"Computes the remainder of dividing " { $snippet "x" } " by " { $snippet "y" } ", with the remainder being negative if " { $snippet "x" } " is negative."
"Computes the quotient " { $snippet "z" } " and remainder " { $snippet "w" } " of dividing " { $snippet "x" } " by " { $snippet "y" } ", with the remainder being negative if " { $snippet "x" } " is negative."
{ $list
"The quotient of two fixnums may overflow and yield a bignum; the remainder is always a fixnum"
"The quotient and remainder of two bignums is always a bignum."
{ $notes "This word implements bitwise inclusive or, so applying it to booleans will throw an error. Boolean inclusive or is the " { $link and } " word." } ;
{ $notes "This word implements bitwise exclusive or, so applying it to booleans will throw an error. Boolean exclusive or is the " { $link xor } " word." } ;
{ $description "Shifts " { $snippet "x" } " to the left by " { $snippet "n" } " bits if " { $snippet "n" } " is positive, or " { $snippet "-n" } " bits to the right if " { $snippet "n" } " is negative. A left shift of a fixnum may overflow, yielding a bignum. A right shift may result in bits “falling off” the right hand side and being discarded." }
{ $notes "This word is not equivalent to " { $snippet "2 /" } " or " { $snippet "2 /i" } "; the name is historic and originates from the Forth programming language." } ;
{ $description "Computes two to the power of " { $snippet "n" } ". This word will only give correct results if " { $snippet "n" } " is greater than zero; for the general case, use " { $snippet "2 swap ^" } "." } ;
HELP:zero?
{ $values { "x"number } { "?""a boolean" } }
{ $description "Tests if the number is equal to zero." } ;
HELP:times
{ $values { "n"integer } { "quot" quotation } }
{ $description "Calls the quotation " { $snippet "n" } " times." }
{ $description "Tests if " { $snippet "x" } " is an IEEE Not-a-Number value. While " { $snippet "x" } " can be any real number, this word will only ever yield true if " { $snippet "x" } " is a " { $link float } "." } ;
{ $description "Tests if " { $snippet "x" } " is an IEEE Infinity value. While " { $snippet "x" } " can be any real number, this word will only ever yield true if " { $snippet "x" } " is a " { $link float } "." }
{ $examples
{ $example "USING: math prettyprint ;""1/0. fp-infinity? .""t" }
{ $description "Applies the quotation to each integer from 0 up to " { $snippet "n" } ", excluding " { $snippet "n" } ". Iterationi stops when the quotation outputs " { $link f } " or the end is reached. If the quotation yields a false value for some integer, this word outputs " { $link f } ". Otherwise, this word outputs " { $link t } "." }
{ $notes "This word is used to implement " { $link all? } "." } ;
{ $description "Applies the quotation to each integer from 0 up to " { $snippet "n" } ", excluding " { $snippet "n" } ". Iterationi stops when the quotation outputs a true value or the end is reached. If the quotation yields a true value for some integer, this word outputs that integer. Otherwise, this word outputs " { $link f } "." }
{ $notes "This word is used to implement " { $link find } "." } ;
{ $description "Applies the quotation to each integer from " { $snippet "n" } " down to 0, inclusive. Iteration stops when the quotation outputs a true value or 0 is reached. If the quotation yields a true value for some integer, the word outputs that integer. Otherwise, the word outputs " { $link f } "." }
{ $notes "This word is used to implement " { $link find-last } "." } ;
{ $description "Converts a byte-array, interpreted as little-endian, into a bignum integer. User code should call " { $link >le } " or " { $link >be } " instead." } ;
"Floating point division never raises an error if the denominator is zero. This means that if at least one of the two inputs to " { $link / } ", " { $link /f } " or " { $link mod } " is a float, the result will be a floating point infinity or not a number value."
$nl
"The behavior of integer division is hardware specific. On x86 processors, " { $link /i } " and " { $link mod } " raise an error if both inputs are integers and the denominator is zero. On PowerPC, integer division by zero yields a result of zero."
$nl
"On the other hand, the " { $link / } " word, when given integer arguments, implements a much more expensive division algorithm which always yields an exact rational answer, and this word always tests for division by zero explicitly.";
ARTICLE: "number-protocol""Number protocol"
"Math operations obey certain numerical upgrade rules. If one of the inputs is a bignum and the other is a fixnum, the latter is first coerced to a bignum; if one of the inputs is a float, the other is coerced to a float."
$nl
"Two examples where you should note the types of the inputs and outputs:"
"The following usual operations are supported by all numbers."
{ $subsection + }
{ $subsection - }
{ $subsection * }
{ $subsection / }
"Non-commutative operations take operands from the stack in the natural order; " { $snippet "6 2 /" } " divides 6 by 2."
{ $subsection "division-by-zero" }
"Real numbers (but not complex numbers) can be ordered:"
{ $subsection < }
{ $subsection <= }
{ $subsection > }
{ $subsection >= }
"Numbers can be compared for equality using " { $link = } ", or a less precise test which disregards types:"
{ $subsection number= } ;
ARTICLE: "modular-arithmetic""Modular arithmetic"
{ $subsection mod }
{ $subsection rem }
{ $subsection /mod }
{ $subsection /i }
{ $see-also "integer-functions" } ;
ARTICLE: "bitwise-arithmetic""Bitwise arithmetic"
"There are two ways of looking at an integer -- as an abstract mathematical entity, or as a string of bits. The latter representation motivates " { $emphasis "bitwise operations" } "."
"Factor attempts to preserve natural mathematical semantics for numbers. Multiplying two large integers never results in overflow, and dividing two integers yields an exact ratio. Floating point numbers are also supported, along with complex numbers."
$nl
"Math words are in the " { $vocab-link "math" } " vocabulary. Implementation details are in the " { $vocab-link "math.private" } " vocabulary."
