lib/mpi/mpih-div.c - kernel/msm-4.9 - Gitiles

 /* mpihelp-div.c  -  MPI helper functions
  *	Copyright (C) 1994, 1996 Free Software Foundation, Inc.
  *	Copyright (C) 1998, 1999 Free Software Foundation, Inc.
  *
  * This file is part of GnuPG.
  *
  * GnuPG is free software; you can redistribute it and/or modify
  * it under the terms of the GNU General Public License as published by
  * the Free Software Foundation; either version 2 of the License, or
  * (at your option) any later version.
  *
  * GnuPG 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 General Public License for more details.
  *
  * You should have received a copy of the GNU General Public License
  * along with this program; if not, write to the Free Software
  * Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA
  *
  * Note: This code is heavily based on the GNU MP Library.
  *	 Actually it's the same code with only minor changes in the
  *	 way the data is stored; this is to support the abstraction
  *	 of an optional secure memory allocation which may be used
  *	 to avoid revealing of sensitive data due to paging etc.
  *	 The GNU MP Library itself is published under the LGPL;
  *	 however I decided to publish this code under the plain GPL.
  */

 #include "mpi-internal.h"
 #include "longlong.h"

 #ifndef UMUL_TIME
 #define UMUL_TIME 1
 #endif
 #ifndef UDIV_TIME
 #define UDIV_TIME UMUL_TIME
 #endif

 /* FIXME: We should be using invert_limb (or invert_normalized_limb)
  * here (not udiv_qrnnd).
  */

 mpi_limb_t
 mpihelp_mod_1(mpi_ptr_t dividend_ptr, mpi_size_t dividend_size,
 	      mpi_limb_t divisor_limb)
 {
 	mpi_size_t i;
 	mpi_limb_t n1, n0, r;
 	int dummy;

 	/* Botch: Should this be handled at all?  Rely on callers?  */
 	if (!dividend_size)
 		return 0;

 	/* If multiplication is much faster than division, and the
 	 * dividend is large, pre-invert the divisor, and use
 	 * only multiplications in the inner loop.
 	 *
 	 * This test should be read:
 	 *   Does it ever help to use udiv_qrnnd_preinv?
 	 *     && Does what we save compensate for the inversion overhead?
 	 */
 	if (UDIV_TIME > (2 * UMUL_TIME + 6)
 	    && (UDIV_TIME - (2 * UMUL_TIME + 6)) * dividend_size > UDIV_TIME) {
 		int normalization_steps;

 		count_leading_zeros(normalization_steps, divisor_limb);
 		if (normalization_steps) {
 			mpi_limb_t divisor_limb_inverted;

 			divisor_limb <<= normalization_steps;

 			/* Compute (2**2N - 2**N * DIVISOR_LIMB) / DIVISOR_LIMB.  The
 			 * result is a (N+1)-bit approximation to 1/DIVISOR_LIMB, with the
 			 * most significant bit (with weight 2**N) implicit.
 			 *
 			 * Special case for DIVISOR_LIMB == 100...000.
 			 */
 			if (!(divisor_limb << 1))
 				divisor_limb_inverted = ~(mpi_limb_t) 0;
 			else
 				udiv_qrnnd(divisor_limb_inverted, dummy,
 					   -divisor_limb, 0, divisor_limb);

 			n1 = dividend_ptr[dividend_size - 1];
 			r = n1 >> (BITS_PER_MPI_LIMB - normalization_steps);

 			/* Possible optimization:
 			 * if (r == 0
 			 * && divisor_limb > ((n1 << normalization_steps)
 			 *                 | (dividend_ptr[dividend_size - 2] >> ...)))
 			 * ...one division less...
 			 */
 			for (i = dividend_size - 2; i >= 0; i--) {
 				n0 = dividend_ptr[i];
 				UDIV_QRNND_PREINV(dummy, r, r,
 						  ((n1 << normalization_steps)
 						   | (n0 >>
 						      (BITS_PER_MPI_LIMB -
 						       normalization_steps))),
 						  divisor_limb,
 						  divisor_limb_inverted);
 				n1 = n0;
 			}
 			UDIV_QRNND_PREINV(dummy, r, r,
 					  n1 << normalization_steps,
 					  divisor_limb, divisor_limb_inverted);
 			return r >> normalization_steps;
 		} else {
 			mpi_limb_t divisor_limb_inverted;

