"Computes the remainder of dividing " { $snippet "x" } " by " { $snippet "y" } ", with the remainder being negative if " { $snippet "x" } " is negative."
- { $list
+ { $list
"Modulus of fixnums always yields a fixnum."
- "Modulus of bignums always yields a bignum."
+ "Modulus of bignums always yields a bignum."
{ "Modulus of rationals always yields a rational. In this case, the remainder is computed using the formula " { $snippet "x - (x mod y) * y" } "." }
"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
+ { $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."
+ "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." } ;
HELP: shift
{ $values { "x" integer } { "n" integer } { "y" integer } }
{ $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." }
{ $values { "n" "a positive integer" } { "2^n" "a positive integer" } }
-{ $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 "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 ^" } "." } ;
{ $values { "m" "a non-negative integer" } { "n" "an integer" } }
-{ $description "Outputs the smallest power of 2 greater than " { $snippet "m" } ". The output value is always at least 1." } ;
+{ $description "Outputs the smallest power of 2 greater than " { $snippet "m" } ". The output value is always at least 2." } ;
HELP: power-of-2?
{ $values { "n" integer } { "?" "a boolean" } }
{ $description "Tests if " { $snippet "n" } " is a power of 2." } ;
HELP: each-integer
-{ $values { "n" integer } { "quot" { $quotation "( i -- )" } } }
+{ $values { "n" integer } { "quot" { $quotation "( ... i -- ... )" } } }
{ $description "Applies the quotation to each integer from 0 up to " { $snippet "n" } ", excluding " { $snippet "n" } "." }
{ $notes "This word is used to implement " { $link each } "." } ;
HELP: all-integers?
-{ $values { "n" integer } { "quot" { $quotation "( i -- ? )" } } { "?" "a boolean" } }
+{ $values { "n" integer } { "quot" { $quotation "( ... i -- ... ? )" } } { "?" "a boolean" } }
{ $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 } "." }
{ $notes "This word is used to implement " { $link all? } "." } ;
HELP: find-integer
-{ $values { "n" integer } { "quot" { $quotation "( i -- ? )" } } { "i" "an integer or " { $link f } } }
-{ $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 } "." }
+{ $values { "n" integer } { "quot" { $quotation "( ... i -- ... ? )" } } { "i" "an integer or " { $link f } } }
+{ $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 } "." }
{ $notes "This word is used to implement " { $link find } "." } ;
HELP: find-last-integer
-{ $values { "n" integer } { "quot" { $quotation "( i -- ? )" } } { "i" "an integer or " { $link f } } }
+{ $values { "n" integer } { "quot" { $quotation "( ... i -- ... ? )" } } { "i" "an integer or " { $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 } "." } ;
-HELP: byte-array>bignum
-{ $values { "x" byte-array } { "y" bignum } }
-{ $description "Converts a byte-array, interpreted as little-endian, into a bignum integer. User code should call " { $link le> } " or " { $link be> } " instead." } ;
-
ARTICLE: "division-by-zero" "Division by zero"
-"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 " { $link mod } " is a float, the result will be a floating point infinity or not a number value."
+"Behavior of division operations when a denominator of zero is used depends on the data types in question, as well as the platform being used."
+$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 } "."
$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." ;
+"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:"
"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" } "."
-{ $subsection bitand }
-{ $subsection bitor }
-{ $subsection bitxor }
-{ $subsection bitnot }
-{ $subsection shift }
-{ $subsection 2/ }
-{ $subsection 2^ }
-{ $subsection bit? }
+{ $subsections
+ bitand
+ bitor
+ bitxor
+ bitnot
+ shift
+ 2/
+ 2^
+ bit?
+}
"Advanced topics:"
-{ $subsection "math.bitwise" }
-{ $subsection "math.bits" }
+{ $subsections
+ "math.bitwise"
+ "math.bits"
+}
{ $see-also "booleans" } ;
ARTICLE: "arithmetic" "Arithmetic"
"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."