366 lines
12 KiB
Factor
366 lines
12 KiB
Factor
USING: help.markup help.syntax kernel math math.order
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sequences quotations math.functions.private math.constants ;
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IN: math.functions
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ARTICLE: "integer-functions" "Integer functions"
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{ $subsections
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align
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gcd
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log2
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next-power-of-2
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}
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"Modular exponentiation:"
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{ $subsections ^mod mod-inv }
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"Tests:"
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{ $subsections
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power-of-2?
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even?
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odd?
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divisor?
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} ;
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ARTICLE: "arithmetic-functions" "Arithmetic functions"
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"Computing additive and multiplicative inverses:"
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{ $subsections neg recip }
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"Complex conjugation:"
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{ $subsections conjugate }
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"Tests:"
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{ $subsections zero? between? }
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"Control flow:"
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{ $subsections
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if-zero
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when-zero
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unless-zero
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}
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"Sign:"
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{ $subsections sgn }
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"Rounding:"
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{ $subsections
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ceiling
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floor
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truncate
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round
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}
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"Inexact comparison:"
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{ $subsections ~ }
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"Numbers implement the " { $link "math.order" } ", therefore operations such as " { $link min } " and " { $link max } " can be used with numbers." ;
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ARTICLE: "power-functions" "Powers and logarithms"
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"Squares:"
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{ $subsections sq sqrt }
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"Exponential and natural logarithm:"
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{ $subsections e^ cis log }
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"Other logarithms:"
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{ $subsections log1+ log10 logn }
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"Raising a number to a power:"
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{ $subsections ^ e^ 10^ }
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"Logistics functions:"
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{ $subsections sigmoid }
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"Finding the root of a number:"
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{ $subsections nth-root }
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"Converting between rectangular and polar form:"
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{ $subsections
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abs
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absq
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arg
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>polar
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polar>
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} ;
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ARTICLE: "trig-hyp-functions" "Trigonometric and hyperbolic functions"
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"Trigonometric functions:"
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{ $subsections cos sin tan }
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"Reciprocals:"
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{ $subsections sec cosec cot }
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"Inverses:"
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{ $subsections acos asin atan }
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"Inverse reciprocals:"
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{ $subsections asec acosec acot }
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"Hyperbolic functions:"
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{ $subsections cosh sinh tanh }
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"Reciprocals:"
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{ $subsections sech cosech coth }
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"Inverses:"
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{ $subsections acosh asinh atanh }
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"Inverse reciprocals:"
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{ $subsections asech acosech acoth } ;
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ARTICLE: "math-functions" "Mathematical functions"
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{ $subsections
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"integer-functions"
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"arithmetic-functions"
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"power-functions"
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"trig-hyp-functions"
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} ;
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ABOUT: "math-functions"
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HELP: align
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{ $values { "m" integer } { "w" "a power of 2" } { "n" "an integer multiple of " { $snippet "w" } } }
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{ $description "Outputs the least multiple of " { $snippet "w" } " greater than " { $snippet "m" } "." }
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{ $notes "This word will give an incorrect result if " { $snippet "w" } " is not a power of 2." } ;
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HELP: e^
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{ $values { "x" number } { "e^x" number } }
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{ $description "Exponential function, raises " { $link e } " to the power of " { $snippet "x" } "." } ;
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HELP: frexp
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{ $values { "x" number } { "y" float } { "exp" integer } }
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{ $description "Break the number " { $snippet "x" } " into a normalized fraction " { $snippet "y" } " and an integral power of 2 " { $snippet "e^" } "." $nl "The function returns a number " { $snippet "y" } " in the interval [1/2, 1) or 0, and a number " { $snippet "exp" } " such that " { $snippet "x = y*(2**exp)" } "." } ;
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HELP: ldexp
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{ $values { "x" number } { "exp" number } { "y" number } }
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{ $description "Multiply " { $snippet "x" } " by " { $snippet "2^exp" } "." }
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{ $notes { $link ldexp } " is the inverse of " { $link frexp } "." } ;
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HELP: log
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{ $values { "x" number } { "y" number } }
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{ $description "Natural logarithm function. Outputs negative infinity if " { $snippet "x" } " is 0." } ;
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HELP: logn
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{ $values { "x" number } { "n" number } { "y" number } }
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{ $description "Finds the base " { $snippet "n" } " logarithm of " { $snippet "x" } "." } ;
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HELP: log1+
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{ $values { "x" number } { "y" number } }
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{ $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" } "." } ;
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HELP: log10
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{ $values { "x" number } { "y" number } }
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{ $description "Logarithm function base 10. Outputs negative infinity if " { $snippet "x" } " is 0." } ;