 			/* Compute (2**2N - 2**N * DIVISOR_LIMB) / DIVISOR_LIMB.  The
 			 * result is a (N+1)-bit approximation to 1/DIVISOR_LIMB, with the
 			 * most significant bit (with weight 2**N) implicit.
 			 *
 			 * Special case for DIVISOR_LIMB == 100...000.
 			 */
 			if (!(divisor_limb << 1))
 				divisor_limb_inverted = ~(mpi_limb_t) 0;
 			else
 				udiv_qrnnd(divisor_limb_inverted, dummy,
 					   -divisor_limb, 0, divisor_limb);

 			i = dividend_size - 1;
 			r = dividend_ptr[i];

 			if (r >= divisor_limb)
 				r = 0;
 			else
 				i--;

 			for (; i >= 0; i--) {
 				n0 = dividend_ptr[i];
 				UDIV_QRNND_PREINV(dummy, r, r,
 						  n0, divisor_limb,
 						  divisor_limb_inverted);
 			}
 			return r;
 		}
 	} else {
 		if (UDIV_NEEDS_NORMALIZATION) {
 			int normalization_steps;

 			count_leading_zeros(normalization_steps, divisor_limb);
 			if (normalization_steps) {
 				divisor_limb <<= normalization_steps;

 				n1 = dividend_ptr[dividend_size - 1];
 				r = n1 >> (BITS_PER_MPI_LIMB -
 					   normalization_steps);

 				/* Possible optimization:
 				 * if (r == 0
 				 * && divisor_limb > ((n1 << normalization_steps)
 				 *                 | (dividend_ptr[dividend_size - 2] >> ...)))
 				 * ...one division less...
 				 */
 				for (i = dividend_size - 2; i >= 0; i--) {
 					n0 = dividend_ptr[i];
 					udiv_qrnnd(dummy, r, r,
 						   ((n1 << normalization_steps)
 						    | (n0 >>
 						       (BITS_PER_MPI_LIMB -
 							normalization_steps))),
 						   divisor_limb);
 					n1 = n0;
 				}
 				udiv_qrnnd(dummy, r, r,
 					   n1 << normalization_steps,
 					   divisor_limb);
 				return r >> normalization_steps;
 			}
 		}
 		/* No normalization needed, either because udiv_qrnnd doesn't require
 		 * it, or because DIVISOR_LIMB is already normalized.  */
 		i = dividend_size - 1;
 		r = dividend_ptr[i];

 		if (r >= divisor_limb)
 			r = 0;
 		else
 			i--;

 		for (; i >= 0; i--) {
 			n0 = dividend_ptr[i];
 			udiv_qrnnd(dummy, r, r, n0, divisor_limb);
 		}
 		return r;
 	}
 }

 /* Divide num (NP/NSIZE) by den (DP/DSIZE) and write
  * the NSIZE-DSIZE least significant quotient limbs at QP
  * and the DSIZE long remainder at NP.	If QEXTRA_LIMBS is
  * non-zero, generate that many fraction bits and append them after the
  * other quotient limbs.
  * Return the most significant limb of the quotient, this is always 0 or 1.
  *
  * Preconditions:
  * 0. NSIZE >= DSIZE.
  * 1. The most significant bit of the divisor must be set.
  * 2. QP must either not overlap with the input operands at all, or
  *    QP + DSIZE >= NP must hold true.	(This means that it's
  *    possible to put the quotient in the high part of NUM, right after the
  *    remainder in NUM.
  * 3. NSIZE >= DSIZE, even if QEXTRA_LIMBS is non-zero.
  */

 mpi_limb_t
 mpihelp_divrem(mpi_ptr_t qp, mpi_size_t qextra_limbs,
 	       mpi_ptr_t np, mpi_size_t nsize, mpi_ptr_t dp, mpi_size_t dsize)
 {
 	mpi_limb_t most_significant_q_limb = 0;

 	switch (dsize) {
 	case 0:
 		/* We are asked to divide by zero, so go ahead and do it!  (To make
 		   the compiler not remove this statement, return the value.)  */
 		return 1 / dsize;

 	case 1:
 		{
 			mpi_size_t i;
 			mpi_limb_t n1;
 			mpi_limb_t d;

 			d = dp[0];
 			n1 = np[nsize - 1];

 			if (n1 >= d) {
 				n1 -= d;
 				most_significant_q_limb = 1;
 			}

 			qp += qextra_limbs;
 			for (i = nsize - 2; i >= 0; i--)
 				udiv_qrnnd(qp[i], n1, n1, np[i], d);
 			qp -= qextra_limbs;

 			for (i = qextra_limbs - 1; i >= 0; i--)
 				udiv_qrnnd(qp[i], n1, n1, 0, d);

 			np[0] = n1;
 		}
 		break;

 	case 2:
 		{
 			mpi_size_t i;
 			mpi_limb_t n1, n0, n2;
 			mpi_limb_t d1, d0;

 			np += nsize - 2;
 			d1 = dp[1];
 			d0 = dp[0];
 			n1 = np[1];
 			n0 = np[0];

