math.c

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/* -*-  Mode:C; c-basic-offset:4; tab-width:4 -*-
 ****************************************************************************
 * (C) 2003 - Rolf Neugebauer - Intel Research Cambridge
 ****************************************************************************
 *
 *        File: math.c
 *      Author: Rolf Neugebauer ([e-mail])
 *     Changes: 
 *              
 *        Date: Aug 2003
 * 
 * Environment: Xen Minimal OS
 * Description:  Library functions for 64bit arith and other
 *               from freebsd, files in sys/libkern/ (qdivrem.c, etc)
 *
 ****************************************************************************
 * $Id: c-insert.c,v 1.7 2002/11/08 16:04:34 rn Exp $
 ****************************************************************************
 *-
 * Copyright (c) 1992, 1993
 *	The Regents of the University of California.  All rights reserved.
 *
 * This software was developed by the Computer Systems Engineering group
 * at Lawrence Berkeley Laboratory under DARPA contract BG 91-66 and
 * contributed to Berkeley.
 *
 * Redistribution and use in source and binary forms, with or without
 * modification, are permitted provided that the following conditions
 * are met:
 * 1. Redistributions of source code must retain the above copyright
 *    notice, this list of conditions and the following disclaimer.
 * 2. Redistributions in binary form must reproduce the above copyright
 *    notice, this list of conditions and the following disclaimer in the
 *    documentation and/or other materials provided with the distribution.
 * 3. All advertising materials mentioning features or use of this software
 *    must display the following acknowledgement:
 *	This product includes software developed by the University of
 *	California, Berkeley and its contributors.
 * 4. Neither the name of the University nor the names of its contributors
 *    may be used to endorse or promote products derived from this software
 *    without specific prior written permission.
 *
 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
 * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
 * SUCH DAMAGE.
 *
 * $FreeBSD: src/sys/libkern/divdi3.c,v 1.6 1999/08/28 00:46:31 peter Exp $
*/

#include <types.h>
#include <lib.h>
#include <time.h>

	/* On ia64 these functions lead to crashes. These are replaced by
	 * assembler functions. */
#if !defined(__ia64__)

/*
 * Depending on the desired operation, we view a `long long' (aka quad_t) in
 * one or more of the following formats.
 */
union uu {
        s64            q;              /* as a (signed) quad */
        s64            uq;             /* as an unsigned quad */
        long           sl[2];          /* as two signed longs */
        unsigned long  ul[2];          /* as two unsigned longs */
};
/* XXX RN: Yuck hardcoded endianess :) */
#define _QUAD_HIGHWORD 1
#define _QUAD_LOWWORD 0
/*
 * Define high and low longwords.
 */
#define H               _QUAD_HIGHWORD
#define L               _QUAD_LOWWORD

/*
 * Total number of bits in a quad_t and in the pieces that make it up.
 * These are used for shifting, and also below for halfword extraction
 * and assembly.
 */
#ifndef HAVE_LIBC
#define CHAR_BIT        8               /* number of bits in a char */
#endif
#define QUAD_BITS       (sizeof(s64) * CHAR_BIT)
#define LONG_BITS       (sizeof(long) * CHAR_BIT)
#define HALF_BITS       (sizeof(long) * CHAR_BIT / 2)

/*
 * Extract high and low shortwords from longword, and move low shortword of
 * longword to upper half of long, i.e., produce the upper longword of
 * ((quad_t)(x) << (number_of_bits_in_long/2)).  (`x' must actually be u_long.)
 *
 * These are used in the multiply code, to split a longword into upper
 * and lower halves, and to reassemble a product as a quad_t, shifted left
 * (sizeof(long)*CHAR_BIT/2).
 */
#define HHALF(x)        ((x) >> HALF_BITS)
#define LHALF(x)        ((x) & ((1UL << HALF_BITS) - 1))
#define LHUP(x)         ((x) << HALF_BITS)

/*
 * Multiprecision divide.  This algorithm is from Knuth vol. 2 (2nd ed),
 * section 4.3.1, pp. 257--259.
 */
#define	B	(1UL << HALF_BITS)	/* digit base */

/* Combine two `digits' to make a single two-digit number. */
#define	COMBINE(a, b) (((u_long)(a) << HALF_BITS) | (b))

/* select a type for digits in base B: use unsigned short if they fit */
#if ULONG_MAX == 0xffffffff && USHRT_MAX >= 0xffff
typedef unsigned short digit;
#else
typedef u_long digit;
#endif


