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Line 1: {{~~draft~~ task}} The '''Minkowski question-mark function''' converts the continued fraction representation {{math\|[a<sub>0</sub>; a<sub>1</sub>, a<sub>2</sub>, a<sub>3</sub>, ...]}} of a number into a binary decimal representation in which the integer part {{math\|a<sub>0</sub>}} is unchanged and the {{math\|a<sub>1</sub>, a<sub>2</sub>, ...}} become alternating runs of binary zeroes and ones of those lengths. The decimal point takes the place of the first zero. Line 21: Don't worry about precision error in the last few digits. ~~;See also:~~ ;See also: * Wikipedia entry: [[wp:Minkowski%27s_question-mark_function\|Minkowski's question-mark function]] <br><br> =={{header\|11l}}== {{trans\|Python}} <syntaxhighlight lang="11l">-V MAXITER = 151 F minkowski(x) -> Float I x > 1 \| x < 0 R floor(x) + minkowski(x - floor(x)) V p = Int(x) V q = 1 V r = p + 1 V s = 1 V d = 1.0 V y = Float(p) L d /= 2 I y + d == y L.break V m = p + r I m < 0 \| p < 0 L.break V n = q + s I n < 0 L.break I x < Float(m) / n r = m s = n E y += d p = m q = n R y + d F minkowski_inv(=x) -> Float I x > 1 \| x < 0 R floor(x) + minkowski_inv(x - floor(x)) I x == 1 \| x == 0 R x V cont_frac = [0] V current = 0 V count = 1 V i = 0 L x = 2 I current == 0 I x < 1 count++ E cont_frac.append(0) cont_frac[i] = count i++ count = 1 current = 1 x-- E I x > 1 count++ x-- E cont_frac.append(0) cont_frac[i] = count i++ count = 1 current = 0 I x == floor(x) cont_frac[i] = count L.break I i == :MAXITER L.break V ret = 1.0 / cont_frac[i] L(j) (i-1 .. 0).step(-1) ret = cont_frac[j] + 1.0 / ret R 1.0 / ret print(‘#2.16 #2.16’.format(minkowski(0.5 (1 + sqrt(5))), 5.0 / 3.0)) print(‘#2.16 #2.16’.format(minkowski_inv(-5.0 / 9.0), (sqrt(13) - 7) / 6)) print(‘#2.16 #2.16’.format(minkowski(minkowski_inv(0.718281828)), minkowski_inv(minkowski(0.1213141516171819))))</syntaxhighlight> {{out}} <pre> 1.6666666666696983 1.6666666666666667 -0.5657414540893350 -0.5657414540893351 0.7182818280000091 0.1213141516171819 </pre> =={{header\|C#}}== {{trans\|Go}} <syntaxhighlight lang="C#"> using System; class Program { const int MAXITER = 151; static double Minkowski(double x) { if (x > 1 \|\| x < 0) { return Math.Floor(x) + Minkowski(x - Math.Floor(x)); } ulong p = (ulong)x; ulong q = 1; ulong r = p + 1; ulong s = 1; double d = 1.0; double y = (double)p; while (true) { d = d / 2; if (y + d == y) { break; } ulong m = p + r; if (m < 0 \|\| p < 0) { break; } ulong n = q + s; if (n < 0) { break; } if (x < (double)m / (double)n) { r = m; s = n; } else { y = y + d; p = m; q = n; } } return y + d; } static double MinkowskiInv(double x) { if (x > 1 \|\| x < 0) { return Math.Floor(x) + MinkowskiInv(x - Math.Floor(x)); } if (x == 1 \|\| x == 0) { return x; } uint[] contFrac = new uint[] { 0 }; uint curr = 0; uint count = 1; int i = 0; while (true) { x = 2; if (curr == 0) { if (x < 1) { count++; } else { i++; Array.Resize(ref contFrac, i + 1); contFrac[i - 1] = count; count = 1; curr = 1; x--; } } else { if (x > 1) { count++; x--; } else { i++; Array.Resize(ref contFrac, i + 1); contFrac[i - 1] = count; count = 1; curr = 0; } } if (x == Math.Floor(x)) { contFrac[i] = count; break; } if (i == MAXITER) { break; } } double ret = 1.0 / contFrac[i]; for (int j = i - 1; j >= 0; j--) { ret = contFrac[j] + 1.0 / ret; } return 1.0 / ret; } static void Main(string[] args) { Console.WriteLine("{0,19:0.0000000000000000} {1,19:0.0000000000000000}", Minkowski(0.5 (1 + Math.Sqrt(5))), 5.0 / 3.0); Console.WriteLine("{0,19:0.0000000000000000} {1,19:0.0000000000000000}", MinkowskiInv(-5.0 / 9.0), (Math.Sqrt(13) - 7) / 6); Console.WriteLine("{0,19:0.0000000000000000} {1,19:0.0000000000000000}", Minkowski(MinkowskiInv(0.718281828)), MinkowskiInv(Minkowski(0.1213141516171819))); } } </syntaxhighlight> {{out}} <pre> 1.6666666666697000 1.6666666666666700 -0.5657414540893350 -0.5657414540893350 0.7182818280000090 0.1213141516171820 </pre> =={{header\|C++}}== <syntaxhighlight lang="c++"> #include <cmath> #include <cstdint> #include <iomanip> #include <iostream> #include <vector> constexpr int32_t MAX_ITERATIONS = 151; double minkowski(const double& x) { if ( x < 0 \|\| x > 1 ) { return floor(x) + minkowski(x - floor(x)); } int64_t p = (int64_t) x; int64_t q = 1; int64_t r = p + 1; int64_t s = 1; double d = 1.0; double y = (double) p; while ( true ) { d /= 2; if ( d == 0.0 ) { break; } int64_t m = p + r; if ( m < 0 \|\| p < 0 ) { break; } int64_t n = q + s; if ( n < 0 ) { break; } if ( x < (double) m / n ) { r = m; s = n; } else { y += d; p = m; q = n; } } return y + d; } double minkowski_inverse(double x) { if ( x < 0 \|\| x > 1 ) { return floor(x) + minkowski_inverse(x - floor(x)); } if ( x == 0 \|\| x == 1 ) { return x; } std::vector<int32_t> continued_fraction(1, 0); int32_t current = 0; int32_t count = 1; int32_t i = 0; while ( true ) { x = 2; if ( current == 0 ) { if ( x < 1 ) { count += 1; } else { continued_fraction.emplace_back(0); continued_fraction[i] = count; i += 1; count = 1; current = 1; x -= 1; } } else { if ( x > 1 ) { count += 1; x -= 1; } else { continued_fraction.emplace_back(0); continued_fraction[i] = count; i += 1; count = 1; current = 0; } } if ( x == floor(x) ) { continued_fraction[i] = count; break; } if ( i == MAX_ITERATIONS ) { break; } } double reciprocal = 1.0 / continued_fraction[i]; for ( int32_t j = i - 1; j >= 0; --j ) { reciprocal = continued_fraction[j] + 1.0 / reciprocal; } return 1.0 / reciprocal; } int main() { std::cout << std::setw(20) << std::fixed << std::setprecision(16) << minkowski(0.5 ( 1 + sqrt(5) )) << std::setw(20) << 5.0 / 3.0 << std::endl; std::cout << std::setw(20) << minkowski_inverse(-5.0 / 9.0) << std::setw(20) << ( sqrt(13) - 7 ) / 6 << std::endl; std::cout << std::setw(20) << minkowski(minkowski_inverse(0.718281828182818)) << std::setw(20) << 0.718281828182818 << std::endl; std::cout << std::setw(20) << minkowski_inverse(minkowski(0.1213141516271819)) << std::setw(20) << 0.1213141516171819 << std::endl; } </syntaxhighlight> {{ out }} <pre> 1.6666666666696983 1.6666666666666667 -0.5657414540893351 -0.5657414540893352 0.7182818281828269 0.7182818281828180 0.1213141516171819 0.1213141516171819 </pre> =={{header\|F_Sharp\|F#}}== <syntaxhighlight lang="fsharp"> // Minkowski question-mark function. Nigel Galloway: July 14th., 2023 let fN g=let n=(int>>float)g in ((if g<0.0 then -1.0 else 1.0),abs n,abs (g-n)) let fI(n,_,(nl,nh))(g,_,(gl,gh))=let l,h=nl+gl,nh+gh in ((n+g)/2.0,(float l)/(float h),(l,h)) let fG n g=(max n g)-(min n g) let fE(s,z,l)=Seq.unfold(fun(i,e)->let (n,g,_) as r=fI i e in Some((s(z+n),s(g+z)),if l<n then (i,r) else if l=n then (r,r) else (r,e)))((0.0,0.0,(0,1)),(1.0,1.0,(1,1))) let fL(s,z,l)=Seq.unfold(fun(i,e)->let (n,g,_) as r=fI i e in Some((s(z+n),s(g+z)),if l<g then (i,r) else if l=g then (r,r) else (r,e)))((0.0,0.0,(0,1)),(1.0,1.0,(1,1))) let f2M g=let _,(n,_)=fL(fN g)\|>Seq.pairwise\|>Seq.find(fun((n,_),(g,_))->(fG n g)<2.328306437e-11) in n let m2F g=let _,(_,n)=fE(fN g)\|>Seq.pairwise\|>Seq.find(fun((_,n),(_,g))->(fG n g)<2.328306437e-11) in n printfn $"?(φ) = 5/3 is %A{fG(f2M 1.61803398874989490253)(5.0/3.0)<2.328306437e-10}" printfn $"?⁻¹(-5/9) = (√13-7)/6 is %A{fG(m2F(-5.0/9.0))((sqrt(13.0)-7.0)/6.0)<2.328306437e-10}" let n=42.0/23.0 in printfn $"?⁻¹(?(n)) = n is %A{(fG(m2F(f2M n)) n)<2.328306437e-10}" let n= -3.0/13.0 in printfn $"?(?⁻¹(n)) = n is %A{(fG(f2M(m2F n)) n)<2.328306437e-10}" </syntaxhighlight> {{out}} <pre> ?(φ) = 5/3 is true ?⁻¹(-5/9) = (√13-7)/6 is true ?⁻¹(?(n)) = n is true ?(?⁻¹(n)) = n is true </pre> =={{header\|Factor}}== {{works with\|Factor\|0.99 2020-08-14}} ~~<lang factor>USING: formatting kernel make math math.constants~~ <syntaxhighlight lang="factor">USING: formatting kernel make math math.constants math.continued-fractions math.functions math.parser math.statistics sequences sequences.extras splitting.monotonic Line 63 ⟶ 454: phi ? 5 3 /f compare -5/9 ?⁻¹ 13 sqrt 7 - 6 /f compare 0.718281828 ?⁻¹ ? 0.1213141516171819 ? ?⁻¹ compare</~~lang~~syntaxhighlight> {{out}} <pre> Line 73 ⟶ 464: =={{header\|FreeBASIC}}== <~~lang~~syntaxhighlight lang="freebasic">#define MAXITER 151 function minkowski( x as double ) as double Line 144 ⟶ 535: print minkowski_inv( -5./9 ), (sqr(13)-7)/6 print minkowski(minkowski_inv(0.718281828)), minkowski_inv(minkowski(0.1213141516171819)) </syntaxhighlight> ~~</lang>~~ {{out}} <pre> 1.666666666669698 1.666666666666667 Line 152 ⟶ 543: =={{header\|Go}}== {{trans\|FreeBASIC}} <~~lang~~syntaxhighlight lang="go">package main import ( Line 256 ⟶ 647: fmt.Printf("%19.16f %19.16f\n", minkowski(minkowskiInv(0.718281828)), minkowskiInv(minkowski(0.1213141516171819))) }</~~lang~~syntaxhighlight> {{out}} Line 263 ⟶ 654: -0.5657414540893351 -0.5657414540893352 0.7182818280000092 0.1213141516171819 </pre> =={{header\|Haskell}}== In a lazy functional language Minkowski question mark function can be implemented using one of it's basic properties: ?(p+r)/(q+s) = 1/2 * ( ?(p/q) + ?(r/s) ), ?(0) = 0, ?(1) = 1. where p/q and r/s are fractions, such that \|ps - rq\| = 1. This recursive definition can be implemented as lazy corecursion, i.e. by generating two infinite binary trees: '''mediant'''-based Stern-Brocot tree, containing all rationals, and '''mean'''-based tree with corresponding values of Minkowsky ?-function. There is one-to-one correspondence between these two trees so both {{math\|?(x)}} and {{math\|?