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<H2>Issue CONTAGION-ON-NUMERICAL-COMPARISONS Writeup</H2>
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<PRE><B>Issue:</B> <A HREF="iss086.htm">CONTAGION-ON-NUMERICAL-COMPARISONS</A><P>
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<P>
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<B>References:</B> CLtL p.194<P>
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<P>
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<B>Category:</B> CHANGE<P>
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<P>
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<B>Edit history:</B> Version 1, 14-Sep-88, Moon<P>
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<P>
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<B>Problem description:<P>
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</B><P>
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The numerical contagion rules specified on CLtL p.194 make it impossible<P>
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for the numerical comparison functions to be transitive when given<P>
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arguments of mixed floating and rational types (see example below).<P>
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<P>
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<B>Proposal (CONTAGION-ON-NUMERICAL-COMPARISONS:TRANSITIVE):<P>
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</B> <P>
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Instead of using the same contagion rule both for combining functions<P>
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and for comparing functions, make the present rule apply only to<P>
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combining functions and add the following rule: When rational and<P>
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floating-point numbers are compared by a numerical function, the<P>
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function <A REL=DEFINITION HREF="../Body/a_ration.htm#rational"><B>RATIONAL</B></A> is called to convert the floating-point number to a<P>
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rational and then an exact comparison is performed. In the case of<P>
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complex numbers (<A REL=DEFINITION HREF="../Body/a_ration.htm#rational"><B>RATIONAL</B></A> is for some unknown reason only allowed on<P>
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reals), the real and imaginary parts are handled individually.<P>
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<P>
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It is of course valid to use a more efficient implementation than<P>
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actually calling <A REL=DEFINITION HREF="../Body/a_ration.htm#rational"><B>RATIONAL</B></A>, as long as the result of the comparison is<P>
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the same.<P>
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<P>
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<B>Test Cases/Examples:<P>
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</B><P>
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(<A REL=DEFINITION HREF="../Body/m_defpar.htm#defvar"><B>defvar</B></A> a (/ 10.0 <A REL=DEFINITION HREF="../Body/v_short_.htm#single-float-epsilon"><B>single-float-epsilon</B></A>))<P>
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<P>
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(= a (<A REL=DEFINITION HREF="../Body/f_1pl_1_.htm#1PL"><B>1+</B></A> (<A REL=DEFINITION HREF="../Body/f_floorc.htm#floor"><B>floor</B></A> a)))<P>
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should be <A REL=DEFINITION HREF="../Body/a_nil.htm#nil"><B>NIL</B></A>, since <P>
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(= a (<A REL=DEFINITION HREF="../Body/f_floorc.htm#floor"><B>floor</B></A> a))<P>
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is certainly T and<P>
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(= (<A REL=DEFINITION HREF="../Body/f_floorc.htm#floor"><B>floor</B></A> a) (<A REL=DEFINITION HREF="../Body/f_1pl_1_.htm#1PL"><B>1+</B></A> (<A REL=DEFINITION HREF="../Body/f_floorc.htm#floor"><B>floor</B></A> a)))<P>
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is certainly <A REL=DEFINITION HREF="../Body/a_nil.htm#nil"><B>NIL</B></A>. However, by the rule of floating-point contagion,<P>
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(= a (<A REL=DEFINITION HREF="../Body/f_1pl_1_.htm#1PL"><B>1+</B></A> (<A REL=DEFINITION HREF="../Body/f_floorc.htm#floor"><B>floor</B></A> a)))<P>
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is the same as <P>
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(= a (<A REL=DEFINITION HREF="../Body/a_float.htm#float"><B>float</B></A> (<A REL=DEFINITION HREF="../Body/f_1pl_1_.htm#1PL"><B>1+</B></A> (<A REL=DEFINITION HREF="../Body/f_floorc.htm#floor"><B>floor</B></A> a)) a))<P>
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and the latter form is certainly T.<P>
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<P>
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To understand this example better, it helps to realize that<P>
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(= a (+ a 1.0))<P>
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is always true, by the definition of <A REL=DEFINITION HREF="../Body/v_short_.htm#single-float-epsilon"><B>single-float-epsilon</B></A>.<P>
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<P>
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Here is a second example:<P>
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<P>
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(<A REL=DEFINITION HREF="../Body/m_defpar.htm#defvar"><B>defvar</B></A> i (<A REL=DEFINITION HREF="../Body/f_floorc.htm#floor"><B>floor</B></A> a))<P>
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<P>
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(<A REL=DEFINITION HREF="../Body/f_eq_sle.htm#LTEQ"><B><=</B></A> a (+ i 1)) is T.<P>
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(< (+ i 1) (+ i 2)) is T.<P>
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(<A REL=DEFINITION HREF="../Body/f_eq_sle.htm#LTEQ"><B><=</B></A> (+ i 2) a) is T by CLtL, <A REL=DEFINITION HREF="../Body/a_nil.htm#nil"><B>NIL</B></A> by this proposal.<P>
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Thus CLtL requires<P>
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a <A REL=DEFINITION HREF="../Body/f_eq_sle.htm#LTEQ"><B><=</B></A> i+1 < i+2 <A REL=DEFINITION HREF="../Body/f_eq_sle.htm#LTEQ"><B><=</B></A> a<P>
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which ought to imply<P>
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a < a<P>
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which is absurd.<P>
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<P>
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<B>Rationale:<P>
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</B><P>
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Transitivity of = and of < are important to many algorithms. What CLtL<P>
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says now was probably not intentional, but just the result of thinking<P>
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that comparing and combining could be lumped together without really<P>
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thinking about it.<P>
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<P>
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Without this change, it is impossible to extend the :TEST argument to<P>
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<A REL=DEFINITION HREF="../Body/f_mk_has.htm#make-hash-table"><B>MAKE-HASH-TABLE</B></A> to allow = or <A REL=DEFINITION HREF="../Body/f_equalp.htm#equalp"><B>EQUALP</B></A>, since there could be two table<P>
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entries with rational keys that are not =, then <A REL=DEFINITION HREF="../Body/f_gethas.htm#gethash"><B>GETHASH</B></A> could be<P>
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presented with a floating-point argument that is = to both keys.<P>
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<P>
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<B>Current practice:<P>
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</B><P>
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Lucid is said to implement the proposal. As far as I know all other<P>
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implementations do what CLtL currently says.<P>
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<P>
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<B>Cost to Implementors:<P>
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</B><P>
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This requires a change in what is likely to be intricate and<P>
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implementation-dependent code. However, the total effort should<P>
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be small compared to the cost of writing that code originally.<P>
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<P>
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<B>Cost to Users:<P>
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</B><P>
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This is an incompatible change. It's difficult to know if any users<P>
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are depending on the current behavior. It seems likely that most users<P>
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would expect the proposed behavior, and may be wondering why their<P>
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programs don't quite work when the numbers get large.<P>
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<P>
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<B>Cost of non-adoption:<P>
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</B><P>
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Comparison functions in Common Lisp will be non-transitive.<P>
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<P>
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<B>Benefits:<P>
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</B><P>
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Comparison functions in Common Lisp will be transitive.<P>
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<P>
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<B>Esthetics:<P>
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</B><P>
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Having two rules of floating-point contagion may seem less esthetic,<P>
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but I'm certain that having the comparison functions behave in a more<P>
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mathematically correct fashion outweighs that esthetically.<P>
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<P>
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<B>Discussion:<P>
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</B><P>
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Everyone who has expressed an opinion has thought this was the right<P>
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thing for years, but we never got around to writing it up as a cleanup<P>
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proposal.<P>
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<P>
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</PRE>
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