ();
MPFR_ASSERTN (test_div (q, a, d, MPFR_RNDZ) == 0); /* exact */
MPFR_ASSERTN (mpfr_inf_p (q) && mpfr_sgn (q) < 0);
MPFR_ASSERTN (__gmpfr_flags == 0);
/* -inf/-0 = +inf */
MPFR_SET_INF (a);
MPFR_SET_NEG (a);
mpfr_set_ui (d, 0, MPFR_RNDZ);
mpfr_neg (d, d, MPFR_RNDZ);
mpfr_clear_flags ();
MPFR_ASSERTN (test_div (q, a, d, MPFR_RNDZ) == 0); /* exact */
MPFR_ASSERTN (mpfr_inf_p (q) && mpfr_sgn (q) > 0);
MPFR_ASSERTN (__gmpfr_flags == 0);
/* check overflow */
emax = mpfr_get_emax ();
set_emax (1);
mpfr_set_ui (a, 1, MPFR_RNDZ);
mpfr_set_ui (d, 1, MPFR_RNDZ);
mpfr_div_2exp (d, d, 1, MPFR_RNDZ);
mpfr_clear_flags ();
test_div (q, a, d, MPFR_RNDU); /* 1 / 0.5 = 2 -> overflow */
MPFR_ASSERTN (mpfr_inf_p (q) && mpfr_sgn (q) > 0);
MPFR_ASSERTN (__gmpfr_flags == (MPFR_FLAGS_OVERFLOW | MPFR_FLAGS_INEXACT));
set_emax (emax);
/* check underflow */
emin = mpfr_get_emin ();
set_emin (-1);
mpfr_set_ui (a, 1, MPFR_RNDZ);
mpfr_div_2exp (a, a, 2, MPFR_RNDZ);
mpfr_set_prec (d, mpfr_get_prec (q) + 8);
for (i = -1; i <= 1; i++)
{
int sign;
/* Test 2^(-2) / (+/- (2 + eps)), with eps < 0, eps = 0, eps > 0.
-> underflow.
With div.c r5513, this test fails for eps > 0 in MPFR_RNDN. */
mpfr_set_ui (d, 2, MPFR_RNDZ);
if (i < 0)
mpfr_nextbelow (d);
if (i > 0)
mpfr_nextabove (d);
for (sign = 0; sign <= 1; sign++)
{
mpfr_clear_flags ();
test_div (q, a, d, MPFR_RNDZ); /* result = 0 */
MPFR_ASSERTN (__gmpfr_flags ==
(MPFR_FLAGS_UNDERFLOW | MPFR_FLAGS_INEXACT));
MPFR_ASSERTN (sign ? MPFR_IS_NEG (q) : MPFR_IS_POS (q));
MPFR_ASSERTN (MPFR_IS_ZERO (q));
mpfr_clear_flags ();
test_div (q, a, d, MPFR_RNDN); /* result = 0 iff eps >= 0 */
MPFR_ASSERTN (__gmpfr_flags ==
(MPFR_FLAGS_UNDERFLOW | MPFR_FLAGS_INEXACT));
MPFR_ASSERTN (sign ? MPFR_IS_NEG (q) : MPFR_IS_POS (q));
if (i < 0)
mpfr_nexttozero (q);
MPFR_ASSERTN (MPFR_IS_ZERO (q));
mpfr_neg (d, d, MPFR_RNDN);
}
}
set_emin (emin);
mpfr_clear (a);
mpfr_clear (d);
mpfr_clear (q);
}
int
main (void)
{
mpfr_t x, y, z;
tests_start_mpfr ();
mpfr_init (x);
mpfr_init (y);
mpfr_init (z);
/* case x=NaN && y=NAN */
mpfr_set_nan (x);
mpfr_set_nan (y);
mpfr_min (z, x, y, MPFR_RNDN);
if (!mpfr_nan_p (z))
{
printf ("Error in mpfr_min (NaN, NaN)\n");
exit (1);
}
mpfr_max (z, x, y, MPFR_RNDN);
if (!mpfr_nan_p (z))
{
printf ("Error in mpfr_max (NaN, NaN)\n");
exit (1);
}
/* case x=NaN */
mpfr_set_nan (x);
mpfr_set_ui (y, 0, MPFR_RNDN);
mpfr_min (z, x, y, MPFR_RNDN);
if (mpfr_cmp_ui (z, 0))
{
printf ("Error in mpfr_min (NaN, 0)\n");
exit (1);
}
mpfr_min (z, y, x, MPFR_RNDN);
if (mpfr_cmp_ui (z, 0))
{
printf ("Error in mpfr_min (0, NaN)\n");
exit (1);
}
mpfr_max (z, x, y, MPFR_RNDN);
if (mpfr_cmp_ui (z, 0))
{
printf ("Error in mpfr_max (NaN, 0)\n");
exit (1);
}
mpfr_max (z, y, x, MPFR_RNDN);
if (mpfr_cmp_ui (z, 0))
{
printf ("Error in mpfr_max (0, NaN)\n");
exit (1);
}
/* Case x=0+ and x=0- */
mpfr_set_ui (x, 0, MPFR_RNDN);
mpfr_set_ui (y, 0, MPFR_RNDN); MPFR_SET_NEG(y);
