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|
(.using
[library
[lux (.except)
[abstract
[equivalence (.only Equivalence)]]
[control
["[0]" maybe]
["<>" parser (.only)
["<[0]>" code]]]
[data
[collection
["[0]" list (.open: "[1]#[0]" functor)]]]
[macro
[syntax (.only syntax)]]
[math
[number
["f" frac]
["[0]" int]]]]])
(type: .public Complex
(Record
[#real Frac
#imaginary Frac]))
(def: .public complex
(syntax (_ [real <code>.any
?imaginary (<>.maybe <code>.any)])
(in (list (` [..#real (~ real)
..#imaginary (~ (maybe.else (' +0.0) ?imaginary))])))))
(def: .public i
Complex
(..complex +0.0 +1.0))
(def: .public +one
Complex
(..complex +1.0 +0.0))
(def: .public -one
Complex
(..complex -1.0 +0.0))
(def: .public zero
Complex
(..complex +0.0 +0.0))
(def: .public (not_a_number? complex)
(-> Complex Bit)
(or (f.not_a_number? (the #real complex))
(f.not_a_number? (the #imaginary complex))))
(def: .public (= param input)
(-> Complex Complex Bit)
(and (f.= (the #real param)
(the #real input))
(f.= (the #imaginary param)
(the #imaginary input))))
(with_template [<name> <op>]
[(def: .public (<name> param input)
(-> Complex Complex Complex)
[#real (<op> (the #real param)
(the #real input))
#imaginary (<op> (the #imaginary param)
(the #imaginary input))])]
[+ f.+]
[- f.-]
)
(def: .public equivalence
(Equivalence Complex)
(implementation
(def: = ..=)))
(with_template [<name> <transform>]
[(def: .public <name>
(-> Complex Complex)
(|>> (revised #real <transform>)
(revised #imaginary <transform>)))]
[opposite f.opposite]
[signum f.signum]
)
(def: .public conjugate
(-> Complex Complex)
(revised #imaginary f.opposite))
(def: .public (*' param input)
(-> Frac Complex Complex)
[#real (f.* param
(the #real input))
#imaginary (f.* param
(the #imaginary input))])
(def: .public (* param input)
(-> Complex Complex Complex)
[#real (f.- (f.* (the #imaginary param)
(the #imaginary input))
(f.* (the #real param)
(the #real input)))
#imaginary (f.+ (f.* (the #real param)
(the #imaginary input))
(f.* (the #imaginary param)
(the #real input)))])
(def: .public (/ param input)
(-> Complex Complex Complex)
(let [(open "[0]") param]
(if (f.< (f.abs #imaginary)
(f.abs #real))
(let [quot (f./ #imaginary #real)
denom (|> #real (f.* quot) (f.+ #imaginary))]
[..#real (|> (the ..#real input) (f.* quot) (f.+ (the ..#imaginary input)) (f./ denom))
..#imaginary (|> (the ..#imaginary input) (f.* quot) (f.- (the ..#real input)) (f./ denom))])
(let [quot (f./ #real #imaginary)
denom (|> #imaginary (f.* quot) (f.+ #real))]
[..#real (|> (the ..#imaginary input) (f.* quot) (f.+ (the ..#real input)) (f./ denom))
..#imaginary (|> (the ..#imaginary input) (f.- (f.* quot (the ..#real input))) (f./ denom))]))))
(def: .public (/' param subject)
(-> Frac Complex Complex)
(let [(open "[0]") subject]
[..#real (f./ param #real)
..#imaginary (f./ param #imaginary)]))
(def: .public (% param input)
(-> Complex Complex Complex)
(let [scaled (/ param input)
quotient (|> scaled
(revised #real f.floor)
(revised #imaginary f.floor))]
(- (* quotient param)
input)))
(def: .public (cos subject)
(-> Complex Complex)
(let [(open "[0]") subject]
[..#real (f.* (f.cosh #imaginary)
(f.cos #real))
..#imaginary (f.opposite (f.* (f.sinh #imaginary)
(f.sin #real)))]))
(def: .public (cosh subject)
(-> Complex Complex)
(let [(open "[0]") subject]
[..#real (f.* (f.cos #imaginary)
(f.cosh #real))
..#imaginary (f.* (f.sin #imaginary)
(f.sinh #real))]))
(def: .public (sin subject)
(-> Complex Complex)
(let [(open "[0]") subject]
[..#real (f.* (f.cosh #imaginary)
(f.sin #real))
..#imaginary (f.* (f.sinh #imaginary)
(f.cos #real))]))
