HELP: number=
{ $values { "x" number } { "y" number } { "?" "a boolean" } }
{ $description "Tests if two numbers have the same numeric value." }
-{ $notes "This word differs from " { $link = } " in that it disregards differences in type when comparing numbers." }
+{ $notes "This word differs from " { $link = } " in that it disregards differences in type when comparing numbers."
+$nl
+"This word performs an unordered comparison on floating point numbers. See " { $link "math.floats.compare" } " for an explanation." }
{ $examples
{ $example "USING: math prettyprint ;" "3.0 3 number= ." "t" }
{ $example "USING: kernel math prettyprint ;" "3.0 3 = ." "f" }
HELP: <
{ $values { "x" real } { "y" real } { "?" boolean } }
-{ $description "Tests if " { $snippet "x" } " is less than " { $snippet "y" } "." } ;
+{ $description "Tests if " { $snippet "x" } " is less than " { $snippet "y" } "." }
+{ $notes "This word performs an ordered comparison on floating point numbers. See " { $link "math.floats.compare" } " for an explanation." } ;
HELP: <=
{ $values { "x" real } { "y" real } { "?" boolean } }
-{ $description "Tests if " { $snippet "x" } " is less than or equal to " { $snippet "y" } "." } ;
+{ $description "Tests if " { $snippet "x" } " is less than or equal to " { $snippet "y" } "." }
+{ $notes "This word performs an ordered comparison on floating point numbers. See " { $link "math.floats.compare" } " for an explanation." } ;
HELP: >
{ $values { "x" real } { "y" real } { "?" boolean } }
-{ $description "Tests if " { $snippet "x" } " is greater than " { $snippet "y" } "." } ;
+{ $description "Tests if " { $snippet "x" } " is greater than " { $snippet "y" } "." }
+{ $notes "This word performs an ordered comparison on floating point numbers. See " { $link "math.floats.compare" } " for an explanation." } ;
HELP: >=
{ $values { "x" real } { "y" real } { "?" boolean } }
-{ $description "Tests if " { $snippet "x" } " is greater than or equal to " { $snippet "y" } "." } ;
+{ $description "Tests if " { $snippet "x" } " is greater than or equal to " { $snippet "y" } "." }
+{ $notes "This word performs an ordered comparison on floating point numbers. See " { $link "math.floats.compare" } " for an explanation." } ;
+
+HELP: unordered?
+{ $values { "x" real } { "y" real } { "?" boolean } }
+{ $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." } ;
+HELP: u<
+{ $values { "x" real } { "y" real } { "?" boolean } }
+{ $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." } ;
+
+HELP: u<=
+{ $values { "x" real } { "y" real } { "?" boolean } }
+{ $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." } ;
+
+HELP: u>
+{ $values { "x" real } { "y" real } { "?" boolean } }
+{ $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." } ;
+
+HELP: u>=
+{ $values { "x" real } { "y" real } { "?" boolean } }
+{ $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." } ;
HELP: +
{ $values { "x" number } { "y" number } { "z" number } }
{ $values { "x" rational } { "y" rational } { "z" rational } }
{ $description
"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" } "." }
}
}
{ $values { "x" integer } { "y" integer } { "z" integer } { "w" integer } }
{ $description
"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."
}
}
{ $see-also "division-by-zero" } ;
{ $values { "x" integer } { "y" integer } { "z" integer } }
{ $description "Outputs a new integer where each bit is set if and only if the corresponding bit is set in both inputs." }
{ $examples
- { $example "USING: math prettyprint ;" "BIN: 101 BIN: 10 bitand .b" "0" }
- { $example "USING: math prettyprint ;" "BIN: 110 BIN: 10 bitand .b" "10" }
+ { $example "USING: math prettyprint ;" "0b101 0b10 bitand .b" "0" }
+ { $example "USING: math prettyprint ;" "0b110 0b10 bitand .b" "10" }
}
{ $notes "This word implements bitwise and, so applying it to booleans will throw an error. Boolean and is the " { $link and } " word." } ;
{ $values { "x" integer } { "y" integer } { "z" integer } }
{ $description "Outputs a new integer where each bit is set if and only if the corresponding bit is set in at least one of the inputs." }
{ $examples
- { $example "USING: math prettyprint ;" "BIN: 101 BIN: 10 bitor .b" "111" }
- { $example "USING: math prettyprint ;" "BIN: 110 BIN: 10 bitor .b" "110" }
+ { $example "USING: math prettyprint ;" "0b101 0b10 bitor .b" "111" }
+ { $example "USING: math prettyprint ;" "0b110 0b10 bitor .b" "110" }
}
{ $notes "This word implements bitwise inclusive or, so applying it to booleans will throw an error. Boolean inclusive or is the " { $link and } " word." } ;
{ $values { "x" integer } { "y" integer } { "z" integer } }
{ $description "Outputs a new integer where each bit is set if and only if the corresponding bit is set in exactly one of the inputs." }
{ $examples
- { $example "USING: math prettyprint ;" "BIN: 101 BIN: 10 bitxor .b" "111" }
- { $example "USING: math prettyprint ;" "BIN: 110 BIN: 10 bitxor .b" "100" }
+ { $example "USING: math prettyprint ;" "0b101 0b10 bitxor .b" "111" }
+ { $example "USING: math prettyprint ;" "0b110 0b10 bitxor .b" "100" }
}
{ $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." }
-{ $examples { $example "USING: math prettyprint ;" "BIN: 101 5 shift .b" "10100000" } { $example "USING: math prettyprint ;" "BIN: 11111 -2 shift .b" "111" } } ;
+{ $examples { $example "USING: math prettyprint ;" "0b101 5 shift .b" "10100000" } { $example "USING: math prettyprint ;" "0b11111 -2 shift .b" "111" } } ;
HELP: bitnot
{ $values { "x" integer } { "y" integer } }
{ $description "Computes the bitwise complement of the input; that is, each bit in the input number is flipped." }
{ $notes "This word implements bitwise not, so applying it to booleans will throw an error. Boolean not is the " { $link not } " word."
