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Ingy döt Net 2023-07-01 11:58:00 -04:00
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2
Task/Minkowski-question-mark-function/00-META.yaml Normal file
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---
from: http://rosettacode.org/wiki/Minkowski_question-mark_function

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Task/Minkowski-question-mark-function/00-TASK.txt Normal file
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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.
Thus, {{math|?}}(31/7) = 71/16 because 31/7 has the continued fraction representation {{math|[4;2,3]}} giving the binary expansion {{math|4 + 0.0111<sub>2</sub>}}.
Among its interesting properties is that it maps roots of quadratic equations, which have repeating continued fractions, to rational numbers, which have repeating binary digits.
The question-mark function is continuous and monotonically increasing, so it has an inverse.
* Produce a function for {{math|?(x)}}. &nbsp; Be careful: rational numbers have two possible continued fraction representations:
:::* &nbsp; {{math|[a<sub>0</sub>;a<sub>1</sub>,... a<sub>n−1</sub>,a<sub>n</sub>]}} &nbsp; &nbsp; and
:::* &nbsp; {{math|[a<sub>0</sub>;a<sub>1</sub>,... a<sub>n−1</sub>,a<sub>n</sub>−1,1]}}
* Choose one of the above that will give a binary expansion ending with a &nbsp; '''1'''.
* Produce the inverse function {{math|?<sup>-1</sup>(x)}}
* Verify that {{math|?(φ)}} = 5/3, where {{math|φ}} is the Greek golden ratio.
* Verify that {{math|?<sup>-1</sup>(-5/9)}} = (√13 - 7)/6
* Verify that the two functions are inverses of each other by showing that {{math|?<sup>-1</sup>(?(x))}}={{math|x}} and {{math|?(?<sup>-1</sup>(y))}}={{math|y}} for {{math|x, y}} of your choice
Don't worry about precision error in the last few digits.
;See also:
* Wikipedia entry: [[wp:Minkowski%27s_question-mark_function|Minkowski's question-mark function]]
<br><br>

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Task/Minkowski-question-mark-function/11l/minkowski-question-mark-function.11l Normal file
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-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))))

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Task/Minkowski-question-mark-function/Factor/minkowski-question-mark-function.factor Normal file
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USING: formatting kernel make math math.constants
math.continued-fractions math.functions math.parser
math.statistics sequences sequences.extras splitting.monotonic
vectors ;
CONSTANT: max-iter 151
: >continued-fraction ( x -- seq )
0 swap 1vector
[ dup last integer? pick max-iter > or ]
[ dup next-approx [ 1 + ] dip ] until nip
dup last integer? [ but-last-slice ] unless ;
: ? ( x -- y )
>continued-fraction unclip swap cum-sum
[ max-iter < ] take-while
[ even? 1 -1 kernel:? swap 2^ / ] map-index
sum 2 * + >float ;
: (float>bin) ( x -- y )
[ dup 0 > ]
[ 2 * dup >integer # dup 1 >= [ 1 - ] when ] while ;
: float>bin ( x -- n str )
>float dup >integer [ - ] keep swap abs
[ 0 # (float>bin) ] "" make nip ;
: ?⁻¹ ( x -- y )
dup float>bin [ = ] monotonic-split
[ length ] map swap prefix >ratio swap copysign ;
: compare ( x y -- ) "%-25u%-25u\n" printf ;
phi ? 5 3 /f compare
-5/9 ?⁻¹ 13 sqrt 7 - 6 /f compare
0.718281828 ?⁻¹ ? 0.1213141516171819 ? ?⁻¹ compare