mpfr_max (z, x, y, MPFR_RNDN);
if (!MPFR_IS_ZERO(z) || MPFR_IS_NEG(z))
{
printf ("Error in mpfr_max (0+, 0-)\n");
exit (1);
}
mpfr_min (z, x, y, MPFR_RNDN);
if (!MPFR_IS_ZERO(z) || MPFR_IS_POS(z))
{
printf ("Error in mpfr_min (0+, 0-)\n");
exit (1);
}
/* Case x=0- and y=0+ */
mpfr_set_ui (x, 0, MPFR_RNDN); MPFR_SET_NEG(x);
mpfr_set_ui (y, 0, MPFR_RNDN);
mpfr_max (z, x, y, MPFR_RNDN);
if (!MPFR_IS_ZERO(z) || MPFR_IS_NEG(z))
{
printf ("Error in mpfr_max (0+, 0-)\n");
exit (1);
}
mpfr_min (z, x, y, MPFR_RNDN);
if (!MPFR_IS_ZERO(z) || MPFR_IS_POS(z))
{
printf ("Error in mpfr_min (0+, 0-)\n");
exit (1);
}
/* case x=+Inf */
mpfr_set_inf (x, 1);
mpfr_set_si (y, -12, MPFR_RNDN);
mpfr_min (z, x, y, MPFR_RNDN);
if ( mpfr_cmp_si (z, -12) )
{
printf ("Error in mpfr_min (+Inf, -12)\n");
exit (1);
}
mpfr_max (z, x, y, MPFR_RNDN);
if ( !MPFR_IS_INF(z) || MPFR_IS_NEG(z) )
{
printf ("Error in mpfr_max (+Inf, 12)\n");
exit (1);
}
/* case x=-Inf */
mpfr_set_inf (x, -1);
mpfr_set_ui (y, 12, MPFR_RNDN);
mpfr_max (z, x, y, MPFR_RNDN);
if ( mpfr_cmp_ui (z, 12) )
{
printf ("Error in mpfr_max (-Inf, 12)\n");
exit (1);
}
mpfr_min (z, x, y, MPFR_RNDN);
if ( !MPFR_IS_INF(z) || MPFR_IS_POS(z) )
{
printf ("Error in mpfr_min (-Inf, 12)\n");
exit (1);
}
/* case x=17 and y=42 */
mpfr_set_ui (x, 17, MPFR_RNDN);
mpfr_set_ui (y, 42, MPFR_RNDN);
mpfr_max (z, x, y, MPFR_RNDN);
if ( mpfr_cmp_ui (z, 42) )
{
printf ("Error in mpfr_max (17, 42)\n");
exit (1);
}
mpfr_max (z, y, x, MPFR_RNDN);
if ( mpfr_cmp_ui (z, 42) )
{
printf ("Error in mpfr_max (42, 17)\n");
exit (1);
}
mpfr_min (z, y, x, MPFR_RNDN);
if ( mpfr_cmp_ui (z, 17) )
{
printf ("Error in mpfr_min (42, 17)\n");
exit (1);
}
mpfr_clear (x);
mpfr_clear (y);
mpfr_clear (z);
tests_end_mpfr ();
return 0;
}
int
mpfr_sub1sp (mpfr_ptr a, mpfr_srcptr b, mpfr_srcptr c, mpfr_rnd_t rnd_mode)
{
mpfr_exp_t bx,cx;
mpfr_uexp_t d;
mpfr_prec_t p, sh, cnt;
mp_size_t n;
mp_limb_t *ap, *bp, *cp;
mp_limb_t limb;
int inexact;
mp_limb_t bcp,bcp1; /* Cp and C'p+1 */
mp_limb_t bbcp = (mp_limb_t) -1, bbcp1 = (mp_limb_t) -1; /* Cp+1 and C'p+2,
gcc claims that they might be used uninitialized. We fill them with invalid
values, which should produce a failure if so. See README.dev file. */
MPFR_TMP_DECL(marker);
MPFR_TMP_MARK(marker);
MPFR_ASSERTD(MPFR_PREC(a) == MPFR_PREC(b) && MPFR_PREC(b) == MPFR_PREC(c));
MPFR_ASSERTD(MPFR_IS_PURE_FP(b));
MPFR_ASSERTD(MPFR_IS_PURE_FP(c));
/* Read prec and num of limbs */
p = MPFR_PREC (b);
n = MPFR_PREC2LIMBS (p);
/* Fast cmp of |b| and |c|*/
bx = MPFR_GET_EXP (b);
cx = MPFR_GET_EXP (c);
if (MPFR_UNLIKELY(bx == cx))
{
mp_size_t k = n - 1;
/* Check mantissa since exponent are equals */
bp = MPFR_MANT(b);
cp = MPFR_MANT(c);
while (k>=0 && MPFR_UNLIKELY(bp[k] == cp[k]))
k--;
if (MPFR_UNLIKELY(k < 0))
/* b == c ! */
{
/* Return exact number 0 */
if (rnd_mode == MPFR_RNDD)
MPFR_SET_NEG(a);
else
