Zeckendorf arithmetic
You are encouraged to solve this task according to the task description, using any language you may know.
This task is a total immersion zeckendorf task; using decimal numbers will attract serious disapprobation.
The task is to implement addition, subtraction, multiplication, and division using Zeckendorf number representation. Optionally provide decrement, increment and comparitive operation functions.
- Addition
Like binary 1 + 1 = 10, note carry 1 left. There the similarity ends. 10 + 10 = 101, note carry 1 left and 1 right. 100 + 100 = 1001, note carry 1 left and 2 right, this is the general case.
Occurrences of 11 must be changed to 100. Occurrences of 111 may be changed from the right by replacing 11 with 100, or from the left converting 111 to 100 + 100;
- Subtraction
10 - 1 = 1. The general rule is borrow 1 right carry 1 left. eg:
abcde
10100 -
1000
_____
100 borrow 1 from a leaves 100
+ 100 add the carry
_____
1001
A larger example:
abcdef
100100 -
1000
______
1*0100 borrow 1 from b
+ 100 add the carry
______
1*1001
Sadly we borrowed 1 from b which didn't have it to lend. So now b borrows from a:
1001
+ 1000 add the carry
____
10100
- Multiplication
Here you teach your computer its zeckendorf tables. eg. 101 * 1001:
a = 1 * 101 = 101 b = 10 * 101 = a + a = 10000 c = 100 * 101 = b + a = 10101 d = 1000 * 101 = c + b = 101010 1001 = d + a therefore 101 * 1001 = 101010 + 101 ______ 1000100
- Division
Lets try 1000101 divided by 101, so we can use the same table used for multiplication.
1000101 -
101010 subtract d (1000 * 101)
_______
1000 -
101 b and c are too large to subtract, so subtract a
____
1 so 1000101 divided by 101 is d + a (1001) remainder 1
Efficient algorithms for Zeckendorf arithmetic is interesting. The sections on addition and subtraction are particularly relevant for this task.
11l
T Zeckendorf
Int dLen
dVal = 0
F (x = ‘0’)
V q = 1
V i = x.len - 1
.dLen = i I/ 2
L i >= 0
.dVal = .dVal + (x[i].code - ‘0’.code) * q
q = q * 2
i = i - 1
F a(n)
V i = n
L
I .dLen < i
.dLen = i
V j = (.dVal >> (i * 2)) [&] 3
I j == 0 | j == 1
R
I j == 2
I (.dVal >> ((i + 1) * 2) [&] 1) != 1
R
.dVal = .dVal + (1 << (i * 2 + 1))
R
I j == 3
V temp = 3 << (i * 2)
temp = temp (+) -1
.dVal = .dVal [&] temp
.b((i + 1) * 2)
i = i + 1
F b(pos)
I pos == 0
.inc()
R
I (.dVal >> pos) [&] 1 == 0
.dVal = .dVal + (1 << pos)
.a(Int(pos / 2))
I pos > 1
.a(Int(pos / 2) - 1)
E
V temp = 1 << pos
temp = temp (+) -1
.dVal = .dVal [&] temp
.b(pos + 1)
.b(pos - (I pos > 1 {2} E 1))
F c(pos)
I (.dVal >> pos) [&] 1 == 1
V temp = 1 << pos
temp = temp (+) -1
.dVal = .dVal [&] temp
R
.c(pos + 1)
I pos > 0
.b(pos - 1)
E
.inc()
F inc() -> Void
.dVal = .dVal + 1
.a(0)
F +(rhs)
V copy = (.)
V rhs_dVal = rhs.dVal
V limit = (rhs.dLen + 1) * 2
L(gn) 0 .< limit
I ((rhs_dVal >> gn) [&] 1) == 1
copy.b(gn)
R copy
F -(rhs)
V copy = (.)
V rhs_dVal = rhs.dVal
V limit = (rhs.dLen + 1) * 2
L(gn) 0 .< limit
I (rhs_dVal >> gn) [&] 1 == 1
copy.c(gn)
L (((copy.dVal >> ((copy.dLen * 2) [&] 31)) [&] 3) == 0) | (copy.dLen == 0)
copy.dLen = copy.dLen - 1
R copy
F *(rhs)
V na = copy(rhs)
V nb = copy(rhs)
V nr = Zeckendorf()
V dVal = .dVal
L(i) 0 .< (.dLen + 1) * 2
I ((dVal >> i) [&] 1) > 0
nr = nr + nb
V nt = copy(nb)
nb = nb + na
na = copy(nt)
R nr
F String()
V dig = [‘00’, ‘01’, ‘10’]
V dig1 = [‘’, ‘1’, ‘10’]
I .dVal == 0
R ‘0’
V idx = (.dVal >> ((.dLen * 2) [&] 31)) [&] 3
String sb = dig1[idx]
V i = .dLen - 1
L i >= 0
idx = (.dVal >> (i * 2)) [&] 3
sb ‘’= dig[idx]
i = i - 1
R sb
print(‘Addition:’)
V g = Zeckendorf(‘10’)
g = g + Zeckendorf(‘10’)
print(g)
g = g + Zeckendorf(‘10’)
print(g)
g = g + Zeckendorf(‘1001’)
print(g)
g = g + Zeckendorf(‘1000’)
print(g)
g = g + Zeckendorf(‘10101’)
print(g)
print()
print(‘Subtraction:’)
g = Zeckendorf(‘1000’)
g = g - Zeckendorf(‘101’)
print(g)
g = Zeckendorf(‘10101010’)
g = g - Zeckendorf(‘1010101’)
print(g)
print()
print(‘Multiplication:’)
g = Zeckendorf(‘1001’)
g = g * Zeckendorf(‘101’)
print(g)
g = Zeckendorf(‘101010’)
g = g + Zeckendorf(‘101’)
print(g)- Output:
Addition: 101 1001 10101 100101 1010000 Subtraction: 1 1000000 Multiplication: 1000100 1000100
C
#include <stdbool.h>
#include <stdio.h>
#include <string.h>
int inv(int a) {
return a ^ -1;
}
struct Zeckendorf {
int dVal, dLen;
};
void a(struct Zeckendorf *self, int n) {
void b(struct Zeckendorf *, int); // forward declare
int i = n;
while (true) {
if (self->dLen < i) self->dLen = i;
int j = (self->dVal >> (i * 2)) & 3;
switch (j) {
case 0:
case 1:
return;
case 2:
if (((self->dVal >> ((i + 1) * 2)) & 1) != 1) return;
self->dVal += 1 << (i * 2 + 1);
return;
case 3:
self->dVal = self->dVal & inv(3 << (i * 2));
b(self, (i + 1) * 2);
break;
default:
break;
}
i++;
}
}
void b(struct Zeckendorf *self, int pos) {
void increment(struct Zeckendorf *); // forward declare
if (pos == 0) {
increment(self);
return;
}
if (((self->dVal >> pos) & 1) == 0) {
self->dVal += 1 << pos;
a(self, pos / 2);
if (pos > 1) a(self, pos / 2 - 1);
} else {
self->dVal = self->dVal & inv(1 << pos);
b(self, pos + 1);
b(self, pos - (pos > 1 ? 2 : 1));
}
}
void c(struct Zeckendorf *self, int pos) {
if (((self->dVal >> pos) & 1) == 1) {
self->dVal = self->dVal & inv(1 << pos);
return;
}
c(self, pos + 1);
if (pos > 0) {
b(self, pos - 1);
} else {
increment(self);
}
}
struct Zeckendorf makeZeckendorf(char *x) {
struct Zeckendorf z = { 0, 0 };
int i = strlen(x) - 1;
int q = 1;
z.dLen = i / 2;
while (i >= 0) {
z.dVal += (x[i] - '0') * q;
q *= 2;
i--;
}
return z;
}
void increment(struct Zeckendorf *self) {
self->dVal++;
a(self, 0);
}
void addAssign(struct Zeckendorf *self, struct Zeckendorf rhs) {
int gn;
for (gn = 0; gn < (rhs.dLen + 1) * 2; gn++) {
if (((rhs.dVal >> gn) & 1) == 1) {
b(self, gn);
}
}
}
void subAssign(struct Zeckendorf *self, struct Zeckendorf rhs) {
int gn;
for (gn = 0; gn < (rhs.dLen + 1) * 2; gn++) {
if (((rhs.dVal >> gn) & 1) == 1) {
c(self, gn);
}
}
while ((((self->dVal >> self->dLen * 2) & 3) == 0) || (self->dLen == 0)) {
self->dLen--;
}
}
void mulAssign(struct Zeckendorf *self, struct Zeckendorf rhs) {
struct Zeckendorf na = rhs;
struct Zeckendorf nb = rhs;
struct Zeckendorf nr = makeZeckendorf("0");
struct Zeckendorf nt;
int i;
for (i = 0; i < (self->dLen + 1) * 2; i++) {
if (((self->dVal >> i) & 1) > 0) addAssign(&nr, nb);
nt = nb;
addAssign(&nb, na);
na = nt;
}
*self = nr;
}
void printZeckendorf(struct Zeckendorf z) {
static const char *const dig[3] = { "00", "01", "10" };
static const char *const dig1[3] = { "", "1", "10" };
if (z.dVal == 0) {
printf("0");
return;
} else {
int idx = (z.dVal >> (z.dLen * 2)) & 3;
int i;
printf(dig1[idx]);
for (i = z.dLen - 1; i >= 0; i--) {
idx = (z.dVal >> (i * 2)) & 3;
printf(dig[idx]);
}
}
}
int main() {
struct Zeckendorf g;
printf("Addition:\n");
g = makeZeckendorf("10");
addAssign(&g, makeZeckendorf("10"));
printZeckendorf(g);
printf("\n");
addAssign(&g, makeZeckendorf("10"));
printZeckendorf(g);
printf("\n");
addAssign(&g, makeZeckendorf("1001"));
printZeckendorf(g);
printf("\n");
addAssign(&g, makeZeckendorf("1000"));
printZeckendorf(g);
printf("\n");
addAssign(&g, makeZeckendorf("10101"));
printZeckendorf(g);
printf("\n\n");
printf("Subtraction:\n");
g = makeZeckendorf("1000");
subAssign(&g, makeZeckendorf("101"));
printZeckendorf(g);
printf("\n");
g = makeZeckendorf("10101010");
subAssign(&g, makeZeckendorf("1010101"));
printZeckendorf(g);
printf("\n\n");
printf("Multiplication:\n");
g = makeZeckendorf("1001");
mulAssign(&g, makeZeckendorf("101"));
printZeckendorf(g);
printf("\n");
g = makeZeckendorf("101010");
