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Diffstat (limited to 'std/Number.bruijn')
| -rw-r--r-- | std/Number.bruijn | 456 |
1 files changed, 4 insertions, 452 deletions
diff --git a/std/Number.bruijn b/std/Number.bruijn index dfed84c..e17d7d6 100644 --- a/std/Number.bruijn +++ b/std/Number.bruijn @@ -1,453 +1,5 @@ -# MIT License, Copyright (c) 2022 Marvin Borner -# This file defines the most basic mathematical operations -# → refer to std/Math for more advanced functions -# Heavily inspired by the works of T.Æ. Mogensen and Douglas W. Jones (see refs in README) +# MIT License, Copyright (c) 2023 Marvin Borner +# this is just a reference to the ternary implementation +# read the readme for the reasoning of using balanced ternary by default -:import std/Combinator . -:import std/Logic . -:import std/Pair . - -# negative trit indicating coeffecient of (-1) -t⁻ [[[2]]] ⧗ Trit - -# positive trit indicating coeffecient of (+1) -t⁺ [[[1]]] ⧗ Trit - -# zero trit indicating coeffecient of 0 -t⁰ [[[0]]] ⧗ Trit - -# returns true if a trit is negative -t⁻? [0 true false false] ⧗ Trit → Boolean - -:test (t⁻? t⁻) (true) -:test (t⁻? t⁺) (false) -:test (t⁻? t⁰) (false) - -# returns true if a trit is positive -t⁺? [0 false true false] ⧗ Trit → Boolean - -:test (t⁺? t⁻) (false) -:test (t⁺? t⁺) (true) -:test (t⁺? t⁰) (false) - -# returns true if a trit is zero -t⁰? [0 false false true] ⧗ Trit → Boolean - -:test (t⁰? t⁻) (false) -:test (t⁰? t⁺) (false) -:test (t⁰? t⁰) (true) - -# shifts a negative trit into a balanced ternary number -↑⁻‣ [[[[[2 (4 3 2 1 0)]]]]] ⧗ Number → Number - -:test (↑⁻(+0)) ((-1)) -:test (↑⁻(-1)) ((-4)) -:test (↑⁻(+42)) ((+125)) - -# shifts a positive trit into a balanced ternary number -↑⁺‣ [[[[[1 (4 3 2 1 0)]]]]] ⧗ Number → Number - -:test (↑⁺(+0)) ((+1)) -:test (↑⁺(-1)) ((-2)) -:test (↑⁺(+42)) ((+127)) - -# shifts a zero trit into a balanced ternary number -↑⁰‣ [[[[[0 (4 3 2 1 0)]]]]] ⧗ Number → Number - -:test (↑⁰(+0)) ([[[[0 3]]]]) -:test (↑⁰(+1)) ((+3)) -:test (↑⁰(+42)) ((+126)) - -# shifts a specified trit into a balanced ternary number -up [[[[[[5 2 1 0 (4 3 2 1 0)]]]]]] ⧗ Trit → Number → Number - -:test (up t⁻ (+42)) (↑⁻(+42)) -:test (up t⁺ (+42)) (↑⁺(+42)) -:test (up t⁰ (+42)) (↑⁰(+42)) - -# infinity -# WARNING: using this mostly results in undefined behavior! (TODO?) -infty z [[[[[1 (4 1)]]]]] ⧗ Number - -# negates a balanced ternary number -negate [[[[[4 3 1 2 0]]]]] ⧗ Number → Number - --‣ negate - -:test (-(+0)) ((+0)) -:test (-(-1)) ((+1)) -:test (-(+42)) ((-42)) - -# converts a balanced ternary number to a list of trits -list! [0 z a⁻ a⁺ a⁰] ⧗ Number → List - z [[0]] - a⁻ [t⁻ : 0] - a⁺ [t⁺ : 0] - a⁰ [t⁰ : 0] - -# TODO: Tests! - -# strips leading 0s from a balanced ternary number -strip [^(0 z a⁻ a⁺ a⁰)] ⧗ Number → Number - z (+0) : true - a⁻ [0 [[↑⁻1 : false]]] - a⁺ [0 [[↑⁺1 : false]]] - a⁰ [0 [[(0 (+0) ↑⁰1) : 0]]] - -%‣ strip - -:test (%[[[[0 3]]]]) ((+0)) -:test (%[[[[2 (0 (0 (0 (0 3))))]]]]) ((-1)) -:test (%(+42)) ((+42)) - -# returns true if balanced ternary number is zero -zero? [0 true [false] [false] i] ⧗ Number → Boolean - -=?‣ zero? - -:test (=?