MPFR_SET_POS(a);
MPFR_SET_ZERO(a);
MPFR_RET(0);
}
else if (bp[k] > cp[k])
goto BGreater;
else
{
MPFR_ASSERTD(bp[k]<cp[k]);
goto CGreater;
}
}
else if (MPFR_UNLIKELY(bx < cx))
{
/* Swap b and c and set sign */
mpfr_srcptr t;
mpfr_exp_t tx;
CGreater:
MPFR_SET_OPPOSITE_SIGN(a,b);
t = b; b = c; c = t;
tx = bx; bx = cx; cx = tx;
}
else
{
/* b > c */
BGreater:
MPFR_SET_SAME_SIGN(a,b);
}
/* Now b > c */
MPFR_ASSERTD(bx >= cx);
d = (mpfr_uexp_t) bx - cx;
DEBUG (printf ("New with diff=%lu\n", (unsigned long) d));
if (MPFR_UNLIKELY(d <= 1))
{
if (MPFR_LIKELY(d < 1))
{
/* <-- b -->
<-- c --> : exact sub */
ap = MPFR_MANT(a);
mpn_sub_n (ap, MPFR_MANT(b), MPFR_MANT(c), n);
/* Normalize */
ExactNormalize:
limb = ap[n-1];
if (MPFR_LIKELY(limb))
{
/* First limb is not zero. */
count_leading_zeros(cnt, limb);
/* cnt could be == 0 <= SubD1Lose */
if (MPFR_LIKELY(cnt))
{
mpn_lshift(ap, ap, n, cnt); /* Normalize number */
bx -= cnt; /* Update final expo */
}
/* Last limb should be ok */
MPFR_ASSERTD(!(ap[0] & MPFR_LIMB_MASK((unsigned int) (-p)
% GMP_NUMB_BITS)));
}
else
{
/* First limb is zero */
mp_size_t k = n-1, len;
/* Find the first limb not equal to zero.
FIXME:It is assume it exists (since |b| > |c| and same prec)*/
do
{
MPFR_ASSERTD( k > 0 );
limb = ap[--k];
}
while (limb == 0);
MPFR_ASSERTD(limb != 0);
count_leading_zeros(cnt, limb);
k++;
len = n - k; /* Number of last limb */
MPFR_ASSERTD(k >= 0);
if (MPFR_LIKELY(cnt))
mpn_lshift(ap+len, ap, k, cnt); /* Normalize the High Limb*/
else
{
/* Must use DECR since src and dest may overlap & dest>=src*/
MPN_COPY_DECR(ap+len, ap, k);
}
MPN_ZERO(ap, len); /* Zeroing the last limbs */
bx -= cnt + len*GMP_NUMB_BITS; /* Update Expo */
/* Last limb should be ok */
MPFR_ASSERTD(!(ap[len]&MPFR_LIMB_MASK((unsigned int) (-p)
% GMP_NUMB_BITS)));
}
/* Check expo underflow */
if (MPFR_UNLIKELY(bx < __gmpfr_emin))
{
MPFR_TMP_FREE(marker);
/* inexact=0 */
DEBUG( printf("(D==0 Underflow)\n") );
if (rnd_mode == MPFR_RNDN &&
(bx < __gmpfr_emin - 1 ||
(/*inexact >= 0 &&*/ mpfr_powerof2_raw (a))))
rnd_mode = MPFR_RNDZ;
return mpfr_underflow (a, rnd_mode, MPFR_SIGN(a));
}
MPFR_SET_EXP (a, bx);
/* No rounding is necessary since the result is exact */
MPFR_ASSERTD(ap[n-1] > ~ap[n-1]);
MPFR_TMP_FREE(marker);
return 0;
}
else /* if (d == 1) */
{
/* | <-- b -->
| <-- c --> */
mp_limb_t c0, mask;
mp_size_t k;
MPFR_UNSIGNED_MINUS_MODULO(sh, p);
/* If we lose at least one bit, compute 2*b-c (Exact)
* else compute b-c/2 */
bp = MPFR_MANT(b);
cp = MPFR_MANT(c);
k = n-1;
limb = bp[k] - cp[k]/2;
if (limb > MPFR_LIMB_HIGHBIT)
{
/* We can't lose precision: compute b-c/2 */
/* Shift c in the allocated temporary block */
SubD1NoLose:
c0 = cp[0] & (MPFR_LIMB_ONE<<sh);
cp = MPFR_TMP_LIMBS_ALLOC (n);
mpn_rshift(cp, MPFR_MANT(c), n, 1);
if (MPFR_LIKELY(c0 == 0))
{
/* Result is exact: no need of rounding! */
ap = MPFR_MANT(a);
mpn_sub_n (ap, bp, cp, n);
