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{{Built-in|Floor|⌊}} is a [[monadic]] [[scalar function]] that gives the [[wikipedia:floor and ceiling functions|floor]] of a real number, that is, the greatest integer [[Comparison tolerance|tolerantly]] [[less than or equal to]] the given value. This operation is also known as '''integral part''', '''entier''', and '''round down'''. Floor shares the [[glyph]] <source lang=apl inline>⌊</source> with the dyadic arithmetic function [[Minimum]]. [[Comparison_with_traditional_mathematics#Prefix|Traditional mathematics]] derives [[Ken_Iverson#Floor_and_Ceiling|its notation]] and name for floor from APL.
{{Built-in|Floor|⌊}} is a [[monadic]] [[scalar function]] that gives the [[wikipedia:floor and ceiling functions|floor]] of a real number, that is, the greatest integer tolerantly<ref>[[Robert Bernecky|Bernecky, Robert]]. [https://www.jsoftware.com/papers/satn23.htm "Comparison Tolerance"]. Sharp APL Technical Notes. 1977-06-10;.</ref> [[less than or equal to]] the given value. This operation is also known as '''integral part''', '''entier''', and '''round down'''. Floor shares the [[glyph]] <syntaxhighlight lang=apl inline>⌊</syntaxhighlight> with the dyadic arithmetic function [[Minimum]]. [[Comparison_with_traditional_mathematics#Prefix|Traditional mathematics]] derives [[Ken_Iverson#Floor_and_Ceiling|its notation]] and name for floor from APL.


== Examples ==
== Examples ==
Line 5: Line 5:
Floor rounds down the given numbers to the nearest integers.
Floor rounds down the given numbers to the nearest integers.


<source lang=apl>
<syntaxhighlight lang=apl>
       ⌊2 2.8 ¯2 ¯2.8
       ⌊2 2.8 ¯2 ¯2.8
2 2 ¯2 ¯3
2 2 ¯2 ¯3
</source>
</syntaxhighlight>


Rounding to the ''nearest'' integer (rounding up on half) can be achieved by [[add|adding]] 0.5 before applying Floor.
Rounding to the ''nearest'' integer (rounding up on half) can be achieved by [[add|adding]] 0.5 before applying Floor.


<source lang=apl>
<syntaxhighlight lang=apl>
       ⌊0.5+2 2.3 2.5 2.8
       ⌊0.5+2 2.3 2.5 2.8
2 2 3 3
2 2 3 3
</source>
</syntaxhighlight>


Integral quotient of division can be found with [[divide|division]] followed by Floor.
Integral quotient of division can be found with [[divide|division]] followed by Floor.


<source lang=apl>
<syntaxhighlight lang=apl>
       ⌊10 20 30÷3
       ⌊10 20 30÷3
3 6 10
3 6 10
</source>
</syntaxhighlight>


== Properties ==
== Properties ==
Line 30: Line 30:
Floor is affected by [[comparison tolerance]]. If the given number is [[tolerant comparison|tolerantly equal]] to its [[ceiling]], it is rounded to that number instead.
Floor is affected by [[comparison tolerance]]. If the given number is [[tolerant comparison|tolerantly equal]] to its [[ceiling]], it is rounded to that number instead.


<source lang=apl>
<syntaxhighlight lang=apl>
       ⎕PP←16
       ⎕PP←16
       ⎕←v←1+0.6×⎕CTׯ2 ¯1 0
       ⎕←v←1+0.6×⎕CTׯ2 ¯1 0
Line 36: Line 36:
       ⌊v
       ⌊v
0 1 1
0 1 1
</source>
</syntaxhighlight>
 
== Model ==
 
Floor can very easily be modelled using residue like:
 
<syntaxhighlight lang=apl>
      model←{⍵-1|⍵}
</syntaxhighlight>
 
To model it without using residue, because residue uses floor under the hood, approaches like converting to strings and then stripping the decimal component or converting to binary and stripping the decimal component can be used.
 
<syntaxhighlight lang=apl>
      model←{
        ⎕pp←34                  ⍝ set to max as we are using strings, the execute and format primitives round the number to the ⎕pp value
        dotPos←⍸,'.'⍷⍕⍵        ⍝ convert num to string and get the position of the decimal point
        int←⍎(⍕⍵)↑⍨¯1+dotPos    ⍝ strip integer based on the decimal point
        int-(⍵<0)∧(~0∊⍴dotPos)  ⍝ Subtract 1 only when negative+non int component exists. eg: ¯123.32→¯124
    }
</syntaxhighlight>
 
Converting to the exponent/scientific notation (123E¯2) and then using the exponent and mantissa to strip the decimal points can be used.
 
Warning: However, the method present here has issues dealing with larger values due to the loss in precision because of the ⍎ operator.
 
<syntaxhighlight lang=apl>
      model←{
        fmt←{⎕FR≡1287:¯33⍕⍵ ⋄ ¯16⍕⍵}⍵
        (m e)←'E'(≠⊆⊢)fmt
        en←⍎e
        diff←-(⍵<0)∧('.'∊⍕⍵)
        en<0:diff+0
        m↑⍨←3+(⍵<0)+en
        diff+⍎m,'E',e
    }
</syntaxhighlight>
 
Other approaches could include writing a hungry loop to evaluate the closest integer value and evaluate from there.