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HELP: sqrt
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{ $values { "x" number } { "y" number } }
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{ $description "Square root function." } ;
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HELP: cosh
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$values-x/y
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{ $description "Hyperbolic cosine." } ;
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HELP: sech
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$values-x/y
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{ $description "Hyperbolic secant." } ;
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HELP: sinh
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$values-x/y
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{ $description "Hyperbolic sine." } ;
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HELP: cosech
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$values-x/y
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{ $description "Hyperbolic cosecant." } ;
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HELP: tanh
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$values-x/y
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{ $description "Hyperbolic tangent." } ;
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HELP: coth
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$values-x/y
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{ $description "Hyperbolic cotangent." } ;
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HELP: cos
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$values-x/y
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{ $description "Trigonometric cosine." } ;
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HELP: sec
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$values-x/y
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{ $description "Trigonometric secant." } ;
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HELP: sin
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$values-x/y
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{ $description "Trigonometric sine." } ;
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HELP: cosec
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$values-x/y
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{ $description "Trigonometric cosecant." } ;
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HELP: tan
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$values-x/y
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{ $description "Trigonometric tangent." } ;
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HELP: cot
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$values-x/y
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{ $description "Trigonometric cotangent." } ;
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HELP: acosh
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$values-x/y
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{ $description "Inverse hyperbolic cosine." } ;
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HELP: asech
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$values-x/y
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{ $description "Inverse hyperbolic secant." } ;
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HELP: asinh
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$values-x/y
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{ $description "Inverse hyperbolic sine." } ;
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HELP: acosech
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$values-x/y
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{ $description "Inverse hyperbolic cosecant." } ;
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HELP: atanh
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$values-x/y
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{ $description "Inverse hyperbolic tangent." } ;
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HELP: acoth
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$values-x/y
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{ $description "Inverse hyperbolic cotangent." } ;
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HELP: acos
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$values-x/y
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{ $description "Inverse trigonometric cosine." } ;
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HELP: asec
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$values-x/y
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{ $description "Inverse trigonometric secant." } ;
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HELP: asin
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$values-x/y
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{ $description "Inverse trigonometric sine." } ;
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HELP: acosec
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$values-x/y
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{ $description "Inverse trigonometric cosecant." } ;
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HELP: atan
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$values-x/y
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{ $description "Inverse trigonometric tangent." } ;
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HELP: acot
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$values-x/y
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{ $description "Inverse trigonometric cotangent." } ;
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HELP: conjugate
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{ $values { "z" number } { "z*" number } }
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{ $description "Computes the complex conjugate by flipping the sign of the imaginary part of " { $snippet "z" } "." } ;
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HELP: arg
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{ $values { "z" number } { "arg" "a number in the interval " { $snippet "(-pi,pi]" } } }
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{ $description "Computes the complex argument." } ;
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HELP: >polar
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{ $values { "z" number } { "abs" "a non-negative real number" } { "arg" "a number in the interval " { $snippet "(-pi,pi]" } } }
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{ $description "Converts a complex number into an absolute value and argument (polar form)." } ;
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HELP: cis
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{ $values { "arg" "a real number" } { "z" "a complex number on the unit circle" } }
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{ $description "Computes a point on the unit circle using Euler's formula for " { $snippet "e^(arg*i)" } "." } ;
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{ cis e^ } related-words
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HELP: polar>
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{ $values { "abs" "a non-negative real number" } { "arg" real } { "z" number } }
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{ $description "Converts an absolute value and argument (polar form) to a complex number." } ;
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HELP: [-1,1]?
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{ $values { "x" number } { "?" boolean } }
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{ $description "Tests if " { $snippet "x" } " is a real number between -1 and 1, inclusive." } ;
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HELP: abs
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{ $values { "x" number } { "y" "a non-negative real number" } }
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{ $description "Computes the absolute value of a number." }
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{ $see-also absq } ;
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HELP: absq
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{ $values { "x" number } { "y" "a non-negative real number" } }
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{ $description "Computes the squared absolute value of a number. For complex numbers this is marginally more efficient than " { $link abs } "." } ;
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HELP: ^
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{ $values { "x" number } { "y" number } { "x^y" number } }
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{ $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." }
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{ $errors "Throws an error if " { $snippet "x" } " and " { $snippet "y" } " are both integer 0." } ;
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HELP: nth-root
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{ $values { "n" integer } { "x" number } { "y" number } }
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{ $description "Calculates the nth root of a number, such that " { $snippet "y^n=x" } "." } ;
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HELP: 10^
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{ $values { "x" number } { "10^x" number } }
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{ $description "Raises 10 to the power of " { $snippet "x" } ". If " { $snippet "x" } " is an integer the answer is computed exactly, otherwise a floating point approximation is used." } ;
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HELP: divisor?
