v3_1_6/src/root.c - mpfr - Git at Google

 /* mpfr_root -- kth root.

 Copyright 2005-2017 Free Software Foundation, Inc.
 Contributed by the AriC and Caramba projects, INRIA.

 This file is part of the GNU MPFR Library.

 The GNU MPFR Library is free software; you can redistribute it and/or modify
 it under the terms of the GNU Lesser General Public License as published by
 the Free Software Foundation; either version 3 of the License, or (at your
 option) any later version.

 The GNU MPFR Library is distributed in the hope that it will be useful, but
 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
 or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU Lesser General Public
 License for more details.

 You should have received a copy of the GNU Lesser General Public License
 along with the GNU MPFR Library; see the file COPYING.LESSER.  If not, see
 http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */

 #define MPFR_NEED_LONGLONG_H
 #include "third_party/mpfr/v3_1_6/src/mpfr-impl.h"

  /* 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 + 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

     then, not taking into account the sign, the result will be
     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, f;
   mpfr_prec_t n, size_m, tmp;
   int inexact, negative;
   MPFR_SAVE_EXPO_DECL (expo);

   MPFR_LOG_FUNC
     (("x[%Pu]=%.*Rg k=%lu rnd=%d",
       mpfr_get_prec (x), mpfr_log_prec, x, k, rnd_mode),
      ("y[%Pu]=%.*Rg inexact=%d",
       mpfr_get_prec (y), mpfr_log_prec, y, inexact));

   if (MPFR_UNLIKELY (k <= 1))
     {
       if (k == 0)
         {
           MPFR_SET_NAN (y);
           MPFR_RET_NAN;
         }
       else /* y = x^(1/1) = x */
         return mpfr_set (y, x, rnd_mode);
     }

   /* Singular values */
   if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
     {
       if (MPFR_IS_NAN (x))
         {
           MPFR_SET_NAN (y); /* NaN^(1/k) = NaN */
           MPFR_RET_NAN;
         }

       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))
             {
               MPFR_SET_NAN (y);
               MPFR_RET_NAN;
             }
           MPFR_SET_INF (y);
         }
       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);
     }

   /* Returns NAN for x < 0 and k even */
   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);

   e = mpfr_get_z_2exp (m, x);                /* x = m * 2^e */
   if ((negative = MPFR_IS_NEG(x)))
     mpz_neg (m, m);
   r = e % (mpfr_exp_t) k;
   if (r < 0)
     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^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
     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 */

   /* we reuse the variable m to store the kth root, since it is not needed
      any more: we just need to know if the root is exact */
   inexact = mpz_root (m, m, k) == 0;

   MPFR_MPZ_SIZEINBASE2 (tmp, m);
   sh = tmp - n;
   if (sh > 0) /* we have to flush to 0 the last sh bits from m */
     {
       inexact = inexact || ((mpfr_exp_t) mpz_scan1 (m, 0) < sh);
       mpz_fdiv_q_2exp (m, m, sh);
       e += k * sh;
     }

   if (inexact)
     {
       if (negative)
         rnd_mode = MPFR_INVERT_RND (rnd_mode);
       if (rnd_mode == MPFR_RNDU || rnd_mode == MPFR_RNDA
           || (rnd_mode == MPFR_RNDN && mpz_tstbit (m, 0)))
         inexact = 1, mpz_add_ui (m, m, 1);
       else
         inexact = -1;
     }

   /* either inexact is not zero, and the conversion is exact, i.e. inexact
      is not changed; or inexact=0, and inexact is set only when
      rnd_mode=MPFR_RNDN and bit (n+1) from m is 1 */
   inexact += mpfr_set_z (y, m, MPFR_RNDN);
   MPFR_SET_EXP (y, MPFR_GET_EXP (y) + e / (mpfr_exp_t) k);

   if (negative)
     {
       MPFR_CHANGE_SIGN (y);
       inexact = -inexact;
     }

   mpz_clear (m);
   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);
 }
	/* mpfr_root -- kth root.

	Copyright 2005-2017 Free Software Foundation, Inc.
	Contributed by the AriC and Caramba projects, INRIA.