<sup>-1</sup>(x)}} may be implemented as mapping between them. For details see the paper [[https://habr.com/ru/post/591949/]] (in Russian). <syntaxhighlight lang="haskell">import Data.Tree import Data.Ratio import Data.List intervalTree :: (a -> a -> a) -> (a, a) -> Tree a intervalTree node = unfoldTree $ \(a, b) -> let m = node a b in (m, [(a,m), (m,b)]) Node a _ ==> Node b [] = const b Node a [] ==> Node b _ = const b Node a [l1, r1] ==> Node b [l2, r2] = \x -> case x `compare` a of LT -> (l1 ==> l2) x EQ -> b GT -> (r1 ==> r2) x mirror :: Num a => Tree a -> Tree a mirror t = Node 0 [reflect (negate <$> t), t] where reflect (Node a [l,r]) = Node a [reflect r, reflect l] ------------------------------------------------------------ sternBrocot :: Tree Rational sternBrocot = toRatio <$> intervalTree mediant ((0,1), (1,0)) where mediant (p, q) (r, s) = (p + r, q + s) toRatio (p, q) = p % q minkowski :: Tree Rational minkowski = toRatio <$> intervalTree mean ((0,1), (1,0)) mean (p, q) (1, 0) = (p+1, q) mean (p, q) (r, s) = (ps + qr, 2qs) questionMark, invQuestionMark :: Rational -> Rational questionMark = mirror sternBrocot ==> mirror minkowski invQuestionMark = mirror minkowski ==> mirror sternBrocot ------------------------------------------------------------ -- Floating point trees and functions sternBrocotF :: Tree Double sternBrocotF = mirror $ fromRational <$> sternBrocot minkowskiF :: Tree Double minkowskiF = mirror $ intervalTree mean (0, 1/0) where mean a b \| isInfinite b = a + 1 \| otherwise = (a + b) / 2 questionMarkF, invQuestionMarkF :: Double -> Double questionMarkF = sternBrocotF ==> minkowskiF invQuestionMarkF = minkowskiF ==> sternBrocotF</syntaxhighlight> <pre>λ> mapM_ print $ take 4 $ levels farey [1 % 2] [1 % 3,2 % 3] [1 % 4,2 % 5,3 % 5,3 % 4] [1 % 5,2 % 7,3 % 8,3 % 7,4 % 7,5 % 8,5 % 7,4 % 5] λ> mapM_ print $ take 4 $ levels minkowski [1 % 2] [1 % 4,3 % 4] [1 % 8,3 % 8,5 % 8,7 % 8] [1 % 16,3 % 16,5 % 16,7 % 16,9 % 16,11 % 16,13 % 16,15 % 16]</pre> λ> questionMark (1/2) 1 % 2 λ> questionMark (2/7) 3 % 16 λ> questionMark (-22/7) (-193) % 64 λ> invQuestionMark (3/16) 2 % 7 λ> invQuestionMark (13/256) 5 % 27</pre> <pre>λ> questionMark $ (sqrt 5 + 1) / 2 1.6666666666678793 λ> 5/3 1.6666666666666667 λ> invQuestionMark (-5/9) -0.5657414540893351 λ> (sqrt 13 - 7)/6 -0.5657414540893352</pre> =={{header\|J}}== Implementation: <syntaxhighlight lang="j">ITERCOUNT=: 52 minkowski=: {{ f=. 1\|y node=. i.2 2 NB. node of Stern-Brocot tree B=. '' for. i.ITERCOUNT do. B=. B, b=. f>:%/t=. +/node node=. t (1-b)} node end. (<.y)+B+/ .2^-1+i.ITERCOUNT }} invmink=: {{ f=. 1\|y cf=. i.0 cur=. 0 NB. 1 if generating "top" side of cf cnt=. 1 NB. proposed continued fraction element for. i.ITERCOUNT do. if. f=<. f do. cf=. cf,%cnt break. end. f=. f2 b=. 1 >`<@.cur f cf=. cf,(-.b)#cnt cnt=. 1+bcnt cur=. cur=b f=. f-cur end. (+%)/(<.y),cf }}</syntaxhighlight> That said, note that this algorithm introduces significant numeric instability for √7 divided by 3: <syntaxhighlight lang="j"> (minkowski@invmink - invmink@minkowski) (p:%%:)3 1.10713e_6</syntaxhighlight> I see this same instability using the python implementation and appending: <syntaxhighlight lang="python"> print( "{:19.16f} {:19.16f}".format( minkowski(minkowski_inv(4.04145188432738056)), minkowski_inv(minkowski(4.04145188432738056)), ) )</syntaxhighlight> Using an exact fraction for 4.04145188432738056 and bumping the iteration count from 52 up to 200 changes that difference to 1.43622e_12. =={{header\|Java}}== <syntaxhighlight lang="java"> import java.util.ArrayList; import java.util.List; public final class MinkowskiQuestionMarkFunction { public static void main(String[] aArgs) { System.out.println(String.format("%20.16f%20.16f", minkowski(0.5 * ( 1 + Math.sqrt(5) )), 5.0 / 3.0)); System.out.println(String.format("%20.16f%20.16f", minkowskiInverse(-5.0 / 9.0), ( Math.sqrt(13) - 7 ) / 6 )); System.out.println(String.format("%20.16f%20.16f", minkowski(minkowskiInverse(0.718281828)), 0.718281828)); System.out.println(String.format("%20.16f%20.16f", minkowskiInverse(minkowski(0.1213141516271819)), 0.1213141516171819)); } private static double minkowski(double aX) { if ( aX < 0 \|\| aX > 1 ) { return Math.floor(aX) + minkowski(aX - Math.floor(aX)); } long p = (long) aX; long q = 1; long r = p + 1; long s = 1; double d = 1.0; double y = (double) p; while ( true ) { d /= 2; if ( d == 0.0 ) { break; } long m = p + r; if ( m < 0 \|\| p < 0 ) { break; } long n = q + s; if ( n < 0 ) { break; } if ( aX < (double) m / n ) { r = m; s = n; } else { y += d; p = m; q = n; } } return y + d; } private static double minkowskiInverse(double aX) { if ( aX < 0 \|\| aX > 1 ) { return Math.floor(aX) + minkowskiInverse(aX - Math.floor(aX)); } if ( aX == 0 \|\| aX == 1 ) { return aX; } List<Integer> continuedFraction = new ArrayList<Integer>(); continuedFraction.add(0); int current = 0; int count = 1; int i = 0; while ( true ) { aX = 2; if ( current == 0 ) { if ( aX < 1 ) { count += 1; } else { continuedFraction.add(0); continuedFraction.set(i, count); i += 1; count = 1; current = 1; aX -= 1; } } else { if ( aX > 1 ) { count += 1; aX -= 1; } else { continuedFraction.add(0); continuedFraction.set(i, count); i += 1; count = 1; current = 0; } } if ( aX == Math.floor(aX) ) { continuedFraction.set(i, count); break; } if ( i == MAX_ITERATIONS ) { break; } } double reciprocal = 1.0 / continuedFraction.get(i); for ( int j = i - 1; j >= 0; j-- ) { reciprocal = continuedFraction.get(j) + 1.0 / reciprocal; } return 1.0 / reciprocal; } private static final int MAX_ITERATIONS = 150; } </syntaxhighlight> {{ out }} <pre> 1.6666666666696983 1.6666666666666667 -0.5657414540893351 -0.5657414540893352 0.7182818280000092 0.7182818280000000 0.1213141516271825 0.1213141516171819 </pre> =={{header\|Julia}}== ~~{{trans\|FreeBASIC}}~~ <~~lang~~syntaxhighlight lang="julia">function ~~minkowski~~questionmark(x) y, p = ~~Int(floor~~fldmod(x), 1) (xq, >d = 1 ~~\|\| x < 0) && return~~- p, ~~+ minkowski(x)~~.5 q,while ~~r, s, m, n = 1, p~~y + ~~1, 1,~~d 0,> 0y d, y p < q ? (q -= ~~1.0,~~p) : ~~Float64~~(p -= q; y += d) ~~while~~ ~~true~~ d /= 2 ~~d /= 2.0~~ ~~y + d == y && break~~ ~~m = p + r~~ ~~(m < 0 \|\| p < 0) && break~~ ~~n = q + s~~ ~~n < 0 && break~~ ~~if x < (m / n)~~ ~~r, s = m, n~~ ~~else~~ ~~y, p, q = y + d, m, n~~ ~~end~~ end ~~return~~ y ~~+ d~~ end function ~~minkowski_inv~~questionmark_inv(x~~, maxiter=151~~) py, bits = ~~Int(floor~~fldmod(x), 1) lo, hi = [0, 1], [1, 1] ~~(x > 1 \|\| x < 0) && return p + minkowski_inv(x - p, maxiter)~~ while (xy ==+ 1/(lo...)) \|\|< x(y ==+ 0/(hi...)) ~~&& return x~~ bit, bits = fldmod(2bits, 1) ~~contfrac = [0]~~ bit > 0 ? (lo .+= hi) : (hi .+= lo) ~~curr, coun, i = 0, 1, 0~~ ~~while i < maxiter~~ ~~x = 2~~ ~~if curr == 0~~ ~~if x < 1~~ ~~coun += 1~~ ~~else~~ ~~i += 1~~ ~~push!(contfrac, 0)~~ ~~contfrac[i] = coun~~ ~~coun = 1~~ ~~curr = 1~~ ~~x -= 1~~ ~~end~~ ~~else~~ ~~if x > 1~~ ~~coun += 1~~ ~~x -= 1~~ ~~else~~ ~~i += 1~~ ~~push!(contfrac, 0)~~ ~~contfrac[i] = coun~~ ~~coun = 1~~ ~~curr = 0~~ ~~end~~ ~~end~~ ~~if x == Int(floor(x))~~ ~~contfrac[i + 1] = coun~~ ~~break~~ ~~end~~ end ~~ret~~y =+ 1/(lo.~~0 / contfrac[i + 1]~~..) ~~for j in i:-1:1~~ ~~ret = contfrac[j] + 1.0 / ret~~ ~~end~~ ~~return 1.0 / ret~~ end x, y = 0.7182818281828, 0.1213141516171819 ~~println(" ", minkowski((1 + sqrt(5)) / 2), " ", 5 / 3)~~ for (a, b) ∈ [ ~~println(minkowski_inv(-5/9), " ", (sqrt(13) - 7) / 6)~~ (5/3, questionmark((1 + √5)/2)), ~~println(" ", minkowski(minkowski_inv(0.718281828)), " ",~~ ((√13-7)/6, questionmark_inv(-5/9)), ~~minkowski_inv(minkowski(0.1213141516171819)))~~ (x, questionmark_inv(questionmark(x))), ~~</lang>{{out}}~~ (y, questionmark(questionmark_inv(y)))] println(a, a ≈ b ? " ≈ " : " != ", b) end </syntaxhighlight>{{out}} <pre> 1.6666666666666667 ≈ 1.666666666667894 ~~1.6666666666696983 1.6666666666666667~~ -0.5657414540893352 ≈ -0.5657414540893351 ~~-0.5657414540893351 -0.5657414540893352~~ 0.7182818281828 ≈ 0.7182818281828183 ~~0.7182818280000092 0.12131415161718191~~ 0.1213141516171819 ≈ 0.12131415161718095 </pre> =={{header\|Mathematica}} / {{header\|Wolfram Language}}== <syntaxhighlight lang="mathematica">ClearAll[InverseMinkowskiQuestionMark] InverseMinkowskiQuestionMark[val_] := Module[{x}, (x /. FindRoot[MinkowskiQuestionMark[x] == val, {x, Floor[val], Ceiling[val]}])] MinkowskiQuestionMark[GoldenRatio] InverseMinkowskiQuestionMark[-5/9] // RootApproximant MinkowskiQuestionMark[InverseMinkowskiQuestionMark[0.1213141516171819]] InverseMinkowskiQuestionMark[MinkowskiQuestionMark[0.1213141516171819]]</syntaxhighlight> {{out}} <pre>5/3 1/6 (-7+Sqrt[13]) 0.121314 0.121314</pre> =={{header\|Nim}}== {{trans\|Go}} <syntaxhighlight lang="nim">import math, strformat const MaxIter = 151 func minkowski(x: float): float = if x notin 0.0..1.0: return floor(x) + minkowski(x - floor(x)) var p = x.uint64 r = p + 1 q, s = 1u64 d = 1.0 y = p.float while true: d /= 2 if y + d == y: break let m = p + r if m < 0 or p < 0: break let n = q + s if n < 0: break if x < m.float / n.float: r = m s = n else: y += d p = m q = n result = y + d func minkowskiInv(x: float): float = if x notin 0.0..1.0: return floor(x) + minkowskiInv(x - floor(x)) if x == 1 or x == 0: return x var contFrac: seq[uint32] curr = 0u32 count = 1u32 i = 0 x = x while true: x = 2 if curr == 0: if x < 1: inc count else: inc i contFrac.setLen(i + 1) contFrac[i - 1] = count count = 1 curr = 1 x -= 1 else: if x > 1: inc count x -= 1 else: inc i contFrac.setLen(i + 1) contFrac[i - 1] = count count = 1 curr = 0 if x == floor(x): contFrac[i] = count break if i == MaxIter: break var ret = 1 / contFrac[i].float for j in countdown(i - 1, 0): ret = contFrac[j].float + 1 / ret result = 1 / ret echo &"{minkowski(0.5(1+sqrt(5.0))):19.16f}, {5/3:19.16f}" echo &"{minkowskiInv(-5/9):19.16f}, {(sqrt(13.0)-7)/6:19.16f}" echo &"{minkowski(minkowskiInv(0.718281828)):19.16f}, " & &"{minkowskiInv(minkowski(0.1213141516171819)):19.16f}"</syntaxhighlight> {{out}} <pre> 1.6666666666696983, 1.6666666666666667 -0.5657414540893351, -0.5657414540893352 0.7182818280000092, 0.1213141516171819</pre> =={{header\|Perl}}== {{trans\|Raku}} <syntaxhighlight lang="perl">use strict; use warnings; use feature 'say'; use POSIX qw(floor); my $MAXITER = 50; sub minkowski { my($x) = @_; return