addAssign(&g, makeZeckendorf("101"));
printZeckendorf(g);
printf("\n");
return 0;
}
- Output:
Addition: 101 1001 10101 100101 1010000 Subtraction: 1 1000000 Multiplication: 1000100 1000100
C#
using System;
using System.Text;
namespace ZeckendorfArithmetic {
class Zeckendorf : IComparable<Zeckendorf> {
private static readonly string[] dig = { "00", "01", "10" };
private static readonly string[] dig1 = { "", "1", "10" };
private int dVal = 0;
private int dLen = 0;
public Zeckendorf() : this("0") {
// empty
}
public Zeckendorf(string x) {
int q = 1;
int i = x.Length - 1;
dLen = i / 2;
while (i >= 0) {
dVal += (x[i] - '0') * q;
q *= 2;
i--;
}
}
private void A(int n) {
int i = n;
while (true) {
if (dLen < i) dLen = i;
int j = (dVal >> (i * 2)) & 3;
switch (j) {
case 0:
case 1:
return;
case 2:
if (((dVal >> ((i + 1) * 2)) & 1) != 1) return;
dVal += 1 << (i * 2 + 1);
return;
case 3:
int temp = 3 << (i * 2);
temp ^= -1;
dVal = dVal & temp;
B((i + 1) * 2);
break;
}
i++;
}
}
private void B(int pos) {
if (pos == 0) {
Inc();
return;
}
if (((dVal >> pos) & 1) == 0) {
dVal += 1 << pos;
A(pos / 2);
if (pos > 1) A(pos / 2 - 1);
}
else {
int temp = 1 << pos;
temp ^= -1;
dVal = dVal & temp;
B(pos + 1);
B(pos - (pos > 1 ? 2 : 1));
}
}
private void C(int pos) {
if (((dVal >> pos) & 1) == 1) {
int temp = 1 << pos;
temp ^= -1;
dVal = dVal & temp;
return;
}
C(pos + 1);
if (pos > 0) {
B(pos - 1);
}
else {
Inc();
}
}
public Zeckendorf Inc() {
dVal++;
A(0);
return this;
}
public Zeckendorf Copy() {
Zeckendorf z = new Zeckendorf {
dVal = dVal,
dLen = dLen
};
return z;
}
public void PlusAssign(Zeckendorf other) {
for (int gn = 0; gn < (other.dLen + 1) * 2; gn++) {
if (((other.dVal >> gn) & 1) == 1) {
B(gn);
}
}
}
public void MinusAssign(Zeckendorf other) {
for (int gn = 0; gn < (other.dLen + 1) * 2; gn++) {
if (((other.dVal >> gn) & 1) == 1) {
C(gn);
}
}
while ((((dVal >> dLen * 2) & 3) == 0) || (dLen == 0)) {
dLen--;
}
}
public void TimesAssign(Zeckendorf other) {
Zeckendorf na = other.Copy();
Zeckendorf nb = other.Copy();
Zeckendorf nt;
Zeckendorf nr = new Zeckendorf();
for (int i = 0; i < (dLen + 1) * 2; i++) {
if (((dVal >> i) & 1) > 0) {
nr.PlusAssign(nb);
}
nt = nb.Copy();
nb.PlusAssign(na);
na = nt.Copy();
}
dVal = nr.dVal;
dLen = nr.dLen;
}
public int CompareTo(Zeckendorf other) {
return dVal.CompareTo(other.dVal);
}
public override string ToString() {
if (dVal == 0) {
return "0";
}
int idx = (dVal >> (dLen * 2)) & 3;
StringBuilder sb = new StringBuilder(dig1[idx]);
for (int i = dLen - 1; i >= 0; i--) {
idx = (dVal >> (i * 2)) & 3;
sb.Append(dig[idx]);
}
return sb.ToString();
}
}
class Program {
static void Main(string[] args) {
Console.WriteLine("Addition:");
Zeckendorf g = new Zeckendorf("10");
g.PlusAssign(new Zeckendorf("10"));
Console.WriteLine(g);
g.PlusAssign(new Zeckendorf("10"));
Console.WriteLine(g);
g.PlusAssign(new Zeckendorf("1001"));
Console.WriteLine(g);
g.PlusAssign(new Zeckendorf("1000"));
Console.WriteLine(g);
g.PlusAssign(new Zeckendorf("10101"));
Console.WriteLine(g);
Console.WriteLine();
Console.WriteLine("Subtraction:");
g = new Zeckendorf("1000");
g.MinusAssign(new Zeckendorf("101"));
Console.WriteLine(g);
g = new Zeckendorf("10101010");
g.MinusAssign(new Zeckendorf("1010101"));
Console.WriteLine(g);
Console.WriteLine();
Console.WriteLine("Multiplication:");
g = new Zeckendorf("1001");
g.TimesAssign(new Zeckendorf("101"));
Console.WriteLine(g);
g = new Zeckendorf("101010");
g.PlusAssign(new Zeckendorf("101"));
Console.WriteLine(g);
}
}
}
- Output:
Addition: 101 1001 10101 100101 1010000 Subtraction: 1 1000000 Multiplication: 1000100 1000100
C++
// For a class N which implements Zeckendorf numbers:
// I define an increment operation ++()
// I define a comparison operation <=(other N)
// I define an addition operation +=(other N)
// I define a subtraction operation -=(other N)
// Nigel Galloway October 28th., 2012
#include <iostream>
enum class zd {N00,N01,N10,N11};
class N {
private:
int dVal = 0, dLen;
void _a(int i) {
for (;; i++) {
if (dLen < i) dLen = i;
switch ((zd)((dVal >> (i*2)) & 3)) {
case zd::N00: case zd::N01: return;
case zd::N10: if (((dVal >> ((i+1)*2)) & 1) != 1) return;
dVal += (1 << (i*2+1)); return;
case zd::N11: dVal &= ~(3 << (i*2)); _b((i+1)*2);
}}}
void _b(int pos) {
if (pos == 0) {++*this; return;}
if (((dVal >> pos) & 1) == 0) {
dVal += 1 << pos;
_a(pos/2);
if (pos > 1) _a((pos/2)-1);
} else {
dVal &= ~(1 << pos);
_b(pos + 1);
_b(pos - ((pos > 1)? 2:1));
}}
void _c(int pos) {
if (((dVal >> pos) & 1) == 1) {dVal &= ~(1 << pos); return;}
_c(pos + 1);
if (pos > 0) _b(pos - 1); else ++*this;
return;
}
public:
N(char const* x = "0") {
int i = 0, q = 1;
for (; x[i] > 0; i++);
for (dLen = --i/2; i >= 0; i--) {dVal+=(x[i]-48)*q; q*=2;
}}
const N& operator++() {dVal += 1; _a(0); return *this;}
const N& operator+=(const N& other) {
for (int GN = 0; GN < (other.dLen + 1) * 2; GN++) if ((other.dVal >> GN) & 1 == 1) _b(GN);
return *this;
}
const N& operator-=(const N& other) {
for (int GN = 0; GN < (other.dLen + 1) * 2; GN++) if ((other.dVal >> GN) & 1 == 1) _c(GN);
for (;((dVal >> dLen*2) & 3) == 0 or dLen == 0; dLen--);
return *this;
}
const N& operator*=(const N& other) {
N Na = other, Nb = other, Nt, Nr;
for (int i = 0; i <= (dLen + 1) * 2; i++) {
if (((dVal >> i) & 1) > 0) Nr += Nb;
Nt = Nb; Nb += Na; Na = Nt;
}
return *this = Nr;
}
const bool operator<=(const N& other) const {return dVal <= other.dVal;}
friend std::ostream& operator<<(std::ostream&, const N&);
};
N operator "" N(char const* x) {return N(x);}
std::ostream &operator<<(std::ostream &os, const N &G) {
const static std::string dig[] {"00","01","10"}, dig1[] {"","1","10"};
if (G.dVal == 0) return os << "0";
os << dig1[(G.dVal >> (G.dLen*2)) & 3];
for (int i = G.dLen-1; i >= 0; i--) os << dig[(G.dVal >> (i*2)) & 3];
return os;
}
Testing
The following tests addtition:
int main(void) {
N G;
G = 10N;
G += 10N;
std::cout << G << std::endl;
G += 10N;
std::cout << G << std::endl;
G += 1001N;
std::cout << G << std::endl;
G += 1000N;
std::cout << G << std::endl;
G += 10101N;
std::cout << G << std::endl;
return 0;
}
- Output:
101 1001 10101 100101 1010000
The following tests subtraction:
int main(void) {
N G;
G = 1000N;
G -= 101N;
std::cout << G << std::endl;
G = 10101010N;
G -= 1010101N;
std::cout << G << std::endl;
return 0;
}
- Output:
1 1000000
The following tests multiplication:
int main(void) {
N G = 1001N;
G *= 101N;
std::cout << G << std::endl;
G = 101010N;
G += 101N;
std::cout << G << std::endl;
return 0;
}
- Output:
1000100 1000100
D
import std.stdio;
int inv(int a) {
return a ^ -1;
}
class Zeckendorf {
private int dVal;
private int dLen;
private void a(int n) {
auto i = n;
while (true) {
if (dLen < i) dLen = i;
auto j = (dVal >> (i * 2)) & 3;
switch(j) {
case 0:
case 1:
return;
case 2:
if (((dVal >> ((i + 1) * 2)) & 1) != 1) return;
dVal += 1 << (i * 2 + 1);
return;
case 3:
dVal = dVal & (3 << (i * 2)).inv();
b((i + 1) * 2);
break;
default:
assert(false);
}
i++;
}
}
private void b(int pos) {
if (pos == 0) {
this++;
return;
}
if (((dVal >> pos) & 1) == 0) {
dVal += 1 << pos;
a(pos / 2);
if (pos > 1) a(pos / 2 - 1);
} else {
dVal = dVal & (1 << pos).inv();
b(pos + 1);
b(pos - (pos > 1 ? 2 : 1));
}
}
private void c(int pos) {
if (((dVal >> pos) & 1) == 1) {
dVal = dVal & (1 << pos).inv();
return;
}
c(pos + 1);
if (pos > 0) {
b(pos - 1);
} else {
++this;
}
}
this(string x = "0") {
int q = 1;
int i = x.length - 1;
dLen = i / 2;
while (i >= 0) {
dVal += (x[i] - '0') * q;
q *= 2;
i--;
}
}
auto opUnary(string op : "++")() {
