Handle (64-bit) / (32-bit) = (64-bit), i.e. one step beyond that
provided by the divl instruction.
This commit is contained in:
Michael Brown
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/*
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* Copyright (C) 2007 Michael Brown <mbrown@fensystems.co.uk>.
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*
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* This program is free software; you can redistribute it and/or
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* modify it under the terms of the GNU General Public License as
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* published by the Free Software Foundation; either version 2 of the
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* License, or any later version.
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*
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* This program is distributed in the hope that it will be useful, but
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* WITHOUT ANY WARRANTY; without even the implied warranty of
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* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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* General Public License for more details.
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*
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* You should have received a copy of the GNU General Public License
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* along with this program; if not, write to the Free Software
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* Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA.
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*/
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/** @file
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*
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* 64-bit division
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*
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* The x86 CPU (386 upwards) has a divl instruction which will perform
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* unsigned division of a 64-bit dividend by a 32-bit divisor. If the
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* resulting quotient does not fit in 32 bits, then a CPU exception
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* will occur.
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*
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* Unsigned integer division is expressed as solving
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*
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* x = d.q + r 0 <= q, 0 <= r < d
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*
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* given the dividend (x) and divisor (d), to find the quotient (q)
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* and remainder (r).
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*
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* The x86 divl instruction will solve
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*
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* x = d.q + r 0 <= q, 0 <= r < d
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*
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* given x in the range 0 <= x < 2^64 and 1 <= d < 2^32, and causing a
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* hardware exception if the resulting q >= 2^32.
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*
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* We can therefore use divl only if we can prove that the conditions
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*
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* 0 <= x < 2^64
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* 1 <= d < 2^32
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* q < 2^32
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*
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* are satisfied.
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*
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*
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* Case 1 : 1 <= d < 2^32
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* ======================
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*
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* We express x as
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*
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* x = xh.2^32 + xl 0 <= xh < 2^32, 0 <= xl < 2^32 (1)
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*
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* i.e. split x into low and high dwords. We then solve
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*
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* xh = d.qh + r' 0 <= qh, 0 <= r' < d (2)
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*
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* which we can do using a divl instruction since
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*
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* 0 <= xh < 2^64 since 0 <= xh < 2^32 from (1) (3)
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*
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* and
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*
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* 1 <= d < 2^32 by definition of this Case (4)
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*
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* and
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*
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* d.qh = xh - r' from (2)
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* d.qh <= xh since r' >= 0 from (2)
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* qh <= xh since d >= 1 from (2)
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* qh < 2^32 since xh < 2^32 from (1) (5)
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*
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* Having obtained qh and r', we then solve
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*
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* ( r'.2^32 + xl ) = d.ql + r 0 <= ql, 0 <= r < d (6)
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*
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* which we can do using another divl instruction since
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*
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* xl <= 2^32 - 1 from (1), so
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* r'.2^32 + xl <= ( r' + 1 ).2^32 - 1
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* r'.2^32 + xl <= d.2^32 - 1 since r' < d from (2)
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* r'.2^32 + xl < d.2^32 (7)
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* r'.2^32 + xl < 2^64 since d < 2^32 from (4) (8)
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*
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* and
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*
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* 1 <= d < 2^32 by definition of this Case (9)
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*
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* and
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*
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* d.ql = ( r'.2^32 + xl ) - r from (6)
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* d.ql <= r'.2^32 + xl since r >= 0 from (6)
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* d.ql < d.2^32 from (7)
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* ql < 2^32 since d >= 1 from (2) (10)
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*
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* This then gives us
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*
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* x = xh.2^32 + xl from (1)
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* x = ( d.qh + r' ).2^32 + xl from (2)
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* x = d.qh.2^32 + ( r'.2^32 + xl )
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* x = d.qh.2^32 + d.ql + r from (3)
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* x = d.( qh.2^32 + ql ) + r (11)
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*
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* Letting
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*
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* q = qh.2^32 + ql (12)
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*
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* gives
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*
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* x = d.q + r from (11) and (12)
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*
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* which is the solution.
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*
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*
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* This therefore gives us a two-step algorithm:
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*
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* xh = d.qh + r' 0 <= qh, 0 <= r' < d (2)
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* ( r'.2^32 + xl ) = d.ql + r 0 <= ql, 0 <= r < d (6)
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*
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* which translates to
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*
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* %edx:%eax = 0:xh
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* divl d
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* qh = %eax
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* r' = %edx
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*
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* %edx:%eax = r':xl
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* divl d
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* ql = %eax
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* r = %edx
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*
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* Note that if
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*
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* xh < d
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*
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* (which is a fast dword comparison) then the first divl instruction
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* can be omitted, since the answer will be
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*
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* qh = 0
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* r = xh
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*
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*
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* Case 2 : 2^32 <= d < 2^64
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* =========================
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*
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* We first express d as
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*
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* d = dh.2^k + dl 2^31 <= dh < 2^32,
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* 0 <= dl < 2^k, 1 <= k <= 32 (1)
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*
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* i.e. find the highest bit set in d, subtract 32, and split d into
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* dh and dl at that point.