$nl
-"Due to the two's complement representation of signed integers, the following two lines are equivalent:" { $code "bitnot" "neg 1-" } } ;
+"Due to the two's complement representation of signed integers, the following two lines are equivalent:" { $code "bitnot" "neg 1 -" } } ;
HELP: bit?
{ $values { "x" integer } { "n" integer } { "?" "a boolean" } }
{ $description "Tests if the " { $snippet "n" } "th bit of " { $snippet "x" } " is set." }
-{ $examples { $example "USING: math prettyprint ;" "BIN: 101 2 bit? ." "t" } } ;
+{ $examples { $example "USING: math prettyprint ;" "0b101 2 bit? ." "t" } } ;
HELP: log2
{ $values { "x" "a positive integer" } { "n" integer } }
{ $description "Outputs the largest integer " { $snippet "n" } " such that " { $snippet "2^n" } " is less than or equal to " { $snippet "x" } "." }
{ $errors "Throws an error if " { $snippet "x" } " is zero or negative." } ;
-HELP: 1+
-{ $values { "x" number } { "y" number } }
-{ $description
- "Increments a number by 1. The following two lines are equivalent:"
- { $code "1+" "1 +" }
- "There is no difference in behavior or efficiency."
-} ;
-
-HELP: 1-
-{ $values { "x" number } { "y" number } }
-{ $description
- "Decrements a number by 1. The following two lines are equivalent:"
- { $code "1-" "1 -" }
- "There is no difference in behavior or efficiency."
-} ;
-
HELP: ?1+
{ $values { "x" { $maybe number } } { "y" number } }
{ $description "If the input is not " { $link f } ", adds one. Otherwise, outputs a " { $snippet "0" } "." } ;
HELP: rem
{ $values { "x" rational } { "y" rational } { "z" rational } }
{ $description
- "Computes the remainder of dividing " { $snippet "x" } " by " { $snippet "y" } ", with the remainder always positive."
- { $list
+ "Computes the remainder of dividing " { $snippet "x" } " by " { $snippet "y" } ", with the remainder always positive or zero."
+ { $list
"Given fixnums, always yields a fixnum."
"Given bignums, always yields a bignum."
- "Given rationals, always yields a rational."
+ "Given rationals, always yields a rational."
}
}
{ $see-also "division-by-zero" mod } ;
HELP: 2^
{ $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 ^" } "." } ;
HELP: zero?
{ $values { "x" number } { "?" "a boolean" } }
{ $description "Tests if the number is equal to zero." } ;
+HELP: if-zero
+{ $values { "n" number } { "quot1" quotation } { "quot2" quotation } }
+{ $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 "Makes an implicit check if the number is zero. A zero is dropped and the " { $snippet "quot" } " is called." }
+{ $examples "This word is equivalent to " { $link if-zero } " with an empty second quotation:"
+ { $example
+ "USING: math prettyprint ;"
+ "0 [ 4 ] [ ] if-zero ."
+ "4"
+ }
+ { $example
+ "USING: math prettyprint ;"
+ "0 [ 4 ] when-zero ."
+ "4"
+ }
+} ;
+
+HELP: unless-zero
+{ $values
+ { "n" number } { "quot" "the second quotation of an " { $link if-empty } } }
+{ $description "Makes an implicit check if the number is zero. A zero is dropped. Otherwise, the " { $snippet "quot" } " is called on the number." }
+{ $examples "This word is equivalent to " { $link if-zero } " with an empty first quotation:"
+ { $example
+ "USING: sequences math prettyprint ;"
+ "3 [ ] [ sq ] if-zero ."