=== Complex floor ===
=== Complex floor ===
Line 44: Line 81:
[[Eugene McDonnell]] designed the domain extension of Floor to [[complex number|complex numbers]].<ref>McDonnell, Eugene. [https://www.jsoftware.com/papers/eem/complexfloor.htm "Complex Floor"].</ref> Complex floor maps every complex number to a [[wikipedia:Gaussian integer|Gaussian integer]], a complex number whose real and imaginary parts are integers. It has an important property that the [[magnitude]] of [[subtract|difference]] between any complex number Z and its floor is [[less than]] 1. This extension is currently implemented in [[Dyalog APL]], [[J]], and [[NARS2000]], and is internally used to implement complex [[ceiling]], [[residue]], and [[GCD]].
[[Eugene McDonnell]] designed the domain extension of Floor to [[complex number|complex numbers]].<ref>McDonnell, Eugene. [https://www.jsoftware.com/papers/eem/complexfloor.htm "Complex Floor"].</ref> Complex floor maps every complex number to a [[wikipedia:Gaussian integer|Gaussian integer]], a complex number whose real and imaginary parts are integers. It has an important property that the [[magnitude]] of [[subtract|difference]] between any complex number Z and its floor is [[less than]] 1. This extension is currently implemented in [[Dyalog APL]], [[J]], and [[NARS2000]], and is internally used to implement complex [[ceiling]], [[residue]], and [[GCD]].


<source lang=apl>
<syntaxhighlight lang=apl>
       v←1.8J2.5 2.2J2.5 2.5J2.2 2.5J1.8
       v←1.8J2.5 2.2J2.5 2.5J2.2 2.5J1.8
       ⌊v
       ⌊v
Line 50: Line 87:
       1>|v-⌊v
       1>|v-⌊v
1 1 1 1
1 1 1 1
</source>{{Works in|[[Dyalog APL]]}}
</syntaxhighlight>{{Works in|[[Dyalog APL]]}}


== External links ==
== External links ==

Latest revision as of 16:27, 24 December 2023

⌊

Floor (⌊) is a monadic scalar function that gives the floor of a real number, that is, the greatest integer tolerantly[1] less than or equal to the given value. This operation is also known as integral part, entier, and round down. Floor shares the glyph ⌊ with the dyadic arithmetic function Minimum. Traditional mathematics derives its notation and name for floor from APL.

Examples

Floor rounds down the given numbers to the nearest integers. 

      ⌊2 2.8 ¯2 ¯2.8
2 2 ¯2 ¯3

Rounding to the nearest integer (rounding up on half) can be achieved by adding 0.5 before applying Floor. 

      ⌊0.5+2 2.3 2.5 2.8
2 2 3 3

Integral quotient of division can be found with division followed by Floor. 

      ⌊10 20 30÷3
3 6 10

Properties

The floor of any real number is an integer.

Floor is affected by comparison tolerance. If the given number is tolerantly equal to its ceiling, it is rounded to that number instead. 

      ⎕PP←16
      ⎕←v←1+0.6×⎕CTׯ2 ¯1 0
0.999999999999988 0.999999999999994 1
      ⌊v
0 1 1

Model

Floor can very easily be modelled using residue like: 

      model←{⍵-1|⍵}

To model it without using residue, because residue uses floor under the hood, approaches like converting to strings and then stripping the decimal component or converting to binary and stripping the decimal component can be used. 

      model←{
        ⎕pp←34                  ⍝ set to max as we are using strings, the execute and format primitives round the number to the ⎕pp value
        dotPos←⍸,'.'⍷⍕⍵         ⍝ convert num to string and get the position of the decimal point
        int←⍎(⍕⍵)↑⍨¯1+dotPos    ⍝ strip integer based on the decimal point
        int-(⍵<0)∧(~0∊⍴dotPos)  ⍝ Subtract 1 only when negative+non int component exists. eg: ¯123.32→¯124
    }

Converting to the exponent/scientific notation (123E¯2) and then using the exponent and mantissa to strip the decimal points can be used.

Warning: However, the method present here has issues dealing with larger values due to the loss in precision because of the ⍎ operator. 

      model←{
        fmt←{⎕FR≡1287:¯33⍕⍵ ⋄ ¯16⍕⍵}⍵
        (m e)←'E'(≠⊆⊢)fmt
        en←⍎e
        diff←-(⍵<0)∧('.'∊⍕⍵)
        en<0:diff+0
        m↑⍨←3+(⍵<0)+en
        diff+⍎m,'E',e
    }

Other approaches could include writing a hungry loop to evaluate the closest integer value and evaluate from there.

Complex floor

Main article: Complex Floor

Eugene McDonnell designed the domain extension of Floor to complex numbers.[2] Complex floor maps every complex number to a Gaussian integer, a complex number whose real and imaginary parts are integers. It has an important property that the magnitude of difference between any complex number Z and its floor is less than 1. This extension is currently implemented in Dyalog APL, J, and NARS2000, and is internally used to implement complex ceiling, residue, and GCD. 

      v←1.8J2.5 2.2J2.5 2.5J2.2 2.5J1.8
      ⌊v
2J2 2J2 2J2 2J2
      1>|v-⌊v
1 1 1 1
Works in: Dyalog APL

External links

Documentation

References

  1. ↑ Bernecky, Robert. "Comparison Tolerance". Sharp APL Technical Notes. 1977-06-10;.
  2. ↑ McDonnell, Eugene. "Complex Floor".
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