(+0)) (true) -:test (=?(-1)) (false) -:test (=?(+1)) (false) -:test (=?(+42)) (false) - -# returns true if balanced ternary number is not -not-zero? [0 false [true] [true] i] ⧗ Number → Boolean - -≠?‣ not-zero? - -:test (≠?(+0)) (false) -:test (≠?(-1)) (true) -:test (≠?(+1)) (true) -:test (≠?(+42)) (true) - -# extracts least significant trit from a balanced ternary number -lst [0 t⁰ [t⁻] [t⁺] [t⁰]] ⧗ Number → Trit - -:test (lst (-1)) (t⁻) -:test (lst (+0)) (t⁰) -:test (lst (+1)) (t⁺) -:test (lst (+42)) (t⁰) - -# extracts most significant trit from a balanced ternary number -# <~>/<>? are hardcoded because list import would be recursive (TODO?) -# while this looks incredibly inefficient it's actually fairly fast because laziness -# TODO: find way of removing requirement of stripping first -# (or better solution in general) -mst [=?0 t⁰ ^(<~>(list! %0))] ⧗ Number → Trit - <~>‣ z [[[[<>?0 1 (3 2 (2 1 ^0) ~0)]]]] f false - <>?‣ [0 [[[false]]] true] - -:test (mst (-1)) (t⁻) -:test (mst (+0)) (t⁰) -:test (mst (+1)) (t⁺) -:test (mst (+42)) (t⁺) - -# returns true if balanced ternary number is negative -negative? [t⁻? (mst 0)] ⧗ Number → Boolean - -<?‣ negative? - -:test (<?(+0)) (false) -:test (<?(-1)) (true) -:test (<?(+1)) (false) -:test (<?(+42)) (false) - -# returns true if balanced ternary number is positive -positive? [t⁺? (mst 0)] ⧗ Number → Boolean - ->?‣ positive? - -:test (>?(+0)) (false) -:test (>?(-1)) (false) -:test (>?(+1)) (true) -:test (>?(+42)) (true) - -# converts the normal balanced ternary representation into abstract -# infinity can't be abstracted in finite time -# → the abstract representation is used in eq?/add/sub/mul -abstract! [0 z a⁻ a⁺ a⁰] ⧗ Number → AbstractNumber - z (+0) - a⁻ [[[[[2 4]]]]] - a⁺ [[[[[1 4]]]]] - a⁰ [[[[[0 4]]]]] - -→^‣ abstract! - -:test (→^(-3)) ([[[[0 [[[[2 [[[[3]]]]]]]]]]]]) -:test (→^(+0)) ([[[[3]]]]) -:test (→^(+3)) ([[[[0 [[[[1 [[[[3]]]]]]]]]]]]) - -# converts the abstracted balanced ternary representation back to normal -normal! ω [[0 z a⁻ a⁺ a⁰]] ⧗ AbstractNumber → Number - z (+0) - a⁻ [↑⁻([3 3 0] 0)] - a⁺ [↑⁺([3 3 0] 0)] - a⁰ [↑⁰([3 3 0] 0)] - -→_‣ normal! - -:test (→_[[[[3]]]]) ((+0)) -:test (→_(→^(+42))) ((+42)) -:test (→_(→^(-42))) ((-42)) - -# returns true if two balanced ternary numbers are equal -# → ignores leading 0s! -eq? [[abs 1 →^0]] ⧗ Number → Number → Boolean - abs [0 z a⁻ a⁺ a⁰] - z [=?(→_0)] - a⁻ [[0 false [2 0] [false] [false]]] - a⁺ [[0 false [false] [2 0] [false]]] - a⁰ [[0 (1 0) [false] [false] [2 0]]] - -…=?… eq? - -# returns true if two balanced ternary numbers are not equal -not-eq? not! ∘∘ eq? ⧗ Number → Number → Boolean - -…≠?… not-eq? - -:test ((-42) =? (-42)) (true) -:test ((-1) =? (-1)) (true) -:test ((-1) =? (+0)) (false) -:test ((+0) =? (+0)) (true) -:test ((+1) =? (+0)) (false) -:test ((+1) =? (+1)) (true) -:test ((+42) =? (+42)) (true) -:test ([[[[(1 (0 (0 (0 (0 3)))))]]]] =? (+1)) (true) -:test ((+1) ≠? (+0)) (true) -:test ((-42) ≠? (+42)) (true) - -# I believe Mogensen's Paper has an error in its inc/dec/add/mul/eq definitions. -# They use 3 instead of 2 abstractions in the functions, also we use