319 lines
9.8 KiB
Factor
319 lines
9.8 KiB
Factor
USING: help.markup help.syntax kernel math math.order
|
|
sequences quotations math.functions.private ;
|
|
IN: math.functions
|
|
|
|
ARTICLE: "integer-functions" "Integer functions"
|
|
{ $subsections
|
|
align
|
|
gcd
|
|
log2
|
|
next-power-of-2
|
|
}
|
|
"Modular exponentiation:"
|
|
{ $subsections ^mod mod-inv }
|
|
"Tests:"
|
|
{ $subsections
|
|
power-of-2?
|
|
even?
|
|
odd?
|
|
divisor?
|
|
} ;
|
|
|
|
ARTICLE: "arithmetic-functions" "Arithmetic functions"
|
|
"Computing additive and multiplicative inverses:"
|
|
{ $subsections neg recip }
|
|
"Complex conjugation:"
|
|
{ $subsections conjugate }
|
|
"Tests:"
|
|
{ $subsections zero? between? }
|
|
"Control flow:"
|
|
{ $subsections
|
|
if-zero
|
|
when-zero
|
|
unless-zero
|
|
}
|
|
"Sign:"
|
|
{ $subsections sgn }
|
|
"Rounding:"
|
|
{ $subsections
|
|
ceiling
|
|
floor
|
|
truncate
|
|
round
|
|
}
|
|
"Inexact comparison:"
|
|
{ $subsections ~ }
|
|
"Numbers implement the " { $link "math.order" } ", therefore operations such as " { $link min } " and " { $link max } " can be used with numbers." ;
|
|
|
|
ARTICLE: "power-functions" "Powers and logarithms"
|
|
"Squares:"
|
|
{ $subsections sq sqrt }
|
|
"Exponential and natural logarithm:"
|
|
{ $subsections
|
|
exp
|
|
cis
|
|
log
|
|
log1+
|
|
log10
|
|
}
|
|
"Raising a number to a power:"
|
|
{ $subsections ^ 10^ }
|
|
"Converting between rectangular and polar form:"
|
|
{ $subsections
|
|
abs
|
|
absq
|
|
arg
|
|
>polar
|
|
polar>
|
|
} ;
|
|
|
|
ARTICLE: "trig-hyp-functions" "Trigonometric and hyperbolic functions"
|
|
"Trigonometric functions:"
|
|
{ $subsections cos sin tan }
|
|
"Reciprocals:"
|
|
{ $subsections sec cosec cot }
|
|
"Inverses:"
|
|
{ $subsections acos asin atan }
|
|
"Inverse reciprocals:"
|
|
{ $subsections asec acosec acot }
|
|
"Hyperbolic functions:"
|
|
{ $subsections cosh sinh tanh }
|
|
"Reciprocals:"
|
|
{ $subsections sech cosech coth }
|
|
"Inverses:"
|
|
{ $subsections acosh asinh atanh }
|
|
"Inverse reciprocals:"
|
|
{ $subsections asech acosech acoth } ;
|
|
|
|
ARTICLE: "math-functions" "Mathematical functions"
|
|
{ $subsections
|
|
"integer-functions"
|
|
"arithmetic-functions"
|
|
"power-functions"
|
|
"trig-hyp-functions"
|
|
} ;
|
|
|
|
ABOUT: "math-functions"
|
|
|
|
HELP: rect>
|
|
{ $values { "x" real } { "y" real } { "z" number } }
|
|
{ $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." } ;
|
|
|
|
HELP: align
|
|
{ $values { "m" integer } { "w" "a power of 2" } { "n" "an integer multiple of " { $snippet "w" } } }
|
|
{ $description "Outputs the least multiple of " { $snippet "w" } " greater than " { $snippet "m" } "." }
|
|
{ $notes "This word will give an incorrect result if " { $snippet "w" } " is not a power of 2." } ;
|
|
|
|
HELP: exp
|
|
{ $values { "x" number } { "y" number } }
|
|
{ $description "Exponential function, " { $snippet "y=e^x" } "." } ;
|
|
|
|
HELP: log
|
|
{ $values { "x" number } { "y" number } }
|
|
{ $description "Natural logarithm function. Outputs negative infinity if " { $snippet "x" } " is 0." } ;
|
|
|
|
HELP: log1+
|
|
{ $values { "x" number } { "y" number } }
|
|
{ $description "Takes the natural logarithm of " { $snippet "1 + x" } ". Outputs negative infinity if " { $snippet "1 + x" } " is zero. This word may be more accurate than " { $snippet "1 + log" } " for very small values of " { $snippet "x" } "." } ;
|
|
|
|