/*
 * Shift p[0]..p[len] left `sh' bits, ignoring any bits that
 * `fall out' the left (there never will be any such anyway).
 * We may assume len >= 0.  NOTE THAT THIS WRITES len+1 DIGITS.
 */
static void
shl(register digit *p, register int len, register int sh)
{
	register int i;

	for (i = 0; i < len; i++)
		p[i] = LHALF(p[i] << sh) | (p[i + 1] >> (HALF_BITS - sh));
	p[i] = LHALF(p[i] << sh);
}

/*
 * __qdivrem(u, v, rem) returns u/v and, optionally, sets *rem to u%v.
 *
 * We do this in base 2-sup-HALF_BITS, so that all intermediate products
 * fit within u_long.  As a consequence, the maximum length dividend and
 * divisor are 4 `digits' in this base (they are shorter if they have
 * leading zeros).
 */
u64
__qdivrem(u64 uq, u64 vq, u64 *arq)
{
	union uu tmp;
	digit *u, *v, *q;
	register digit v1, v2;
	u_long qhat, rhat, t;
	int m, n, d, j, i;
	digit uspace[5], vspace[5], qspace[5];

	/*
	 * Take care of special cases: divide by zero, and u < v.
	 */
	if (vq == 0) {
		/* divide by zero. */
		static volatile const unsigned int zero = 0;

		tmp.ul[H] = tmp.ul[L] = 1 / zero;
		if (arq)
			*arq = uq;
		return (tmp.q);
	}
	if (uq < vq) {
		if (arq)
			*arq = uq;
		return (0);
	}
	u = &uspace[0];
	v = &vspace[0];
	q = &qspace[0];

	/*
	 * Break dividend and divisor into digits in base B, then
	 * count leading zeros to determine m and n.  When done, we
	 * will have:
	 *	u = (u[1]u[2]...u[m+n]) sub B
	 *	v = (v[1]v[2]...v[n]) sub B
	 *	v[1] != 0
	 *	1 < n <= 4 (if n = 1, we use a different division algorithm)
	 *	m >= 0 (otherwise u < v, which we already checked)
	 *	m + n = 4
	 * and thus
	 *	m = 4 - n <= 2
	 */
	tmp.uq = uq;
	u[0] = 0;
	u[1] = HHALF(tmp.ul[H]);
	u[2] = LHALF(tmp.ul[H]);
	u[3] = HHALF(tmp.ul[L]);
	u[4] = LHALF(tmp.ul[L]);
	tmp.uq = vq;
	v[1] = HHALF(tmp.ul[H]);
	v[2] = LHALF(tmp.ul[H]);
	v[3] = HHALF(tmp.ul[L]);
	v[4] = LHALF(tmp.ul[L]);
	for (n = 4; v[1] == 0; v++) {
		if (--n == 1) {
			u_long rbj;	/* r*B+u[j] (not root boy jim) */
			digit q1, q2, q3, q4;

			/*
			 * Change of plan, per exercise 16.
			 *	r = 0;
			 *	for j = 1..4:
			 *		q[j] = floor((r*B + u[j]) / v),
			 *		r = (r*B + u[j]) % v;
			 * We unroll this completely here.
			 */
			t = v[2];	/* nonzero, by definition */
			q1 = u[1] / t;
			rbj = COMBINE(u[1] % t, u[2]);
			q2 = rbj / t;
			rbj = COMBINE(rbj % t, u[3]);
			q3 = rbj / t;
			rbj = COMBINE(rbj % t, u[4]);
			q4 = rbj / t;
			if (arq)
				*arq = rbj % t;
			tmp.ul[H] = COMBINE(q1, q2);
			tmp.ul[L] = COMBINE(q3, q4);
			return (tmp.q);
		}
	}

	/*
	 * By adjusting q once we determine m, we can guarantee that
	 * there is a complete four-digit quotient at &qspace[1] when
	 * we finally stop.
	 */
	for (m = 4 - n; u[1] == 0; u++)
		m--;
	for (i = 4 - m; --i >= 0;)
		q[i] = 0;
	q += 4 - m;

	/*
	 * Here we run Program D, translated from MIX to C and acquiring
	 * a few minor changes.
	 *
	 * D1: choose multiplier 1 << d to ensure v[1] >= B/2.
	 */
	d = 0;
	for (t = v[1]; t < B / 2; t <<= 1)
		d++;
	if (d > 0) {
		shl(&u[0], m + n, d);		/* u <<= d */
		shl(&v[1], n - 1, d);		/* v <<= d */
	}
	/*
	 * D2: j = 0.
	 */
	j = 0;
	v1 = v[1];	/* for D3 -- note that v[1..n] are constant */
	v2 = v[2];	/* for D3 */
	do {
		register digit uj0, uj1, uj2;