dVal += 1;
a(0);
return this;
}
void opOpAssign(string op : "+")(Zeckendorf rhs) {
foreach (gn; 0..(rhs.dLen + 1) * 2) {
if (((rhs.dVal >> gn) & 1) == 1) {
b(gn);
}
}
}
void opOpAssign(string op : "-")(Zeckendorf rhs) {
foreach (gn; 0..(rhs.dLen + 1) * 2) {
if (((rhs.dVal >> gn) & 1) == 1) {
c(gn);
}
}
while ((((dVal >> dLen * 2) & 3) == 0) || (dLen == 0)) {
dLen--;
}
}
void opOpAssign(string op : "*")(Zeckendorf rhs) {
auto na = rhs.dup;
auto nb = rhs.dup;
Zeckendorf nt;
auto nr = "0".Z;
foreach (i; 0..(dLen + 1) * 2) {
if (((dVal >> i) & 1) > 0) nr += nb;
nt = nb.dup;
nb += na;
na = nt.dup;
}
dVal = nr.dVal;
dLen = nr.dLen;
}
void toString(scope void delegate(const(char)[]) sink) const {
if (dVal == 0) {
sink("0");
return;
}
sink(dig1[(dVal >> (dLen * 2)) & 3]);
foreach_reverse (i; 0..dLen) {
sink(dig[(dVal >> (i * 2)) & 3]);
}
}
Zeckendorf dup() {
auto z = "0".Z;
z.dVal = dVal;
z.dLen = dLen;
return z;
}
enum dig = ["00", "01", "10"];
enum dig1 = ["", "1", "10"];
}
auto Z(string val) {
return new Zeckendorf(val);
}
void main() {
writeln("Addition:");
auto g = "10".Z;
g += "10".Z;
writeln(g);
g += "10".Z;
writeln(g);
g += "1001".Z;
writeln(g);
g += "1000".Z;
writeln(g);
g += "10101".Z;
writeln(g);
writeln();
writeln("Subtraction:");
g = "1000".Z;
g -= "101".Z;
writeln(g);
g = "10101010".Z;
g -= "1010101".Z;
writeln(g);
writeln();
writeln("Multiplication:");
g = "1001".Z;
g *= "101".Z;
writeln(g);
g = "101010".Z;
g += "101".Z;
writeln(g);
}
- Output:
Addition: 101 1001 10101 100101 1010000 Subtraction: 1 1000000 Multiplication: 1000100 1000100
Dart
class Zeckendorf {
int dVal = 0;
int dLen = 0;
Zeckendorf(String x) {
var q = 1;
var i = x.length - 1;
dLen = i ~/ 2;
while (i >= 0) {
dVal += (x[i].codeUnitAt(0) - '0'.codeUnitAt(0)) * q;
q *= 2;
i--;
}
}
void a(int n) {
var i = n;
while (true) {
if (dLen < i) dLen = i;
var j = (dVal >> (i * 2)) & 3;
switch (j) {
case 0:
case 1:
return;
case 2:
if (((dVal >> ((i + 1) * 2)) & 1) != 1) return;
dVal += 1 << (i * 2 + 1);
return;
case 3:
dVal &= ~(3 << (i * 2));
b((i + 1) * 2);
break;
}
i++;
}
}
void b(int pos) {
if (pos == 0) {
this.increment();
return;
}
if (((dVal >> pos) & 1) == 0) {
dVal += 1 << pos;
a(pos ~/ 2);
if (pos > 1) a(pos ~/ 2 - 1);
} else {
dVal &= ~(1 << pos);
b(pos + 1);
b(pos - (pos > 1 ? 2 : 1));
}
}
void c(int pos) {
if (((dVal >> pos) & 1) == 1) {
dVal &= ~(1 << pos);
return;
}
c(pos + 1);
if (pos > 0)
b(pos - 1);
else
this.increment();
}
Zeckendorf increment() {
dVal += 1;
a(0);
return this;
}
void operator + (Zeckendorf other) {
for (var gn = 0; gn < (other.dLen + 1) * 2; gn++) {
if (((other.dVal >> gn) & 1) == 1) b(gn);
}
}
void operator - (Zeckendorf other) {
for (var gn = 0; gn < (other.dLen + 1) * 2; gn++) {
if (((other.dVal >> gn) & 1) == 1) c(gn);
}
while (dLen > 0 && (((dVal >> dLen * 2) & 3) == 0)) dLen--;
}
void operator * (Zeckendorf other) {
var na = other.copy();
var nb = other.copy();
Zeckendorf nt;
var nr = Zeckendorf("0");
for (var i = 0; i <= (dLen + 1) * 2; i++) {
if (((dVal >> i) & 1) > 0) nr + nb;
nt = nb.copy();
nb + na;
na = nt.copy();
}
dVal = nr.dVal;
dLen = nr.dLen;
}
int compareTo(Zeckendorf other) {
return dVal.compareTo(other.dVal);
}
@override
String toString() {
if (dVal == 0) return "0";
var sb = StringBuffer(dig1[(dVal >> (dLen * 2)) & 3]);
for (var i = dLen - 1; i >= 0; i--) {
sb.write(dig[(dVal >> (i * 2)) & 3]);
}
return sb.toString();
}
Zeckendorf copy() {
var z = Zeckendorf("0");
z.dVal = dVal;
z.dLen = dLen;
return z;
}
static final List<String> dig = ["00", "01", "10"];
static final List<String> dig1 = ["", "1", "10"];
}
void main() {
print("Addition:");
var g = Zeckendorf("10");
g + Zeckendorf("10");
print(g);
g + Zeckendorf("10");
print(g);
g + Zeckendorf("1001");
print(g);
g + Zeckendorf("1000");
print(g);
g + Zeckendorf("10101");
print(g);
print("\nSubtraction:");
g = Zeckendorf("1000");
g - Zeckendorf("101");
print(g);
g = Zeckendorf("10101010");
g - Zeckendorf("1010101");
print(g);
print("\nMultiplication:");
g = Zeckendorf("1001");
g * Zeckendorf("101");
print(g);
g = Zeckendorf("101010");
g + Zeckendorf("101");
print(g);
}
- Output:
Addition: 101 1001 10101 100101 1010000 Subtraction: 1 1000000 Multiplication: 1000100 1000100
Elena
ELENA 6.x :
import extensions;
const dig = new string[]{"00","01","10"};
const dig1 = new string[]{"","1","10"};
sealed struct ZeckendorfNumber
{
int dVal;
int dLen;
clone()
= ZeckendorfNumber.newInternal(dVal,dLen);
cast n(string s)
{
int i := s.Length - 1;
int q := 1;
dLen := i / 2;
dVal := 0;
while (i >= 0)
{
dVal += ((intConvertor.convert(s[i]) - 48) * q);
q *= 2;
i -= 1
}
}
internal readContent(ref int val, ref int len)
{
val := dVal;
len := dLen;
}
private a(int n)
{
int i := n;
while (true)
{
if (dLen < i)
{
dLen := i
};
int v := (dVal $shr (i * 2)) & 3;
v =>
0 { ^ self }
1 { ^ self }
2 {
ifnot ((dVal $shr ((i + 1) * 2)).allMask(1))
{
^ self
};
dVal += (1 $shl (i*2 + 1));
^ self
}
3 {
int tmp := 3 $shl (i * 2);
tmp := tmp.bxor(-1);
dVal := dVal & tmp;
self.b((i+1)*2)
};
i += 1
}
}
inc()
{
dVal += 1;
self.a(0)
}
private b(int pos)
{
if (pos == 0) { ^ self.inc() };
ifnot((dVal $shr pos).allMask(1))
{
dVal += (1 $shl pos);
self.a(pos / 2);
if (pos > 1) { self.a((pos / 2) - 1) }
}
else
{
dVal := dVal & (1 $shl pos).BInverted;
self.b(pos + 1);
int arg := pos - ((pos > 1) ? 2 : 1);
self.b(/*pos - ((pos > 1) ? 2 : 1)*/arg)
}
}
private c(int pos)
{
if ((dVal $shr pos).allMask(1))
{
int tmp := 1 $shl pos;
tmp := tmp.bxor(-1);
dVal := dVal & tmp;
^ self
};
self.c(pos + 1);
if (pos > 0)
{
self.b(pos - 1)
}
else
{
self.inc()
}
}
internal constructor sum(ZeckendorfNumber n, ZeckendorfNumber m)
{
int mVal := 0;
int mLen := 0;
n.readContent(ref int v, ref int l);
m.readContent(ref mVal, ref mLen);
dVal := v;
dLen := l;
for(int GN := 0; GN < (mLen + 1) * 2; GN += 1)
{
if ((mVal $shr GN).allMask(1))
{
self.b(GN)
}
}
}
internal constructor difference(ZeckendorfNumber n, ZeckendorfNumber m)
{
int mVal := 0;
int mLen := 0;
n.readContent(ref int v, ref int l);
m.readContent(ref mVal, ref mLen);
dVal := v;
dLen := l;
for(int GN := 0; GN < (mLen + 1) * 2; GN += 1)
{
if ((mVal $shr GN).allMask(1))
{
self.c(GN)
}
};
while (((dVal $shr (dLen*2)) & 3) == 0 || dLen == 0)
{
dLen -= 1
}
}
internal constructor product(ZeckendorfNumber n, ZeckendorfNumber m)
{
n.readContent(ref int v, ref int l);
dVal := v;
dLen := l;
ZeckendorfNumber Na := m;
ZeckendorfNumber Nb := m;
ZeckendorfNumber Nr := 0n;
ZeckendorfNumber Nt := 0n;
for(int i := 0; i < (dLen + 1) * 2; i += 1)
{
if (((dVal $shr i) & 1) > 0)
{
Nr += Nb
};
Nt := Nb;
Nb += Na;
Na := Nt
};
Nr.readContent(ref v, ref l);
dVal := v;
dLen := l;
}
internal constructor newInternal(int v, int l)
{
dVal := v;
dLen := l
}
string toPrintable()
{
if (dVal == 0)
{ ^ "0" };
string s := dig1[(dVal $shr (dLen * 2)) & 3];
int i := dLen - 1;
while (i >= 0)
{
s := s + dig[(dVal $shr (i * 2)) & 3];
i-=1
};
^ s
}
add(ZeckendorfNumber n)
= ZeckendorfNumber.sum(self, n);
subtract(ZeckendorfNumber n)
= ZeckendorfNumber.difference(self, n);
multiply(ZeckendorfNumber n)
= ZeckendorfNumber.product(self, n);
}
public program()
{
console.printLine("Addition:");
var n := 10n;
n += 10n;
console.printLine(n);
n += 10n;
console.printLine(n);
n += 1001n;
console.printLine(n);
n += 1000n;
console.printLine(n);
n += 10101n;
console.printLine(n);
console.printLine("Subtraction:");
n := 1000n;
n -= 101n;
console.printLine(n);
n := 10101010n;
n -= 1010101n;
console.printLine(n);
console.printLine("Multiplication:");
n := 1001n;
n *= 101n;
console.printLine(n);
