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119 lines
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<!-- Common Lisp HyperSpec (TM), version 7.0 generated by Kent M. Pitman on Mon, 11-Apr-2005 2:31am EDT -->
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<TITLE>CLHS: Function FLOOR, FFLOOR, CEILING, FCEILING...</TITLE>
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<H1><A REV=MADE HREF="http://www.lispworks.com/"><IMG WIDTH=80 HEIGHT=65 ALT="[LISPWORKS]" SRC="../Graphics/LWSmall.gif" ALIGN=Bottom></A><A REL=TOP HREF="../Front/index.htm"><IMG WIDTH=237 HEIGHT=65 ALT="[Common Lisp HyperSpec (TM)]" SRC="../Graphics/CLHS_Sm.gif" ALIGN=Bottom></A> <A REL=PREV HREF="f_zerop.htm"><IMG WIDTH=40 HEIGHT=40 ALT="[Previous]" SRC="../Graphics/Prev.gif" ALIGN=Bottom></A><A REL=UP HREF="c_number.htm"><IMG WIDTH=40 HEIGHT=40 ALT="[Up]" SRC="../Graphics/Up.gif" ALIGN=Bottom></A><A REL=NEXT HREF="f_sin_c.htm"><IMG WIDTH=40 HEIGHT=40 ALT="[Next]" SRC="../Graphics/Next.gif" ALIGN=Bottom></A></H1>
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<HR>
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<A NAME="floor"><A NAME="ffloor"><A NAME="ceiling"><A NAME="fceiling"><A NAME="truncate"><A NAME="ftruncate"><A NAME="round"><A NAME="fround"><I>Function</I> <B>FLOOR, FFLOOR, CEILING, FCEILING, TRUNCATE, FTRUNCATE, ROUND, FROUND</B></A></A></A></A></A></A></A></A> <P>
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<P><B>Syntax:</B><P>
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<P>
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<B>floor</B> <I>number </I><TT>&optional</TT><I> divisor</I> => <I>quotient, remainder</I><P>
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<B>ffloor</B> <I>number </I><TT>&optional</TT><I> divisor</I> => <I>quotient, remainder</I><P>
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<B>ceiling</B> <I>number </I><TT>&optional</TT><I> divisor</I> => <I>quotient, remainder</I><P>
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<B>fceiling</B> <I>number </I><TT>&optional</TT><I> divisor</I> => <I>quotient, remainder</I><P>
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<B>truncate</B> <I>number </I><TT>&optional</TT><I> divisor</I> => <I>quotient, remainder</I><P>
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<B>ftruncate</B> <I>number </I><TT>&optional</TT><I> divisor</I> => <I>quotient, remainder</I><P>
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<B>round</B> <I>number </I><TT>&optional</TT><I> divisor</I> => <I>quotient, remainder</I><P>
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<B>fround</B> <I>number </I><TT>&optional</TT><I> divisor</I> => <I>quotient, remainder</I><P>
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<P>
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<P><B>Arguments and Values:</B><P>
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<P>
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<I>number</I>---a <A REL=DEFINITION HREF="t_real.htm#real"><I>real</I></A>. <P>
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<I>divisor</I>---a non-zero <A REL=DEFINITION HREF="t_real.htm#real"><I>real</I></A>. The default is the <A REL=DEFINITION HREF="26_glo_i.htm#integer"><I>integer</I></A> <TT>1</TT>. <P>
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<I>quotient</I>---for <A REL=DEFINITION HREF="#floor"><B>floor</B></A>, <A REL=DEFINITION HREF="#ceiling"><B>ceiling</B></A>, <A REL=DEFINITION HREF="#truncate"><B>truncate</B></A>, and <A REL=DEFINITION HREF="#round"><B>round</B></A>: an <A REL=DEFINITION HREF="26_glo_i.htm#integer"><I>integer</I></A>; for <A REL=DEFINITION HREF="#ffloor"><B>ffloor</B></A>, <A REL=DEFINITION HREF="#fceiling"><B>fceiling</B></A>, <A REL=DEFINITION HREF="#ftruncate"><B>ftruncate</B></A>, and <A REL=DEFINITION HREF="#fround"><B>fround</B></A>: a <A REL=DEFINITION HREF="26_glo_f.htm#float"><I>float</I></A>. <P>
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<I>remainder</I>---a <A REL=DEFINITION HREF="t_real.htm#real"><I>real</I></A>. <P>
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<P><B>Description:</B><P>
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<P>
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These functions divide <I>number</I> by <I>divisor</I>, returning a <I>quotient</I> and <I>remainder</I>, such that <P>
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<I>quotient</I>*<I>divisor</I>+<I>remainder</I>=<I>number</I> <P>
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The <I>quotient</I> always represents a mathematical integer. When more than one mathematical integer might be possible (i.e., when the remainder is not zero), the kind of rounding or truncation depends on the <A REL=DEFINITION HREF="26_glo_o.htm#operator"><I>operator</I></A>: <P>
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<P><DL><P>
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<DT><A REL=DEFINITION HREF="#floor"><B>floor</B></A>, <A REL=DEFINITION HREF="#ffloor"><B>ffloor</B></A> <P><DD>
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<A REL=DEFINITION HREF="#floor"><B>floor</B></A> and <A REL=DEFINITION HREF="#ffloor"><B>ffloor</B></A> produce a <I>quotient</I> that has been truncated toward negative infinity; that is, the <I>quotient</I> represents the largest mathematical integer that is not larger than the mathematical quotient. <P>
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<DT><A REL=DEFINITION HREF="#ceiling"><B>ceiling</B></A>, <A REL=DEFINITION HREF="#fceiling"><B>fceiling</B></A> <P><DD>
