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|
(.require
[library
[lux (.except)
[abstract
[monad (.only do)]
[\\specification
["$[0]" equivalence]]]
[data
[collection
["[0]" list (.use "[1]#[0]" functor)]]]
[math
["[0]" random (.only Random)]]
[test
["_" property (.only Test)]]]]
[\\library
["[0]" / (.only)
[//
["n" nat]
["f" frac]
["[0]" int]]]])
... This margin of error is necessary because floating-point arithmetic is not exact.
(def margin_of_error
+0.000000001)
(def dimension
(Random Frac)
(do [! random.monad]
[factor (|> random.nat (at ! each (|>> (n.% 1000) (n.max 1))))
measure (|> random.safe_frac (random.only (f.> +0.0)))]
(in (f.* (|> factor .int int.frac)
measure))))
(def .public random
(Random /.Complex)
(do random.monad
[real ..dimension
imaginary ..dimension]
(in (/.complex real imaginary))))
(def angle
(Random /.Complex)
(at random.monad each
(|>> (revised /.#real (f.% +1.0))
(revised /.#imaginary (f.% +1.0)))
..random))
(def construction
Test
(do random.monad
[real ..dimension
imaginary ..dimension]
(all _.and
(_.coverage [/.complex]
(and (let [r+i (/.complex real imaginary)]
(and (f.= real (the /.#real r+i))
(f.= imaginary (the /.#imaginary r+i))))
(let [r+i (/.complex real)]
(and (f.= real (the /.#real r+i))
(f.= +0.0 (the /.#imaginary r+i))))))
(_.coverage [/.approximately?]
(/.approximately? ..margin_of_error
(/.complex real imaginary)
(/.complex real imaginary)))
(_.coverage [/.not_a_number?]
(and (/.not_a_number? (/.complex f.not_a_number imaginary))
(/.not_a_number? (/.complex real f.not_a_number))))
)))
(def constant
Test
(do random.monad
[sample ..random
dimension ..dimension]
(all _.and
(_.coverage [/.zero]
(/.= /.zero (/.* /.zero sample)))
(_.coverage [/.+one]
(/.= sample (/.* /.+one sample)))
(_.coverage [/.-one]
(and (/.= /.zero
(/.+ sample
(/.* /.-one sample)))
(/.= sample (/.* /.-one (/.* /.-one sample)))))
(_.coverage [/.i]
(and (/.= (/.complex +0.0 dimension)
(/.* /.i (/.complex dimension)))
(/.= (/.* /.-one sample)
(/.* /.i (/.* /.i sample)))))
)))
(def absolute_value&argument
Test
(do random.monad
[real ..dimension
imaginary ..dimension]
(all _.and
(_.coverage [/.abs]
(let [normal!
(let [r+i (/.complex real imaginary)]
(and (f.>= (f.abs real) (/.abs r+i))
(f.>= (f.abs imaginary) (/.abs r+i))))
not_a_number!
(and (f.not_a_number? (/.abs (/.complex f.not_a_number imaginary)))
(f.not_a_number? (/.abs (/.complex real f.not_a_number))))
infinity!
(and (f.= f.positive_infinity (/.abs (/.complex f.positive_infinity imaginary)))
(f.= f.positive_infinity (/.abs (/.complex real f.positive_infinity)))
(f.= f.positive_infinity (/.abs (/.complex f.negative_infinity imaginary)))
(f.= f.positive_infinity (/.abs (/.complex real f.negative_infinity))))]
(and normal!
not_a_number!
infinity!)))
... https://en.wikipedia.org/wiki/Argument_(complex_analysis)#Identities
(_.coverage [/.argument]
(let [sample (/.complex real imaginary)]
(or (/.= /.zero sample)
(/.approximately? ..margin_of_error
sample
(/.*' (/.abs sample)
(/.exp (/.* /.i (/.complex (/.argument sample)))))))))
)))
(def number
Test
(do random.monad
[x ..random
y ..random
factor ..dimension]
(all _.and
(_.coverage [/.+]
(let [z (/.+ y x)]
(and (/.= z
(/.complex (f.+ (the /.#real y)
(the /.#real x))
(f.+ (the /.#imaginary y)
(the /.#imaginary x)))))))
(_.coverage [/.-]
(let [normal!
(let [z (/.- y x)]
(and (/.= z
(/.complex (f.- (the /.#real y)
(the /.#real x))
(f.- (the /.#imaginary y)
(the /.#imaginary x))))))
inverse!
