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====== Function FLOOR, FFLOOR, CEILING, FCEILING, TRUNCATE, FTRUNCATE, ROUND, FROUND ====== ====Syntax==== * **floor** //number ''&optional'' divisor// → //quotient, remainder// * **ffloor** //number ''&optional'' divisor// → //quotient, remainder// * **ceiling** //number ''&optional'' divisor// → //quotient, remainder// * **fceiling** //number ''&optional'' divisor// → //quotient, remainder// * **truncate** //number ''&optional'' divisor// → //quotient, remainder// * **ftruncate** //number ''&optional'' divisor// → //quotient, remainder// * **round** //number ''&optional'' divisor// → //quotient, remainder// * **fround** //number ''&optional'' divisor// → //quotient, remainder// ====Arguments and Values==== * //number// - a //[[CL:Glossary:real]]//. * //divisor// - a non-zero //[[CL:Glossary:real]]//. The default is the //[[CL:Glossary:integer]]// ''1''. * //quotient// - for **floor**, **ceiling**, **truncate**, and **round**: an //[[CL:Glossary:integer]]//; for **ffloor**, **fceiling**, **ftruncate**, and **fround**: a //[[CL:Glossary:float]]//. * //remainder// - a //[[CL:Glossary:real]]//. ====Description==== These functions divide //number// by //divisor//, returning a //quotient// and //remainder//, such that: <blockquote> //quotient// * //divisor// + //remainder// = //number// </blockquote> The //quotient// 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 //[[CL:Glossary:operator]]//: ===floor, ffloor=== **floor** and **ffloor** produce a //quotient// that has been truncated toward negative infinity; that is, the //quotient// represents the largest mathematical integer that is not larger than the mathematical quotient. ===ceiling, fceiling=== **ceiling** and **fceiling** produce a //quotient// that has been truncated toward positive infinity; that is, the //quotient// represents the smallest mathematical integer that is not smaller than the mathematical result. ===truncate, ftruncate=== **truncate** and **ftruncate** produce a //quotient// that has been truncated towards zero; that is, the //quotient// 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. ===round, fround=== **round** and **fround** produce a //quotient// 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 ''integer + 1/2''), then the //quotient// has been rounded to the even (divisible by two) integer. ---- All of these functions perform type conversion operations on //numbers//. The //remainder// is an //[[CL:Glossary:integer]]// if both //x// and //y// are //[[CL:Glossary:integers]]//, is a //[[CL:Glossary:rational]]// if both //x// and //y// are //[[CL:Glossary:rationals]]//, and is a //[[CL:Glossary:float]]// if either //x// or //y// is a //[[CL:Glossary:float]]//. **ffloor**, **fceiling**, **ftruncate**, and **fround** handle arguments of different //[[CL:Glossary:type|types]]// in the following way: If //number// is a //[[CL:Glossary:float]]//, and //divisor// is not a //[[CL:Glossary:float]]// of longer format, then the first result is a //[[CL:Glossary:float]]// of the same //[[CL:Glossary:type]]// as //number//. Otherwise, the first result is of the //[[CL:Glossary:type]]// determined by //[[CL:Glossary:contagion]]// rules; see section {\secref\NumericContagionRules}. ====Examples==== <blockquote> (floor 3/2) <r>1 1/2 </r> (ceiling 3 2) <r>2 -1 </r> (ffloor 3 2) <r>1.0 1 </r> (ffloor -4.7) <r>-5.0 0.3 </r> (ffloor 3.5d0) <r>3.0d0 0.5d0 </r> (fceiling 3/2) <r>2.0 -1/2 </r> (truncate 1) <r>1 0 </r> (truncate .5) <r>0 0.5 </r> (round .5) <r>0 0.5 </r> (ftruncate -7 2) <r>-3.0 -1 </r> (fround -7 2) <r>-4.0 1 </r> ([[CL:Macros:dolist]] (n '(2.6 2.5 2.4 0.7 0.3 -0.3 -0.7 -2.4 -2.5 -2.6)) ([[CL:Functions:format]] [[CL:Constant Variables:t]] "~&~4,[email protected] ~2,' D ~2,' D ~2,' D ~2,' D" n (floor n) (ceiling n) (truncate n) (round n))) <o>+2.6 2 3 2 3 +2.5 2 3 2 2 +2.4 2 3 2 2 +0.7 0 1 0 1 +0.3 0 1 0 0 -0.3 -1 0 0 0 -0.7 -1 0 0 -1 -2.4 -3 -2 -2 -2 -2.5 -3 -2 -2 -2 -2.6 -3 -2 -2 -3 </o> <r>NIL </r> </blockquote> ====Side Effects==== None. ====Affected By==== None. ====Exceptional Situations==== None. ====See Also==== None. ====Notes==== When only //number// is given, the two results are exact; the mathematical sum of the two results is always equal to the mathematical value of //number//. ''(function //number// //divisor//)'' and ''(function (/ //number// //divisor//))'' (where ''function'' is any of one of **floor**, **ceiling**, **ffloor**, **fceiling**, **truncate**, **round**, **ftruncate**, and **fround**) return the same first value, but they return different remainders as the second value. For example: <blockquote> (floor 5 2) <r>2 1</r> (floor (/ 5 2)) <r>2 1/2</r> </blockquote> If an effect is desired that is similar to **round**, 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 ''(floor ([[CL:Functions:math-add|+]] x 1/2))'' or ''(ceiling ([[CL:Functions:math-subtract|-]] x 1/2))''.