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{ $values { "m" integer } { "n" integer } { "?" boolean } }
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{ $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" } "." }
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{ $notes "Returns t for both negative and positive divisors, as well as for trivial and non-trivial divisors." } ;
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HELP: mod-inv
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{ $values { "x" integer } { "n" integer } { "y" integer } }
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{ $description "Outputs an integer " { $snippet "y" } " such that " { $snippet "xy = 1 (mod n)" } "." }
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{ $errors "Throws an error if " { $snippet "n" } " is not invertible modulo " { $snippet "n" } "." }
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{ $examples
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{ $example "USING: math.functions prettyprint ;" "173 1119 mod-inv ." "815" }
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{ $example "USING: math prettyprint ;" "173 815 * 1119 mod ." "1" }
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} ;
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HELP: ^mod
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{ $values { "x" real } { "y" real } { "n" real } { "z" real } }
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{ $description "Outputs the result of computing " { $snippet "x^y mod n" } "." } ;
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HELP: ~
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{ $values { "x" real } { "y" real } { "epsilon" real } { "?" boolean } }
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{ $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" } ":"
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{ $list
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{ { $snippet "epsilon" } " is zero: exact comparison." }
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{ { $snippet "epsilon" } " is positive: absolute distance test." }
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{ { $snippet "epsilon" } " is negative: relative distance test." }
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}
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} ;
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HELP: truncate
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{ $values { "x" real } { "y" "a whole real number" } }
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{ $description "Outputs the number that results from subtracting the fractional component of " { $snippet "x" } "." }
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{ $notes "The result is not necessarily an integer." } ;
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HELP: floor
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{ $values { "x" real } { "y" "a whole real number" } }
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{ $description "Outputs the greatest whole number smaller than or equal to " { $snippet "x" } "." }
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{ $notes "The result is not necessarily an integer." } ;
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HELP: ceiling
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{ $values { "x" real } { "y" "a whole real number" } }
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{ $description "Outputs the least whole number greater than or equal to " { $snippet "x" } "." }
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{ $notes "The result is not necessarily an integer." } ;
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HELP: round
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{ $values { "x" real } { "y" "a whole real number" } }
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{ $description "Outputs the whole number closest to " { $snippet "x" } ", rounding out at half." }
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{ $notes "The result is not necessarily an integer." }
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{ $examples
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{ $example "USING: math.functions prettyprint ;" "4.5 round ." "5.0" }
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{ $example "USING: math.functions prettyprint ;" "4.4 round ." "4.0" }
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} ;
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HELP: round-to-even
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{ $values { "x" real } { "y" "a whole real number" } }
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{ $description "Outputs the whole number closest to " { $snippet "x" } ", rounding out at half, breaking ties towards even numbers. This is also known as banker's rounding or unbiased rounding." }
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{ $notes "The result is not necessarily an integer." }
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{ $examples
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{ $example "USING: math.functions prettyprint ;" "0.5 round-to-even ." "0.0" }
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{ $example "USING: math.functions prettyprint ;" "1.5 round-to-even ." "2.0" }
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} ;
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HELP: round-to-odd
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{ $values { "x" real } { "y" "a whole real number" } }
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{ $description "Outputs the whole number closest to " { $snippet "x" } ", rounding out at half, breaking ties towards odd numbers." }
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{ $notes "The result is not necessarily an integer." }
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{ $examples
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{ $example "USING: math.functions prettyprint ;" "0.5 round-to-odd ." "1.0" }
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{ $example "USING: math.functions prettyprint ;" "1.5 round-to-odd ." "1.0" }
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} ;
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HELP: roots
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{ $values { "x" number } { "t" integer } { "seq" sequence } }
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{ $description "Outputs the " { $snippet "t" } " roots of a number " { $snippet "x" } "." }
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{ $notes "The results are not necessarily real." } ;
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HELP: sigmoid
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{ $values { "x" number } { "y" number } }
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{ $description "Outputs the sigmoid, an S-shaped \"logistic\" function, from 0 to 1, of the number " { $snippet "x" } "." } ;
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HELP: signum
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{ $values { "x" number } { "y" number } }
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{ $description "Calculates the signum value. For a real number, " { $snippet "x" } ", this is its sign (-1, 0, or 1). For a complex number, " { $snippet "x" } ", this is the point on the unit circle of the complex plane that is nearest to " { $snippet "x" } "." } ;
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HELP: copysign
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{ $values { "x" number } { "y" number } { "x'" number } }
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{ $description "Returns " { $snippet "x" } " with the sign of " { $snippet "y" } ", as a " { $link float } "." } ;
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