USING: help.markup help.syntax kernel math math.order sequences quotations math.functions.private math.constants ; 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 e^ cis log } "Other logarithms:" { $subsections log1+ log10 logn } "Raising a number to a power:" { $subsections ^ e^ 10^ } "Logistics functions:" { $subsections sigmoid } "Finding the root of a number:" { $subsections nth-root } "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: 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: e^ { $values { "x" number } { "e^x" number } } { $description "Exponential function, raises " { $link e } " to the power of " { $snippet "x" } "." } ; HELP: frexp { $values { "x" number } { "y" float } { "exp" integer } } { $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)" } "." } ; HELP: ldexp { $values { "x" number } { "exp" number } { "y" number } } { $description "Multiply " { $snippet "x" } " by " { $snippet "2^exp" } "." } { $notes { $link ldexp } " is the inverse of " { $link frexp } "." } ; HELP: log { $values { "x" number } { "y" number } } { $description "Natural logarithm function. Outputs negative infinity if " { $snippet "x" } " is 0." } ; HELP: logn { $values { "x" number } { "n" number } { "y" number } } { $description "Finds the base " { $snippet "n" } " logarithm of " { $snippet "x" } "." } ; 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 "e^(arg*i)" } "." } ; { cis e^ } related-words HELP: polar> { $values { "abs" "a non-negative real number" } { "arg" real } { "z" number } } { $description "Converts an absolute value and argument (polar form) to a complex number." } ; HELP: [-1,1]? { $values { "x" number } { "?" 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 number." } { $see-also absq } ; HELP: absq { $values { "x" number } { "y" "a non-negative real number" } } { $description "Computes the squared absolute value of a number. For complex numbers this is marginally more efficient than " { $link abs } "." } ; HELP: ^ { $values { "x" number } { "y" number } { "x^y" 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: nth-root { $values { "n" integer } { "x" number } { "y" number } } { $description "Calculates the nth root of a number, such that " { $snippet "y^n=x" } "." } ; HELP: 10^ { $values { "x" number } { "10^x" number } } { $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." } ; HELP: divisor? { $values { "m" integer } { "n" integer } { "?" 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: ^mod { $values { "x" real } { "y" real } { "n" real } { "z" real } } { $description "Outputs the result of computing " { $snippet "x^y mod n" } "." } ; HELP: ~ { $values { "x" real } { "y" real } { "epsilon" real } { "?" 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" } ", rounding out at half." } { $notes "The result is not necessarily an integer." } { $examples { $example "USING: math.functions prettyprint ;" "4.5 round ." "5.0" } { $example "USING: math.functions prettyprint ;" "4.4 round ." "4.0" } } ; HELP: round-to-even { $values { "x" real } { "y" "a whole real number" } } { $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." } { $notes "The result is not necessarily an integer." } { $examples { $example "USING: math.functions prettyprint ;" "0.5 round-to-even ." "0.0" } { $example "USING: math.functions prettyprint ;" "1.5 round-to-even ." "2.0" } } ; HELP: round-to-odd { $values { "x" real } { "y" "a whole real number" } } { $description "Outputs the whole number closest to " { $snippet "x" } ", rounding out at half, breaking ties towards odd numbers." } { $notes "The result is not necessarily an integer." } { $examples { $example "USING: math.functions prettyprint ;" "0.5 round-to-odd ." "1.0" } { $example "USING: math.functions prettyprint ;" "1.5 round-to-odd ." "1.0" } } ; HELP: roots { $values { "x" number } { "t" integer } { "seq" sequence } } { $description "Outputs the " { $snippet "t" } " roots of a number " { $snippet "x" } "." } { $notes "The results are not necessarily real." } ; HELP: sigmoid { $values { "x" number } { "y" number } } { $description "Outputs the sigmoid, an S-shaped \"logistic\" function, from 0 to 1, of the number " { $snippet "x" } "." } ; HELP: signum { $values { "x" number } { "y" number } } { $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" } "." } ; HELP: copysign { $values { "x" number } { "y" number } { "x'" number } } { $description "Returns " { $snippet "x" } " with the sign of " { $snippet "y" } ", as a " { $link float } "." } ;