<html><head></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; ">C specifies truncated division.<div><br><div>
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<br><div><div>On 19 Jan 2010, at 21:59, Jay K wrote:</div><br class="Apple-interchange-newline"><blockquote type="cite"><div><br>That paper also suggests a more efficient algorithm we should probably adopt:<br><br><br>/* Floored division */<br>long divF( long D, long d )<br>{<br>long q = D/d;<br>long r = D%d;<br>if ((r> 0 && d < 0) || (r < 0 && d> 0)) q = q-1;<br>return q;<br>}<br><br><br>long modF( long D, long d )<br>{<br>long r = D%d;<br>if ((r> 0 && d < 0) || (r < 0 && d> 0)) r = r+d;<br>return r;<br>}<br><br><br>Though the condition should be perhaps the equivalent (r < 0) != (d < 0)<br>or (r < 0) ^ (d < 0)<br><br><br>We'd probably want to be sure the / followed by % got optimized into just one divide though.<br><br><br>or maybe what I have is fine.<br>As long as the inputs are the same sign, the current code does just one / or one %. It is only for mixed sign inputs that you need more care.<br>I am assuming C never "rounds", but either truncates or "floors".<br>e.g. 1/2 in the face of rounding is 1.<br><br><br>Integer division probably not a big concern anyway, esp. with any negative numbers. The main place I ever see division/mod used is hash tables, and the numbers are always positive (I rarely see negative numbers used anywhere in "systems" programming actually -- file sizes, array indices, array sizes..all positive; "math" needs floating point often...)<br><br><br> - Jay<br><br><br>----------------------------------------<br><blockquote type="cite">From: <a href="mailto:jay.krell@cornell.edu">jay.krell@cornell.edu</a><br></blockquote><blockquote type="cite">To: <a href="mailto:rodney_bates@lcwb.coop">rodney_bates@lcwb.coop</a>; <a href="mailto:m3devel@elegosoft.com">m3devel@elegosoft.com</a><br></blockquote><blockquote type="cite">Date: Wed, 20 Jan 2010 02:45:43 +0000<br></blockquote><blockquote type="cite">Subject: Re: [M3devel] integer division/mod questions?<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite"><blockquote type="cite">There is a positive zero and a negative zero, and which one you get<br></blockquote></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">Rodney you reminded me of something I forgot to ask:<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">Is 0 div -1 = 0 or -1?<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">In particular, in floating point, 0 / -1 is presumably -0 and 0 / 1 is presumably +0. Modula-3 is specified as returning the largest integer not greater than the floating point result. So the question then is, is 0> -0? Or are they equal? Integer-wise, there is no -0.<br></blockquote><blockquote type="cite">If 0> -0, then 0 div -1 should equal -1 instead of 0.<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">My current code I believe returns 0 for 0 div anything.<br></blockquote><blockquote type="cite">And I believe "the mod formula" holds as correct, since my testing checks that. I'll have to double double check.<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">The previous code probably also but I'll have to double check.<br></blockquote><blockquote type="cite">The previous code isn't necessarily the definition of correct though, since it was pretty clearly wrong for some cases.<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">Which is imply the question: Everyone agrees with me that the previous code was wrong? In particular, for LONG_MIN div negative, it was giving negative results, but I believe the result should be positive. Mod had a similar problem.<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">Nice if anyone can reproduce the problem with K&R + long long as well.<br></blockquote><blockquote type="cite">I don't like making changes here without some kind of independent confirmation, though I'm also pretty sure of myself..a contradiction I realize.<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">The previous code and I believe the current code are inconsistent as to when it would "trap", depending on compiler and optimization level. I assume that is ok for now, though in future we might want to make it predictable and honor the FloatMode stuff, but until that time, "unpredictable" is ok.<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">In particular, LONG_MIN div or mod -1 can trap.<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">div clearly has little choice. Mod I have to think about -- mod should be zero, right? LONG_MIN "evenly" divides by -1, the answer can't be represented, but the remainder can (it is zero). So maybe we should special case that to ensure it doesn't trap.<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">(anywhere I say "LONG_MIN", INT64_MIN has similar problem)<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">- Jay<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">----------------------------------------<br></blockquote><blockquote type="cite"><blockquote type="cite">Date: Tue, 19 Jan 2010 18:39:22 -0600<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">From: <a href="mailto:rodney_bates@lcwb.coop">rodney_bates@lcwb.coop</a><br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">To: <a href="mailto:m3devel@elegosoft.com">m3devel@elegosoft.com</a><br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">Subject: Re: [M3devel] integer division/mod questions?<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><a href="mailto:hendrik@topoi.pooq.com">hendrik@topoi.pooq.com</a> wrote:<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">On Sun, Jan 17, 2010 at 10:44:53AM +0000, Jay K wrote:<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">-2147483648 div 2147483647 ?