floor($x) + minkowski( $x - floor($x) ) if $x > 1 \|\| $x < 0 ; my $y = my $p = floor($x); my ($q,$s,$d) = (1,1,1); my $r = $p + 1; while () { last if ( $y + ($d /= 2) == $y ) or ( my $m = $p + $r) < 0 or ( my $n = $q + $s) < 0; $x < $m/$n ? ($r,$s) = ($m, $n) : ($y += $d and ($p,$q) = ($m, $n) ); } return $y + $d } sub minkowskiInv { my($x) = @_; return floor($x) + minkowskiInv($x - floor($x)) if $x > 1 \|\| $x < 0; return $x if $x == 1 \|\| $x == 0 ; my @contFrac = 0; my $i = my $curr = 0 ; my $count = 1; while () { $x = 2; if ($curr == 0) { if ($x < 1) { $count++ } else { $i++; push @contFrac, 0; $contFrac[$i-1] = $count; ($count,$curr) = (1,1); $x--; } } else { if ($x > 1) { $count++; $x--; } else { $i++; push @contFrac, 0; @contFrac[$i-1] = $count; ($count,$curr) = (1,0); } } if ($x == floor($x)) { @contFrac[$i] = $count; last } last if $i == $MAXITER; } my $ret = 1 / $contFrac[$i]; for (my $j = $i - 1; $j >= 0; $j--) { $ret = $contFrac[$j] + 1/$ret } return 1 / $ret } printf "%19.16f %19.16f\n", minkowski(0.5(1 + sqrt(5))), 5/3; printf "%19.16f %19.16f\n", minkowskiInv(-5/9), (sqrt(13)-7)/6; printf "%19.16f %19.16f\n", minkowski(minkowskiInv(0.718281828)), minkowskiInv(minkowski(0.1213141516171819));</syntaxhighlight> {{out}} <pre> 1.6666666666696983 1.6666666666666667 -0.5657414540893351 -0.5657414540893352 0.7182818280000092 0.1213141516171819</pre> =={{header\|Phix}}== {{trans\|FreeBASIC}} <!--<syntaxhighlight lang="phix">(phixonline)--> ~~<lang Euphoria>constant MAXITER = 151~~ <span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span> <span style="color: #008080;">constant</span> <span style="color: #000000;">MAXITER</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">151</span> ~~function minkowski(atom x)~~ ~~atom p = floor(x)~~ <span style="color: #008080;">function</span> <span style="color: #000000;">minkowski</span><span style="color: #0000FF;">(</span><span style="color: #004080;">atom</span> <span style="color: #000000;">x</span><span style="color: #0000FF;">)</span> ~~if x>1 or x<0 then return p+minkowski(x-p) end if~~ <span style="color: #004080;">atom</span> <span style="color: #000000;">p</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">)</span> ~~atom q = 1, r = p + 1, s = 1, m, n, d = 1, y = p~~ <span style="color: #008080;">if</span> <span style="color: #000000;">x</span><span style="color: #0000FF;">></span><span style="color: #000000;">1</span> <span style="color: #008080;">or</span> <span style="color: #000000;">x</span><span style="color: #0000FF;"><</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span> <span style="color: #008080;">return</span> <span style="color: #000000;">p</span><span style="color: #0000FF;">+</span><span style="color: #000000;">minkowski</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">-</span><span style="color: #000000;">p</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span> ~~while true do~~ <span style="color: #004080;">atom</span> <span style="color: #000000;">q</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">r</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">p</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">m</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">d</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">y</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">p</span> ~~d = d/2~~ <span style="color: #008080;">while</span> <span style="color: #004600;">true</span> <span style="color: #008080;">do</span> ~~if y + d = y then exit end if~~ <span style="color: #000000;">d</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">d</span><span style="color: #0000FF;">/</span><span style="color: #000000;">2</span> <span style="color: #008080;">if</span> <span style="color: #000000;">y</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">d</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">y</span> <span style="color: #008080;">then</span> <span style="color: #008080;">exit</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span> <span style="color: #000000;">m</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">p</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">r</span> <span style="color: #008080;">if</span> <span style="color: #000000;">m</span> <span style="color: #0000FF;"><</span> <span style="color: #000000;">0</span> <span style="color: #008080;">or</span> <span style="color: #000000;">p</span> <span style="color: #0000FF;"><</span> <span style="color: #000000;">0</span> <span style="color: #008080;">then</span> <span style="color: #008080;">exit</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span> <span style="color: #000000;">n</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">q</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">s</span> <span style="color: #008080;">if</span> <span style="color: #000000;">n</span> <span style="color: #0000FF;"><</span> <span style="color: #000000;">0</span> <span style="color: #008080;">then</span> <span style="color: #008080;">exit</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span> <span style="color: #008080;">if</span> <span style="color: #000000;">x</span> <span style="color: #0000FF;"><</span> <span style="color: #000000;">m</span><span style="color: #0000FF;">/</span><span style="color: #000000;">n</span> <span style="color: #008080;">then</span> <span style="color: #000000;">r</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">m</span> <span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">n</span> <span style="color: #008080;">else</span> <span style="color: #000000;">y</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">y</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">d</span> <span style="color: #000000;">p</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">m</span> <span style="color: #000000;">q</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">n</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span> <span style="color: #008080;">end</span> <span style="color: #008080;">while</span> <span style="color: #008080;">return</span> <span style="color: #000000;">y</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">d</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span> <span style="color: #008080;">function</span> <span style="color: #000000;">minkowski_inv</span><span style="color: #0000FF;">(</span><span style="color: #004080;">atom</span> <span style="color: #000000;">x</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">if</span> <span style="color: #000000;">x</span><span style="color: #0000FF;">></span><span style="color: #000000;">1</span> <span style="color: #008080;">or</span> <span style="color: #000000;">x</span><span style="color: #0000FF;"><</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span> <span style="color: #008080;">return</span> <span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">)+</span><span style="color: #000000;">minkowski_inv</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">-</span><span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">))</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span> <span style="color: #008080;">if</span> <span style="color: #000000;">x</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">or</span> <span style="color: #000000;">x</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span> <span style="color: #008080;">return</span> <span style="color: #000000;">x</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span> <span style="color: #004080;">sequence</span> <span style="color: #000000;">contfrac</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{}</span> <span style="color: #004080;">integer</span> <span style="color: #000000;">curr</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">count</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span> <span style="color: #008080;">while</span> <span style="color: #004600;">true</span> <span style="color: #008080;">do</span> <span style="color: #000000;">x</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">2</span> <span style="color: #008080;">if</span> <span style="color: #000000;">curr</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span> <span style="color: #008080;">then</span> <span style="color: #008080;">if</span> <span style="color: #000000;">x</span><span style="color: #0000FF;"><</span><span style="color: #000000;">1</span> <span style="color: #008080;">then</span> <span style="color: #000000;">count</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span> <span style="color: #008080;">else</span> <span style="color: #000000;">contfrac</span> <span style="color: #0000FF;">&=</span> <span style="color: #000000;">count</span> <span style="color: #000000;">count</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span> <span style="color: #000000;">curr</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span> <span style="color: #000000;">x</span> <span style="color: #0000FF;">-=</span> <span style="color: #000000;">1</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span> <span style="color: #008080;">else</span> <span style="color: #008080;">if</span> <span style="color: #000000;">x</span><span style="color: #0000FF;">></span><span style="color: #000000;">1</span> <span style="color: #008080;">then</span> <span style="color: #000000;">count</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span> <span style="color: #000000;">x</span> <span style="color: #0000FF;">-=</span> <span style="color: #000000;">1</span> <span style="color: #008080;">else</span> <span style="color: #000000;">contfrac</span> <span style="color: #0000FF;">&=</span> <span style="color: #000000;">count</span> <span style="color: #000000;">count</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span> <span style="color: #000000;">curr</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span> <span