View source for Help:Mod, round, floor, ceil, trunc

{{h:h}}[[Category:Editor handbook]]

The MediaWiki extension [[mw:ParserFunctions|ParserFunctions]] enables users to perform simple mathematical [[Help:Calculation|computations]]; #expr and #ifexpr allow various discontinuous functions.

Uniform w.r.t. 0, non-strict monotonic:
*ceil
*floor
Symmetric w.r.t. 0, on each side of 0 non-strict monotonic:
*trunc
*round
Otherwise related to 0, on each side of 0 periodic:
*mod

See also [[Help:Mod, round, floor, ceil, trunc/table]].

==Trunc==
Trunc converts a float to a number of integer type by truncation of the fraction. For 2^63 <= x <= 2^64 we get x - 2^64, for larger x we get 0; for x < -2^63 we get -2^63.

For example:
*{{xpsoc|#expr:trunc(-2*2^63-2^12)}}
*{{xpsoc|#expr:trunc(-2*2^63+2^12)}}
*{{xpsoc|#expr:trunc(-1*2^63-2^12)}}
*{{xpsoc|#expr:trunc(-1*2^63+2^12)}}
*{{xpsoc|#expr:trunc(0*2^63-2^12)}}
*{{xpsoc|#expr:trunc(0*2^63+2^12)}}
*{{xpsoc|#expr:trunc(1*2^63-2^12)}}
*{{xpsoc|#expr:trunc(1*2^63+2^12)}}
*{{xpsoc|#expr:trunc(2*2^63-2^12)}}
*{{xpsoc|#expr:trunc(2*2^63+2^12)}}
*{{xpsoc|#expr:trunc(2^64+1024)}}
*{{xpsoc|#expr:trunc(3*2^63-2^12)}}
*{{xpsoc|#expr:trunc(3*2^63+2^12)}}

Conversion to a number of integer type is useful because the result, and subsequent results of type integer, are displayed exactly, instead of being rounded to 14 digits.

x = trunc x if and only if x is of type integer or a float with a value that a number of type integer also can have.

==Mod==
[[w:en:Modulo operation|mod]] (modulo, PHP operator [http://www.php.net/manual/en/language.operators.arithmetic.php %]): if a is nonnegative, a mod b is the remainder by division after truncating both operands to an integer (more accurately: apply the trunc function), while (-a) mod b = - ( a mod b) and a mod (-b) = a mod b.<ref>mod with non-integer arguments is different from all programming languages, see [[bugzilla:6068]] (marked as won't fix).</ref>

{{evaldemo|#expr: 30 mod 7}}<br />{{evaldemo|#expr: -8 mod -3}}<br />{{evaldemo|#expr: -8 mod +3}}<br />{{evaldemo|#expr: 8 mod 2.7}} (elsewhere 2.6)<br />{{evaldemo|#expr: 8 mod 3.2}} (elsewhere 1.6)<br />{{evaldemo|#expr: 8.9 mod 3}} (elsewhere 2.9)

To get a positive mod even for a negative number, use "(p mod q + q) mod q" instead of "p mod q".

Rounding to a float before applying mod can be avoided by constructing a type-integer form of the number:
*{{xpsoc|#expr:(1e16+1)mod3}}
*{{xpsoc|#expr:(trunc1e16+trunc1)mod3}}

For large integers there are errors (see below), use [[Template:Modint]] instead.

===Details===
p mod 0 gives an error message (produced by MediaWiki, not php operator %):
*{{xpsoc|#expr:-27mod0}}
Otherwise php operator % is applied. This involves applying the trunc function to the arguments first. The result of p % 0 is the empty string; this applies for 0 < |q| < 1 and q >= 2^64:
*{{xpsoocc|formatnum:|#expr:-123 mod .9}}
*{{xpsoocc|formatnum:|#expr:-123 mod -.9}}
*{{xpsoocc|formatnum:|#expr:-123 mod (2^64)}}
*{{xpsoocc|formatnum:|#expr:-123 mod 1e20}}
*Compare:
**{{xpsoocc|formatnum:|#expr:-123 mod (2^64-2048)}}

[[Help:Mod, round, floor, ceil, trunc/mod demo 1]] illustrates that the rules after applying trunc, (-a) mod b = - ( a mod b) and a mod (-b) = a mod b,  apply even in the case of erratic results, so we can concentrate on p mod q for integers (integer type or float with integer value) p and q, with 0 <= p <= 2^63 and 0 < q <= 2^63.