(and (|> x (/.+ y) (/.- y) (/.approximately? ..margin_of_error x))
(|> x (/.- y) (/.+ y) (/.approximately? ..margin_of_error x)))]
(and normal!
inverse!)))
(_.coverage [/.* /./]
(|> x (/.* y) (/./ y) (/.approximately? ..margin_of_error x)))
(_.coverage [/.*' /./']
(|> x (/.*' factor) (/./' factor) (/.approximately? ..margin_of_error x)))
(_.coverage [/.%]
(let [rem (/.% y x)
quotient (|> x (/.- rem) (/./ y))
floored (|> quotient
(revised /.#real f.floor)
(revised /.#imaginary f.floor))]
(/.approximately? +0.000000000001
x
(|> quotient (/.* y) (/.+ rem)))))
)))
(def conjugate&reciprocal&signum&negation
Test
(do random.monad
[x ..random]
(all _.and
(_.coverage [/.conjugate]
(let [cx (/.conjugate x)]
(and (f.= (the /.#real x)
(the /.#real cx))
(f.= (f.opposite (the /.#imaginary x))
(the /.#imaginary cx)))))
(_.coverage [/.reciprocal]
(let [reciprocal!
(|> x (/.* (/.reciprocal x)) (/.approximately? ..margin_of_error /.+one))
own_inverse!
(|> x /.reciprocal /.reciprocal (/.approximately? ..margin_of_error x))]
(and reciprocal!
own_inverse!)))
(_.coverage [/.signum]
... Absolute value of signum is always root_2(2), 1 or 0.
(let [signum_abs (|> x /.signum /.abs)]
(or (f.= +0.0 signum_abs)
(f.= +1.0 signum_abs)
(f.= (f.pow +0.5 +2.0) signum_abs))))
(_.coverage [/.opposite]
(let [own_inverse!
(let [there (/.opposite x)
back_again (/.opposite there)]
(and (not (/.= there x))
(/.= back_again x)))
absolute!
(f.= (/.abs x)
(/.abs (/.opposite x)))]
(and own_inverse!
absolute!)))
)))
(def (trigonometric_symmetry forward backward angle)
(-> (-> /.Complex /.Complex) (-> /.Complex /.Complex) /.Complex Bit)
(let [normal (|> angle forward backward)]
(|> normal forward backward (/.approximately? ..margin_of_error normal))))
(def trigonometry
Test
(do [! random.monad]
[angle ..angle]
(all _.and
(_.coverage [/.sin /.asin]
(trigonometric_symmetry /.sin /.asin angle))
(_.coverage [/.cos /.acos]
(trigonometric_symmetry /.cos /.acos angle))
(_.coverage [/.tan /.atan]
(trigonometric_symmetry /.tan /.atan angle)))))
(def hyperbolic
Test
(do [! random.monad]
[angle ..angle]
(all _.and
(_.coverage [/.sinh]
(/.approximately? ..margin_of_error
(|> angle (/.* /.i) /.sin (/.* /.i) (/.* /.-one))
(/.sinh angle)))
(_.coverage [/.cosh]
(/.approximately? ..margin_of_error
(|> angle (/.* /.i) /.cos)
(/.cosh angle)))
(_.coverage [/.tanh]
(/.approximately? ..margin_of_error
(|> angle (/.* /.i) /.tan (/.* /.i) (/.* /.-one))
(/.tanh angle)))
)))
(def exponentiation&logarithm
Test
(do random.monad
[x ..random]
(all _.and
(_.coverage [/.pow /.root_2]
(|> x (/.pow (/.complex +2.0)) /.root_2 (/.approximately? ..margin_of_error x)))
(_.coverage [/.pow']
(|> x (/.pow' +2.0) (/.pow' +0.5) (/.approximately? ..margin_of_error x)))
(_.coverage [/.log /.exp]
(|> x /.log /.exp (/.approximately? ..margin_of_error x)))
)))
(def root
Test
(do [! random.monad]
[sample ..random
degree (|> random.nat (at ! each (|>> (n.max 1) (n.% 5))))]
(_.coverage [/.roots]
(|> sample
(/.roots degree)
(list#each (/.pow' (|> degree .int int.frac)))
(list.every? (/.approximately? ..margin_of_error sample))))))
(def .public test
Test
(<| (_.covering /._)
(_.for [/.Complex])
(all _.and
(_.for [/.= /.equivalence]
($equivalence.spec /.equivalence ..random))
..construction
..constant
..absolute_value&argument
..number
..conjugate&reciprocal&signum&negation
..trigonometry
..hyperbolic
..exponentiation&logarithm
..root
)))
|