HELP: log10
|
|
{ $values { "x" number } { "y" number } }
|
|
{ $description "Logarithm function base 10. Outputs negative infinity if " { $snippet "x" } " is 0." } ;
|
|
|
|
HELP: sqrt
|
|
{ $values { "x" number } { "y" number } }
|
|
{ $description "Square root function." } ;
|
|
|
|
HELP: cosh
|
|
$values-x/y
|
|
{ $description "Hyperbolic cosine." } ;
|
|
|
|
HELP: sech
|
|
$values-x/y
|
|
{ $description "Hyperbolic secant." } ;
|
|
|
|
HELP: sinh
|
|
$values-x/y
|
|
{ $description "Hyperbolic sine." } ;
|
|
|
|
HELP: cosech
|
|
$values-x/y
|
|
{ $description "Hyperbolic cosecant." } ;
|
|
|
|
HELP: tanh
|
|
$values-x/y
|
|
{ $description "Hyperbolic tangent." } ;
|
|
|
|
HELP: coth
|
|
$values-x/y
|
|
{ $description "Hyperbolic cotangent." } ;
|
|
|
|
HELP: cos
|
|
$values-x/y
|
|
{ $description "Trigonometric cosine." } ;
|
|
|
|
HELP: sec
|
|
$values-x/y
|
|
{ $description "Trigonometric secant." } ;
|
|
|
|
HELP: sin
|
|
$values-x/y
|
|
{ $description "Trigonometric sine." } ;
|
|
|
|
HELP: cosec
|
|
$values-x/y
|
|
{ $description "Trigonometric cosecant." } ;
|
|
|
|
HELP: tan
|
|
$values-x/y
|
|
{ $description "Trigonometric tangent." } ;
|
|
|
|
HELP: cot
|
|
$values-x/y
|
|
{ $description "Trigonometric cotangent." } ;
|
|
|
|
HELP: acosh
|
|
$values-x/y
|
|
{ $description "Inverse hyperbolic cosine." } ;
|
|
|
|
HELP: asech
|
|
$values-x/y
|
|
{ $description "Inverse hyperbolic secant." } ;
|
|
|
|
HELP: asinh
|
|
$values-x/y
|
|
{ $description "Inverse hyperbolic sine." } ;
|
|
|
|
HELP: acosech
|
|
$values-x/y
|
|
{ $description "Inverse hyperbolic cosecant." } ;
|
|
|
|
HELP: atanh
|
|
$values-x/y
|
|
{ $description "Inverse hyperbolic tangent." } ;
|
|
|
|
HELP: acoth
|
|
$values-x/y
|
|
{ $description "Inverse hyperbolic cotangent." } ;
|
|
|
|
HELP: acos
|
|
$values-x/y
|
|
{ $description "Inverse trigonometric cosine." } ;
|
|
|
|
HELP: asec
|
|
$values-x/y
|
|
{ $description "Inverse trigonometric secant." } ;
|
|
|
|
HELP: asin
|
|
$values-x/y
|
|
{ $description "Inverse trigonometric sine." } ;
|
|
|
|
HELP: acosec
|
|
$values-x/y
|
|
{ $description "Inverse trigonometric cosecant." } ;
|
|
|
|
HELP: atan
|
|
$values-x/y
|
|
{ $description "Inverse trigonometric tangent." } ;
|
|
|
|
HELP: acot
|
|
$values-x/y
|
|
{ $description "Inverse trigonometric cotangent." } ;
|
|
|
|
HELP: conjugate
|
|
{ $values { "z" number } { "z*" number } }
|
|
{ $description "Computes the complex conjugate by flipping the sign of the imaginary part of " { $snippet "z" } "." } ;
|
|
|
|
HELP: arg
|
|
{ $values { "z" number } { "arg" "a number in the interval " { $snippet "(-pi,pi]" } } }
|
|
{ $description "Computes the complex argument." } ;
|
|
|
|
HELP: >polar
|
|
{ $values { "z" number } { "abs" "a non-negative real number" } { "arg" "a number in the interval " { $snippet "(-pi,pi]" } } }
|
|
{ $description "Converts a complex number into an absolute value and argument (polar form)." } ;
|
|
|
|
HELP: cis
|
|
{ $values { "arg" "a real number" } { "z" "a complex number on the unit circle" } }
|
|
{ $description "Computes a point on the unit circle using Euler's formula for " { $snippet "exp(arg*i)" } "." } ;
|
|
|
|
{ cis exp } related-words
|
|
|
|
HELP: polar>
|
|
{ $values { "z" number } { "abs" "a non-negative real number" } { "arg" real } }
|
|
{ $description "Converts an absolute value and argument (polar form) to a complex number." } ;
|
|
|
|
HELP: [-1,1]?