n := 101010n;
n += 101n;
console.printLine(n)
}- Output:
Addition: 101 1001 10101 100101 1010000 Subtraction: 1 1000000 Multiplication: 1000100 1000100
FreeBASIC
Type Zeckendorf
As Integer dLen
As Ulongint dVal
Declare Constructor()
Declare Constructor(x As String)
Declare Sub a(n As Integer)
Declare Sub b(Pos As Integer)
Declare Sub c(Pos As Integer)
Declare Sub inc()
Declare Function suma(rhs As Zeckendorf) As Zeckendorf
Declare Function resta(rhs As Zeckendorf) As Zeckendorf
Declare Function producto(rhs As Zeckendorf) As Zeckendorf
Declare Function toString() As String
End Type
Constructor Zeckendorf()
This.dLen = 0
This.dVal = 0
End Constructor
Constructor Zeckendorf(x As String)
Dim As Ulongint q = 1
Dim As Integer i = Len(x) - 1
This.dLen = Int(i / 2)
This.dVal = 0
While i >= 0
This.dVal += (Asc(Mid(x, i+1, 1)) - Asc("0")) * q
q *= 2
i -= 1
Wend
End Constructor
Sub Zeckendorf.a(n As Integer)
Dim As Integer i = n
Do
If This.dLen < i Then This.dLen = i
Dim As Integer j = (This.dVal Shr (i * 2)) And 3
If j = 0 Or j = 1 Then Exit Sub
If j = 2 Then
If ((This.dVal Shr ((i + 1) * 2)) And 1) <> 1 Then Exit Sub
This.dVal += 1ULL Shl (i * 2 + 1)
Exit Sub
End If
If j = 3 Then
Dim As Ulongint temp = 3ULL Shl (i * 2)
temp Xor= &HFFFFFFFFFFFFFFFFULL
This.dVal And= temp
This.b((i + 1) * 2)
End If
i += 1
Loop
End Sub
Sub Zeckendorf.b(posic As Integer)
If posic = 0 Then
This.inc()
Exit Sub
End If
If ((This.dVal Shr posic) And 1) = 0 Then
This.dVal += 1ULL Shl posic
This.a(Int(posic / 2))
If posic > 1 Then This.a(Int(posic / 2) - 1)
Else
Dim As Ulongint temp = 1ULL Shl posic
temp Xor= &HFFFFFFFFFFFFFFFFULL
This.dVal And= temp
This.b(posic + 1)
This.b(posic - Iif(posic > 1, 2, 1))
End If
End Sub
Sub Zeckendorf.c(posic As Integer)
If ((This.dVal Shr posic) And 1) = 1 Then
Dim As Ulongint temp = 1ULL Shl posic
temp Xor= &HFFFFFFFFFFFFFFFFULL
This.dVal And= temp
Exit Sub
End If
This.c(posic + 1)
If posic > 0 Then
This.b(posic - 1)
Else
This.inc()
End If
End Sub
Sub Zeckendorf.inc()
This.dVal += 1
This.a(0)
End Sub
Function Zeckendorf.suma(rhs As Zeckendorf) As Zeckendorf
Dim As Zeckendorf copy = This
Dim As Ulongint rhs_dVal = rhs.dVal
Dim As Integer limit = (rhs.dLen + 1) * 2
For gn As Integer = 0 To limit - 1
If ((rhs_dVal Shr gn) And 1) = 1 Then copy.b(gn)
Next
Return copy
End Function
Function Zeckendorf.resta(rhs As Zeckendorf) As Zeckendorf
Dim As Zeckendorf copy = This
Dim As Ulongint rhs_dVal = rhs.dVal
Dim As Integer limit = (rhs.dLen + 1) * 2
For gn As Integer = 0 To limit - 1
If ((rhs_dVal Shr gn) And 1) = 1 Then copy.c(gn)
Next
While (((copy.dVal Shr ((copy.dLen * 2) And 31)) And 3) = 0) Or (copy.dLen = 0)
copy.dLen -= 1
Wend
Return copy
End Function
Function Zeckendorf.producto(rhs As Zeckendorf) As Zeckendorf
Dim As Zeckendorf na = rhs, nb = rhs, nr
Dim As Ulongint dVal = This.dVal
For i As Integer = 0 To (This.dLen + 1) * 2 - 1
If ((dVal Shr i) And 1) > 0 Then nr = nr.suma(nb)
Dim As Zeckendorf nt = nb
nb = nb.suma(na)
na = nt
Next
Return nr
End Function
Function Zeckendorf.toString() As String
Dim As String dig(2) = {"00", "01", "10"}
Dim As String dig1(2) = {"", "1", "10"}
If This.dVal = 0 Then Return "0"
Dim As Integer idx = (This.dVal Shr ((This.dLen * 2) And 31)) And 3
Dim As String sb = dig1(idx)
For i As Integer = This.dLen - 1 To 0 Step -1
idx = (This.dVal Shr (i * 2)) And 3
sb &= dig(idx)
Next
Return sb
End Function
' Main
Print "Addition:"
Dim As Zeckendorf g = Zeckendorf("10")
g = g.suma(Zeckendorf("10"))
Print g.toString()
g = g.suma(Zeckendorf("10"))
Print g.toString()
g = g.suma(Zeckendorf("1001"))
Print g.toString()
g = g.suma(Zeckendorf("1000"))
Print g.toString()
g = g.suma(Zeckendorf("10101"))
Print g.toString()
Print
Print "Subtraction:"
g = Zeckendorf("1000")
g = g.resta(Zeckendorf("101"))
Print g.toString()
g = Zeckendorf("10101010")
g = g.resta(Zeckendorf("1010101"))
Print g.toString()
Print
Print "Multiplication:"
g = Zeckendorf("1001")
g = g.producto(Zeckendorf("101"))
Print g.toString()
g = Zeckendorf("101010")
g = g.suma(Zeckendorf("101"))
Print g.toString()
Sleep
- Output:
Addition: 101 1001 10101 100101 1010000 Subtraction: 1 1000000 Multiplication: 1000100 1000100
Go
package main
import (
"fmt"
"strings"
)
var (
dig = [3]string{"00", "01", "10"}
dig1 = [3]string{"", "1", "10"}
)
type Zeckendorf struct{ dVal, dLen int }
func NewZeck(x string) *Zeckendorf {
z := new(Zeckendorf)
if x == "" {
x = "0"
}
q := 1
i := len(x) - 1
z.dLen = i / 2
for ; i >= 0; i-- {
z.dVal += int(x[i]-'0') * q
q *= 2
}
return z
}
func (z *Zeckendorf) a(i int) {
for ; ; i++ {
if z.dLen < i {
z.dLen = i
}
j := (z.dVal >> uint(i*2)) & 3
switch j {
case 0, 1:
return
case 2:
if ((z.dVal >> (uint(i+1) * 2)) & 1) != 1 {
return
}
z.dVal += 1 << uint(i*2+1)
return
case 3:
z.dVal &= ^(3 << uint(i*2))
z.b((i + 1) * 2)
}
}
}
func (z *Zeckendorf) b(pos int) {
if pos == 0 {
z.Inc()
return
}
if ((z.dVal >> uint(pos)) & 1) == 0 {
z.dVal += 1 << uint(pos)
z.a(pos / 2)
if pos > 1 {
z.a(pos/2 - 1)
}
} else {
z.dVal &= ^(1 << uint(pos))
z.b(pos + 1)
temp := 1
if pos > 1 {
temp = 2
}
z.b(pos - temp)
}
}
func (z *Zeckendorf) c(pos int) {
if ((z.dVal >> uint(pos)) & 1) == 1 {
z.dVal &= ^(1 << uint(pos))
return
}
z.c(pos + 1)
if pos > 0 {
z.b(pos - 1)
} else {
z.Inc()
}
}
func (z *Zeckendorf) Inc() {
z.dVal++
z.a(0)
}
func (z1 *Zeckendorf) PlusAssign(z2 *Zeckendorf) {
for gn := 0; gn < (z2.dLen+1)*2; gn++ {
if ((z2.dVal >> uint(gn)) & 1) == 1 {
z1.b(gn)
}
}
}
func (z1 *Zeckendorf) MinusAssign(z2 *Zeckendorf) {
for gn := 0; gn < (z2.dLen+1)*2; gn++ {
if ((z2.dVal >> uint(gn)) & 1) == 1 {
z1.c(gn)
}
}
for z1.dLen > 0 && ((z1.dVal>>uint(z1.dLen*2))&3) == 0 {
z1.dLen--
}
}
func (z1 *Zeckendorf) TimesAssign(z2 *Zeckendorf) {
na := z2.Copy()
nb := z2.Copy()
nr := new(Zeckendorf)
for i := 0; i <= (z1.dLen+1)*2; i++ {
if ((z1.dVal >> uint(i)) & 1) > 0 {
nr.PlusAssign(nb)
}
nt := nb.Copy()
nb.PlusAssign(na)
na = nt.Copy()
}
z1.dVal = nr.dVal
z1.dLen = nr.dLen
}
func (z *Zeckendorf) Copy() *Zeckendorf {
return &Zeckendorf{z.dVal, z.dLen}
}
func (z1 *Zeckendorf) Compare(z2 *Zeckendorf) int {
switch {
case z1.dVal < z2.dVal:
return -1
case z1.dVal > z2.dVal:
return 1
default:
return 0
}
}
func (z *Zeckendorf) String() string {
if z.dVal == 0 {
return "0"
}
var sb strings.Builder
sb.WriteString(dig1[(z.dVal>>uint(z.dLen*2))&3])
for i := z.dLen - 1; i >= 0; i-- {
sb.WriteString(dig[(z.dVal>>uint(i*2))&3])
}
return sb.String()
}
func main() {
fmt.Println("Addition:")
g := NewZeck("10")
g.PlusAssign(NewZeck("10"))
fmt.Println(g)
g.PlusAssign(NewZeck("10"))
fmt.Println(g)
g.PlusAssign(NewZeck("1001"))
fmt.Println(g)
g.PlusAssign(NewZeck("1000"))
fmt.Println(g)
g.PlusAssign(NewZeck("10101"))
fmt.Println(g)
fmt.Println("\nSubtraction:")
g = NewZeck("1000")
g.MinusAssign(NewZeck("101"))
fmt.Println(g)
g = NewZeck("10101010")
g.MinusAssign(NewZeck("1010101"))
fmt.Println(g)
fmt.Println("\nMultiplication:")
g = NewZeck("1001")
g.TimesAssign(NewZeck("101"))
fmt.Println(g)
g = NewZeck("101010")
g.PlusAssign(NewZeck("101"))
fmt.Println(g)
}
- Output:
Addition: 101 1001 10101 100101 1010000 Subtraction: 1 1000000 Multiplication: 1000100 1000100
Haskell
We make Zeckendorf numbers first class citizens implementing instances of Eq, Ord, Num, Enum, Real and Integral classes. So everything that could be done with integral numbers is applicable with Zeckendorf numbers.
Addition and subtraction are done using cellular automata. Conversion from integers, multiplication and division are implemented via generalized Fibonacci series (Zeckendorf tables).