style="color: #008080;">if</span> <span style="color: #000000;">x</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span> <span style="color: #000000;">contfrac</span> <span style="color: #0000FF;">&=</span> <span style="color: #000000;">count</span> <span style="color: #008080;">exit</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span> <span style="color: #008080;">if</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">contfrac</span><span style="color: #0000FF;">)=</span><span style="color: #000000;">MAXITER</span> <span style="color: #008080;">then</span> <span style="color: #008080;">exit</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span> <span style="color: #008080;">end</span> <span style="color: #008080;">while</span> <span style="color: #004080;">atom</span> <span style="color: #000000;">ret</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">/</span><span style="color: #000000;">contfrac</span><span style="color: #0000FF;">[$]</span> <span style="color: #008080;">for</span> <span style="color: #000000;">i</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">contfrac</span><span style="color: #0000FF;">)-</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">1</span> <span style="color: #008080;">by</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">1</span> <span style="color: #008080;">do</span> <span style="color: #000000;">ret</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">contfrac</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">1.0</span><span style="color: #0000FF;">/</span><span style="color: #000000;">ret</span> <span style="color: #008080;">end</span> <span style="color: #008080;">for</span> <span style="color: #008080;">return</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">/</span><span style="color: #000000;">ret</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span> <span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"%20.16f %20.16f\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">minkowski</span><span style="color: #0000FF;">(</span><span style="color: #000000;">0.5</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">+</span><span style="color: #7060A8;">sqrt</span><span style="color: #0000FF;">(</span><span style="color: #000000;">5</span><span style="color: #0000FF;">))),</span> <span style="color: #000000;">5</span><span style="color: #0000FF;">/</span><span style="color: #000000;">3</span><span style="color: #0000FF;">})</span> <span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"%20.16f %20.16f\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">minkowski_inv</span><span style="color: #0000FF;">(-</span><span style="color: #000000;">5</span><span style="color: #0000FF;">/</span><span style="color: #000000;">9</span><span style="color: #0000FF;">),</span> <span style="color: #0000FF;">(</span><span style="color: #7060A8;">sqrt</span><span style="color: #0000FF;">(</span><span style="color: #000000;">13</span><span style="color: #0000FF;">)-</span><span style="color: #000000;">7</span><span style="color: #0000FF;">)/</span><span style="color: #000000;">6</span><span style="color: #0000FF;">})</span> <span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"%20.16f %20.16f\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">minkowski</span><span style="color: #0000FF;">(</span><span style="color: #000000;">minkowski_inv</span><span style="color: #0000FF;">(</span><span style="color: #000000;">0.718281828</span><span style="color: #0000FF;">)),</span> <span style="color: #000000;">minkowski_inv</span><span style="color: #0000FF;">(</span><span style="color: #000000;">minkowski</span><span style="color: #0000FF;">(</span><span style="color: #000000;">0.1213141516171819</span><span style="color: #0000FF;">))})</span> <!--</syntaxhighlight>--> {{out}} <pre> 1.6666666666696983 1.6666666666666668 -0.5657414540893351 -0.5657414540893352 0.7182818280000092 0.1213141516171819 </pre> =={{header\|Python}}== {{trans\|Go}} <syntaxhighlight lang="python">import math MAXITER = 151 def minkowski(x): if x > 1 or x < 0: return math.floor(x) + minkowski(x - math.floor(x)) p = int(x) q = 1 r = p + 1 s = 1 d = 1.0 y = float(p) while True: d /= 2 if y + d == y: break m = p + r if m < 0 or p < 0 ~~then exit end if~~: break n = q + s if n < 0 ~~then exit end if~~: if x < ~~m/n~~ ~~then~~break if x < m / n: r = m s = n else: ~~y =~~ y += d p = m q = n ~~end if~~ ~~end while~~ return y + d ~~end function~~ ~~function~~def minkowski_inv(~~atom~~ x): if x > 1 or x < 0 ~~then return floor(x)+minkowski_inv(x-floor(x)) end if~~: if ~~x=1~~ or ~~x=0 then~~ return math.floor(x) ~~end~~+ ifminkowski_inv(x - math.floor(x)) ~~sequence contfrac = {}~~ ~~integer~~if ~~curr~~x == 0,1 ~~count~~or x == 10: ~~while~~ ~~true~~ do return x cont_frac = [0] current = 0 count = 1 i = 0 while True: x = 2 ~~if curr = 0 then~~ if current == ~~if x<1 then~~0: if