<br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">-2147483648 mod 2147483647 ?<br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">quotient = -1, remainer = -1 seems reasonable.<br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">2147483647 * -1 + -1 == -2147483648<br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">There are two kinds of binary integer arithmetic -- two's complement<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">and one's complement. In my experience, two's complement machines tend<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">to have the remainder being the same sign as the dividend; one's<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">complement machines semm to have a remainder the same sign as the<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">divisor.<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">One's complement machines are all but extinct these days.<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">Tony, you were concerned about showing your age, but 20 years is but an<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">evening past. I remember programming ones-complement arithmetic<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">in assembly language, definitely more decades ago than two.<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">And that was not my first job.<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">There is a positive zero and a negative zero, and which one you get<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">can depend on the operation and operand values that produced the<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">result, so you have to check for both of them, often with two<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">separate conditional branches.<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">Anyone remember nines- or tens-complement arithmetic on decimal<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">machines? Electromechanical accounting machines?<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">However, Modula-3 specifies div as rounding down, so<br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">-2147483648 div 2147483647 == -2<br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">and then<br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><a href="http://www.modula3.com/cm3/doc/reference/arithmetic.html">http://www.modula3.com/cm3/doc/reference/arithmetic.html</a><br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">The value x DIV y is the floor of<br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">the quotient of x and y; that is, the maximum integer<br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">not exceeding the real number z such that z * y = x.<br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">For integers x and y, the value of x MOD y is<br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">defined to be x - y * (x DIV y).<br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">This means that for positive y, the value of x MOD y<br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">lies in the interval [0 .. y-1], regardless of<br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">the sign of x. For negative y, the value of<br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">x MOD y lies in the interval [y+1 .. 0], regardless<br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">of the sign of x.<br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">And this is consistent with MOD producing a canonical representative of<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">an equivalence class modulo y. It's what's wanted for a lot of<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">algorithms. What it returns for negative y isn't as important, but<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">it is important to have positive MODuli for positive y no matter what<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">the sign of x is.<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">But it's not what most two's-complement processors produce naturally.<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">Having a MODulo operation that is mathematically well-behaved takes<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">extra effort on these machines.<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">And it's also important to have a remainder that corresponds to the<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">division operator. On two's complement machines you have to either<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">* use extra instructions to correct the result of the divide<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">instruction to correspond to the mathematically desirable MOD<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">operator, or<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">* Have a separate remainder operation that does correspond to<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">division.<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">On one's complement machines MOD will have the same effect as remainder.<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">On two's complement, different.<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">-- hendrik<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><span class="Apple-tab-span" style="white-space:pre"> </span><span class="Apple-tab-span" style="white-space:pre"> </span> <span class="Apple-tab-span" style="white-space:pre"> </span> <span class="Apple-tab-span" style="white-space:pre"> </span><span class="Apple-tab-span" style="white-space:pre"> </span> <br></blockquote></blockquote></blockquote></div></blockquote></div><br></div></body></html>