{-# LANGUAGE LambdaCase #-}
import Data.List (find, mapAccumL)
import Control.Arrow (first, second)
-- Generalized Fibonacci series defined for any Num instance, and for Zeckendorf numbers as well.
-- Used to build Zeckendorf tables.
fibs :: Num a => a -> a -> [a]
fibs a b = res
where
res = a : b : zipWith (+) res (tail res)
data Fib = Fib { sign :: Int, digits :: [Int]}
-- smart constructor
mkFib s ds =
case dropWhile (==0) ds of
[] -> 0
ds -> Fib s (reverse ds)
-- Textual representation
instance Show Fib where
show (Fib s ds) = sig s ++ foldMap show (reverse ds)
where sig = \case { -1 -> "-"; s -> "" }
-- Equivalence relation
instance Eq Fib where
Fib sa a == Fib sb b = sa == sb && a == b
-- Order relation
instance Ord Fib where
a `compare` b =
sign a `compare` sign b <>
case find (/= 0) $ alignWith (-) (digits a) (digits b) of
Nothing -> EQ
Just 1 -> if sign a > 0 then GT else LT
Just (-1) -> if sign a > 0 then LT else GT
-- Arithmetic
instance Num Fib where
negate (Fib s ds) = Fib (negate s) ds
abs (Fib s ds) = Fib 1 ds
signum (Fib s _) = fromIntegral s
fromInteger n =
case compare n 0 of
LT -> negate $ fromInteger (- n)
EQ -> Fib 0 [0]
GT -> Fib 1 . reverse . fst $ divModFib n 1
0 + a = a
a + 0 = a
a + b =
case (sign a, sign b) of
( 1, 1) -> res
(-1, 1) -> b - (-a)
( 1,-1) -> a - (-b)
(-1,-1) -> - ((- a) + (- b))
where
res = mkFib 1 . process $ 0:0:c
c = alignWith (+) (digits a) (digits b)
-- use cellular automata
process =
runRight 3 r2 . runLeftR 3 r2 . runRightR 4 r1
0 - a = -a
a - 0 = a
a - b =
case (sign a, sign b) of
( 1, 1) -> res
(-1, 1) -> - ((-a) + b)
( 1,-1) -> a + (-b)
(-1,-1) -> - ((-a) - (-b))
where
res = case find (/= 0) c of
Just 1 -> mkFib 1 . process $ c
Just (-1) -> - (b - a)
Nothing -> 0
c = alignWith (-) (digits a) (digits b)
-- use cellular automata
process =
runRight 3 r2 . runLeftR 3 r2 . runRightR 4 r1 . runRight 3 r3
0 * a = 0
a * 0 = 0
1 * a = a
a * 1 = a
a * b =
case (sign a, sign b) of
(1, 1) -> res
(-1, 1) -> - ((-a) * b)
( 1,-1) -> - (a * (-b))
(-1,-1) -> ((-a) * (-b))
where
-- use Zeckendorf table
table = fibs a (a + a)
res = sum $ onlyOnes $ zip (digits b) table
onlyOnes = map snd . filter ((==1) . fst)
-- Enumeration
instance Enum Fib where
toEnum = fromInteger . fromIntegral
fromEnum = fromIntegral . toInteger
instance Real Fib where
toRational = fromInteger . toInteger
-- Integral division
instance Integral Fib where
toInteger (Fib s ds) = signum (fromIntegral s) * res
where
res = sum (zipWith (*) (fibs 1 2) (fromIntegral <$> ds))
quotRem 0 _ = (0, 0)
quotRem a 0 = error "divide by zero"
quotRem a b = case (sign a, sign b) of
(1, 1) -> first (mkFib 1) $ divModFib a b
(-1, 1) -> second negate . first negate $ quotRem (-a) b
( 1,-1) -> first negate $ quotRem a (-b)
(-1,-1) -> second negate $ quotRem (-a) (-b)
------------------------------------------------------------
-- helper funtions
-- general division using Zeckendorf table
divModFib :: (Ord a, Num c, Num a) => a -> a -> ([c], a)
divModFib a b = (q, r)
where
(r, q) = mapAccumL f a $ reverse $ takeWhile (<= a) table
table = fibs b (b+b)
f n x = if n < x then (n, 0) else (n - x, 1)
-- application of rewriting rules
-- runs window from left to right
runRight n f = go
where
go [] = []
go lst = let (w, r) = splitAt n lst
(h: t) = f w
in h : go (t ++ r)
-- runs window from left to right and reverses the result
runRightR n f = go []
where
go res [] = res
go res lst = let (w, r) = splitAt n lst
(h: t) = f w
in go (h : res) (t ++ r)
-- runs reversed window and reverses the result
runLeftR n f = runRightR n (reverse . f . reverse)
-- rewriting rules from [C. Ahlbach et. all]
r1 = \case [0,3,0] -> [1,1,1]
[0,2,0] -> [1,0,1]
[0,1,2] -> [1,0,1]
[0,2,1] -> [1,1,0]
[x,0,2] -> [x,1,0]
[x,0,3] -> [x,1,1]
[0,1,2,0] -> [1,0,1,0]
[0,2,0,x] -> [1,0,0,x+1]
[0,3,0,x] -> [1,1,0,x+1]
[0,2,1,x] -> [1,1,0,x ]
[0,1,2,x] -> [1,0,1,x ]
l -> l
r2 = \case [0,1,1] -> [1,0,0]
l -> l
r3 = \case [1,-1] -> [0,1]
[2,-1] -> [1,1]
[1, 0, 0] -> [0,1,1]
[1,-1, 0] -> [0,0,1]
[1,-1, 1] -> [0,0,2]
[1, 0,-1] -> [0,1,0]
[2, 0, 0] -> [1,1,1]
[2,-1, 0] -> [1,0,1]
[2,-1, 1] -> [1,0,2]
[2, 0,-1] -> [1,1,0]
l -> l
alignWith :: (Int -> Int -> a) -> [Int] -> [Int] -> [a]
alignWith f a b = go [] a b
where
go res as [] = ((`f` 0) <$> reverse as) ++ res
go res [] bs = ((0 `f`) <$> reverse bs) ++ res
go res (a:as) (b:bs) = go (f a b : res) as bs
λ> 15 :: Fib 100010 λ> 153 :: Fib 10000010001 λ> [1..13] :: [Fib] [1,10,100,101,1000,1001,1010,10000,10001,10010,10100,10101,100000] λ> 15 + 47 :: Fib 100001010 λ> toInteger it 62 λ> 15 - 47 :: Fib -1010100 λ> toInteger it -32 λ> 15 * 47 :: Fib 10001000001001 λ> toInteger it 705 λ> 47 `div` 15 :: Fib 100 λ> 47 `mod` 15 :: Fib 10
J
Loosely based on the perl implementation:
zform=: {{ 10 |."1@(#.inv) y }} :. (10#.|."1) NB. use decimal numbers for representation
zinc=: {{ carry ({.,2}.])carry 1,y }}
zdec=: {{ (|.k$0 1),y }.~k=. 1+y i.1 }}
zadd=: {{ x while. 1 e. y do. x=. zinc x [ y=. zdec y end. }}
zsub=: {{ x while. 1 e. y do. x=. zdec x [ y=. zdec y end. }} NB. intended for unsigned arithmetic
zmul=: {{ t=. 0 0 while. 1 e. y do. t=. t zadd x [ y=. zdec y end. }}
zdiv=: {{ t=. 0 0 while. x zge y do. t=. zinc t [ x=. x zsub y end. }} NB. discards remainder
carry=: {{
s=. 0
for_b. y do.
if. (1+b) = s=. s-_1^b do. y=. (-.b) (b_index-0,b)} y end.
end.
if. 2=s do. y,1 else. y end.
}}
zge=: {{ cmp=. x -/@,: y while. (#cmp)*0={:cmp do. cmp=. }:cmp end. 0<:{:cmp }}
For example, we use the decimal number 10100 to represent 11 in base 10, and 1010 would represent 7. We convert these numbers to an internal zeckendorf representation and add them, then convert the result back to decimal 101000 which represents 18 in base 10.
Task examples:
1 zadd&.zform 1
10
10 zadd&.zform 10
101
10100 zadd&.zform 1010
101000
10100 zsub&.zform 1010
101
10100 zmul&.zform 100101
10010010001
10100 zdiv&.zform 1010
1
10100 zdiv&.zform 1000
10
100001000001 zdiv&.zform 100010
100101
100001000001 zdiv&.zform 100101
100010
Java
import java.util.List;
public class Zeckendorf implements Comparable<Zeckendorf> {
private static List<String> dig = List.of("00", "01", "10");
private static List<String> dig1 = List.of("", "1", "10");
private String x;
private int dVal = 0;
private int dLen = 0;
public Zeckendorf() {
this("0");
}
public Zeckendorf(String x) {
this.x = x;
int q = 1;
int i = x.length() - 1;
dLen = i / 2;
while (i >= 0) {
dVal += (x.charAt(i) - '0') * q;
q *= 2;
i--;
}
}
private void a(int n) {
int i = n;
while (true) {
if (dLen < i) dLen = i;
int j = (dVal >> (i * 2)) & 3;
switch (j) {
case 0:
case 1:
return;
case 2:
if (((dVal >> ((i + 1) * 2)) & 1) != 1) return;
dVal += 1 << (i * 2 + 1);
return;
case 3:
int temp = 3 << (i * 2);
temp ^= -1;
dVal = dVal & temp;
b((i + 1) * 2);
break;
}
i++;
}
}
private void b(int pos) {
if (pos == 0) {
Zeckendorf thiz = this;
thiz.inc();
return;
}
if (((dVal >> pos) & 1) == 0) {
dVal += 1 << pos;
a(pos / 2);
if (pos > 1) a(pos / 2 - 1);
} else {
int temp = 1 << pos;
temp ^= -1;
dVal = dVal & temp;
b(pos + 1);
b(pos - (pos > 1 ? 2 : 1));
}
}
private void c(int pos) {
if (((dVal >> pos) & 1) == 1) {
int temp = 1 << pos;
temp ^= -1;
dVal = dVal & temp;
return;
}
c(pos + 1);
if (pos > 0) {
b(pos - 1);
} else {
Zeckendorf thiz = this;
thiz.inc();
}
}
public Zeckendorf inc() {
dVal++;
a(0);
return this;
}
public void plusAssign(Zeckendorf other) {
for (int gn = 0; gn < (other.dLen + 1) * 2; gn++) {
if (((other.dVal >> gn) & 1) == 1) {
b(gn);
}
}
}
public void minusAssign(Zeckendorf other) {
for (int gn = 0; gn < (other.dLen + 1) * 2; gn++) {
if (((other.dVal >> gn) & 1) == 1) {
c(gn);
}
}
while ((((dVal >> dLen * 2) & 3) == 0) || (dLen == 0)) {
dLen--;
}
}
public void timesAssign(Zeckendorf other) {
Zeckendorf na = other.copy();
Zeckendorf nb = other.copy();
Zeckendorf nt;
Zeckendorf nr = new Zeckendorf();
for (int i = 0; i < (dLen + 1) * 2; i++) {
if (((dVal >> i) & 1) > 0) {
nr.plusAssign(nb);
}
nt = nb.copy();
nb.plusAssign(na);
na = nt.copy();
}
dVal = nr.dVal;
dLen = nr.dLen;
}
private Zeckendorf copy() {
Zeckendorf z = new Zeckendorf();
z.dVal = dVal;
z.dLen = dLen;
return z;
}
@Override
public int compareTo(Zeckendorf other) {
return ((Integer) dVal).compareTo(other.dVal);
}
@Override
public String toString() {
if (dVal == 0) {
return "0";
}
int idx = (dVal >> (dLen * 2)) & 3;
StringBuilder stringBuilder = new StringBuilder(dig1.get(idx));
for (int i = dLen - 1; i >= 0; i--) {
idx = (dVal >> (i * 2)) & 3;
stringBuilder.append(dig.get(idx));
}
return stringBuilder.toString();
}
public static void main(String[] args) {
System.out.println("Addition:");
Zeckendorf g = new Zeckendorf("10");
g.plusAssign(new Zeckendorf("10"));
System.out.println(g);
g.plusAssign(new Zeckendorf("10"));
System.out.println(g);
g.plusAssign(new Zeckendorf("1001"));
System.out.println(g);
g.plusAssign(new Zeckendorf("1000"));
System.out.println(g);
g.plusAssign(new Zeckendorf("10101"));
System.out.println(g);
System.out.println("\nSubtraction:");
g = new Zeckendorf("1000");
g.minusAssign(new Zeckendorf("101"));
System.out.println(g);
g = new Zeckendorf("10101010");
g.minusAssign(new Zeckendorf("1010101"));
System.out.println(g);
System.out.println("\nMultiplication:");
g = new Zeckendorf("1001");
g.timesAssign(new Zeckendorf("101"));
System.out.println(g);
g = new Zeckendorf("101010");
g.plusAssign(new Zeckendorf("101"));
System.out.println(g);
}
}
- Output:
Addition: 101 1001 10101 100101 1010000 Subtraction: 1 1000000 Multiplication: 1000100 1000100
Julia
Influenced by the format of the Tcl and Raku versions, but added other functionality.