x < 1: count += 1 else: ~~contfrac &= count~~cont_frac.append(0) cont_frac[i] = count i += 1 count = 1 ~~curr~~current = 1 x -= 1 ~~end if~~else: ~~else~~ if x > 1: ~~if x>1 then~~ count += 1 x -= 1 else: ~~contfrac &= count~~cont_frac.append(0) cont_frac[i] = count i += 1 count = 1 ~~curr~~current = 0 ~~end if~~ ~~end~~ if x == math.floor(x): if x = ~~floor(x)~~ ~~then~~cont_frac[i] = count ~~contfrac &= count~~break ~~exit~~ ~~end~~ if i == MAXITER: if ~~length(contfrac)=MAXITER~~ ~~then~~ ~~exit~~ ~~end if~~break ~~end while~~ ~~atom~~ ret = 1.0 /~~contfrac~~ cont_frac[$i] for ij =in ~~length~~range(~~contfrac)~~i -~~1 to~~ 1 by, -1, do-1): ret = ~~contfrac~~cont_frac[ij] + 1.0 / ret ~~end for~~ return 1.0 / ret ~~end function~~ if __name__ == "__main__": ~~printf(1,"%20.16f %20.16f\n",{minkowski(0.5(1+sqrt(5))), 5/3})~~ print( ~~printf(1,"%20.16f %20.16f\n",{minkowski_inv(-5/9), (sqrt(13)-7)/6})~~ "{:19.16f} {:19.16f}".format( ~~printf(1,"%20.16f %20.16f\n",{minkowski(minkowski_inv(0.718281828)),~~ minkowski(0.5 * (1 + math.sqrt(5))), ~~minkowski_inv(minkowski(0.1213141516171819))})</lang>~~ 5.0 / 3.0, ) ) print( "{:19.16f} {:19.16f}".format( minkowski_inv(-5.0 / 9.0), (math.sqrt(13) - 7) / 6, ) ) print( "{:19.16f} {:19.16f}".format( minkowski(minkowski_inv(0.718281828)), minkowski_inv(minkowski(0.1213141516171819)), ) ) </syntaxhighlight> {{out}} <pre> 1.6666666666696983 1.~~6666666666666668~~6666666666666667 -0.5657414540893351 -0.5657414540893352 0.7182818280000092 0.1213141516171819 </pre> =={{header\|Raku}}== {{trans\|Go}} <syntaxhighlight lang="raku" ~~perl6~~line># 20201120 Raku programming solution my \MAXITER = 151; Line 485 ⟶ 1,425: printf "%19.16f %19.16f\n", minkowskiInv(-5/9), (13.sqrt-7)/6; printf "%19.16f %19.16f\n", minkowski(minkowskiInv(0.718281828)), minkowskiInv(minkowski(0.1213141516171819))</~~lang~~syntaxhighlight> {{out}} <pre> 1.6666666666696983 1.6666666666666667 Line 494 ⟶ 1,434: {{trans\|FreeBASIC}} {{trans\|Phix}} <~~lang~~syntaxhighlight lang="rexx">/REXX program uses the Minkowski question─mark function to convert a continued fraction/ numeric digits 2040 /use enough dec. digits for precision./ say fmt( mink( 0.5 * (1+sqrt(5) ) ) ) fmt( 5/3 ) say fmt( minkI(-5/9) ) fmt( (sqrt(13) - 7) / 6) say fmt( mink( minkI(0.718281828) ) ) fmt( mink( minkI(.1213141516171819) ) ) exit 0 /stick a fork in it, we're all done. / /──────────────────────────────────────────────────────────────────────────────────────/ floor: procedure; parse arg x; _t= trunc(x); return t - ~~return _ -~~ (x<0) * (x\=_t) fmt: procedure: parse arg za; d= digits(); return right( format(za, , ~~digits()~~ d- 2, 0), ~~digits()~~ d+5) /──────────────────────────────────────────────────────────────────────────────────────/ mink: procedure: parse arg x; p= x % 1; if x>1 \| x<0 then return p + mink(x-p) Line 517 ⟶ 1,457: minkI: procedure; parse arg x; p= floor(x); if x>1 \| x<0 then return p + minkI(x-p) if x=1 \| x=0 then return x ~~curr~~cur= 0; ~~count=~~ 1; ~~maxIter~~ limit= 200; $= /limit: max iterations/ #= 1 /#: is the count. / do until ~~count~~#==~~maxIter~~limit \| words($)==~~maxIter~~limit; x= x + x ~~/a~~ ~~fast~~ ~~double~~ x= x / 2 if ~~curr~~cur==0 then if x<1 then ~~count~~ #= ~~count~~# + 1 else do; $= $ ~~count~~#; ~~count~~#= 1; ~~curr~~ cur= 1; x= x-1; end else if x>1 then do; ~~count~~#= ~~count~~# + 1; x= x-1; end else do; $= $ ~~count~~#; ~~count~~#= 1; ~~curr~~ cur= 0; end if x==floor(x) ~~then do;~~ then ~~$= $ count~~do; ~~leave;~~ $= $ #; leave; end end /until/ z= words($) #ret= ~~words~~1 / word($, z) ~~ret~~ do j=z for z by -1; / ret= word($, #j) + 1 / ret do ~~j=#~~end ~~for # by -1; ret= word($,~~ /j~~) + 1~~ / ~~ret~~ ~~end /j/~~ return 1 / ret /──────────────────────────────────────────────────────────────────────────────────────/ Line 536 ⟶ 1,475: numeric form; m.=9; parse value format(x,2,1,,0) 'E0' with g "E" _ .; g=g .5'e'_ %2 do j=0 while h>9; m.j= h; h= h % 2 + 1; end /j/ do k=j+5 to 0 by -1; numeric digits m.k; g= (g + x/g) .5; end /k/; return g</~~lang~~syntaxhighlight> {{out\|output\|text=  when using the internal default inputs:}} <pre> 1.66666666666666666666666666673007566392 1.66666666666666666666666666666666666667 ~~1.666666666666666963 1.666666666666666667~~ -0.56574145408933511781346312208825067563 -0.56574145408933511781346312208825067562 ~~-0.565741454089335118 -0.565741454089335118~~ 0.71828182799999999999999999999999992890 0.12131415161718190000000000000000000833 ~~0.718281828000000011 0.121314151617181900~~ </pre> Line 547 ⟶ 1,486: {{trans\|FreeBASIC}} {{libheader\|Wren-fmt}} <~~lang~~syntaxhighlight ~~ecmascript~~lang="wren">import "./fmt" for Fmt var MAXITER = 151 Line 630 ⟶ 1,569: Fmt.print("$17.14f $17.14f", minkowskiInv.call(-5/9), (13.sqrt - 7)/6) Fmt.print("$17.14f $17.14f", minkowski.call(minkowskiInv.call(0.718281828)), minkowskiInv.call(minkowski.call(0.1213141516171819)))</~~lang~~syntaxhighlight> {{out}}