 			if (n1 >= d1 && (n1 > d1 || n0 >= d0)) {
 				sub_ddmmss(n1, n0, n1, n0, d1, d0);
 				most_significant_q_limb = 1;
 			}

 			for (i = qextra_limbs + nsize - 2 - 1; i >= 0; i--) {
 				mpi_limb_t q;
 				mpi_limb_t r;

 				if (i >= qextra_limbs)
 					np--;
 				else
 					np[0] = 0;

 				if (n1 == d1) {
 					/* Q should be either 111..111 or 111..110.  Need special
 					 * treatment of this rare case as normal division would
 					 * give overflow.  */
 					q = ~(mpi_limb_t) 0;

 					r = n0 + d1;
 					if (r < d1) {	/* Carry in the addition? */
 						add_ssaaaa(n1, n0, r - d0,
 							   np[0], 0, d0);
 						qp[i] = q;
 						continue;
 					}
 					n1 = d0 - (d0 != 0 ? 1 : 0);
 					n0 = -d0;
 				} else {
 					udiv_qrnnd(q, r, n1, n0, d1);
 					umul_ppmm(n1, n0, d0, q);
 				}

 				n2 = np[0];
 q_test:
 				if (n1 > r || (n1 == r && n0 > n2)) {
 					/* The estimated Q was too large.  */
 					q--;
 					sub_ddmmss(n1, n0, n1, n0, 0, d0);
 					r += d1;
 					if (r >= d1)	/* If not carry, test Q again.  */
 						goto q_test;
 				}

 				qp[i] = q;
 				sub_ddmmss(n1, n0, r, n2, n1, n0);
 			}
 			np[1] = n1;
 			np[0] = n0;
 		}
 		break;

 	default:
 		{
 			mpi_size_t i;
 			mpi_limb_t dX, d1, n0;

 			np += nsize - dsize;
 			dX = dp[dsize - 1];
 			d1 = dp[dsize - 2];
 			n0 = np[dsize - 1];

 			if (n0 >= dX) {
 				if (n0 > dX
 				    || mpihelp_cmp(np, dp, dsize - 1) >= 0) {
 					mpihelp_sub_n(np, np, dp, dsize);
 					n0 = np[dsize - 1];
 					most_significant_q_limb = 1;
 				}
 			}

 			for (i = qextra_limbs + nsize - dsize - 1; i >= 0; i--) {
 				mpi_limb_t q;
 				mpi_limb_t n1, n2;
 				mpi_limb_t cy_limb;

 				if (i >= qextra_limbs) {
 					np--;
 					n2 = np[dsize];
 				} else {
 					n2 = np[dsize - 1];
 					MPN_COPY_DECR(np + 1, np, dsize - 1);
 					np[0] = 0;
 				}

 				if (n0 == dX) {
 					/* This might over-estimate q, but it's probably not worth
 					 * the extra code here to find out.  */
 					q = ~(mpi_limb_t) 0;
 				} else {
 					mpi_limb_t r;

 					udiv_qrnnd(q, r, n0, np[dsize - 1], dX);
 					umul_ppmm(n1, n0, d1, q);

 					while (n1 > r
 					       || (n1 == r
 						   && n0 > np[dsize - 2])) {
 						q--;
 						r += dX;
 						if (r < dX)	/* I.e. "carry in previous addition?" */
 							break;
 						n1 -= n0 < d1;
 						n0 -= d1;
 					}
 				}

 				/* Possible optimization: We already have (q * n0) and (1 * n1)
 				 * after the calculation of q.  Taking advantage of that, we
 				 * could make this loop make two iterations less.  */
 				cy_limb = mpihelp_submul_1(np, dp, dsize, q);

 				if (n2 != cy_limb) {
 					mpihelp_add_n(np, np, dp, dsize);
 					q--;
 				}

 				qp[i] = q;
 				n0 = np[dsize - 1];
 			}
 		}
 	}

 	return most_significant_q_limb;
 }

 /****************
  * Divide (DIVIDEND_PTR,,DIVIDEND_SIZE) by DIVISOR_LIMB.
  * Write DIVIDEND_SIZE limbs of quotient at QUOT_PTR.
  * Return the single-limb remainder.
  * There are no constraints on the value of the divisor.
  *
  * QUOT_PTR and DIVIDEND_PTR might point to the same limb.
  */

 mpi_limb_t
 mpihelp_divmod_1(mpi_ptr_t quot_ptr,
 		 mpi_ptr_t dividend_ptr, mpi_size_t dividend_size,
 		 mpi_limb_t divisor_limb)
 {
 	mpi_size_t i;
 	mpi_limb_t n1, n0, r;
 	int dummy;

 	if (!dividend_size)
 		return 0;

