Minkowski question-mark function: Difference between revisions
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=={{header|Mathematica}} / {{header|Wolfram Language}}== |
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<lang Mathematica>ClearAll[InverseMinkowskiQuestionMark] |
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InverseMinkowskiQuestionMark[val_] := Module[{x}, (x /. FindRoot[MinkowskiQuestionMark[x] == val, {x, Floor[val], Ceiling[val]}])] |
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MinkowskiQuestionMark[GoldenRatio] |
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InverseMinkowskiQuestionMark[-5/9] // RootApproximant |
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MinkowskiQuestionMark[InverseMinkowskiQuestionMark[0.1213141516171819]] |
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InverseMinkowskiQuestionMark[MinkowskiQuestionMark[0.1213141516171819]]</lang> |
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{{out}} |
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<pre>5/3 |
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1/6 (-7+Sqrt[13]) |
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0.121314 |
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0.121314</pre> |
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=={{header|Nim}}== |
=={{header|Nim}}== |
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Revision as of 09:11, 8 June 2021
You are encouraged to solve this task according to the task description, using any language you may know.
The Minkowski question-mark function converts the continued fraction representation [a0; a1, a2, a3, ...] of a number into a binary decimal representation in which the integer part a0 is unchanged and the a1, a2, ... become alternating runs of binary zeroes and ones of those lengths. The decimal point takes the place of the first zero.
Thus, ?(31/7) = 71/16 because 31/7 has the continued fraction representation [4;2,3] giving the binary expansion 4 + 0.01112.
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 ?(x). Be careful: rational numbers have two possible continued fraction representations:
- [a0;a1,... an−1,an] and
- [a0;a1,... an−1,an−1,1]
- Choose one of the above that will give a binary expansion ending with a 1.
- Produce the inverse function ?-1(x)
- Verify that ?(φ) = 5/3, where φ is the Greek golden ratio.
- Verify that ?-1(-5/9) = (√13 - 7)/6
- Verify that the two functions are inverses of each other by showing that ?-1(?(x))=x and ?(?-1(y))=y for x, y of your choice
Don't worry about precision error in the last few digits.
- See also
- Wikipedia entry: Minkowski's question-mark function
Factor
<lang factor>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</lang>
- Output:
1.666666666668335 1.666666666666667 -0.5657414540893351 -0.5657414540893352 0.718281828000002 0.1213141516171819
FreeBASIC
<lang freebasic>#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)) </lang>
- Output:
1.666666666669698 1.666666666666667 -0.5657414540893351 -0.5657414540893352 0.7182818280000092 0.1213141516171819
Go
<lang go>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)))
}</lang>
- Output:
1.6666666666696983 1.6666666666666667 -0.5657414540893351 -0.5657414540893352 0.7182818280000092 0.1213141516171819
Julia
<lang julia>function minkowski(x)
p = Int(floor(x))
(x > 1 || x < 0) && return p + minkowski(x)
q, r, s, m, n = 1, p + 1, 1, 0, 0
d, y = 1.0, Float64(p)
while true
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(x, maxiter=151)
p = Int(floor(x))
(x > 1 || x < 0) && return p + minkowski_inv(x - p, maxiter)
(x == 1 || x == 0) && return x
contfrac = [0]
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 = 1.0 / contfrac[i + 1]
for j in i:-1:1
ret = contfrac[j] + 1.0 / ret
end
return 1.0 / ret
end
println(" ", minkowski((1 + sqrt(5)) / 2), " ", 5 / 3) println(minkowski_inv(-5/9), " ", (sqrt(13) - 7) / 6) println(" ", minkowski(minkowski_inv(0.718281828)), " ",
minkowski_inv(minkowski(0.1213141516171819)))
