{ $description "Tests if " { $snippet "x" } " is unordered with respect to " { $snippet "y" } ". This can only occur if one or both values is a floating-point Not-a-Number value." } ;
{ $description "Tests if " { $snippet "x" } " is less than " { $snippet "y" } "." }
{ $notes "This word performs an unordered comparison on floating point numbers. On rational numbers it is equivalent to " { $link < } ". See " { $link "math.floats.compare" } " for an explanation." } ;
{ $description "Tests if " { $snippet "x" } " is less than or equal to " { $snippet "y" } "." }
{ $notes "This word performs an unordered comparison on floating point numbers. On rational numbers it is equivalent to " { $link <= } ". See " { $link "math.floats.compare" } " for an explanation." } ;
{ $description "Tests if " { $snippet "x" } " is greater than " { $snippet "y" } "." }
{ $notes "This word performs an unordered comparison on floating point numbers. On rational numbers it is equivalent to " { $link > } ". See " { $link "math.floats.compare" } " for an explanation." } ;
{ $description "Tests if " { $snippet "x" } " is greater than or equal to " { $snippet "y" } "." }
{ $notes "This word performs an unordered comparison on floating point numbers. On rational numbers it is equivalent to " { $link >= } ". See " { $link "math.floats.compare" } " for an explanation." } ;
"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."
{ $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." }
{ $description "Creates a complex number from real and imaginary components. If " { $snippet "z" } " is an integer zero, this will simply output " { $snippet "x" } "." } ;
HELP:>rect
{ $values { "z"number } { "x"real } { "y"real } }
{ $description "Extracts the real and imaginary components of a complex number." } ;
{ $description "Computes the positive greatest common divisor " { $snippet "d" } " of " { $snippet "x" } " and " { $snippet "y" } ", and another value " { $snippet "a" } " satisfying:" { $code "a*y = d mod x" } }
{ $notes "If " { $snippet "d" } " is 1, then " { $snippet "a" } " is the inverse of " { $snippet "y" } " modulo " { $snippet "x" } "." } ;
{ $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 ^" } "." } ;
{ $description "Makes an implicit check if the number is zero. A zero is dropped and " { $snippet "quot1" } " is called. Otherwise, if the number is not zero, " { $snippet "quot2" } " is called on it." }
{ $example
"USING: kernel math prettyprint sequences ;"
"3 [ \"zero\" ] [ sq ] if-zero ."
"9"
} ;
HELP:when-zero
{ $values
{ "n"number } { "quot""the first quotation of an " { $link if-zero } } }
{ $description "Compares two floating point numbers for bit equality." }
{ $notes "Unlike " { $link = } " or " { $link number= } ", this word will consider NaNs with equal payloads to be equal, and positive zero and negative zero to be not equal." }
{ $description "Tests if " { $snippet "x" } " is an IEEE special value (Not-a-Number or Infinity). 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 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 Quiet 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 Signaling 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 "Outputs the sign bit of " { $snippet "x" } ". For ordered non-zero values, this is equivalent to calling " { $snippet "0 <" } ". For zero values, this outputs the zero's sign bit." } ;
{ $description "If " { $snippet "x" } " is an IEEE Not-a-Number value, returns the payload encoded in the value. Returns " { $link f } " if " { $snippet "x" } " is not a " { $link float } "." } ;
{ $description "Returns the least representable " { $link float } " value greater than " { $snippet "m" } ", or in the case of " { $snippet "-0.0" } ", returns " { $snippet "+0.0" } "." } ;
{ $description "Returns the greatest representable " { $link float } " value less than " { $snippet "m" } ", or in the case of " { $snippet "+0.0" } ", returns " { $snippet "-0.0" } "." } ;
{ $description "Applies the quotation to each integer from 0 up to " { $snippet "n" } ", excluding " { $snippet "n" } ". Iteration 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 } "." }
{ $description "Applies the quotation to each integer from 0 up to " { $snippet "n" } ", excluding " { $snippet "n" } ". Iteration 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 } "." }
{ $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 } "." } ;
"Behavior of division operations when a denominator of zero is used depends on the data types in question, as well as the platform being used."
$nl
"Floating point division only throws an error if the appropriate traps are enabled in the floating point environment. If traps are disabled, a Not-a-number value or an infinity is output, depending on whether the numerator is zero or non-zero."
$nl
"Floating point traps are disabled by default and the " { $vocab-link "math.floats.env" } " vocabulary provides words to enable them. Floating point division is performed by " { $link / } ", " { $link /f } " or " { $link mod } " if at least one of the two inputs is a float. Floating point division is always performed by " { $link /f } "."
"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."
"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.";
"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:"
"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."