 	/* If multiplication is much faster than division, and the
 	 * dividend is large, pre-invert the divisor, and use
 	 * only multiplications in the inner loop.
 	 *
 	 * This test should be read:
 	 * Does it ever help to use udiv_qrnnd_preinv?
 	 * && Does what we save compensate for the inversion overhead?
 	 */
 	if (UDIV_TIME > (2 * UMUL_TIME + 6)
 	    && (UDIV_TIME - (2 * UMUL_TIME + 6)) * dividend_size > UDIV_TIME) {
 		int normalization_steps;

 		count_leading_zeros(normalization_steps, divisor_limb);
 		if (normalization_steps) {
 			mpi_limb_t divisor_limb_inverted;

 			divisor_limb <<= normalization_steps;

 			/* Compute (2**2N - 2**N * DIVISOR_LIMB) / DIVISOR_LIMB.  The
 			 * result is a (N+1)-bit approximation to 1/DIVISOR_LIMB, with the
 			 * most significant bit (with weight 2**N) implicit.
 			 */
 			/* Special case for DIVISOR_LIMB == 100...000.  */
 			if (!(divisor_limb << 1))
 				divisor_limb_inverted = ~(mpi_limb_t) 0;
 			else
 				udiv_qrnnd(divisor_limb_inverted, dummy,
 					   -divisor_limb, 0, divisor_limb);

 			n1 = dividend_ptr[dividend_size - 1];
 			r = n1 >> (BITS_PER_MPI_LIMB - normalization_steps);

 			/* Possible optimization:
 			 * if (r == 0
 			 * && divisor_limb > ((n1 << normalization_steps)
 			 *                 | (dividend_ptr[dividend_size - 2] >> ...)))
 			 * ...one division less...
 			 */
 			for (i = dividend_size - 2; i >= 0; i--) {
 				n0 = dividend_ptr[i];
 				UDIV_QRNND_PREINV(quot_ptr[i + 1], r, r,
 						  ((n1 << normalization_steps)
 						   | (n0 >>
 						      (BITS_PER_MPI_LIMB -
 						       normalization_steps))),
 						  divisor_limb,
 						  divisor_limb_inverted);
 				n1 = n0;
 			}
 			UDIV_QRNND_PREINV(quot_ptr[0], r, r,
 					  n1 << normalization_steps,
 					  divisor_limb, divisor_limb_inverted);
 			return r >> normalization_steps;
 		} else {
 			mpi_limb_t divisor_limb_inverted;

 			/* Compute (2**2N - 2**N * DIVISOR_LIMB) / DIVISOR_LIMB.  The
 			 * result is a (N+1)-bit approximation to 1/DIVISOR_LIMB, with the
 			 * most significant bit (with weight 2**N) implicit.
 			 */
 			/* Special case for DIVISOR_LIMB == 100...000.  */
 			if (!(divisor_limb << 1))
 				divisor_limb_inverted = ~(mpi_limb_t) 0;
 			else
 				udiv_qrnnd(divisor_limb_inverted, dummy,
 					   -divisor_limb, 0, divisor_limb);

 			i = dividend_size - 1;
 			r = dividend_ptr[i];

 			if (r >= divisor_limb)
 				r = 0;
 			else
 				quot_ptr[i--] = 0;

 			for (; i >= 0; i--) {
 				n0 = dividend_ptr[i];
 				UDIV_QRNND_PREINV(quot_ptr[i], r, r,
 						  n0, divisor_limb,
 						  divisor_limb_inverted);
 			}
 			return r;
 		}
 	} else {
 		if (UDIV_NEEDS_NORMALIZATION) {
 			int normalization_steps;

 			count_leading_zeros(normalization_steps, divisor_limb);
 			if (normalization_steps) {
 				divisor_limb <<= normalization_steps;

 				n1 = dividend_ptr[dividend_size - 1];
 				r = n1 >> (BITS_PER_MPI_LIMB -
 					   normalization_steps);