(def: .public (sinh subject)
(-> Complex Complex)
(let [(open "[0]") subject]
[..#real (f.* (f.cos #imaginary)
(f.sinh #real))
..#imaginary (f.* (f.sin #imaginary)
(f.cosh #real))]))
(def: .public (tan subject)
(-> Complex Complex)
(let [(open "[0]") subject
r2 (f.* +2.0 #real)
i2 (f.* +2.0 #imaginary)
d (f.+ (f.cos r2) (f.cosh i2))]
[..#real (f./ d (f.sin r2))
..#imaginary (f./ d (f.sinh i2))]))
(def: .public (tanh subject)
(-> Complex Complex)
(let [(open "[0]") subject
r2 (f.* +2.0 #real)
i2 (f.* +2.0 #imaginary)
d (f.+ (f.cosh r2) (f.cos i2))]
[..#real (f./ d (f.sinh r2))
..#imaginary (f./ d (f.sin i2))]))
(def: .public (abs subject)
(-> Complex Frac)
(let [(open "[0]") subject]
(if (f.< (f.abs #imaginary)
(f.abs #real))
(if (f.= +0.0 #imaginary)
(f.abs #real)
(let [q (f./ #imaginary #real)]
(f.* (f.pow +0.5 (f.+ +1.0 (f.* q q)))
(f.abs #imaginary))))
(if (f.= +0.0 #real)
(f.abs #imaginary)
(let [q (f./ #real #imaginary)]
(f.* (f.pow +0.5 (f.+ +1.0 (f.* q q)))
(f.abs #real)))))))
(def: .public (exp subject)
(-> Complex Complex)
(let [(open "[0]") subject
r_exp (f.exp #real)]
[..#real (f.* r_exp (f.cos #imaginary))
..#imaginary (f.* r_exp (f.sin #imaginary))]))
(def: .public (log subject)
(-> Complex Complex)
(let [(open "[0]") subject]
[..#real (|> subject ..abs f.log)
..#imaginary (f.atan_2 #real #imaginary)]))
(with_template [<name> <type> <op>]
[(def: .public (<name> param input)
(-> <type> Complex Complex)
(|> input log (<op> param) exp))]
[pow Complex ..*]
[pow' Frac ..*']
)
(def: (with_sign sign magnitude)
(-> Frac Frac Frac)
(f.* (f.signum sign) magnitude))
(def: .public (root_2 input)
(-> Complex Complex)
(let [(open "[0]") input
t (|> input ..abs (f.+ (f.abs #real)) (f./ +2.0) (f.pow +0.5))]
(if (f.< +0.0 #real)
[..#real (f./ (f.* +2.0 t)
(f.abs #imaginary))
..#imaginary (f.* t (..with_sign #imaginary +1.0))]
[..#real t
..#imaginary (f./ (f.* +2.0 t)
#imaginary)])))
(def: (root_2-1z input)
(-> Complex Complex)
(|> (complex +1.0) (- (* input input)) ..root_2))
(def: .public (reciprocal (open "[0]"))
(-> Complex Complex)
(if (f.< (f.abs #imaginary)
(f.abs #real))
(let [q (f./ #imaginary #real)
scale (f./ (|> #real (f.* q) (f.+ #imaginary))
+1.0)]
[..#real (f.* q scale)
..#imaginary (f.opposite scale)])
(let [q (f./ #real #imaginary)
scale (f./ (|> #imaginary (f.* q) (f.+ #real))
+1.0)]
[..#real scale
..#imaginary (|> scale f.opposite (f.* q))])))
(def: .public (acos input)
(-> Complex Complex)
(|> input
(..+ (|> input ..root_2-1z (..* ..i)))
..log
(..* (..opposite ..i))))
(def: .public (asin input)
(-> Complex Complex)
(|> input
..root_2-1z
(..+ (..* ..i input))
..log
(..* (..opposite ..i))))
(def: .public (atan input)
(-> Complex Complex)
(|> input
(..+ ..i)
(../ (..- input ..i))
..log
(..* (../ (..complex +2.0) ..i))))
(def: .public (argument (open "[0]"))
(-> Complex Frac)
(f.atan_2 #real #imaginary))
(def: .public (roots nth input)
(-> Nat Complex (List Complex))
(case nth
0 (list)
_ (let [r_nth (|> nth .int int.frac)
nth_root_of_abs (|> input ..abs (f.pow (f./ r_nth +1.0)))
nth_phi (|> input ..argument (f./ r_nth))
slice (|> f.pi (f.* +2.0) (f./ r_nth))]
(|> (list.indices nth)
(list#each (function (_ nth')
(let [inner (|> nth' .int int.frac
(f.* slice)
(f.+ nth_phi))
real (f.* nth_root_of_abs
(f.cos inner))
imaginary (f.* nth_root_of_abs
(f.sin inner))]
[..#real real
..#imaginary imaginary])))))))
(def: .public (approximately? margin_of_error standard value)
(-> Frac Complex Complex Bit)
(and (f.approximately? margin_of_error
(the ..#real standard)
(the ..#real value))
(f.approximately? margin_of_error
(the ..#imaginary standard)
(the ..#imaginary value))))
|