import Base.*, Base.+, Base.-, Base./, Base.show, Base.!=, Base.==, Base.<=, Base.<, Base.>, Base.>=, Base.divrem
const z0 = "0"
const z1 = "1"
const flipordered = (z1 < z0)
mutable struct Z s::String end
Z() = Z(z0)
Z(z::Z) = Z(z.s)
pairlen(x::Z, y::Z) = max(length(x.s), length(y.s))
tolen(x::Z, n::Int) = (s = x.s; while length(s) < n s = z0 * s end; s)
<(x::Z, y::Z) = (l = pairlen(x, y); flipordered ? tolen(x, l) > tolen(y, l) : tolen(x, l) < tolen(y, l))
>(x::Z, y::Z) = (l = pairlen(x, y); flipordered ? tolen(x, l) < tolen(y, l) : tolen(x, l) > tolen(y, l))
==(x::Z, y::Z) = (l = pairlen(x, y); tolen(x, l) == tolen(y, l))
<=(x::Z, y::Z) = (l = pairlen(x, y); flipordered ? tolen(x, l) >= tolen(y, l) : tolen(x, l) <= tolen(y, l))
>=(x::Z, y::Z) = (l = pairlen(x, y); flipordered ? tolen(x, l) <= tolen(y, l) : tolen(x, l) >= tolen(y, l))
!=(x::Z, y::Z) = (l = pairlen(x, y); tolen(x, l) != tolen(y, l))
function tocanonical(z::Z)
while occursin(z0 * z1 * z1, z.s)
z.s = replace(z.s, z0 * z1 * z1 => z1 * z0 * z0)
end
len = length(z.s)
if len > 1 && z.s[1:2] == z1 * z1
z.s = z1 * z0 * z0 * ((len > 2) ? z.s[3:end] : "")
end
while (len = length(z.s)) > 1 && string(z.s[1]) == z0
if len == 2
if z.s == z0 * z0
z.s = z0
elseif z.s == z0 * z1
z.s = z1
end
else
z.s = z.s[2:end]
end
end
z
end
function inc(z)
if z.s[end] == z0[1]
z.s = z.s[1:end-1] * z1[1]
elseif z.s[end] == z1[1]
if length(z.s) > 1
if z.s[end-1:end] == z0 * z1
z.s = z.s[1:end-2] * z1 * z0
end
else
z.s = z1 * z0
end
end
tocanonical(z)
end
function dec(z)
if z.s[end] == z1[1]
z.s = z.s[1:end-1] * z0
else
if (m = match(Regex(z1 * z0 * '+' * '$'), z.s)) != nothing
len = length(m.match)
if iseven(len)
z.s = z.s[1:end-len] * (z0 * z1) ^ div(len, 2)
else
z.s = z.s[1:end-len] * (z0 * z1) ^ div(len, 2) * z0
end
end
end
tocanonical(z)
z
end
function +(x::Z, y::Z)
a = Z(x.s)
b = Z(y.s)
while b.s != z0
inc(a)
dec(b)
end
a
end
function -(x::Z, y::Z)
a = Z(x.s)
b = Z(y.s)
while b.s != z0
dec(a)
dec(b)
end
a
end
function *(x::Z, y::Z)
if (x.s == z0) || (y.s == z0)
return Z(z0)
elseif x.s == z1
return Z(y.s)
elseif y.s == z1
return Z(x.s)
end
a = Z(x.s)
b = Z(z1)
while b != y
c = Z(z0)
while c != x
inc(a)
inc(c)
end
inc(b)
end
a
end
function divrem(x::Z, y::Z)
if y.s == z0
throw("Zeckendorf division by 0")
elseif (y.s == z1) || (x.s == z0)
return Z(x.s)
end
a = Z(x.s)
b = Z(y.s)
c = Z(z0)
while a > b
a = a - b
inc(c)
end
tocanonical(c), tocanonical(a)
end
function /(x::Z, y::Z)
a, _ = divrem(x, y)
a
end
show(io::IO, z::Z) = show(io, parse(BigInt, tocanonical(z).s))
function zeckendorftest()
a = Z("10")
b = Z("1001")
c = Z("1000")
d = Z("10101")
println("Addition:")
x = a
println(x += a)
println(x += a)
println(x += b)
println(x += c)
println(x += d)
println("\nSubtraction:")
x = Z("1000")
println(x - Z("101"))
x = Z("10101010")
println(x - Z("1010101"))
println("\nMultiplication:")
x = Z("1001")
y = Z("101")
println(x * y)
println(Z("101010") * y)
println("\nDivision:")
x = Z("1000101")
y = Z("101")
println(x / y)
println(divrem(x, y))
end
zeckendorftest()
- Output:
Addition: 101 1001 10101 100101 1010000 Subtraction: 1 1000000 Multiplication: 1000100 101000101 Division: 1001 (1001, 1)
Kotlin
// version 1.1.51
class Zeckendorf(x: String = "0") : Comparable<Zeckendorf> {
var dVal = 0
var dLen = 0
private fun a(n: Int) {
var i = n
while (true) {
if (dLen < i) dLen = i
val j = (dVal shr (i * 2)) and 3
when (j) {
0, 1 -> return
2 -> {
if (((dVal shr ((i + 1) * 2)) and 1) != 1) return
dVal += 1 shl (i * 2 + 1)
return
}
3 -> {
dVal = dVal and (3 shl (i * 2)).inv()
b((i + 1) * 2)
}
}
i++
}
}
private fun b(pos: Int) {
if (pos == 0) {
var thiz = this
++thiz
return
}
if (((dVal shr pos) and 1) == 0) {
dVal += 1 shl pos
a(pos / 2)
if (pos > 1) a(pos / 2 - 1)
}
else {
dVal = dVal and (1 shl pos).inv()
b(pos + 1)
b(pos - (if (pos > 1) 2 else 1))
}
}
private fun c(pos: Int) {
if (((dVal shr pos) and 1) == 1) {
dVal = dVal and (1 shl pos).inv()
return
}
c(pos + 1)
if (pos > 0) b(pos - 1) else { var thiz = this; ++thiz }
}
init {
var q = 1
var i = x.length - 1
dLen = i / 2
while (i >= 0) {
dVal += (x[i] - '0').toInt() * q
q *= 2
i--
}
}
operator fun inc(): Zeckendorf {
dVal += 1
a(0)
return this
}
operator fun plusAssign(other: Zeckendorf) {
for (gn in 0 until (other.dLen + 1) * 2) {
if (((other.dVal shr gn) and 1) == 1) b(gn)
}
}
operator fun minusAssign(other: Zeckendorf) {
for (gn in 0 until (other.dLen + 1) * 2) {
if (((other.dVal shr gn) and 1) == 1) c(gn)
}
while ((((dVal shr dLen * 2) and 3) == 0) || (dLen == 0)) dLen--
}
operator fun timesAssign(other: Zeckendorf) {
var na = other.copy()
var nb = other.copy()
var nt: Zeckendorf
var nr = "0".Z
for (i in 0..(dLen + 1) * 2) {
if (((dVal shr i) and 1) > 0) nr += nb
nt = nb.copy()
nb += na
na = nt.copy()
}
dVal = nr.dVal
dLen = nr.dLen
}
override operator fun compareTo(other: Zeckendorf) = dVal.compareTo(other.dVal)
override fun toString(): String {
if (dVal == 0) return "0"
val sb = StringBuilder(dig1[(dVal shr (dLen * 2)) and 3])
for (i in dLen - 1 downTo 0) {
sb.append(dig[(dVal shr (i * 2)) and 3])
}
return sb.toString()
}
fun copy(): Zeckendorf {
val z = "0".Z
z.dVal = dVal
z.dLen = dLen
return z
}
companion object {
val dig = listOf("00", "01", "10")
val dig1 = listOf("", "1", "10")
}
}
val String.Z get() = Zeckendorf(this)
fun main(args: Array<String>) {
println("Addition:")
var g = "10".Z
g += "10".Z
println(g)
g += "10".Z
println(g)
g += "1001".Z
println(g)
g += "1000".Z
println(g)
g += "10101".Z
println(g)
println("\nSubtraction:")
g = "1000".Z
g -= "101".Z
println(g)
g = "10101010".Z
g -= "1010101".Z
println(g)
println("\nMultiplication:")
g = "1001".Z
g *= "101".Z
println(g)
g = "101010".Z
g += "101".Z
println(g)
}
- Output:
Addition: 101 1001 10101 100101 1010000 Subtraction: 1 1000000 Multiplication: 1000100 1000100
Nim
type Zeckendorf = object
dVal: Natural
dLen: Natural
const
Dig = ["00", "01", "10"]
Dig1 = ["", "1", "10"]