		/*
		 * D3: Calculate qhat (\^q, in TeX notation).
		 * Let qhat = min((u[j]*B + u[j+1])/v[1], B-1), and
		 * let rhat = (u[j]*B + u[j+1]) mod v[1].
		 * While rhat < B and v[2]*qhat > rhat*B+u[j+2],
		 * decrement qhat and increase rhat correspondingly.
		 * Note that if rhat >= B, v[2]*qhat < rhat*B.
		 */
		uj0 = u[j + 0];	/* for D3 only -- note that u[j+...] change */
		uj1 = u[j + 1];	/* for D3 only */
		uj2 = u[j + 2];	/* for D3 only */
		if (uj0 == v1) {
			qhat = B;
			rhat = uj1;
			goto qhat_too_big;
		} else {
			u_long nn = COMBINE(uj0, uj1);
			qhat = nn / v1;
			rhat = nn % v1;
		}
		while (v2 * qhat > COMBINE(rhat, uj2)) {
	qhat_too_big:
			qhat--;
			if ((rhat += v1) >= B)
				break;
		}
		/*
		 * D4: Multiply and subtract.
		 * The variable `t' holds any borrows across the loop.
		 * We split this up so that we do not require v[0] = 0,
		 * and to eliminate a final special case.
		 */
		for (t = 0, i = n; i > 0; i--) {
			t = u[i + j] - v[i] * qhat - t;
			u[i + j] = LHALF(t);
			t = (B - HHALF(t)) & (B - 1);
		}
		t = u[j] - t;
		u[j] = LHALF(t);
		/*
		 * D5: test remainder.
		 * There is a borrow if and only if HHALF(t) is nonzero;
		 * in that (rare) case, qhat was too large (by exactly 1).
		 * Fix it by adding v[1..n] to u[j..j+n].
		 */
		if (HHALF(t)) {
			qhat--;
			for (t = 0, i = n; i > 0; i--) { /* D6: add back. */
				t += u[i + j] + v[i];
				u[i + j] = LHALF(t);
				t = HHALF(t);
			}
			u[j] = LHALF(u[j] + t);
		}
		q[j] = qhat;
	} while (++j <= m);		/* D7: loop on j. */

	/*
	 * If caller wants the remainder, we have to calculate it as
	 * u[m..m+n] >> d (this is at most n digits and thus fits in
	 * u[m+1..m+n], but we may need more source digits).
	 */
	if (arq) {
		if (d) {
			for (i = m + n; i > m; --i)
				u[i] = (u[i] >> d) |
				    LHALF(u[i - 1] << (HALF_BITS - d));
			u[i] = 0;
		}
		tmp.ul[H] = COMBINE(uspace[1], uspace[2]);
		tmp.ul[L] = COMBINE(uspace[3], uspace[4]);
		*arq = tmp.q;
	}

	tmp.ul[H] = COMBINE(qspace[1], qspace[2]);
	tmp.ul[L] = COMBINE(qspace[3], qspace[4]);
	return (tmp.q);
}


/*
 * Divide two signed quads.
 * ??? if -1/2 should produce -1 on this machine, this code is wrong
 */
s64
__divdi3(s64 a, s64 b)
{
	u64 ua, ub, uq;
	int neg;

	if (a < 0)
		ua = -(u64)a, neg = 1;
	else
		ua = a, neg = 0;
	if (b < 0)
		ub = -(u64)b, neg ^= 1;
	else
		ub = b;
	uq = __qdivrem(ua, ub, (u64 *)0);
	return (neg ? -uq : uq);
}

/*
 * Divide two unsigned quads.
 */
u64
__udivdi3(u64 a, u64 b)
{
        return (__qdivrem(a, b, (u64 *)0));
}


/*
 * Return remainder after dividing two unsigned quads.
 */
u_quad_t
__umoddi3(u_quad_t a, u_quad_t b)
{
        u_quad_t r;

        (void)__qdivrem(a, b, &r);
        return (r);
}

/*
 * Return remainder after dividing two signed quads.
 *
 * XXX
 * If -1/2 should produce -1 on this machine, this code is wrong.
 */
quad_t
__moddi3(quad_t a, quad_t b)
{
	u_quad_t ua, ub, ur;
	int neg;

	if (a < 0)
		ua = -(u_quad_t)a, neg = 1;
	else
		ua = a, neg = 0;
	if (b < 0)
		ub = -(u_quad_t)b;
	else
		ub = b;
	(void)__qdivrem(ua, ub, &ur);
	return (neg ? -ur : ur);
}
#endif /* !defined(__ia64__) */

#ifndef HAVE_LIBC
/* Should be random enough for our uses */
int rand(void)
{
    static unsigned int previous;
    struct timeval tv;
    gettimeofday(&tv, NULL);
    previous += tv.tv_sec + tv.tv_usec;
    previous *= RAND_MIX;
    return previous;
}
#endif