switched -# +/0 in comparison to their implementation, yet the order of neg/pos/zero is -# the same. Something's weird. - -# adds (+1) to a balanced ternary number (can introduce leading 0s) -inc [~(0 z a⁻ a⁺ a⁰)] ⧗ Number → Number - z (+0) : (+1) - a⁻ [0 [[↑⁻1 : ↑⁰1]]] - a⁺ [0 [[↑⁺1 : ↑⁻0]]] - a⁰ [0 [[↑⁰1 : ↑⁺1]]] - -++‣ inc - -:test ((++(-42)) =? (-41)) (true) -:test ((++(-1)) =? (+0)) (true) -:test ((++(+0)) =? (+1)) (true) -:test ((++(++(++(++(++(+0)))))) =? (+5)) (true) -:test ((++(+42)) =? (+43)) (true) - -# subs (+1) from a balanced ternary number (can introduce leading 0s) -dec [~(0 z a⁻ a⁺ a⁰)] ⧗ Number → Number - z (+0) : (-1) - a⁻ [0 [[↑⁻1 : ↑⁺0]]] - a⁺ [0 [[↑⁺1 : ↑⁰1]]] - a⁰ [0 [[↑⁰1 : ↑⁻1]]] - ---‣ dec - -:test ((--(-42)) =? (-43)) (true) -:test ((--(+0)) =? (-1)) (true) -:test ((--(--(--(--(--(+5)))))) =? (+0)) (true) -:test ((--(+1)) =? (+0)) (true) -:test ((--(+42)) =? (+41)) (true) - -# adds two balanced ternary numbers (can introduce leading 0s) -# second argument gets abstracted (performance) -add [[abs 1 →^0]] ⧗ Number → Number → Number - abs [c (0 z a⁻ a⁺ a⁰)] - b⁻ [1 ↑⁺(3 0 t⁻) ↑⁰(3 0 t⁰) ↑⁻(3 0 t⁰)] - b⁰ [up 1 (3 0 t⁰)] - b⁺ [1 ↑⁰(3 0 t⁰) ↑⁻(3 0 t⁺) ↑⁺(3 0 t⁰)] - a⁻ [[[1 (b⁻ 1) b⁻' b⁰ b⁻]]] - b⁻' [1 ↑⁰(3 0 t⁻) ↑⁻(3 0 t⁰) ↑⁺(3 0 t⁻)] - a⁺ [[[1 (b⁺ 1) b⁰ b⁺' b⁺]]] - b⁺' [1 ↑⁺(3 0 t⁰) ↑⁰(3 0 t⁺) ↑⁻(3 0 t⁺)] - a⁰ [[[1 (b⁰ 1) b⁻ b⁺ b⁰]]] - z [[0 --(→_1) ++(→_1) →_1]] - c [[1 0 t⁰]] - -…+… add - -:test ((-42) + (-1) =? (-43)) (true) -:test ((-5) + (+6) =? (+1)) (true) -:test ((-1) + (+0) =? (-1)) (true) -:test ((+0) + (+0) =? (+0)) (true) -:test ((+1) + (+2) =? (+3)) (true) -:test ((+42) + (+1) =? (+43)) (true) - -# subs two balanced ternary numbers (can introduce leading 0s) -# second argument gets abstracted (performance) -sub [[1 + -0]] ⧗ Number → Number → Number - -…-… sub - -:test ((-42) - (-1) =? (-41)) (true) -:test ((-5) - (+6) =? (-11)) (true) -:test ((-1) - (+0) =? (-1)) (true) -:test ((+0) - (+0) =? (+0)) (true) -:test ((+1) - (+2) =? (-1)) (true) -:test ((+42) - (+1) =? (+41)) (true) - -# returns true if number is greater than other number -# larger numbers should be second argument (performance) -gre? [[>?(1 - 0)]] ⧗ Number → Number → Boolean - -…>?… gre? - -:test ((+1) >? (+2)) (false) -:test ((+2) >? (+2)) (false) -:test ((+3) >? (+2)) (true) - -# returns true if number is less than other number -# smaller numbers should be second argument (performance) -les? \gre? ⧗ Number → Number → Boolean - -…<?… les? - -:test ((+1) <? (+2)) (true) -:test ((+2) <? (+2)) (false) -:test ((+3) <? (+2)) (false) - -# returns true if number is less than or equal to other number -# smaller numbers should be second argument (performance) -leq? [[¬(1 >? 0)]] ⧗ Number → Number → Boolean - -…≤?… leq? - -:test ((+1) ≤? (+2)) (true) -:test ((+2) ≤? (+2)) (true) -:test ((+3) ≤? (+2)) (false) - -# returns true if number is greater than or equal to other number -# smaller numbers should be second argument (performance) -geq? \leq? ⧗ Number → Number → Boolean - -…≥?