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Task/Minkowski-question-mark-function/FreeBASIC/minkowski-question-mark-function.basic Normal file
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#define MAXITER 151
function minkowski( x as double ) as double
if x>1 or x<0 then return int(x)+minkowski(x-int(x))
dim as ulongint p = int(x)
dim as ulongint q = 1, r = p + 1, s = 1, m, n
dim as double d = 1, y = p
while true
d = d / 2.0
if y + d = y then exit while
m = p + r
if m < 0 or p < 0 then exit while
n = q + s
if n < 0 then exit while
if x < cast(double,m) / n then
r = m
s = n
else
y = y + d
p = m
q = n
end if
wend
return y + d
end function
function minkowski_inv( byval x as double ) as double
if x>1 or x<0 then return int(x)+minkowski_inv(x-int(x))
if x=1 or x=0 then return x
redim as uinteger contfrac(0 to 0)
dim as uinteger curr=0, count=1, i = 0
do
x *= 2
if curr = 0 then
if x<1 then
count += 1
else
i += 1
redim preserve contfrac(0 to i)
contfrac(i-1)=count
count = 1
curr = 1
x=x-1
endif
else
if x>1 then
count += 1
x=x-1
else
i += 1
redim preserve contfrac(0 to i)
contfrac(i-1)=count
count = 1
curr = 0
endif
end if
if x = int(x) then
contfrac(i)=count
exit do
end if
loop until i = MAXITER
dim as double ret = 1.0/contfrac(i)
for j as integer = i-1 to 0 step -1
ret = contfrac(j) + 1.0/ret
next j
return 1./ret
end function
print minkowski( 0.5*(1+sqr(5)) ), 5./3
print minkowski_inv( -5./9 ), (sqr(13)-7)/6
print minkowski(minkowski_inv(0.718281828)), minkowski_inv(minkowski(0.1213141516171819))

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Task/Minkowski-question-mark-function/Go/minkowski-question-mark-function.go Normal file
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package main
import (
"fmt"
"math"
)
const MAXITER = 151
func minkowski(x float64) float64 {
if x > 1 || x < 0 {
return math.Floor(x) + minkowski(x-math.Floor(x))
}
p := uint64(x)
q := uint64(1)
r := p + 1
s := uint64(1)
d := 1.0
y := float64(p)
for {
d = d / 2
if y+d == y {
break
}
m := p + r
if m < 0 || p < 0 {
break
}
n := q + s
if n < 0 {
break
}
if x < float64(m)/float64(n) {
r = m
s = n
} else {
y = y + d
p = m
q = n
}
}
return y + d
}
func minkowskiInv(x float64) float64 {
if x > 1 || x < 0 {
return math.Floor(x) + minkowskiInv(x-math.Floor(x))
}
if x == 1 || x == 0 {
return x
}
contFrac := []uint32{0}
curr := uint32(0)
count := uint32(1)
i := 0
for {
x *= 2
if curr == 0 {
if x < 1 {
count++
} else {
i++
t := contFrac
contFrac = make([]uint32, i+1)
copy(contFrac, t)
contFrac[i-1] = count
count = 1
curr = 1
x--
}
} else {
if x > 1 {
count++
x--
} else {
i++
t := contFrac
contFrac = make([]uint32, i+1)
copy(contFrac, t)
contFrac[i-1] = count
count = 1
curr = 0
}
}
if x == math.Floor(x) {
contFrac[i] = count
break
}
if i == MAXITER {
break
}
}
ret := 1.0 / float64(contFrac[i])
for j := i - 1; j >= 0; j-- {
ret = float64(contFrac[j]) + 1.0/ret
}
return 1.0 / ret
}
func main() {
fmt.Printf("%19.16f %19.16f\n", minkowski(0.5*(1+math.Sqrt(5))), 5.0/3.0)
fmt.Printf("%19.16f %19.16f\n", minkowskiInv(-5.0/9.0), (math.Sqrt(13)-7)/6)
fmt.Printf("%19.16f %19.16f\n", minkowski(minkowskiInv(0.718281828)),
minkowskiInv(minkowski(0.1213141516171819)))
}

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Task/Minkowski-question-mark-function/Haskell/minkowski-question-mark-function.hs Normal file
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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) = (p*s + q*r, 2*q*s)
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

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Task/Minkowski-question-mark-function/J/minkowski-question-mark-function-1.j Normal file
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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=. f*2
b=. 1 >`<@.cur f
cf=. cf,(-.b)#cnt
cnt=. 1+b*cnt
cur=. cur=b
f=. f-cur
end.
(+%)/(<.y),cf
}}

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Task/Minkowski-question-mark-function/J/minkowski-question-mark-function-2.j Normal file
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(minkowski@invmink - invmink@minkowski) (p:%%:)3
1.10713e_6