# Forward references.
func b(z: var Zeckendorf; pos: Natural)
func inc(z: var Zeckendorf)
func a(z: var Zeckendorf; n: Natural) =
var i = n
while true:
if z.dLen < i: z.dLen = i
let j = z.dVal shr (i * 2) and 3
case j
of 0, 1:
return
of 2:
if (z.dVal shr ((i + 1) * 2) and 1) != 1: return
z.dVal += 1 shl (i * 2 + 1)
return
of 3:
z.dVal = z.dVal and not (3 shl (i * 2))
z.b((i + 1) * 2)
else:
assert(false)
inc i
func b(z: var Zeckendorf; pos: Natural) =
if pos == 0:
inc z
return
if (z.dVal shr pos and 1) == 0:
z.dVal += 1 shl pos
z.a(pos div 2)
if pos > 1: z.a(pos div 2 - 1)
else:
z.dVal = z.dVal and not(1 shl pos)
z.b(pos + 1)
z.b(pos - (if pos > 1: 2 else: 1))
func c(z: var Zeckendorf; pos: Natural) =
if (z.dVal shr pos and 1) == 1:
z.dVal = z.dVal and not(1 shl pos)
return
z.c(pos + 1)
if pos > 0:
z.b(pos - 1)
else:
inc z
func initZeckendorf(s = "0"): Zeckendorf =
var q = 1
var i = s.high
result.dLen = i div 2
while i >= 0:
result.dVal += (ord(s[i]) - ord('0')) * q
q *= 2
dec i
func inc(z: var Zeckendorf) =
inc z.dVal
z.a(0)
func `+=`(z1: var Zeckendorf; z2: Zeckendorf) =
for gn in 0 .. (2 * z2.dLen + 1):
if (z2.dVal shr gn and 1) == 1:
z1.b(gn)
func `-=`(z1: var Zeckendorf; z2: Zeckendorf) =
for gn in 0 .. (2 * z2.dLen + 1):
if (z2.dVal shr gn and 1) == 1:
z1.c(gn)
while z1.dLen > 0 and (z1.dVal shr (z1.dLen * 2) and 3) == 0:
dec z1.dLen
func `*=`(z1: var Zeckendorf; z2: Zeckendorf) =
var na, nb = z2
var nr: Zeckendorf
for i in 0 .. (z1.dLen + 1) * 2:
if (z1.dVal shr i and 1) > 0: nr += nb
let nt = nb
nb += na
na = nt
z1 = nr
func`$`(z: var Zeckendorf): string =
if z.dVal == 0: return "0"
result.add Dig1[z.dVal shr (z.dLen * 2) and 3]
for i in countdown(z.dLen - 1, 0):
result.add Dig[z.dVal shr (i * 2) and 3]
when isMainModule:
var g: Zeckendorf
echo "Addition:"
g = initZeckendorf("10")
g += initZeckendorf("10")
echo g
g += initZeckendorf("10")
echo g
g += initZeckendorf("1001")
echo g
g += initZeckendorf("1000")
echo g
g += initZeckendorf("10101")
echo g
echo "\nSubtraction:"
g = initZeckendorf("1000")
g -= initZeckendorf("101")
echo g
g = initZeckendorf("10101010")
g -= initZeckendorf("1010101")
echo g
echo "\nMultiplication:"
g = initZeckendorf("1001")
g *= initZeckendorf("101")
echo g
g = initZeckendorf("101010")
g += initZeckendorf("101")
echo g
- Output:
Addition: 101 1001 10101 100101 1010000 Subtraction: 1 1000000 Multiplication: 1000100 1000100
Perl
use v5.36;
package Zeckendorf;
use overload qw("" zstring + zadd - zsub ++ zinc -- zdec * zmul / zdiv ge zge);
sub new ($class, $value) {
bless \$value, ref $class || $class;
}
sub zinc ($self, $, $) {
local $_ = $$self;
s/0$/1/ or s/(?:^|0)1$/10/;
1 while s/(?:^|0)11/100/;
$$self = $self->new( s/^0+\B//r )
}
sub zdec ($self, $, $) {
local $_ = $$self;
1 while s/100(?=0*$)/011/;
s/1$/0/ || s/10$/01/;
$$self = $self->new( s/^0+\B//r )
}
sub zadd ($self, $other, $) {
my ($x, $y) = map $self->new($$_), $self, $other;
$x++, $y-- while $$y;
$x
}
sub zsub ($self, $other, $) {
my ($x, $y) = map $self->new($$_), $self, $other;
$x--, $y-- while $$y;
$x
}
sub zmul ($self, $other, $) {
my ($x, $y) = map $self->new($$_), $self, $other;
my $product = Zeckendorf->new(0);
$product = $product + $x, $y-- while $y;
$product
}
sub zdiv ($self, $other, $) {
my ($x, $y) = map $self->new($$_), $self, $other;
my $quotient = Zeckendorf->new(0);
$quotient++, $x = $x - $y while $x ge $y;
$quotient
}
sub zge ($self, $other, $) {
my $l; $l = length $$other if length $other > ($l = length $$self);
0 x ($l - length $$self) . $$self ge 0 x ($l - length $$other) . $$other;
}
sub asdecimal ($self) {
my($aa, $bb, $n) = (1, 1, 0);
for ( reverse split '', $$self ) {
$n += $bb * $_;
($aa, $bb) = ($bb, $aa + $bb);
}
$n
}
sub fromdecimal ($self, $value) {
my $z = $self->new(0);
$z++ for 1 .. $value;
$z
}
sub zstring { ${ shift() } }
package main;
for ( split /\n/, <<END ) # test cases
1 + 1
10 + 10
10100 + 1010
10100 - 1010
10100 * 1010
100010 * 100101
10100 / 1010
101000 / 1000
100001000001 / 100010
100001000001 / 100101
END
{
my ($left, $op, $right) = split;
my ($x, $y) = map Zeckendorf->new($_), $left, $right;
my $answer =
$op eq '+' ? $x + $y :
$op eq '-' ? $x - $y :
$op eq '*' ? $x * $y :
$op eq '/' ? $x / $y :
die "bad op <$op>";
printf "%12s %s %-9s => %12s in Zeckendorf\n", $x, $op, $y, $answer;
printf "%12d %s %-9d => %12d in decimal\n\n",
$x->asdecimal, $op, $y->asdecimal, $answer->asdecimal;
}
- Output:
1 + 1 => 10 in Zeckendorf
1 + 1 => 2 in decimal
10 + 10 => 101 in Zeckendorf
2 + 2 => 4 in decimal
10100 + 1010 => 101000 in Zeckendorf
11 + 7 => 18 in decimal
10100 - 1010 => 101 in Zeckendorf
11 - 7 => 4 in decimal
10100 * 1010 => 101000001 in Zeckendorf
11 * 7 => 77 in decimal
100010 * 100101 => 100001000001 in Zeckendorf
15 * 17 => 255 in decimal
10100 / 1010 => 1 in Zeckendorf
11 / 7 => 1 in decimal
101000 / 1000 => 100 in Zeckendorf
18 / 5 => 3 in decimal
100001000001 / 100010 => 100101 in Zeckendorf
255 / 15 => 17 in decimal
100001000001 / 100101 => 100010 in Zeckendorf
255 / 17 => 15 in decimal
Phix
Uses a binary representation of Zeckendorf numbers, eg decimal 11 is stored as 0b10100, ie meaning 8+3, but actually 20 in decimal.
As such, they can be directly compared using the standard comparison operators, and printed quite trivially just by using the %b format.
They are however (and not all that surprisingly) pulled apart into individual bits for addition/subtraction, etc.
Does not handle negative numbers or anything >139583862445 (-ve probably doable but messy, >1.4e12 requires a total rewrite, probably using string representation).