|
|
{ $values { "x" number } { "?" "a boolean" } }
|
|
{ $description "Tests if " { $snippet "x" } " is a real number between -1 and 1, inclusive." } ;
|
|
|
|
HELP: abs
|
|
{ $values { "x" number } { "y" "a non-negative real number" } }
|
|
{ $description "Computes the absolute value of a complex number." } ;
|
|
|
|
HELP: absq
|
|
{ $values { "x" number } { "y" "a non-negative real number" } }
|
|
{ $description "Computes the squared absolute value of a complex number. This is marginally more efficient than " { $link abs } "." } ;
|
|
|
|
HELP: ^
|
|
{ $values { "x" number } { "y" number } { "z" number } }
|
|
{ $description "Raises " { $snippet "x" } " to the power of " { $snippet "y" } ". If " { $snippet "y" } " is an integer the answer is computed exactly, otherwise a floating point approximation is used." }
|
|
{ $errors "Throws an error if " { $snippet "x" } " and " { $snippet "y" } " are both integer 0." } ;
|
|
|
|
HELP: 10^
|
|
{ $values { "x" number } { "y" number } }
|
|
{ $description "Raises " { $snippet "x" } " to the power of 10. If " { $snippet "x" } " is an integer the answer is computed exactly, otherwise a floating point approximation is used." } ;
|
|
|
|
HELP: gcd
|
|
{ $values { "x" integer } { "y" integer } { "a" integer } { "d" integer } }
|
|
{ $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" } "." } ;
|
|
|
|
HELP: divisor?
|
|
{ $values { "m" integer } { "n" integer } { "?" "a boolean" } }
|
|
{ $description "Tests if " { $snippet "n" } " is a divisor of " { $snippet "m" } ". This is the same thing as asking if " { $snippet "m" } " is divisible by " { $snippet "n" } "." }
|
|
{ $notes "Returns t for both negative and positive divisors, as well as for trivial and non-trivial divisors." } ;
|
|
|
|
HELP: mod-inv
|
|
{ $values { "x" integer } { "n" integer } { "y" integer } }
|
|
{ $description "Outputs an integer " { $snippet "y" } " such that " { $snippet "xy = 1 (mod n)" } "." }
|
|
{ $errors "Throws an error if " { $snippet "n" } " is not invertible modulo " { $snippet "n" } "." }
|
|
{ $examples
|
|
{ $example "USING: math.functions prettyprint ;" "173 1119 mod-inv ." "815" }
|
|
{ $example "USING: math prettyprint ;" "173 815 * 1119 mod ." "1" }
|
|
} ;
|
|
|
|
HELP: ~
|
|
{ $values { "x" real } { "y" real } { "epsilon" real } { "?" "a boolean" } }
|
|
{ $description "Tests if " { $snippet "x" } " and " { $snippet "y" } " are approximately equal to each other. There are three possible comparison tests, chosen based on the sign of " { $snippet "epsilon" } ":"
|
|
{ $list
|
|
{ { $snippet "epsilon" } " is zero: exact comparison." }
|
|
{ { $snippet "epsilon" } " is positive: absolute distance test." }
|
|
{ { $snippet "epsilon" } " is negative: relative distance test." }
|
|
}
|
|
} ;
|
|
|
|
|
|
HELP: truncate
|
|
{ $values { "x" real } { "y" "a whole real number" } }
|
|
{ $description "Outputs the number that results from subtracting the fractional component of " { $snippet "x" } "." }
|
|
{ $notes "The result is not necessarily an integer." } ;
|
|
|
|
HELP: floor
|
|
{ $values { "x" real } { "y" "a whole real number" } }
|
|
{ $description "Outputs the greatest whole number smaller than or equal to " { $snippet "x" } "." }
|
|
{ $notes "The result is not necessarily an integer." } ;
|
|
|
|
HELP: ceiling
|
|
{ $values { "x" real } { "y" "a whole real number" } }
|
|
{ $description "Outputs the least whole number greater than or equal to " { $snippet "x" } "." }
|
|
{ $notes "The result is not necessarily an integer." } ;
|
|
|
|
HELP: round
|
|
{ $values { "x" real } { "y" "a whole real number" } }
|
|
{ $description "Outputs the whole number closest to " { $snippet "x" } "." }
|
|
{ $notes "The result is not necessarily an integer." } ;
|