MPFR_SET_EXP(a, bx); /* No expo overflow! */
/* No truncate or normalize is needed */
MPFR_ASSERTD(ap[n-1] > ~ap[n-1]);
/* No rounding is necessary since the result is exact */
MPFR_TMP_FREE(marker);
return 0;
}
ap = MPFR_MANT(a);
mask = ~MPFR_LIMB_MASK(sh);
cp[0] &= mask; /* Delete last bit of c */
mpn_sub_n (ap, bp, cp, n);
MPFR_SET_EXP(a, bx); /* No expo overflow! */
MPFR_ASSERTD( !(ap[0] & ~mask) ); /* Check last bits */
/* No normalize is needed */
MPFR_ASSERTD(ap[n-1] > ~ap[n-1]);
/* Rounding is necessary since c0 = 1*/
/* Cp =-1 and C'p+1=0 */
bcp = 1; bcp1 = 0;
if (MPFR_LIKELY(rnd_mode == MPFR_RNDN))
{
/* Even Rule apply: Check Ap-1 */
if (MPFR_LIKELY( (ap[0] & (MPFR_LIMB_ONE<<sh)) == 0) )
goto truncate;
else
goto sub_one_ulp;
}
MPFR_UPDATE_RND_MODE(rnd_mode, MPFR_IS_NEG(a));
if (rnd_mode == MPFR_RNDZ)
goto sub_one_ulp;
else
goto truncate;
}
else if (MPFR_LIKELY(limb < MPFR_LIMB_HIGHBIT))
{
/* We lose at least one bit of prec */
/* Calcul of 2*b-c (Exact) */
/* Shift b in the allocated temporary block */
SubD1Lose:
bp = MPFR_TMP_LIMBS_ALLOC (n);
mpn_lshift (bp, MPFR_MANT(b), n, 1);
ap = MPFR_MANT(a);
mpn_sub_n (ap, bp, cp, n);
bx--;
goto ExactNormalize;
}
else
{
/* Case: limb = 100000000000 */
/* Check while b[k] == c'[k] (C' is C shifted by 1) */
/* If b[k]<c'[k] => We lose at least one bit*/
/* If b[k]>c'[k] => We don't lose any bit */
/* If k==-1 => We don't lose any bit
AND the result is 100000000000 0000000000 00000000000 */
mp_limb_t carry;
do {
carry = cp[k]&MPFR_LIMB_ONE;
k--;
} while (k>=0 &&
bp[k]==(carry=cp[k]/2+(carry<<(GMP_NUMB_BITS-1))));
if (MPFR_UNLIKELY(k<0))
{
/*If carry then (sh==0 and Virtual c'[-1] > Virtual b[-1]) */
if (MPFR_UNLIKELY(carry)) /* carry = cp[0]&MPFR_LIMB_ONE */
{
/* FIXME: Can be faster? */
MPFR_ASSERTD(sh == 0);
goto SubD1Lose;
}
/* Result is a power of 2 */
ap = MPFR_MANT (a);
MPN_ZERO (ap, n);
ap[n-1] = MPFR_LIMB_HIGHBIT;
MPFR_SET_EXP (a, bx); /* No expo overflow! */
/* No Normalize is needed*/
/* No Rounding is needed */
MPFR_TMP_FREE (marker);
return 0;
}
/* carry = cp[k]/2+(cp[k-1]&1)<<(GMP_NUMB_BITS-1) = c'[k]*/
else if (bp[k] > carry)
goto SubD1NoLose;
else
{
MPFR_ASSERTD(bp[k]<carry);
goto SubD1Lose;
}
}
}
}
else if (MPFR_UNLIKELY(d >= p))
{
ap = MPFR_MANT(a);
MPFR_UNSIGNED_MINUS_MODULO(sh, p);
/* We can't set A before since we use cp for rounding... */
/* Perform rounding: check if a=b or a=b-ulp(b) */
if (MPFR_UNLIKELY(d == p))
{
/* cp == -1 and c'p+1 = ? */
bcp = 1;
/* We need Cp+1 later for a very improbable case. */
bbcp = (MPFR_MANT(c)[n-1] & (MPFR_LIMB_ONE<<(GMP_NUMB_BITS-2)));
/* We need also C'p+1 for an even more unprobable case... */
if (MPFR_LIKELY( bbcp ))
bcp1 = 1;
else
{
cp = MPFR_MANT(c);
if (MPFR_UNLIKELY(cp[n-1] == MPFR_LIMB_HIGHBIT))
{
mp_size_t k = n-1;
do {
k--;
} while (k>=0 && cp[k]==0);
bcp1 = (k>=0);
}
else
bcp1 = 1;
}
DEBUG( printf("(D=P) Cp=-1 Cp+1=%d C'p+1=%d \n", bbcp!=0, bcp1!=0) );