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Task/Minkowski-question-mark-function/J/minkowski-question-mark-function-3.j Normal file
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print(
"{:19.16f} {:19.16f}".format(
minkowski(minkowski_inv(4.04145188432738056)),
minkowski_inv(minkowski(4.04145188432738056)),
)
)

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Task/Minkowski-question-mark-function/Julia/minkowski-question-mark-function.julia Normal file
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function questionmark(x)
y, p = fldmod(x, 1)
q, d = 1 - p, .5
while y + d > y
p < q ? (q -= p) : (p -= q; y += d)
d /= 2
end
y
end
function questionmark_inv(x)
y, bits = fldmod(x, 1)
lo, hi = [0, 1], [1, 1]
while (y + /(lo...)) < (y + /(hi...))
bit, bits = fldmod(2bits, 1)
bit > 0 ? (lo .+= hi) : (hi .+= lo)
end
y + /(lo...)
end
x, y = 0.7182818281828, 0.1213141516171819
for (a, b) ∈ [
(5/3, questionmark((1 + √5)/2)),
((√13-7)/6, questionmark_inv(-5/9)),
(x, questionmark_inv(questionmark(x))),
(y, questionmark(questionmark_inv(y)))]
println(a, a ≈ b ? " ≈ " : " != ", b)
end

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Task/Minkowski-question-mark-function/Mathematica/minkowski-question-mark-function.math Normal file
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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]]

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Task/Minkowski-question-mark-function/Nim/minkowski-question-mark-function.nim Normal file
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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}"

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Task/Minkowski-question-mark-function/Perl/minkowski-question-mark-function.pl Normal file
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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));

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Task/Minkowski-question-mark-function/Phix/minkowski-question-mark-function.phix Normal file
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(phixonline)-->
<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>
<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>
<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>
<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>
<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>
<span style="color: #008080;">while</span> <span style="color: #004600;">true</span> <span style="color: #008080;">do</span>
<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>
<!--

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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:
break
n = q + s
if n < 0:
break
if x < m / n:
r = m
s = n
else:
y += d
p = m
q = n
return y + d
def minkowski_inv(x):
if x > 1 or x < 0:
return math.floor(x) + minkowski_inv(x - math.floor(x))
if x == 1 or x == 0:
return x
cont_frac = [0]
current = 0
count = 1
i = 0
while True:
x *= 2
if current == 0:
if x < 1:
count += 1
else:
cont_frac.append(0)
cont_frac[i] = count
i += 1
count = 1
current = 1
x -= 1
else:
if x > 1:
count += 1
x -= 1
else:
cont_frac.append(0)
cont_frac[i] = count
i += 1
count = 1
current = 0
if x == math.floor(x):
cont_frac[i] = count
break
if i == MAXITER:
break
ret = 1.0 / cont_frac[i]
for j in range(i - 1, -1, -1):
ret = cont_frac[j] + 1.0 / ret
return 1.0 / ret
if __name__ == "__main__":
print(
"{:19.16f} {:19.16f}".format(
minkowski(0.5 * (1 + math.sqrt(5))),
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)),
)
)

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Task/Minkowski-question-mark-function/REXX/minkowski-question-mark-function.rexx Normal file
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/*REXX program uses the Minkowski question─mark function to convert a continued fraction*/
numeric digits 40 /*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 - (x<0) * (x\=t)
fmt: procedure: parse arg a; d= digits(); return right( format(a, , d-2, 0), d+5)
/*──────────────────────────────────────────────────────────────────────────────────────*/
mink: procedure: parse arg x; p= x % 1; if x>1 | x<0 then return p + mink(x-p)
q= 1; s= 1; m= 0; n= 0; d= 1; y= p
r= p + 1
do forever; d= d * 0.5; if y+d=y | d=0 then leave /*d= d÷2*/
m= p + r; if m<0 | p<0 then leave
n= q + s; if n<0 then leave
if x<m/n then do; r= m; s= n; end
else do; y= y + d; p= m; q= n; end
end /*forever*/
return y + d
/*──────────────────────────────────────────────────────────────────────────────────────*/
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
cur= 0; limit= 200; $= /*limit: max iterations*/
#= 1 /*#: is the count. */
do until #==limit | words($)==limit; x= x * 2
if cur==0 then if x<1 then #= # + 1
else do; $= $ #; #= 1; cur= 1; x= x-1; end
else if x>1 then do; #= # + 1; x= x-1; end
else do; $= $ #; #= 1; cur= 0; end
if x==floor(x) then do; $= $ #; leave; end
end /*until*/
z= words($)
ret= 1 / word($, z)
do j=z for z by -1; ret= word($, j) + 1 / ret
end /*j*/
return 1 / ret
/*──────────────────────────────────────────────────────────────────────────────────────*/
sqrt: procedure; parse arg x; if x=0 then return 0; d=digits(); numeric digits; h=d+6
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