	This file is part of the GNU MPFR Library.

	The GNU MPFR Library is free software; you can redistribute it and/or modify
	it under the terms of the GNU Lesser General Public License as published by
	the Free Software Foundation; either version 3 of the License, or (at your
	option) any later version.

	The GNU MPFR Library is distributed in the hope that it will be useful, but
	WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
	or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
	License for more details.

	You should have received a copy of the GNU Lesser General Public License
	along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see
	http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
	51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */

	#define MPFR_NEED_LONGLONG_H
	#include "third_party/mpfr/v3_1_6/src/mpfr-impl.h"

	/* 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^(kn) where n = PREC(y)

	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

	then, not taking into account the sign, the result will be
	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, f;
	mpfr_prec_t n, size_m, tmp;
	int inexact, negative;
	MPFR_SAVE_EXPO_DECL (expo);

	MPFR_LOG_FUNC
	(("x[%Pu]=%.*Rg k=%lu rnd=%d",
	mpfr_get_prec (x), mpfr_log_prec, x, k, rnd_mode),
	("y[%Pu]=%.*Rg inexact=%d",
	mpfr_get_prec (y), mpfr_log_prec, y, inexact));

	if (MPFR_UNLIKELY (k <= 1))
	{
	if (k == 0)
	{
	MPFR_SET_NAN (y);
	MPFR_RET_NAN;
	}
	else /* y = x^(1/1) = x */
	return mpfr_set (y, x, rnd_mode);
	}

	/* Singular values */
	if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
	{
	if (MPFR_IS_NAN (x))
	{
	MPFR_SET_NAN (y); /* NaN^(1/k) = NaN */
	MPFR_RET_NAN;
	}

	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))
	{
	MPFR_SET_NAN (y);
	MPFR_RET_NAN;
	}
	MPFR_SET_INF (y);
	}
	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);
	}

	/* Returns NAN for x < 0 and k even */
	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);

	e = mpfr_get_z_2exp (m, x); /* x = m * 2^e */
	if ((negative = MPFR_IS_NEG(x)))
	mpz_neg (m, m);
	r = e % (mpfr_exp_t) k;
	if (r < 0)
	r += k; /* now r = e (mod k) with 0 <= r < k */
	MPFR_ASSERTD (0 <= r && r < k);
	/* x = (m2^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^sh so that root(m,k) will give
	exactly n bits: we want k(n-1)+1 <= size_m + sh <= kn
	i.e. sh = kf + r with f = max(floor((kn-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
	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 = m2^e, with e divisible by k /

	/* we reuse the variable m to store the kth root, since it is not needed
	any more: we just need to know if the root is exact */
	inexact = mpz_root (m, m, k) == 0;

	MPFR_MPZ_SIZEINBASE2 (tmp, m);
	sh = tmp - n;
	if (sh > 0) /* we have to flush to 0 the last sh bits from m */
	{
	inexact = inexact \|\| ((mpfr_exp_t) mpz_scan1 (m, 0) < sh);
	mpz_fdiv_q_2exp (m, m, sh);
	e += k * sh;
	}

	if (inexact)
	{
	if (negative)
	rnd_mode = MPFR_INVERT_RND (rnd_mode);
	if (rnd_mode == MPFR_RNDU \|\| rnd_mode == MPFR_RNDA
	\|\| (rnd_mode == MPFR_RNDN && mpz_tstbit (m, 0)))
	inexact = 1, mpz_add_ui (m, m, 1);
	else
	inexact = -1;
	}

	/* either inexact is not zero, and the conversion is exact, i.e. inexact
	is not changed; or inexact=0, and inexact is set only when
	rnd_mode=MPFR_RNDN and bit (n+1) from m is 1 */
	inexact += mpfr_set_z (y, m, MPFR_RNDN);
	MPFR_SET_EXP (y, MPFR_GET_EXP (y) + e / (mpfr_exp_t) k);

	if (negative)
	{
	MPFR_CHANGE_SIGN (y);
	inexact = -inexact;
	}

	mpz_clear (m);
	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);
	}