 				/* Possible optimization:
 				 * if (r == 0
 				 * && divisor_limb > ((n1 << normalization_steps)
 				 *                 | (dividend_ptr[dividend_size - 2] >> ...)))
 				 * ...one division less...
 				 */
 				for (i = dividend_size - 2; i >= 0; i--) {
 					n0 = dividend_ptr[i];
 					udiv_qrnnd(quot_ptr[i + 1], r, r,
 						   ((n1 << normalization_steps)
 						    | (n0 >>
 						       (BITS_PER_MPI_LIMB -
 							normalization_steps))),
 						   divisor_limb);
 					n1 = n0;
 				}
 				udiv_qrnnd(quot_ptr[0], r, r,
 					   n1 << normalization_steps,
 					   divisor_limb);
 				return r >> normalization_steps;
 			}
 		}
 		/* No normalization needed, either because udiv_qrnnd doesn't require
 		 * it, or because DIVISOR_LIMB is already normalized.  */
 		i = dividend_size - 1;
 		r = dividend_ptr[i];

 		if (r >= divisor_limb)
 			r = 0;
 		else
 			quot_ptr[i--] = 0;

 		for (; i >= 0; i--) {
 			n0 = dividend_ptr[i];
 			udiv_qrnnd(quot_ptr[i], r, r, n0, divisor_limb);
 		}
 		return r;
 	}
 }
	/* mpihelp-div.c - MPI helper functions
	* Copyright (C) 1994, 1996 Free Software Foundation, Inc.
	* Copyright (C) 1998, 1999 Free Software Foundation, Inc.
	*
	* This file is part of GnuPG.
	*
	* GnuPG is free software; you can redistribute it and/or modify
	* it under the terms of the GNU General Public License as published by
	* the Free Software Foundation; either version 2 of the License, or
	* (at your option) any later version.
	*
	* GnuPG 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 General Public License for more details.
	*
	* You should have received a copy of the GNU General Public License
	* along with this program; if not, write to the Free Software
	* Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA
	*
	* Note: This code is heavily based on the GNU MP Library.
	* Actually it's the same code with only minor changes in the
	* way the data is stored; this is to support the abstraction
	* of an optional secure memory allocation which may be used
	* to avoid revealing of sensitive data due to paging etc.
	* The GNU MP Library itself is published under the LGPL;
	* however I decided to publish this code under the plain GPL.
	*/

	#include "mpi-internal.h"
	#include "longlong.h"

	#ifndef UMUL_TIME
	#define UMUL_TIME 1
	#endif
	#ifndef UDIV_TIME
	#define UDIV_TIME UMUL_TIME
	#endif

	/* FIXME: We should be using invert_limb (or invert_normalized_limb)
	* here (not udiv_qrnnd).
	*/

	mpi_limb_t
	mpihelp_mod_1(mpi_ptr_t dividend_ptr, mpi_size_t dividend_size,
	mpi_limb_t divisor_limb)
	{
	mpi_size_t i;
	mpi_limb_t n1, n0, r;
	int dummy;

	/* Botch: Should this be handled at all? Rely on callers? */
	if (!dividend_size)
	return 0;

	/* If multiplication is much faster than division, and the
	* dividend is large, pre-invert the divisor, and use
	* only multiplications in the inner loop.
	*
	* This test should be read:
	* Does it ever help to use udiv_qrnnd_preinv?
	* && Does what we save compensate for the inversion overhead?
	*/
	if (UDIV_TIME > (2 * UMUL_TIME + 6)
	&& (UDIV_TIME - (2 * UMUL_TIME + 6)) * dividend_size > UDIV_TIME) {
	int normalization_steps;

	count_leading_zeros(normalization_steps, divisor_limb);
	if (normalization_steps) {
	mpi_limb_t divisor_limb_inverted;

	divisor_limb <<= normalization_steps;

	/* Compute (22N - 2N * DIVISOR_LIMB) / DIVISOR_LIMB. The
	* result is a (N+1)-bit approximation to 1/DIVISOR_LIMB, with the
	* most significant bit (with weight 2**N) implicit.
	*
	* Special case for DIVISOR_LIMB == 100...000.
	*/
	if (!(divisor_limb << 1))
	divisor_limb_inverted = ~(mpi_limb_t) 0;
	else
	udiv_qrnnd(divisor_limb_inverted, dummy,
	-divisor_limb, 0, divisor_limb);

	n1 = dividend_ptr[dividend_size - 1];
	r = n1 >> (BITS_PER_MPI_LIMB - normalization_steps);

	/* Possible optimization:
	* if (r == 0
	* && divisor_limb > ((n1 << normalization_steps)
	* \| (dividend_ptr[dividend_size - 2] >> ...)))
	* ...one division less...
	*/
	for (i = dividend_size - 2; i >= 0; i--) {
	n0 = dividend_ptr[i];
	UDIV_QRNND_PREINV(dummy, r, r,
	((n1 << normalization_steps)
	\| (n0 >>
	(BITS_PER_MPI_LIMB -
	normalization_steps))),
	divisor_limb,
	divisor_limb_inverted);
	n1 = n0;
	}
	UDIV_QRNND_PREINV(dummy, r, r,
	n1 << normalization_steps,
	divisor_limb, divisor_limb_inverted);
	return r >> normalization_steps;
	} else {
	mpi_limb_t divisor_limb_inverted;

