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-rw-r--r--Build/source/libs/mpfr/mpfr-src/src/root.c194
1 files changed, 134 insertions, 60 deletions
diff --git a/Build/source/libs/mpfr/mpfr-src/src/root.c b/Build/source/libs/mpfr/mpfr-src/src/root.c
index c1a141c90bf..5bf45449851 100644
--- a/Build/source/libs/mpfr/mpfr-src/src/root.c
+++ b/Build/source/libs/mpfr/mpfr-src/src/root.c
@@ -1,7 +1,7 @@
/* mpfr_root -- kth root.
-Copyright 2005-2015 Free Software Foundation, Inc.
-Contributed by the AriC and Caramel projects, INRIA.
+Copyright 2005-2016 Free Software Foundation, Inc.
+Contributed by the AriC and Caramba projects, INRIA.
This file is part of the GNU MPFR Library.
@@ -23,13 +23,15 @@ http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
#define MPFR_NEED_LONGLONG_H
#include "mpfr-impl.h"
- /* The computation of y = x^(1/k) is done as follows:
+ /* The computation of y = x^(1/k) is done as follows, except for large
+ values of k, for which this would be inefficient or yield internal
+ integer overflows:
Let x = sign * m * 2^(k*e) where m is an integer
with 2^(k*(n-1)) <= m < 2^(k*n) where n = PREC(y)
- and m = s^k + r where 0 <= r and m < (s+1)^k
+ and m = s^k + t where 0 <= t and m < (s+1)^k
we want that s has n bits i.e. s >= 2^(n-1), or m >= 2^(k*(n-1))
i.e. m must have at least k*(n-1)+1 bits
@@ -38,11 +40,15 @@ http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
x^(1/k) = s * 2^e or (s+1) * 2^e according to the rounding mode.
*/
+static int
+mpfr_root_aux (mpfr_ptr y, mpfr_srcptr x, unsigned long k,
+ mpfr_rnd_t rnd_mode);
+
int
mpfr_root (mpfr_ptr y, mpfr_srcptr x, unsigned long k, mpfr_rnd_t rnd_mode)
{
mpz_t m;
- mpfr_exp_t e, r, sh;
+ mpfr_exp_t e, r, sh, f;
mpfr_prec_t n, size_m, tmp;
int inexact, negative;
MPFR_SAVE_EXPO_DECL (expo);
@@ -55,50 +61,27 @@ mpfr_root (mpfr_ptr y, mpfr_srcptr x, unsigned long k, mpfr_rnd_t rnd_mode)
if (MPFR_UNLIKELY (k <= 1))
{
- if (k < 1) /* k==0 => y=x^(1/0)=x^(+Inf) */
-#if 0
- /* For 0 <= x < 1 => +0.
- For x = 1 => 1.
- For x > 1, => +Inf.
- For x < 0 => NaN.
- */
+ if (k == 0)
{
- if (MPFR_IS_NEG (x) && !MPFR_IS_ZERO (x))
- {
- MPFR_SET_NAN (y);
- MPFR_RET_NAN;
- }
- inexact = mpfr_cmp (x, __gmpfr_one);
- if (inexact == 0)
- return mpfr_set_ui (y, 1, rnd_mode); /* 1 may be Out of Range */
- else if (inexact < 0)
- return mpfr_set_ui (y, 0, rnd_mode); /* 0+ */
- else
- {
- mpfr_set_inf (y, 1);
- return 0;
- }
+ MPFR_SET_NAN (y);
+ MPFR_RET_NAN;
}
-#endif
- {
- MPFR_SET_NAN (y);
- MPFR_RET_NAN;
- }
- else /* y =x^(1/1)=x */
+ else /* y = x^(1/1) = x */
return mpfr_set (y, x, rnd_mode);
}
/* Singular values */
- else if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
+ if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
{
if (MPFR_IS_NAN (x))
{
MPFR_SET_NAN (y); /* NaN^(1/k) = NaN */
MPFR_RET_NAN;
}
- else if (MPFR_IS_INF (x)) /* +Inf^(1/k) = +Inf