bp = MPFR_MANT (b);
/* Even if src and dest overlap, it is ok using MPN_COPY */
if (MPFR_LIKELY(rnd_mode == MPFR_RNDN))
{
if (MPFR_UNLIKELY( bcp && bcp1==0 ))
/* Cp=-1 and C'p+1=0: Even rule Apply! */
/* Check Ap-1 = Bp-1 */
if ((bp[0] & (MPFR_LIMB_ONE<<sh)) == 0)
{
MPN_COPY(ap, bp, n);
goto truncate;
}
MPN_COPY(ap, bp, n);
goto sub_one_ulp;
}
MPFR_UPDATE_RND_MODE(rnd_mode, MPFR_IS_NEG(a));
if (rnd_mode == MPFR_RNDZ)
{
MPN_COPY(ap, bp, n);
goto sub_one_ulp;
}
else
{
MPN_COPY(ap, bp, n);
goto truncate;
}
}
else
{
/* Cp=0, Cp+1=-1 if d==p+1, C'p+1=-1 */
bcp = 0; bbcp = (d==p+1); bcp1 = 1;
DEBUG( printf("(D>P) Cp=%d Cp+1=%d C'p+1=%d\n", bcp!=0,bbcp!=0,bcp1!=0) );
/* Need to compute C'p+2 if d==p+1 and if rnd_mode=NEAREST
(Because of a very improbable case) */
if (MPFR_UNLIKELY(d==p+1 && rnd_mode==MPFR_RNDN))
{
cp = MPFR_MANT(c);
if (MPFR_UNLIKELY(cp[n-1] == MPFR_LIMB_HIGHBIT))
{
mp_size_t k = n-1;
do {
k--;
} while (k>=0 && cp[k]==0);
bbcp1 = (k>=0);
}
else
bbcp1 = 1;
DEBUG( printf("(D>P) C'p+2=%d\n", bbcp1!=0) );
}
/* Copy mantissa B in A */
MPN_COPY(ap, MPFR_MANT(b), n);
/* Round */
if (MPFR_LIKELY(rnd_mode == MPFR_RNDN))
goto truncate;
MPFR_UPDATE_RND_MODE(rnd_mode, MPFR_IS_NEG(a));
if (rnd_mode == MPFR_RNDZ)
goto sub_one_ulp;
else /* rnd_mode = AWAY */
goto truncate;
}
}
else
{
mpfr_uexp_t dm;
mp_size_t m;
mp_limb_t mask;
/* General case: 2 <= d < p */
MPFR_UNSIGNED_MINUS_MODULO(sh, p);
cp = MPFR_TMP_LIMBS_ALLOC (n);
/* Shift c in temporary allocated place */
dm = d % GMP_NUMB_BITS;
m = d / GMP_NUMB_BITS;
if (MPFR_UNLIKELY(dm == 0))
{
/* dm = 0 and m > 0: Just copy */
MPFR_ASSERTD(m!=0);
MPN_COPY(cp, MPFR_MANT(c)+m, n-m);
MPN_ZERO(cp+n-m, m);
}
else if (MPFR_LIKELY(m == 0))
{
/* dm >=2 and m == 0: just shift */
MPFR_ASSERTD(dm >= 2);
mpn_rshift(cp, MPFR_MANT(c), n, dm);
}
else
{
/* dm > 0 and m > 0: shift and zero */
mpn_rshift(cp, MPFR_MANT(c)+m, n-m, dm);
MPN_ZERO(cp+n-m, m);
}
DEBUG( mpfr_print_mant_binary("Before", MPFR_MANT(c), p) );
DEBUG( mpfr_print_mant_binary("B= ", MPFR_MANT(b), p) );
DEBUG( mpfr_print_mant_binary("After ", cp, p) );
/* Compute bcp=Cp and bcp1=C'p+1 */
if (MPFR_LIKELY(sh))
{
/* Try to compute them from C' rather than C (FIXME: Faster?) */
bcp = (cp[0] & (MPFR_LIMB_ONE<<(sh-1))) ;
if (MPFR_LIKELY( cp[0] & MPFR_LIMB_MASK(sh-1) ))
bcp1 = 1;
else
{
/* We can't compute C'p+1 from C'. Compute it from C */
/* Start from bit x=p-d+sh in mantissa C
(+sh since we have already looked sh bits in C'!) */
mpfr_prec_t x = p-d+sh-1;
if (MPFR_LIKELY(x>p))
/* We are already looked at all the bits of c, so C'p+1 = 0*/
bcp1 = 0;
else
{
mp_limb_t *tp = MPFR_MANT(c);
mp_size_t kx = n-1 - (x / GMP_NUMB_BITS);
mpfr_prec_t sx = GMP_NUMB_BITS-1-(x%GMP_NUMB_BITS);
DEBUG (printf ("(First) x=%lu Kx=%ld Sx=%lu\n",
(unsigned long) x, (long) kx,
(unsigned long) sx));
/* Looks at the last bits of limb kx (if sx=0 does nothing)*/