	/* Compute (22N - 2N * DIVISOR_LIMB) / DIVISOR_LIMB. The
	* result is a (N+1)-bit approximation to 1/DIVISOR_LIMB, with the
	* most significant bit (with weight 2**N) implicit.
	*
	* Special case for DIVISOR_LIMB == 100...000.
	*/
	if (!(divisor_limb << 1))
	divisor_limb_inverted = ~(mpi_limb_t) 0;
	else
	udiv_qrnnd(divisor_limb_inverted, dummy,
	-divisor_limb, 0, divisor_limb);

	i = dividend_size - 1;
	r = dividend_ptr[i];

	if (r >= divisor_limb)
	r = 0;
	else
	i--;

	for (; i >= 0; i--) {
	n0 = dividend_ptr[i];
	UDIV_QRNND_PREINV(dummy, r, r,
	n0, divisor_limb,
	divisor_limb_inverted);
	}
	return r;
	}
	} else {
	if (UDIV_NEEDS_NORMALIZATION) {
	int normalization_steps;

	count_leading_zeros(normalization_steps, divisor_limb);
	if (normalization_steps) {
	divisor_limb <<= normalization_steps;

	n1 = dividend_ptr[dividend_size - 1];
	r = n1 >> (BITS_PER_MPI_LIMB -
	normalization_steps);

	/* Possible optimization:
	* if (r == 0
	* && divisor_limb > ((n1 << normalization_steps)
	* \| (dividend_ptr[dividend_size - 2] >> ...)))
	* ...one division less...
	*/
	for (i = dividend_size - 2; i >= 0; i--) {
	n0 = dividend_ptr[i];
	udiv_qrnnd(dummy, r, r,
	((n1 << normalization_steps)
	\| (n0 >>
	(BITS_PER_MPI_LIMB -
	normalization_steps))),
	divisor_limb);
	n1 = n0;
	}
	udiv_qrnnd(dummy, r, r,
	n1 << normalization_steps,
	divisor_limb);
	return r >> normalization_steps;
	}
	}
	/* No normalization needed, either because udiv_qrnnd doesn't require
	* it, or because DIVISOR_LIMB is already normalized. */
	i = dividend_size - 1;
	r = dividend_ptr[i];

	if (r >= divisor_limb)
	r = 0;
	else
	i--;

	for (; i >= 0; i--) {
	n0 = dividend_ptr[i];
	udiv_qrnnd(dummy, r, r, n0, divisor_limb);
	}
	return r;
	}
	}

	/* Divide num (NP/NSIZE) by den (DP/DSIZE) and write
	* the NSIZE-DSIZE least significant quotient limbs at QP
	* and the DSIZE long remainder at NP. If QEXTRA_LIMBS is
	* non-zero, generate that many fraction bits and append them after the
	* other quotient limbs.
	* Return the most significant limb of the quotient, this is always 0 or 1.
	*
	* Preconditions:
	* 0. NSIZE >= DSIZE.
	* 1. The most significant bit of the divisor must be set.
	* 2. QP must either not overlap with the input operands at all, or
	* QP + DSIZE >= NP must hold true. (This means that it's
	* possible to put the quotient in the high part of NUM, right after the
	* remainder in NUM.
	* 3. NSIZE >= DSIZE, even if QEXTRA_LIMBS is non-zero.
	*/

	mpi_limb_t
	mpihelp_divrem(mpi_ptr_t qp, mpi_size_t qextra_limbs,
	mpi_ptr_t np, mpi_size_t nsize, mpi_ptr_t dp, mpi_size_t dsize)
	{
	mpi_limb_t most_significant_q_limb = 0;

	switch (dsize) {
	case 0:
	/* We are asked to divide by zero, so go ahead and do it! (To make
	the compiler not remove this statement, return the value.) */
	return 1 / dsize;

	case 1:
	{
	mpi_size_t i;
	mpi_limb_t n1;
	mpi_limb_t d;

	d = dp[0];
	n1 = np[nsize - 1];

	if (n1 >= d) {
	n1 -= d;
	most_significant_q_limb = 1;
	}

	qp += qextra_limbs;
	for (i = nsize - 2; i >= 0; i--)
	udiv_qrnnd(qp[i], n1, n1, np[i], d);
	qp -= qextra_limbs;

	for (i = qextra_limbs - 1; i >= 0; i--)
	udiv_qrnnd(qp[i], n1, n1, 0, d);

	np[0] = n1;
	}
	break;

	case 2:
	{
	mpi_size_t i;
	mpi_limb_t n1, n0, n2;
	mpi_limb_t d1, d0;