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*
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* We then express x as
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*
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* x = xh.2^k + xl 0 <= xl < 2^k (2)
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*
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* giving
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*
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* xh.2^k = x - xl from (2)
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* xh.2^k <= x since xl >= 0 from (1)
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* xh.2^k < 2^64 since xh < 2^64 from (1)
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* xh < 2^(64-k) (3)
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*
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* We then solve the division
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*
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* xh = dh.q' + r' 0 <= r' < dh (4)
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*
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* which we can do using a divl instruction since
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*
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* 0 <= xh < 2^64 since x < 2^64 and xh < x
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*
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* and
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*
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* 1 <= dh < 2^32 from (1)
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*
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* and
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*
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* dh.q' = xh - r' from (4)
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* dh.q' <= xh since r' >= 0 from (4)
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* dh.q' < 2^(64-k) from (3) (5)
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* q'.2^31 <= dh.q' since dh >= 2^31 from (1) (6)
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* q'.2^31 < 2^(64-k) from (5) and (6)
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* q' < 2^(33-k)
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* q' < 2^32 since k >= 1 from (1) (7)
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*
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* This gives us
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*
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* xh.2^k = dh.q'.2^k + r'.2^k from (4)
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* x - xl = ( d - dl ).q' + r'.2^k from (1) and (2)
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* x = d.q' + ( r'.2^k + xl ) - dl.q' (8)
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*
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* Now
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*
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* r'.2^k + xl < r'.2^k + 2^k since xl < 2^k from (2)
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* r'.2^k + xl < ( r' + 1 ).2^k
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* r'.2^k + xl < dh.2^k since r' < dh from (4)
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* r'.2^k + xl < ( d - dl ) from (1) (9)
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*
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*
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* (missing)
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*
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*
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* This gives us two cases to consider:
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*
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* case (a):
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*
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* dl.q' <= ( r'.2^k + xl ) (15a)
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*
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* in which case
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*
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* x = d.q' + ( r'.2^k + xl - dl.q' )
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*
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* is a direct solution to the division, since
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*
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* r'.2^k + xl < d from (9)
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* ( r'.2^k + xl - dl.q' ) < d since dl >= 0 and q' >= 0
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*
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* and
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*
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* 0 <= ( r'.2^k + xl - dl.q' ) from (15a)
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*
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* case (b):
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*
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* dl.q' > ( r'.2^k + xl ) (15b)
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*
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* Express
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*
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* x = d.(q'-1) + ( r'.2^k + xl ) + ( d - dl.q' )
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*
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*
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* (missing)
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*
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*
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* special case: k = 32 cannot be handled with shifts
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*
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* (missing)
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*
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*/
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#include <stdint.h>
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#include <assert.h>
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typedef uint64_t UDItype;
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struct uint64 {
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uint32_t l;
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uint32_t h;
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};
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static inline void udivmod64_lo ( const struct uint64 *x,
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const struct uint64 *d,
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struct uint64 *q,
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struct uint64 *r ) {
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uint32_t r_dash;
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q->h = 0;
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r->h = 0;
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r_dash = x->h;
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if ( x->h >= d->l ) {
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__asm__ ( "divl %2"
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: "=&a" ( q->h ), "=&d" ( r_dash )
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: "g" ( d->l ), "0" ( x->h ), "1" ( 0 ) );
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}
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__asm__ ( "divl %2"
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: "=&a" ( q->l ), "=&d" ( r->l )
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: "g" ( d->l ), "0" ( x->l ), "1" ( r_dash ) );
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}
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static void udivmod64 ( const struct uint64 *x,
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const struct uint64 *d,
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struct uint64 *q,
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struct uint64 *r ) {
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if ( d->h == 0 ) {
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udivmod64_lo ( x, d, q, r );
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} else {
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assert ( 0 );
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while ( 1 ) {};
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}
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}
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/**
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* 64-bit division with remainder
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*
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* @v x Dividend
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* @v d Divisor
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* @ret r Remainder
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* @ret q Quotient
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*/
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UDItype __udivmoddi4 ( UDItype x, UDItype d, UDItype *r ) {
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UDItype q;
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UDItype *_x = &x;
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UDItype *_d = &d;
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UDItype *_q = &q;
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UDItype *_r = r;
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udivmod64 ( ( struct uint64 * ) _x, ( struct uint64 * ) _d,
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( struct uint64 * ) _q, ( struct uint64 * ) _r );
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return q;
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}
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/**
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* 64-bit division
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*
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* @v x Dividend
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* @v d Divisor
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* @ret q Quotient
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*/
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UDItype __udivdi3 ( UDItype x, UDItype d ) {
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UDItype r;
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return __udivmoddi4 ( x, d, &r );
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}
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