- -Inf^(1/k) = -Inf if k odd
- -Inf^(1/k) = NaN if k even */
+
+ if (MPFR_IS_INF (x)) /* +Inf^(1/k) = +Inf
+ -Inf^(1/k) = -Inf if k odd
+ -Inf^(1/k) = NaN if k even */
{
if (MPFR_IS_NEG(x) && (k % 2 == 0))
{
@@ -106,27 +89,31 @@ mpfr_root (mpfr_ptr y, mpfr_srcptr x, unsigned long k, mpfr_rnd_t rnd_mode)
MPFR_RET_NAN;
}
MPFR_SET_INF (y);
- MPFR_SET_SAME_SIGN (y, x);
- MPFR_RET (0);
}
else /* x is necessarily 0: (+0)^(1/k) = +0
(-0)^(1/k) = -0 */
{
MPFR_ASSERTD (MPFR_IS_ZERO (x));
MPFR_SET_ZERO (y);
- MPFR_SET_SAME_SIGN (y, x);
- MPFR_RET (0);
}
+ MPFR_SET_SAME_SIGN (y, x);
+ MPFR_RET (0);
}
/* Returns NAN for x < 0 and k even */
- else if (MPFR_IS_NEG (x) && (k % 2 == 0))
+ if (MPFR_UNLIKELY (MPFR_IS_NEG (x) && (k % 2 == 0)))
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
/* General case */
+
+ /* For large k, use exp(log(x)/k). The threshold of 100 seems to be quite
+ good when the precision goes to infinity. */
+ if (k > 100)
+ return mpfr_root_aux (y, x, k, rnd_mode);
+
MPFR_SAVE_EXPO_MARK (expo);
mpz_init (m);
@@ -135,31 +122,24 @@ mpfr_root (mpfr_ptr y, mpfr_srcptr x, unsigned long k, mpfr_rnd_t rnd_mode)
mpz_neg (m, m);
r = e % (mpfr_exp_t) k;
if (r < 0)
- r += k; /* now r = e (mod k) with 0 <= e < r */
+ r += k; /* now r = e (mod k) with 0 <= r < k */
+ MPFR_ASSERTD (0 <= r && r < k);
/* x = (m*2^r) * 2^(e-r) where e-r is a multiple of k */
MPFR_MPZ_SIZEINBASE2 (size_m, m);
/* for rounding to nearest, we want the round bit to be in the root */
n = MPFR_PREC (y) + (rnd_mode == MPFR_RNDN);
- /* we now multiply m by 2^(r+k*sh) so that root(m,k) will give
- exactly n bits: we want k*(n-1)+1 <= size_m + k*sh + r <= k*n
- i.e. sh = floor ((kn-size_m-r)/k) */
- if ((mpfr_exp_t) size_m + r > k * (mpfr_exp_t) n)
- sh = 0; /* we already have too many bits */
+ /* we now multiply m by 2^sh so that root(m,k) will give
+ exactly n bits: we want k*(n-1)+1 <= size_m + sh <= k*n
+ i.e. sh = k*f + r with f = max(floor((k*n-size_m-r)/k),0) */
+ if ((mpfr_exp_t) size_m + r >= k * (mpfr_exp_t) n)
+ f = 0; /* we already have too many bits */
else
- sh = (k * (mpfr_exp_t) n - (mpfr_exp_t) size_m - r) / k;
- sh = k * sh + r;
- if (sh >= 0)
- {
- mpz_mul_2exp (m, m, sh);
- e = e - sh;
- }
- else if (r > 0)
- {
- mpz_mul_2exp (m, m, r);
- e = e - r;
- }
+ f = (k * (mpfr_exp_t) n - (mpfr_exp_t) size_m - r) / k;
+ sh = k * f + r;
+ mpz_mul_2exp (m, m, sh);
+ e = e - sh;
/* invariant: x = m*2^e, with e divisible by k */
@@ -203,3 +183,97 @@ mpfr_root (mpfr_ptr y, mpfr_srcptr x, unsigned long k, mpfr_rnd_t rnd_mode)
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inexact, rnd_mode);
}
+
+/* Compute y <- x^(1/k) using exp(log(x)/k).
+ Assume all special cases have been eliminated before.
+ In the extended exponent range, overflows/underflows are not possible.
+ Assume x > 0, or x < 0 and k odd.