if (tp[kx] & MPFR_LIMB_MASK(sx))
bcp1 = 1;
else
{
/*kx += (sx==0);*/
/*If sx==0, tp[kx] hasn't been checked*/
do {
kx--;
} while (kx>=0 && tp[kx]==0);
bcp1 = (kx >= 0);
}
}
}
}
else
{
/* Compute Cp and C'p+1 from C with sh=0 */
mp_limb_t *tp = MPFR_MANT(c);
/* Start from bit x=p-d in mantissa C */
mpfr_prec_t x = p-d;
mp_size_t kx = n-1 - (x / GMP_NUMB_BITS);
mpfr_prec_t sx = GMP_NUMB_BITS-1-(x%GMP_NUMB_BITS);
MPFR_ASSERTD(p >= d);
bcp = (tp[kx] & (MPFR_LIMB_ONE<<sx));
/* Looks at the last bits of limb kx (If sx=0, does nothing)*/
if (tp[kx] & MPFR_LIMB_MASK(sx))
bcp1 = 1;
else
{
/*kx += (sx==0);*/ /*If sx==0, tp[kx] hasn't been checked*/
do {
kx--;
} while (kx>=0 && tp[kx]==0);
bcp1 = (kx>=0);
}
}
DEBUG( printf("sh=%lu Cp=%d C'p+1=%d\n", sh, bcp!=0, bcp1!=0) );
/* Check if we can lose a bit, and if so compute Cp+1 and C'p+2 */
bp = MPFR_MANT(b);
if (MPFR_UNLIKELY((bp[n-1]-cp[n-1]) <= MPFR_LIMB_HIGHBIT))
{
/* We can lose a bit so we precompute Cp+1 and C'p+2 */
/* Test for trivial case: since C'p+1=0, Cp+1=0 and C'p+2 =0 */
if (MPFR_LIKELY(bcp1 == 0))
{
bbcp = 0;
bbcp1 = 0;
}
else /* bcp1 != 0 */
{
/* We can lose a bit:
compute Cp+1 and C'p+2 from mantissa C */
mp_limb_t *tp = MPFR_MANT(c);
/* Start from bit x=(p+1)-d in mantissa C */
mpfr_prec_t x = p+1-d;
mp_size_t kx = n-1 - (x/GMP_NUMB_BITS);
mpfr_prec_t sx = GMP_NUMB_BITS-1-(x%GMP_NUMB_BITS);
MPFR_ASSERTD(p > d);
DEBUG (printf ("(pre) x=%lu Kx=%ld Sx=%lu\n",
(unsigned long) x, (long) kx,
(unsigned long) sx));
bbcp = (tp[kx] & (MPFR_LIMB_ONE<<sx)) ;
/* Looks at the last bits of limb kx (If sx=0, does nothing)*/
/* If Cp+1=0, since C'p+1!=0, C'p+2=1 ! */
if (MPFR_LIKELY(bbcp==0 || (tp[kx]&MPFR_LIMB_MASK(sx))))
bbcp1 = 1;
else
{
/*kx += (sx==0);*/ /*If sx==0, tp[kx] hasn't been checked*/
do {
kx--;
} while (kx>=0 && tp[kx]==0);
bbcp1 = (kx>=0);
DEBUG (printf ("(Pre) Scan done for %ld\n", (long) kx));
}
} /*End of Bcp1 != 0*/
DEBUG( printf("(Pre) Cp+1=%d C'p+2=%d\n", bbcp!=0, bbcp1!=0) );
} /* End of "can lose a bit" */
/* Clean shifted C' */
mask = ~MPFR_LIMB_MASK (sh);
cp[0] &= mask;
/* Subtract the mantissa c from b in a */
ap = MPFR_MANT(a);
mpn_sub_n (ap, bp, cp, n);
DEBUG( mpfr_print_mant_binary("Sub= ", ap, p) );
/* Normalize: we lose at max one bit*/
if (MPFR_UNLIKELY(MPFR_LIMB_MSB(ap[n-1]) == 0))
{
/* High bit is not set and we have to fix it! */
/* Ap >= 010000xxx001 */
mpn_lshift(ap, ap, n, 1);
/* Ap >= 100000xxx010 */
if (MPFR_UNLIKELY(bcp!=0)) /* Check if Cp = -1 */
/* Since Cp == -1, we have to substract one more */
{
mpn_sub_1(ap, ap, n, MPFR_LIMB_ONE<<sh);
MPFR_ASSERTD(MPFR_LIMB_MSB(ap[n-1]) != 0);
}
/* Ap >= 10000xxx001 */
/* Final exponent -1 since we have shifted the mantissa */
bx--;
/* Update bcp and bcp1 */
MPFR_ASSERTN(bbcp != (mp_limb_t) -1);
MPFR_ASSERTN(bbcp1 != (mp_limb_t) -1);
bcp = bbcp;
bcp1 = bbcp1;
/* We dont't have anymore a valid Cp+1!