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<A REL=DEFINITION HREF="#ceiling"><B>ceiling</B></A> and <A REL=DEFINITION HREF="#fceiling"><B>fceiling</B></A> produce a <I>quotient</I> that has been truncated toward positive infinity; that is, the <I>quotient</I> represents the smallest mathematical integer that is not smaller than the mathematical result. <P>
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<DT><A REL=DEFINITION HREF="#truncate"><B>truncate</B></A>, <A REL=DEFINITION HREF="#ftruncate"><B>ftruncate</B></A> <P><DD>
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<A REL=DEFINITION HREF="#truncate"><B>truncate</B></A> and <A REL=DEFINITION HREF="#ftruncate"><B>ftruncate</B></A> produce a <I>quotient</I> that has been truncated towards zero; that is, the <I>quotient</I> represents the mathematical integer of the same sign as the mathematical quotient, and that has the greatest integral magnitude not greater than that of the mathematical quotient. <P>
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<DT><A REL=DEFINITION HREF="#round"><B>round</B></A>, <A REL=DEFINITION HREF="#fround"><B>fround</B></A> <P><DD>
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<A REL=DEFINITION HREF="#round"><B>round</B></A> and <A REL=DEFINITION HREF="#fround"><B>fround</B></A> produce a <I>quotient</I> that has been rounded to the nearest mathematical integer; if the mathematical quotient is exactly halfway between two integers, (that is, it has the form <I>integer</I>+1/2), then the <I>quotient</I> has been rounded to the even (divisible by two) integer. <P>
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<P></DL><P>
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All of these functions perform type conversion operations on <I>numbers</I>. <P>
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The <I>remainder</I> is an <A REL=DEFINITION HREF="26_glo_i.htm#integer"><I>integer</I></A> if both <TT>x</TT> and <TT>y</TT> are <A REL=DEFINITION HREF="26_glo_i.htm#integer"><I>integers</I></A>, is a <A REL=DEFINITION HREF="26_glo_r.htm#rational"><I>rational</I></A> if both <TT>x</TT> and <TT>y</TT> are <A REL=DEFINITION HREF="26_glo_r.htm#rational"><I>rationals</I></A>, and is a <A REL=DEFINITION HREF="26_glo_f.htm#float"><I>float</I></A> if either <TT>x</TT> or <TT>y</TT> is a <A REL=DEFINITION HREF="26_glo_f.htm#float"><I>float</I></A>. <P>
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<A REL=DEFINITION HREF="#ffloor"><B>ffloor</B></A>, <A REL=DEFINITION HREF="#fceiling"><B>fceiling</B></A>, <A REL=DEFINITION HREF="#ftruncate"><B>ftruncate</B></A>, and <A REL=DEFINITION HREF="#fround"><B>fround</B></A> handle arguments of different <A REL=DEFINITION HREF="26_glo_t.htm#type"><I>types</I></A> in the following way: If <I>number</I> is a <A REL=DEFINITION HREF="26_glo_f.htm#float"><I>float</I></A>, and <I>divisor</I> is not a <A REL=DEFINITION HREF="26_glo_f.htm#float"><I>float</I></A> of longer format, then the first result is a <A REL=DEFINITION HREF="26_glo_f.htm#float"><I>float</I></A> of the same <A REL=DEFINITION HREF="26_glo_t.htm#type"><I>type</I></A> as <I>number</I>. Otherwise, the first result is of the <A REL=DEFINITION HREF="26_glo_t.htm#type"><I>type</I></A> determined by <A REL=DEFINITION HREF="26_glo_c.htm#contagion"><I>contagion</I></A> rules; see <A REL=CHILD HREF="12_aab.htm">Section 12.1.1.2 (Contagion in Numeric Operations)</A>. <P>
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<P><B>Examples:</B><P>
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<P>
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<PRE>
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(floor 3/2) => 1, 1/2
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(ceiling 3 2) => 2, -1
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(ffloor 3 2) => 1.0, 1
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(ffloor -4.7) => -5.0, 0.3
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(ffloor 3.5d0) => 3.0d0, 0.5d0
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(fceiling 3/2) => 2.0, -1/2
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(truncate 1) => 1, 0
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(truncate .5) => 0, 0.5
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(round .5) => 0, 0.5
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(ftruncate -7 2) => -3.0, -1
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(fround -7 2) => -4.0, 1
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(dolist (n '(2.6 2.5 2.4 0.7 0.3 -0.3 -0.7 -2.4 -2.5 -2.6))
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(format t "~&~4,1@F ~2,' D ~2,' D ~2,' D ~2,' D"
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n (floor n) (ceiling n) (truncate n) (round n)))
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>> +2.6 2 3 2 3
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>> +2.5 2 3 2 2
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>> +2.4 2 3 2 2
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>> +0.7 0 1 0 1
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>> +0.3 0 1 0 0
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>> -0.3 -1 0 0 0
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>> -0.7 -1 0 0 -1
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>> -2.4 -3 -2 -2 -2
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>> -2.5 -3 -2 -2 -2
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>> -2.6 -3 -2 -2 -3
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=> NIL
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</PRE>
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</TT> <P>
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<P><B>Side Effects:</B> None.