</lang>
- Output:
1.6666666666696983 1.6666666666666667 -0.5657414540893351 -0.5657414540893352 0.7182818280000092 0.12131415161718191
Mathematica / Wolfram Language
<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]]</lang>
- Output:
5/3 1/6 (-7+Sqrt[13]) 0.121314 0.121314
Nim
<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}"</lang>
- Output:
1.6666666666696983, 1.6666666666666667 -0.5657414540893351, -0.5657414540893352 0.7182818280000092, 0.1213141516171819
Perl
<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 / $ret
}
printf "%19.16f %19.16f\n", minkowski(0.5*(1 + sqrt(5))), 5/3; printf "%19.16f %19.16f\n", minkowskiInv(-5/9), (sqrt(13)-7)/6; printf "%19.16f %19.16f\n", minkowski(minkowskiInv(0.718281828)), minkowskiInv(minkowski(0.1213141516171819));</lang>
- Output:
1.6666666666696983 1.6666666666666667 -0.5657414540893351 -0.5657414540893352 0.7182818280000092 0.1213141516171819
Phix
<lang Euphoria>constant MAXITER = 151
function minkowski(atom x)
atom p = floor(x)
if x>1 or x<0 then return p+minkowski(x-p) end if
atom q = 1, r = p + 1, s = 1, m, n, d = 1, y = p
while true do
d = d/2
if y + d = y then exit end if
m = p + r
if m < 0 or p < 0 then exit end if
n = q + s
if n < 0 then exit end if
if x < m/n then
r = m
s = n
else
y = y + d
p = m
q = n
end if
end while
return y + d
end function
function minkowski_inv(atom x)
if x>1 or x<0 then return floor(x)+minkowski_inv(x-floor(x)) end if
if x=1 or x=0 then return x end if
sequence contfrac = {}
integer curr = 0, count = 1
while true do
x *= 2
if curr = 0 then
if x<1 then
count += 1
else
contfrac &= count
count = 1
curr = 1
x -= 1
end if
else
if x>1 then
count += 1
x -= 1
else
contfrac &= count
count = 1
curr = 0
end if
end if
if x = floor(x) then
contfrac &= count
exit
end if
if length(contfrac)=MAXITER then exit end if
end while
atom ret = 1/contfrac[$]
for i = length(contfrac)-1 to 1 by -1 do
ret = contfrac[i] + 1.0/ret
end for
return 1/ret
end function
printf(1,"%20.16f %20.16f\n",{minkowski(0.5*(1+sqrt(5))), 5/3}) printf(1,"%20.16f %20.16f\n",{minkowski_inv(-5/9), (sqrt(13)-7)/6}) printf(1,"%20.16f %20.16f\n",{minkowski(minkowski_inv(0.718281828)),
minkowski_inv(minkowski(0.1213141516171819))})</lang>
- Output:
1.6666666666696983 1.6666666666666668 -0.5657414540893351 -0.5657414540893352 0.7182818280000092 0.1213141516171819
Python
<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:
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)),
)
)
</lang>
- Output:
1.6666666666696983 1.6666666666666667 -0.5657414540893351 -0.5657414540893352 0.7182818280000092 0.1213141516171819
Raku
<lang perl6># 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))</lang>
- Output:
1.6666666666696983 1.6666666666666667 -0.5657414540893351 -0.5657414540893352 0.7182818280000092 0.1213141516171819
REXX
<lang rexx>/*REXX program uses the Minkowski question─mark function to convert a continued fraction*/ numeric digits 20 /*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; _= trunc(x); return _ - (x<0) * (x\=_) fmt: procedure: parse arg z; return right( format(z, , digits() - 2, 0), digits() +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
curr= 0; count= 1; maxIter= 200; $=
do until count==maxIter | words($)==maxIter; x= x + x /*a fast double*/
if curr==0 then if x<1 then count= count + 1
else do; $= $ count; count= 1; curr= 1; x= x-1; end
else if x>1 then do; count= count + 1; x= x-1; end
else do; $= $ count; count= 1; curr= 0; end
if x==floor(x) then do; $= $ count; leave; end
end /*until*/
#= words($)
ret= 1 / word($, #)
do j=# for # 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</lang>
- output when using the internal default inputs:
1.666666666666666963 1.666666666666666667
-0.565741454089335118 -0.565741454089335118
0.718281828000000011 0.121314151617181900
Wren
<lang ecmascript>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)))</lang>
- Output:
1.66666666666970 1.66666666666667 -0.56574145408934 -0.56574145408934 0.71828182800001 0.12131415161718