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# 20201120 Raku programming solution
my \MAXITER = 151;
sub minkowski(\x) {
return x.floor + minkowski( x - x.floor ) if x > 1 || x < 0 ;
my $y = my $p = x.floor;
my ($q,$s,$d) = 1 xx 3;
my $r = $p + 1;
loop {
last if ( $y + ($d /= 2) == $y ) ||
( my $m = $p + $r) < 0 | $p < 0 ||
( my $n = $q + $s) < 0 ;
x < $m/$n ?? ( ($r,$s) = ($m, $n) ) !! ( $y += $d; ($p,$q) = ($m, $n) );
}
return $y + $d
}
sub minkowskiInv($x is copy) {
return $x.floor + minkowskiInv($x - $x.floor) if $x > 1 || $x < 0 ;
return $x if $x == 1 || $x == 0 ;
my @contFrac = 0;
my $i = my $curr = 0 ; my $count = 1;
loop {
$x *= 2;
if $curr == 0 {
if $x < 1 {
$count++
} else {
$i++;
@contFrac.append: 0;
@contFrac[$i-1] = $count;
($count,$curr) = 1,1;
$x--;
}
} else {
if $x > 1 {
$count++;
$x--;
} else {
$i++;
@contFrac.append: 0;
@contFrac[$i-1] = $count;
($count,$curr) = 1,0;
}
}
if $x == $x.floor { @contFrac[$i] = $count ; last }
last if $i == MAXITER;
}
my $ret = 1 / @contFrac[$i];
loop (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 + 5.sqrt)), 5/3;
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))

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import "/fmt" for Fmt
var MAXITER = 151
var minkowski // predeclare as recursive
minkowski = Fn.new { |x|
if (x > 1 || x < 0) return x.floor + minkowski.call(x - x.floor)
var p = x.floor
var q = 1
var r = p + 1
var s = 1
var d = 1
var y = p
while (true) {
d = d / 2
if (y + d == y) break
var m = p + r
if (m < 0 || p < 0 ) break
var n = q + s
if (n < 0) break
if (x < m/n) {
r = m
s = n
} else {
y = y + d
p = m
q = n
}
}
return y + d
}
var minkowskiInv
minkowskiInv = Fn.new { |x|
if (x > 1 || x < 0) return x.floor + minkowskiInv.call(x - x.floor)
if (x == 1 || x == 0) return x
var contFrac = [0]
var curr = 0
var count = 1
var i = 0
while (true) {
x = x * 2
if (curr == 0) {
if (x < 1) {
count = count + 1
} else {
i = i + 1
var t = contFrac
contFrac = List.filled(i + 1, 0)
for (j in 0...t.count) contFrac[j] = t[j]
contFrac[i-1] = count
count = 1
curr = 1
x = x - 1
}
} else {
if (x > 1) {
count = count + 1
x = x - 1
} else {
i = i + 1
var t = contFrac
contFrac = List.filled(i + 1, 0)
for (j in 0...t.count) contFrac[j] = t[j]
contFrac[i-1] = count
count = 1
curr = 0
}
}
if (x == x.floor) {
contFrac[i] = count
break
}
if (i == MAXITER) break
}
var ret = 1/contFrac[i]
for (j in i-1..0) ret = contFrac[j] + 1/ret
return 1/ret
}
Fmt.print("$17.16f $17.14f", minkowski.call(0.5 * (1 + 5.sqrt)), 5/3)
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)))