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<P>
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<P><B>Affected By:</B> None.
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<P>
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<P><B>Exceptional Situations:</B> None.
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<P>
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<P><B>See Also:</B> None.
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<P>
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<P><B>Notes:</B><P>
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<P>
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When only <I>number</I> is given, the two results are exact; the mathematical sum of the two results is always equal to the mathematical value of <I>number</I>. <P>
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<TT>(</TT><I>function</I><TT> </TT><I>number</I><TT> </TT><I>divisor</I><TT>)</TT> and <TT>(</TT><I>function</I><TT> (/ </TT><I>number</I><TT> </TT><I>divisor</I><TT>))</TT> (where <I>function</I> is any of one of <A REL=DEFINITION HREF="#floor"><B>floor</B></A>, <A REL=DEFINITION HREF="#ceiling"><B>ceiling</B></A>, <A REL=DEFINITION HREF="#ffloor"><B>ffloor</B></A>, <A REL=DEFINITION HREF="#fceiling"><B>fceiling</B></A>, <A REL=DEFINITION HREF="#truncate"><B>truncate</B></A>, <A REL=DEFINITION HREF="#round"><B>round</B></A>, <A REL=DEFINITION HREF="#ftruncate"><B>ftruncate</B></A>, and <A REL=DEFINITION HREF="#fround"><B>fround</B></A>) return the same first value, but they return different remainders as the second value. For example: <P>
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<PRE>
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(floor 5 2) => 2, 1
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(floor (/ 5 2)) => 2, 1/2
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</PRE>
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</TT> <P>
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If an effect is desired that is similar to <A REL=DEFINITION HREF="#round"><B>round</B></A>, but that always rounds up or down (rather than toward the nearest even integer) if the mathematical quotient is exactly halfway between two integers, the programmer should consider a construction such as <TT>(floor (+ x 1/2))</TT> or <TT>(ceiling (- x 1/2))</TT>. <P>
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<A REL=NAVIGATOR HREF="../Front/StartPts.htm"><IMG WIDTH=80 HEIGHT=40 ALT="[Starting Points]" SRC="../Graphics/StartPts.gif" ALIGN=Bottom></A><A REL=TOC HREF="../Front/Contents.htm"><IMG WIDTH=80 HEIGHT=40 ALT="[Contents]" SRC="../Graphics/Contents.gif" ALIGN=Bottom></A><A REL=INDEX HREF="../Front/X_Master.htm"><IMG WIDTH=80 HEIGHT=40 ALT="[Index]" SRC="../Graphics/Index.gif" ALIGN=Bottom></A><A REL=INDEX HREF="../Front/X_Symbol.htm"><IMG WIDTH=80 HEIGHT=40 ALT="[Symbols]" SRC="../Graphics/Symbols.gif" ALIGN=Bottom></A><A REL=GLOSSARY HREF="../Body/26_a.htm"><IMG WIDTH=80 HEIGHT=40 ALT="[Glossary]" SRC="../Graphics/Glossary.gif" ALIGN=Bottom></A><A HREF="../Front/X3J13Iss.htm"><IMG WIDTH=80 HEIGHT=40 ALT="[Issues]" SRC="../Graphics/Issues.gif" ALIGN=Bottom></A><BR>
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<A REL=COPYRIGHT HREF="../Front/Help.htm#Legal"><I>Copyright 1996-2005, LispWorks Ltd. All rights reserved.</I></A><P>
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