+ "9"
+ }
+ { $example
+ "USING: sequences math prettyprint ;"
+ "3 [ sq ] unless-zero ."
+ "9"
+ }
+} ;
+
HELP: times
{ $values { "n" integer } { "quot" quotation } }
{ $description "Calls the quotation " { $snippet "n" } " times." }
{ "x" float } { "y" float }
{ "?" boolean }
}
-{ $description "Compares two floating point numbers for bit equality." } ;
+{ $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." }
+{ $examples
+ "Not-a-number equality:"
+ { $example
+ "USING: kernel math prettyprint ;"
+ "0.0 0.0 / dup number= ."
+ "f"
+ }
+ { $example
+ "USING: kernel math prettyprint ;"
+ "0.0 0.0 / dup fp-bitwise= ."
+ "t"
+ }
+ "Signed zero equality:"
+ { $example
+ "USING: math prettyprint ;"
+ "-0.0 0.0 fp-bitwise= ."
+ "f"
+ }
+ { $example
+ "USING: math prettyprint ;"
+ "-0.0 0.0 number= ."
+ "t"
+ }
+} ;
HELP: fp-special?
{ $values { "x" real } { "?" "a boolean" } }
{ $example "USING: io kernel math ;" "-1/0. [ fp-infinity? ] [ 0 < ] bi and [ \"negative infinity\" print ] when" "negative infinity" }
} ;
+HELP: fp-sign
+{ $values { "x" float } { "?" "a boolean" } }
+{ $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." } ;
+
HELP: fp-nan-payload
{ $values { "x" real } { "bits" integer } }
{ $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 } "." } ;
HELP: real-part
{ $values { "z" number } { "x" real } }
{ $description "Outputs the real part of a complex number. This acts as the identity on real numbers." }
-{ $examples { $example "USING: math prettyprint ; C{ 1 2 } real-part ." "1" } } ;
+{ $examples { $example "USING: math prettyprint ;" "C{ 1 2 } real-part ." "1" } } ;
HELP: imaginary-part
{ $values { "z" number } { "y" real } }
{ $description "Outputs the imaginary part of a complex number. This outputs zero for real numbers." }
{ $examples
- { $example "USING: math prettyprint ; C{ 1 2 } imaginary-part ." "2" }
- { $example "USING: math prettyprint ; 3 imaginary-part ." "0" }
+ { $example "USING: math prettyprint ;" "C{ 1 2 } imaginary-part ." "2" }
+ { $example "USING: math prettyprint ;" "3 imaginary-part ." "0" }
} ;
HELP: real
HELP: next-power-of-2
{ $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:"
-{ $example "USE: classes" "3 >fixnum 6 >bignum * class ." "bignum" }
+{ $example "USE: classes" "3 >fixnum 6 >bignum * class-of ." "bignum" }
{ $example "1/2 2.0 + ." "2.5" }
"The following usual operations are supported by all numbers."
-{ $subsection + }
-{ $subsection - }
-{ $subsection * }
-{ $subsection / }
+{ $subsections
+ +
+ -
+ *
+ /
+}
"Non-commutative operations take operands from the stack in the natural order; " { $snippet "6 2 /" } " divides 6 by 2."
-{ $subsection "division-by-zero" }
+{ $subsections "division-by-zero" }
"Real numbers (but not complex numbers) can be ordered:"
-{ $subsection < }
-{ $subsection <= }
-{ $subsection > }
-{ $subsection >= }
+{ $subsections
+ <
+ <=
+ >
+ >=
+}
"Numbers can be compared for equality using " { $link = } ", or a less precise test which disregards types:"
-{ $subsection number= } ;
+{ $subsections number= }
+{ $see-also "math.floats.compare" } ;
ARTICLE: "modular-arithmetic" "Modular arithmetic"
-{ $subsection mod }
-{ $subsection rem }
-{ $subsection /mod }
-{ $subsection /i }
+{ $subsections
+ mod
+ rem
+ /mod
+ /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" } "."
-{ $subsection bitand }
-{ $subsection bitor }
-{ $subsection bitxor }
-{ $subsection bitnot }
-{ $subsection shift }
-{ $subsection 2/ }
-{ $subsection 2^ }
-{ $subsection bit? }
-{ $subsection "math.bitwise" }
-{ $subsection "math.bits" }
+{ $subsections
+ bitand
+ bitor
+ bitxor
+ bitnot
+ shift
+ 2/
+ 2^
+ bit?
+}
+"Advanced topics:"
+{ $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."
-{ $subsection "number-protocol" }
-{ $subsection "modular-arithmetic" }
-{ $subsection "bitwise-arithmetic" }
+{ $subsections
+ "number-protocol"
+ "modular-arithmetic"
+ "bitwise-arithmetic"
+}
{ $see-also "integers" "rationals" "floats" "complex-numbers" } ;
ABOUT: "arithmetic"
projects / factor.git / blobdiff