===Errors===
p mod q with q = 2^n-1 and q = 2^n+1  for n >= 32 always give zero:
*{{for|
*|01|02|03|04|05|06|07|08|09
|10|11|12|13|14|15|16|17|18|19
|20|21|22|23|24|25|26|27|28|29
|30|31|32|33|34|35|36|37|38|39
|40|41|42|43|44|45|46|47|48|49
|50|51|52|53|54|55|56|57|58|59
|60|61|62|call=xpsoc|pc1n=1|pc1v=numf|pv=2|prefix=123mod((trunc2)^trunc|postfix=-trunc1)}}
*{{xpsoc|numf|123mod(trunc(2^63-1024)+trunc1023)}}

*{{for|
*|00|01|02|03|04|05|06|07|08|09
|10|11|12|13|14|15|16|17|18|19
|20|21|22|23|24|25|26|27|28|29
|30|31|32|33|34|35|36|37|38|39
|40|41|42|43|44|45|46|47|48|49
|50|51|52|53|54|55|56|57|58|59
|60|61|62|call=xpsoc|pc1n=1|pc1v=numf|pv=2|prefix=123mod((trunc2)^trunc|postfix=+trunc1)}}

For the important special case that q is a power of 10 no errors have been observed:
*{{for|
*|00|01|02|03|04|05|06|07|08|09
|10|11|12|13|14|15|16|17|18
|call=xpsoc|pc1n=1|pc1v=numf|pv=2|prefix=123mod((trunc10)^trunc|postfix=)}}

==Round==
See [[Help:Round]].

==Floor and ceil==
When the functions floor and ceil are applied to an integer-type number, this number is first rounded to float (in the general way, not specifically downward for floor or upward for ceil) before applying the mathematical function:

*{{xpsoc|numf|floor (trunc1e17+trunc1)}}
*{{xpsoc|numf|ceil (trunc1e17+trunc1)}}

Use [[Template:Floor]] and [[Template:Ceil]] to get in such a case the integer-type expression for the exact result:

*{{xpsopocc|numf|floor|trunc1e17+trunc1}}
*{{xpsopocc|numf|ceil|trunc1e17+trunc1}}

To check whether the internal result of an expression x is mathematically an integer one can simply test "(x) = floor (x)", or similarly with ceil (not with trunc because for large floats we would get false negatives, and not with round0, because for odd numbers between 2^52 and 2^53 we would get false negatives).

==Safety margins==
To round an integer x down to a multiple of 7 we can do one of the following:
*7*((x-3)/7 round 0) 
*7*floor((x+.5)/7)
*7*ceil((x-6.5)/7)
and to round x to the nearest multiple of 7 we can do one of the following:
*7*((x/7) round 0) 
*7*floor((x+3.5)/7)
*7*ceil((x-3.5)/7)
In these cases the position with respect to 0 (the sign of x) is not important, even for round, because the value to be rounded is never halfway between integers. All methods provide the same generous safety margin of .5 in x, or 1/14 after division by 7, for rounding errors.

Note that for arbitrary real x the difference between the two problems corresponds to a shift in x of 3.5, while here, with integers x, the shift is 3.

==Precedence==
*{{evaldemo|#expr:1.234 + 1.234 round 1 + 1 }}

(first additions, then round)

*{{evaldemo|#expr:3 * 4 mod 10 * 10 }}

(mod and multiplication have equal precedence, evaluation from left to right)

;When using spaces where there is precedence, the layout of the expression may be confusing&#58;
:{{evaldemo|#expr:23+45 mod 10}}
;Instead one can write&#58;
:{{evaldemo|#expr:23 + 45 mod10}}
;or use parentheses&#58;
:{{evaldemo|#expr:23 + (45 mod 10)}}

==See also==
*{{tim|mod}}
*[[m:Category:Mathematical templates]]
*[[w:en:Category:Mathematical function templates]]
{{refs}}
{{h:f|enname=Modulo and round}}