	np += nsize - 2;
	d1 = dp[1];
	d0 = dp[0];
	n1 = np[1];
	n0 = np[0];

	if (n1 >= d1 && (n1 > d1 \|\| n0 >= d0)) {
	sub_ddmmss(n1, n0, n1, n0, d1, d0);
	most_significant_q_limb = 1;
	}

	for (i = qextra_limbs + nsize - 2 - 1; i >= 0; i--) {
	mpi_limb_t q;
	mpi_limb_t r;

	if (i >= qextra_limbs)
	np--;
	else
	np[0] = 0;

	if (n1 == d1) {
	/* Q should be either 111..111 or 111..110. Need special
	* treatment of this rare case as normal division would
	* give overflow. */
	q = ~(mpi_limb_t) 0;

	r = n0 + d1;
	if (r < d1) { /* Carry in the addition? */
	add_ssaaaa(n1, n0, r - d0,
	np[0], 0, d0);
	qp[i] = q;
	continue;
	}
	n1 = d0 - (d0 != 0 ? 1 : 0);
	n0 = -d0;
	} else {
	udiv_qrnnd(q, r, n1, n0, d1);
	umul_ppmm(n1, n0, d0, q);
	}

	n2 = np[0];
	q_test:
	if (n1 > r \|\| (n1 == r && n0 > n2)) {
	/* The estimated Q was too large. */
	q--;
	sub_ddmmss(n1, n0, n1, n0, 0, d0);
	r += d1;
	if (r >= d1) /* If not carry, test Q again. */
	goto q_test;
	}

	qp[i] = q;
	sub_ddmmss(n1, n0, r, n2, n1, n0);
	}
	np[1] = n1;
	np[0] = n0;
	}
	break;

	default:
	{
	mpi_size_t i;
	mpi_limb_t dX, d1, n0;

	np += nsize - dsize;
	dX = dp[dsize - 1];
	d1 = dp[dsize - 2];
	n0 = np[dsize - 1];

	if (n0 >= dX) {
	if (n0 > dX
	\|\| mpihelp_cmp(np, dp, dsize - 1) >= 0) {
	mpihelp_sub_n(np, np, dp, dsize);
	n0 = np[dsize - 1];
	most_significant_q_limb = 1;
	}
	}

	for (i = qextra_limbs + nsize - dsize - 1; i >= 0; i--) {
	mpi_limb_t q;
	mpi_limb_t n1, n2;
	mpi_limb_t cy_limb;

	if (i >= qextra_limbs) {
	np--;
	n2 = np[dsize];
	} else {
	n2 = np[dsize - 1];
	MPN_COPY_DECR(np + 1, np, dsize - 1);
	np[0] = 0;
	}

	if (n0 == dX) {
	/* This might over-estimate q, but it's probably not worth
	* the extra code here to find out. */
	q = ~(mpi_limb_t) 0;
	} else {
	mpi_limb_t r;

	udiv_qrnnd(q, r, n0, np[dsize - 1], dX);
	umul_ppmm(n1, n0, d1, q);

	while (n1 > r
	\|\| (n1 == r
	&& n0 > np[dsize - 2])) {
	q--;
	r += dX;
	if (r < dX) /* I.e. "carry in previous addition?" */
	break;
	n1 -= n0 < d1;
	n0 -= d1;
	}
	}

	/* Possible optimization: We already have (q * n0) and (1 * n1)
	* after the calculation of q. Taking advantage of that, we
	* could make this loop make two iterations less. */
	cy_limb = mpihelp_submul_1(np, dp, dsize, q);

	if (n2 != cy_limb) {
	mpihelp_add_n(np, np, dp, dsize);
	q--;
	}

	qp[i] = q;
	n0 = np[dsize - 1];
	}
	}
	}

	return most_significant_q_limb;
	}

	/****************
	* Divide (DIVIDEND_PTR,,DIVIDEND_SIZE) by DIVISOR_LIMB.
	* Write DIVIDEND_SIZE limbs of quotient at QUOT_PTR.
	* Return the single-limb remainder.
	* There are no constraints on the value of the divisor.
	*
	* QUOT_PTR and DIVIDEND_PTR might point to the same limb.
	*/

	mpi_limb_t
	mpihelp_divmod_1(mpi_ptr_t quot_ptr,
	mpi_ptr_t dividend_ptr, mpi_size_t dividend_size,
	mpi_limb_t divisor_limb)
	{
	mpi_size_t i;
	mpi_limb_t n1, n0, r;
	int dummy;

	if (!dividend_size)
	return 0;