But since Ap >= 100000xxx001, the final sub can't unnormalize!*/
}
MPFR_ASSERTD( !(ap[0] & ~mask) );
/* Rounding */
if (MPFR_LIKELY(rnd_mode == MPFR_RNDN))
{
if (MPFR_LIKELY(bcp==0))
goto truncate;
else if ((bcp1) || ((ap[0] & (MPFR_LIMB_ONE<<sh)) != 0))
goto sub_one_ulp;
else
goto truncate;
}
/* Update rounding mode */
MPFR_UPDATE_RND_MODE(rnd_mode, MPFR_IS_NEG(a));
if (rnd_mode == MPFR_RNDZ && (MPFR_LIKELY(bcp || bcp1)))
goto sub_one_ulp;
goto truncate;
}
MPFR_RET_NEVER_GO_HERE ();
/* Sub one ulp to the result */
sub_one_ulp:
mpn_sub_1 (ap, ap, n, MPFR_LIMB_ONE << sh);
/* Result should be smaller than exact value: inexact=-1 */
inexact = -1;
/* Check normalisation */
if (MPFR_UNLIKELY(MPFR_LIMB_MSB(ap[n-1]) == 0))
{
/* ap was a power of 2, and we lose a bit */
/* Now it is 0111111111111111111[00000 */
mpn_lshift(ap, ap, n, 1);
bx--;
/* And the lost bit x depends on Cp+1, and Cp */
/* Compute Cp+1 if it isn't already compute (ie d==1) */
/* FIXME: Is this case possible? */
if (MPFR_UNLIKELY(d == 1))
bbcp = 0;
DEBUG( printf("(SubOneUlp)Cp=%d, Cp+1=%d C'p+1=%d\n", bcp!=0,bbcp!=0,bcp1!=0));
/* Compute the last bit (Since we have shifted the mantissa)
we need one more bit!*/
MPFR_ASSERTN(bbcp != (mp_limb_t) -1);
if ( (rnd_mode == MPFR_RNDZ && bcp==0)
|| (rnd_mode==MPFR_RNDN && bbcp==0)
|| (bcp && bcp1==0) ) /*Exact result*/
{
ap[0] |= MPFR_LIMB_ONE<<sh;
if (rnd_mode == MPFR_RNDN)
inexact = 1;
DEBUG( printf("(SubOneUlp) Last bit set\n") );
}
/* Result could be exact if C'p+1 = 0 and rnd == Zero
since we have had one more bit to the result */
/* Fixme: rnd_mode == MPFR_RNDZ needed ? */
if (bcp1==0 && rnd_mode==MPFR_RNDZ)
{
DEBUG( printf("(SubOneUlp) Exact result\n") );
inexact = 0;
}
}
goto end_of_sub;
truncate:
/* Check if the result is an exact power of 2: 100000000000
in which cases, we could have to do sub_one_ulp due to some nasty reasons:
If Result is a Power of 2:
+ If rnd = AWAY,
| If Cp=-1 and C'p+1 = 0, SubOneUlp and the result is EXACT.
If Cp=-1 and C'p+1 =-1, SubOneUlp and the result is above.
Otherwise truncate
+ If rnd = NEAREST,
If Cp= 0 and Cp+1 =-1 and C'p+2=-1, SubOneUlp and the result is above
If cp=-1 and C'p+1 = 0, SubOneUlp and the result is exact.
Otherwise truncate.
X bit should always be set if SubOneUlp*/
if (MPFR_UNLIKELY(ap[n-1] == MPFR_LIMB_HIGHBIT))
{
mp_size_t k = n-1;
do {
k--;
} while (k>=0 && ap[k]==0);
if (MPFR_UNLIKELY(k<0))
{
/* It is a power of 2! */
/* Compute Cp+1 if it isn't already compute (ie d==1) */
/* FIXME: Is this case possible? */
if (d == 1)
bbcp=0;
DEBUG( printf("(Truncate) Cp=%d, Cp+1=%d C'p+1=%d C'p+2=%d\n", \
bcp!=0, bbcp!=0, bcp1!=0, bbcp1!=0) );
MPFR_ASSERTN(bbcp != (mp_limb_t) -1);
MPFR_ASSERTN((rnd_mode != MPFR_RNDN) || (bcp != 0) || (bbcp == 0) || (bbcp1 != (mp_limb_t) -1));
if (((rnd_mode != MPFR_RNDZ) && bcp)
||
((rnd_mode == MPFR_RNDN) && (bcp == 0) && (bbcp) && (bbcp1)))
{
DEBUG( printf("(Truncate) Do sub\n") );
mpn_sub_1 (ap, ap, n, MPFR_LIMB_ONE << sh);
mpn_lshift(ap, ap, n, 1);
ap[0] |= MPFR_LIMB_ONE<<sh;
bx--;
/* FIXME: Explain why it works (or why not)... */
inexact = (bcp1 == 0) ? 0 : (rnd_mode==MPFR_RNDN) ? -1 : 1;
goto end_of_sub;
}
}
}
/* Calcul of Inexact flag.*/
inexact = MPFR_LIKELY(bcp || bcp1) ? 1 : 0;
end_of_sub:
/* Update Expo */
/* FIXME: Is this test really useful?