+*/
+static int
+mpfr_root_aux (mpfr_ptr y, mpfr_srcptr x, unsigned long k, mpfr_rnd_t rnd_mode)
+{
+ int inexact, exact_root = 0;
+ mpfr_prec_t w; /* working precision */
+ mpfr_t absx, t;
+ MPFR_GROUP_DECL(group);
+ MPFR_TMP_DECL(marker);
+ MPFR_ZIV_DECL(loop);
+ MPFR_SAVE_EXPO_DECL (expo);
+
+ MPFR_TMP_INIT_ABS (absx, x);
+
+ MPFR_TMP_MARK(marker);
+ w = MPFR_PREC(y) + 10;
+ /* Take some guard bits to prepare for the 'expt' lost bits below.
+ If |x| < 2^k, then log|x| < k, thus taking log2(k) bits should be fine. */
+ if (MPFR_GET_EXP(x) > 0)
+ w += MPFR_INT_CEIL_LOG2 (MPFR_GET_EXP(x));
+ MPFR_GROUP_INIT_1(group, w, t);
+ MPFR_SAVE_EXPO_MARK (expo);
+ MPFR_ZIV_INIT (loop, w);
+ for (;;)
+ {
+ mpfr_exp_t expt;
+ unsigned int err;
+
+ mpfr_log (t, absx, MPFR_RNDN);
+ /* t = log|x| * (1 + theta) with |theta| <= 2^(-w) */
+ mpfr_div_ui (t, t, k, MPFR_RNDN);
+ expt = MPFR_GET_EXP (t);
+ /* t = log|x|/k * (1 + theta) + eps with |theta| <= 2^(-w)
+ and |eps| <= 1/2 ulp(t), thus the total error is bounded
+ by 1.5 * 2^(expt - w) */
+ mpfr_exp (t, t, MPFR_RNDN);
+ /* t = |x|^(1/k) * exp(tau) * (1 + theta1) with
+ |tau| <= 1.5 * 2^(expt - w) and |theta1| <= 2^(-w).
+ For |tau| <= 0.5 we have |exp(tau)-1| < 4/3*tau, thus
+ for w >= expt + 2 we have:
+ t = |x|^(1/k) * (1 + 2^(expt+2)*theta2) * (1 + theta1) with
+ |theta1|, |theta2| <= 2^(-w).
+ If expt+2 > 0, as long as w >= 1, we have:
+ t = |x|^(1/k) * (1 + 2^(expt+3)*theta3) with |theta3| < 2^(-w).
+ For expt+2 = 0, we have:
+ t = |x|^(1/k) * (1 + 2^2*theta3) with |theta3| < 2^(-w).
+ Finally for expt+2 < 0 we have:
+ t = |x|^(1/k) * (1 + 2*theta3) with |theta3| < 2^(-w).
+ */
+ err = (expt + 2 > 0) ? expt + 3
+ : (expt + 2 == 0) ? 2 : 1;
+ /* now t = |x|^(1/k) * (1 + 2^(err-w)) thus the error is at most
+ 2^(EXP(t) - w + err) */
+ if (MPFR_LIKELY (MPFR_CAN_ROUND(t, w - err, MPFR_PREC(y), rnd_mode)))
+ break;
+
+ /* If we fail to round correctly, check for an exact result or a
+ midpoint result with MPFR_RNDN (regarded as hard-to-round in
+ all precisions in order to determine the ternary value). */
+ {
+ mpfr_t z, zk;
+
+ mpfr_init2 (z, MPFR_PREC(y) + (rnd_mode == MPFR_RNDN));
+ mpfr_init2 (zk, MPFR_PREC(x));
+ mpfr_set (z, t, MPFR_RNDN);
+ inexact = mpfr_pow_ui (zk, z, k, MPFR_RNDN);
+ exact_root = !inexact && mpfr_equal_p (zk, absx);
+ if (exact_root) /* z is the exact root, thus round z directly */
+ inexact = mpfr_set4 (y, z, rnd_mode, MPFR_SIGN (x));
+ mpfr_clear (zk);
+ mpfr_clear (z);
+ if (exact_root)
+ break;
+ }
+
+ MPFR_ZIV_NEXT (loop, w);
+ MPFR_GROUP_REPREC_1(group, w, t);
+ }
+ MPFR_ZIV_FREE (loop);
+
+ if (!exact_root)
+ inexact = mpfr_set4 (y, t, rnd_mode, MPFR_SIGN (x));
+
+ MPFR_GROUP_CLEAR(group);
+ MPFR_TMP_FREE(marker);
+ MPFR_SAVE_EXPO_FREE (expo);
+
+ return mpfr_check_range (y, inexact, rnd_mode);
+}