	/* If multiplication is much faster than division, and the
	* dividend is large, pre-invert the divisor, and use
	* only multiplications in the inner loop.
	*
	* This test should be read:
	* Does it ever help to use udiv_qrnnd_preinv?
	* && Does what we save compensate for the inversion overhead?
	*/
	if (UDIV_TIME > (2 * UMUL_TIME + 6)
	&& (UDIV_TIME - (2 * UMUL_TIME + 6)) * dividend_size > UDIV_TIME) {
	int normalization_steps;

	count_leading_zeros(normalization_steps, divisor_limb);
	if (normalization_steps) {
	mpi_limb_t divisor_limb_inverted;

	divisor_limb <<= normalization_steps;

	/* Compute (22N - 2N * DIVISOR_LIMB) / DIVISOR_LIMB. The
	* result is a (N+1)-bit approximation to 1/DIVISOR_LIMB, with the
	* most significant bit (with weight 2**N) implicit.
	*/
	/* Special case for DIVISOR_LIMB == 100...000. */
	if (!(divisor_limb << 1))
	divisor_limb_inverted = ~(mpi_limb_t) 0;
	else
	udiv_qrnnd(divisor_limb_inverted, dummy,
	-divisor_limb, 0, divisor_limb);

	n1 = dividend_ptr[dividend_size - 1];
	r = n1 >> (BITS_PER_MPI_LIMB - normalization_steps);

	/* Possible optimization:
	* if (r == 0
	* && divisor_limb > ((n1 << normalization_steps)
	* \| (dividend_ptr[dividend_size - 2] >> ...)))
	* ...one division less...
	*/
	for (i = dividend_size - 2; i >= 0; i--) {
	n0 = dividend_ptr[i];
	UDIV_QRNND_PREINV(quot_ptr[i + 1], r, r,
	((n1 << normalization_steps)
	\| (n0 >>
	(BITS_PER_MPI_LIMB -
	normalization_steps))),
	divisor_limb,
	divisor_limb_inverted);
	n1 = n0;
	}
	UDIV_QRNND_PREINV(quot_ptr[0], r, r,
	n1 << normalization_steps,
	divisor_limb, divisor_limb_inverted);
	return r >> normalization_steps;
	} else {
	mpi_limb_t divisor_limb_inverted;

	/* Compute (22N - 2N * DIVISOR_LIMB) / DIVISOR_LIMB. The
	* result is a (N+1)-bit approximation to 1/DIVISOR_LIMB, with the
	* most significant bit (with weight 2**N) implicit.
	*/
	/* Special case for DIVISOR_LIMB == 100...000. */
	if (!(divisor_limb << 1))
	divisor_limb_inverted = ~(mpi_limb_t) 0;
	else
	udiv_qrnnd(divisor_limb_inverted, dummy,
	-divisor_limb, 0, divisor_limb);

	i = dividend_size - 1;
	r = dividend_ptr[i];

	if (r >= divisor_limb)
	r = 0;
	else
	quot_ptr[i--] = 0;

	for (; i >= 0; i--) {
	n0 = dividend_ptr[i];
	UDIV_QRNND_PREINV(quot_ptr[i], r, r,
	n0, divisor_limb,
	divisor_limb_inverted);
	}
	return r;
	}
	} else {
	if (UDIV_NEEDS_NORMALIZATION) {
	int normalization_steps;

	count_leading_zeros(normalization_steps, divisor_limb);
	if (normalization_steps) {
	divisor_limb <<= normalization_steps;

	n1 = dividend_ptr[dividend_size - 1];
	r = n1 >> (BITS_PER_MPI_LIMB -
	normalization_steps);

	/* Possible optimization:
	* if (r == 0
	* && divisor_limb > ((n1 << normalization_steps)
	* \| (dividend_ptr[dividend_size - 2] >> ...)))
	* ...one division less...
	*/
	for (i = dividend_size - 2; i >= 0; i--) {
	n0 = dividend_ptr[i];
	udiv_qrnnd(quot_ptr[i + 1], r, r,
	((n1 << normalization_steps)
	\| (n0 >>
	(BITS_PER_MPI_LIMB -
	normalization_steps))),
	divisor_limb);
	n1 = n0;
	}
	udiv_qrnnd(quot_ptr[0], r, r,
	n1 << normalization_steps,
	divisor_limb);
	return r >> normalization_steps;
	}
	}
	/* No normalization needed, either because udiv_qrnnd doesn't require
	* it, or because DIVISOR_LIMB is already normalized. */
	i = dividend_size - 1;
	r = dividend_ptr[i];

	if (r >= divisor_limb)
	r = 0;
	else
	quot_ptr[i--] = 0;

	for (; i >= 0; i--) {
	n0 = dividend_ptr[i];
	udiv_qrnnd(quot_ptr[i], r, r, n0, divisor_limb);
	}
	return r;
	}
	}