… geq? - -:test ((+1) ≥? (+2)) (false) -:test ((+2) ≥? (+2)) (true) -:test ((+3) ≥? (+2)) (true) - -# negates a balanced ternary number if <0 -abs [<?0 -0 0] ⧗ Number → Number - -|‣ abs - -:test (|(+0)) ((+0)) -:test (|(-1)) ((+1)) -:test (|(+42)) ((+42)) - -# apply a function n times to a value -# ~> substitute church numbers -apply z [[[rec]]] ⧗ Number → (a → a) → a → a - rec =?1 case-end case-apply - case-apply 0 ∘ (2 --1 0) - case-end i - -:test (apply (+5) ++‣ (+3)) ((+8)) - -# muls two balanced ternary numbers (can introduce leading 0s) -mul [[1 z a⁻ a⁺ a⁰]] ⧗ Number → Number → Number - z (+0) - a⁻ [↑⁰0 - 1] - a⁺ [↑⁰0 + 1] - a⁰ [↑⁰0] - -…⋅… mul - -:test ((+42) ⋅ (+0) =? (+0)) (true) -:test ((-1) ⋅ (+42) =? (-42)) (true) -:test ((+3) ⋅ (+11) =? (+33)) (true) -:test ((+42) ⋅ (-4) =? (-168)) (true) - -# divs a balanced ternary number by three (rshifts least significant trit) -div³ [~(0 z a⁻ a⁺ a⁰)] ⧗ Number → Number - z (+0) : (+0) - a⁻ [0 [[↑⁻1 : 1]]] - a⁺ [0 [[↑⁺1 : 1]]] - a⁰ [0 [[↑⁰1 : 1]]] - -/³‣ div³ - -:test (/³(+6)) ((+2)) -:test (/³(-6)) ((-2)) -:test (/³(+5)) ((+2)) - -# divs a balanced ternary number by two (essentially binary >>1) -div² [z [[[[rec]]]] (+0) 0 0] ⧗ Number → Number - rec =?1 case-end case-div - case-div 3 /³(2 + 0) /³1 0 - case-end 2 - -/²‣ div² - -:test (/²(+6)) ((+3)) -:test (/²(-6)) ((-3)) -:test (/²(+5)) ((+2)) - -# manually counts how many times a balanced ternary number fits into another one -# TODO: quadratic approximation? -# TODO: fix for negative numbers -brute-div \[z [[[[[rec]]]]] (+0) 0 0] ⧗ Number → Number → Number - rec (2 >? 0) case-end case-count - case-count 4 ++3 (2 + 1) 1 0 - case-end 3 - -…/!… brute-div - -:test ((+4) /! (+2)) ((+2)) -:test ((+4) /! (+4)) ((+1)) -:test ((+4) /! (+5)) ((+0)) - -# TODO: fix for negative numbers -brute-mod \[z [[[[[rec]]]]] (+0) 0 0] ⧗ Number → Number → Number - rec (2 >? 0) case-end case-count - case-count 4 ++3 (2 + 1) 1 0 - case-end 0 - (3 ⋅ 1) - -…%!… brute-mod - -# finds quotient and remainder using long division -# WARNING: don't use; incorrect and slow -# TODO: faster algorithm -# dividend -> divisor -> (quot, rem) -# 0 divisor, 1 dividend, 2 (quot, rem) -# align: (quot, divisor) -quot-rem [[z [[[[rec]]]] ((+1) : (+0)) 1 0]] ⧗ Number → Number → (Pair Number Number) - rec (1 =? 0) case-eq ((1 <? 0) case-les case-div) - case-div calc (z [[[align]]] ^2 0) - align (0 ≤? 4) (1 : 0) (2 ↑⁰1 ↑⁰0) - calc [final (4 (^0 : ~3) (2 - ~0) 1)] - final [(^4 + ^0) : (~4 + ~0)] - case-eq (+0) : (+1) - case-les (+1) : 1 - -# divs two balanced ternary numbers -# WARNING: don't use; incorrect and slow -div ^‣ ∘∘ quot-rem ⧗ Number → Number - -…/… div - -# returns remainder of integer division -# WARNING: don't use; incorrect and slow -mod ~‣ ∘∘ quot-rem ⧗ Number → Number - -…%… mod - -# returns max number of two -max [[(1 ≤? 0) 0 1]] ⧗ Number → Number → Number - -:test (max (+5) (+2)) ((+5)) - -# returns min number of two -min [[(1 ≤? 0) 1 0]] ⧗ Number → Number → Number - -:test (min (+5) (+2)) ((+2)) - -# clamps a number between two numbers -clamp [[[min 1 (max 0 2)]]] ⧗ Number → Number → Number - -:test (clamp (+0) (+5) (+3)) ((+3)) -:test (clamp (+0) (+5) (-2)) ((+0)) -:test (clamp (+0) (+5) (+7)) ((+5)) +:input std/Number/Ternary . |