If d==0 : Exact case. This is never called.
if 1 < d < p : bx=MPFR_EXP(b) or MPFR_EXP(b)-1 > MPFR_EXP(c) > emin
if d == 1 : bx=MPFR_EXP(b). If we could lose any bits, the exact
normalisation is called.
if d >= p : bx=MPFR_EXP(b) >= MPFR_EXP(c) + p > emin
After SubOneUlp, we could have one bit less.
if 1 < d < p : bx >= MPFR_EXP(b)-2 >= MPFR_EXP(c) > emin
if d == 1 : bx >= MPFR_EXP(b)-1 = MPFR_EXP(c) > emin.
if d >= p : bx >= MPFR_EXP(b)-1 > emin since p>=2.
*/
MPFR_ASSERTD( bx >= __gmpfr_emin);
/*
if (MPFR_UNLIKELY(bx < __gmpfr_emin))
{
DEBUG( printf("(Final Underflow)\n") );
if (rnd_mode == MPFR_RNDN &&
(bx < __gmpfr_emin - 1 ||
(inexact >= 0 && mpfr_powerof2_raw (a))))
rnd_mode = MPFR_RNDZ;
MPFR_TMP_FREE(marker);
return mpfr_underflow (a, rnd_mode, MPFR_SIGN(a));
}
*/
MPFR_SET_EXP (a, bx);
MPFR_TMP_FREE(marker);
MPFR_RET (inexact * MPFR_INT_SIGN (a));
}
int
mpfr_pow_si (mpfr_ptr y, mpfr_srcptr x, long int n, mp_rnd_t rnd_mode)
{
if (n > 0)
return mpfr_pow_ui(y, x, n, rnd_mode);
else
{
int inexact = 0;
if (MPFR_IS_NAN(x))
{
MPFR_SET_NAN(y);
MPFR_RET_NAN;
}
MPFR_CLEAR_NAN(y);
if (n == 0)
return mpfr_set_ui(y, 1, GMP_RNDN);
if (MPFR_IS_INF(x))
{
MPFR_SET_ZERO(y);
if (MPFR_SIGN(x) > 0 || ((unsigned) n & 1) == 0)
MPFR_SET_POS(y);
else
MPFR_SET_NEG(y);
MPFR_RET(0);
}
if (MPFR_IS_ZERO(x))
{
MPFR_SET_INF(y);
if (MPFR_SIGN(x) > 0 || ((unsigned) n & 1) == 0)
MPFR_SET_POS(y);
else
MPFR_SET_NEG(y);
MPFR_RET(0);
}
MPFR_CLEAR_INF(y);
n = -n;
/* General case */
{
/* Declaration of the intermediary variable */
mpfr_t t, ti;
/* Declaration of the size variable */
mp_prec_t Nx = MPFR_PREC(x); /* Precision of input variable */
mp_prec_t Ny = MPFR_PREC(y); /* Precision of input variable */
mp_prec_t Nt; /* Precision of the intermediary variable */
long int err; /* Precision of error */
/* compute the precision of intermediary variable */
Nt=MAX(Nx,Ny);
/* the optimal number of bits : see algorithms.ps */
Nt=Nt+3+_mpfr_ceil_log2(Nt);
/* initialise of intermediary variable */
mpfr_init(t);
mpfr_init(ti);
do {
/* reactualisation of the precision */
mpfr_set_prec(t,Nt);
mpfr_set_prec(ti,Nt);
/* compute 1/(x^n) n>0*/
mpfr_pow_ui(ti,x,(unsigned long int)(n),GMP_RNDN);
mpfr_ui_div(t,1,ti,GMP_RNDN);
/* estimation of the error -- see pow function in algorithms.ps*/
err = Nt - 3;
/* actualisation of the precision */
Nt += 10;
} while (err<0 || !mpfr_can_round(t,err,GMP_RNDN,rnd_mode,Ny));
inexact = mpfr_set(y,t,rnd_mode);
mpfr_clear(t);
mpfr_clear(ti);
}
return inexact;
}
}