with javascript_semantics sequence fib = {1,1} function zeckendorf(atom n) -- Same as Zeckendorf_number_representation#Phix atom r = 0 while fib[$]<n do fib &= fib[$] + fib[$-1] end while integer k = length(fib) while k>2 and n<fib[k] do k -= 1 end while for i=k to 2 by -1 do integer c = n>=fib[i] r += r+c n -= c*fib[i] end for return r end function function decimal(object z) -- Convert Zeckendorf number(s) to decimal if sequence(z) then sequence res = repeat(0,length(z)) for i=1 to length(z) do res[i] = decimal(z[i]) end for return res end if atom dec = 0, bit = 2 while z do if and_bits(z,1) then dec += fib[bit] end if bit += 1 if bit>length(fib) then fib &= fib[$] + fib[$-1] end if z = floor(z/2) end while return dec end function function to_bits(integer x) -- Simplified copy of int_to_bits(), but in reverse order, -- and +ve only but (also only) as many bits as needed, and -- ensures there are *two* trailing 0 (most significant) if x<0 then ?9/0 end if -- sanity/avoid infinite loop sequence bits = {} while 1 do bits &= remainder(x,2) if x=0 then exit end if x = floor(x/2) end while bits &= 0 -- (since eg 101+101 -> 10000) return bits end function function to_bits2(integer a,b) -- Apply to_bits() to a and b, and pad to the same length sequence sa = to_bits(a), sb = to_bits(b) integer diff = length(sa)-length(sb) if diff!=0 then if diff<0 then sa &= repeat(0,-diff) else sb &= repeat(0,+diff) end if end if return {sa,sb} end function function to_int(sequence bits) -- Copy of bits_to_int(), but in reverse order (lsb last) atom val = 0, p = 1 for i=length(bits) to 1 by -1 do if bits[i] then val += p end if p += p end for return val end function function zstr(object z) if sequence(z) then sequence res = repeat(0,length(z)) for i=1 to length(z) do res[i] = zstr(z[i]) end for return res end if return sprintf("%b",z) end function function rep(sequence res, integer ds, sequence was, wth) -- helper for cleanup, validates replacements integer de = ds+length(was)-1 if res[ds..de]!=was then ?9/0 end if if length(was)!=length(wth) then ?9/0 end if res = deep_copy(res) res[ds..de] = wth return res end function function zcleanup(sequence res) -- (shared by zadd and zsub) integer l = length(res) res = deep_copy(res) -- first stage, left to right, {020x -> 100x', 030x -> 110x', 021x->110x, 012x->101x} for i=1 to l-3 do sequence s3 = res[i..i+2] if s3={0,2,0} then res[i..i+2] = {1,0,0} res[i+3] += 1 elsif s3={0,3,0} then res[i..i+2] = {1,1,0} res[i+3] += 1 elsif s3={0,2,1} then res[i..i+2] = {1,1,0} elsif s3={0,1,2} then res[i..i+2] = {1,0,1} end if end for -- first stage cleanup if l>1 then if res[l-1]=3 then res = rep(res,l-2,{0,3,0},{1,1,1}) -- 030 -> 111 elsif res[l-1]=2 then if res[l-2]=0 then res = rep(res,l-2,{0,2,0},{1,0,1}) -- 020 -> 101 else res = rep(res,l-3,{0,1,2,0},{1,0,1,0}) -- 0120 -> 1010 end if end if end if if res[l]=3 then res = rep(res,l-1,{0,3},{1,1}) -- 03 -> 11 elsif res[l]=2 then if res[l-1]=0 then res = rep(res,l-1,{0,2},{1,0}) -- 02 -> 10 else res = rep(res,l-2,{0,1,2},{1,0,1}) -- 012 -> 101 end if end if -- second stage, pass 1, right to left, 011 -> 100 for i=length(res)-2 to 1 by -1 do if res[i..i+2]={0,1,1} then res[i..i+2] = {1,0,0} end if end for -- second stage, pass 2, left to right, 011 -> 100 for i=1 to length(res)-2 do if res[i..i+2]={0,1,1} then res[i..i+2] = {1,0,0} end if end for return to_int(res) end function function zadd(integer a, b) sequence {sa,sb} = to_bits2(a,b) return zcleanup(reverse(sq_add(sa,sb))) end function function zinc(integer a) return zadd(a,0b1) end function function zsub(integer a, b) sequence {sa,sb} = to_bits2(a,b) sequence res = reverse(sq_sub(sa,sb)) -- (/not/ combined with the first pass of the add routine!) for i=1 to length(res)-2 do sequence s3 = res[i..i+2] if s3={1, 0, 0} then res[i..i+2] = {0,1,1} elsif s3={1,-1, 0} then res[i..i+2] = {0,0,1} elsif s3={1,-1, 1} then res[i..i+2] = {0,0,2} elsif s3={1, 0,-1} then res[i..i+2] = {0,1,0} elsif s3={2, 0, 0} then res[i..i+2] = {1,1,1} elsif s3={2,-1, 0} then res[i..i+2] = {1,0,1} elsif s3={2,-1, 1} then res[i..i+2] = {1,0,2} elsif s3={2, 0,-1} then res[i..i+2] = {1,1,0} end if end for -- copied from PicoLisp: {1,-1} -> {0,1} and {2,-1} -> {1,1} for i=1 to length(res)-1 do sequence s2 = res[i..i+1] if s2={1,-1} then res[i..i+1] = {0,1} elsif s2={2,-1} then res[i..i+1] = {1,1} end if end for if find(-1,res) then ?9/0 end if -- sanity check return zcleanup(res) end function function zdec(integer a) return zsub(a,0b1) end function function zmul(integer a, b) sequence mult = {a,zadd(a,a)} -- (as per task desc) integer bits = 2 while bits<b do mult = append(mult,zadd(mult[$],mult[$-1])) bits *= 2 end while integer res = 0, bit = 1 while b do if and_bits(b,1) then res = zadd(res,mult[bit]) end if b = floor(b/2) bit += 1 end while return res end function function zdiv(integer a, b) sequence mult = {b,zadd(b,b)} integer bits = 2 while mult[$]<a do mult = append(mult,zadd(mult[$],mult[$-1])) bits *= 2 end while integer res = 0 for i=length(mult) to 1 by -1 do integer mi = mult[i] if mi<=a then res = zadd(res,bits) a = zsub(a,mi) if a=0 then exit end if end if bits = floor(bits/2) end for return {res,a} -- (a is the remainder) end function for i=0 to 20 do integer zi = zeckendorf(i) atom d = decimal(zi) printf(1,"%2d: %7b (%d)\n",{i,zi,d}) end for procedure test(atom a, string op, atom b, object res, string expected) string zres = iff(atom(res)?zstr(res):join(zstr(res)," rem ")), dres = sprintf(iff(atom(res)?"%d":"%d rem %d"),decimal(res)), aka = sprintf("aka %d %s %d = %s",{decimal(a),op,decimal(b),dres}), ok = iff(zres=expected?"":" *** ERROR ***!!") printf(1,"%s %s %s = %s, %s %s\n",{zstr(a),op,zstr(b),zres,aka,ok}) end procedure test(0b0,"+",0b0,zadd(0b0,0b0),"0") test(0b101,"+",0b101,zadd(0b101,0b101),"10000") test(0b10100,"-",0b1000,zsub(0b10100,0b1000),"1001") test(0b100100,"-",0b1000,zsub(0b100100,0b1000),"10100") test(0b1001,"*",0b101,zmul(0b1001,0b101),"1000100") test(0b1000101,"/",0b101,zdiv(0b1000101,0b101),"1001 rem 1") test(0b10,"+",0b10,zadd(0b10,0b10),"101") test(0b101,"+",0b10,zadd(0b101,0b10),"1001") test(0b1001,"+",0b1001,zadd(0b1001,0b1001),"10101") test(0b10101,"+",0b1000,zadd(0b10101,0b1000),"100101") test(0b100101,"+",0b10101,zadd(0b100101,0b10101),"1010000") test(0b1000,"-",0b101,zsub(0b1000,0b101),"1") test(0b10101010,"-",0b1010101,zsub(0b10101010,0b1010101),"1000000") test(0b1001,"*",0b101,zmul(0b1001,0b101),"1000100") test(0b101010,"+",0b101,zadd(0b101010,0b101),"1000100") test(0b10100,"+",0b1010,zadd(0b10100,0b1010),"101000") test(0b101000,"-",0b1010,zsub(0b101000,0b1010),"10100") test(0b100010,"*",0b100101,zmul(0b100010,0b100101),"100001000001") test(0b100001000001,"/",0b100,zdiv(0b100001000001,0b100),"101010001 rem 0") test(0b101000101,"*",0b101001,zmul(0b101000101,0b101001),"101010000010101") test(0b101010000010101,"/",0b100,zdiv(0b101010000010101,0b100),"1001010001001 rem 10") test(0b10100010010100,"+",0b1001000001,zadd(0b10100010010100,0b1001000001),"100000000010101") test(0b10100010010100,"-",0b1001000001,zsub(0b10100010010100,0b1001000001),"10010001000010") test(0b10000,"*",0b1001000001,zmul(0b10000,0b1001000001),"10100010010100") test(0b1010001010000001001,"/",0b100000000100000,zdiv(0b1010001010000001001,0b100000000100000),"10001 rem 10100001010101") test(0b10100,"+",0b1010,zadd(0b10100,0b1010),"101000") test(0b10100,"-",0b1010,zsub(0b10100,0b1010),"101") test(0b10100,"*",0b1010,zmul(0b10100,0b1010),"101000001") test(0b10100,"/",0b1010,zdiv(0b10100,0b1010),"1 rem 101") integer m = zmul(0b10100,0b1010) test(m,"/",0b1010,zdiv(m,0b1010),"10100 rem 0")
- Output:
0: 0 (0) 1: 1 (1) 2: 10 (2) 3: 100 (3) 4: 101 (4) 5: 1000 (5) 6: 1001 (6) 7: 1010 (7) 8: 10000 (8) 9: 10001 (9) 10: 10010 (10) 11: 10100 (11) 12: 10101 (12) 13: 100000 (13) 14: 100001 (14) 15: 100010 (15) 16: 100100 (16) 17: 100101 (17) 18: 101000 (18) 19: 101001 (19) 20: 101010 (20) 0 + 0 = 0, aka 0 + 0 = 0 101 + 101 = 10000, aka 4 + 4 = 8 10100 - 1000 = 1001, aka 11 - 5 = 6 100100 - 1000 = 10100, aka 16 - 5 = 11 1001 * 101 = 1000100, aka 6 * 4 = 24 1000101 / 101 = 1001 rem 1, aka 25 / 4 = 6 rem 1 10 + 10 = 101, aka 2 + 2 = 4 101 + 10 = 1001, aka 4 + 2 = 6 1001 + 1001 = 10101, aka 6 + 6 = 12 10101 + 1000 = 100101, aka 12 + 5 = 17 100101 + 10101 = 1010000, aka 17 + 12 = 29 1000 - 101 = 1, aka 5 - 4 = 1 10101010 - 1010101 = 1000000, aka 54 - 33 = 21 1001 * 101 = 1000100, aka 6 * 4 = 24 101010 + 101 = 1000100, aka 20 + 4 = 24 10100 + 1010 = 101000, aka 11 + 7 = 18 101000 - 1010 = 10100, aka 18 - 7 = 11 100010 * 100101 = 100001000001, aka 15 * 17 = 255 100001000001 / 100 = 101010001 rem 0, aka 255 / 3 = 85 rem 0 101000101 * 101001 = 101010000010101, aka 80 * 19 = 1520 101010000010101 / 100 = 1001010001001 rem 10, aka 1520 / 3 = 506 rem 2 10100010010100 + 1001000001 = 100000000010101, aka 888 + 111 = 999 10100010010100 - 1001000001 = 10010001000010, aka 888 - 111 = 777 10000 * 1001000001 = 10100010010100, aka 8 * 111 = 888 1010001010000001001 / 100000000100000 = 10001 rem 10100001010101, aka 9876 / 1000 = 9 rem 876 10100 + 1010 = 101000, aka 11 + 7 = 18 10100 - 1010 = 101, aka 11 - 7 = 4 10100 * 1010 = 101000001, aka 11 * 7 = 77 10100 / 1010 = 1 rem 101, aka 11 / 7 = 1 rem 4 101000001 / 1010 = 10100 rem 0, aka 77 / 7 = 11 rem 0
PicoLisp
(seed (in "/dev/urandom" (rd 8)))
(de unpad (Lst)
(while (=0 (car Lst))
(pop 'Lst) )
Lst )
(de numz (N)
(let Fibs (1 1)
(while (>= N (+ (car Fibs) (cadr Fibs)))
(push 'Fibs (+ (car Fibs) (cadr Fibs))) )
(make
(for I (uniq Fibs)
(if (> I N)
(link 0)
(link 1)
(dec 'N I) ) ) ) ) )
(de znum (Lst)
(let Fibs (1 1)
(do (dec (length Lst))
(push 'Fibs (+ (car Fibs) (cadr Fibs))) )
(sum
'((X Y) (unless (=0 X) Y))
Lst
(uniq Fibs) ) ) )
(de incz (Lst)
(addz Lst (1)) )
(de decz (Lst)
(subz Lst (1)) )
(de addz (Lst1 Lst2)
(let Max (max (length Lst1) (length Lst2))
(reorg
(mapcar + (need Max Lst1 0) (need Max Lst2 0)) ) ) )
(de subz (Lst1 Lst2)
(use (@A @B)
(let
(Max (max (length Lst1) (length Lst2))
Lst (mapcar - (need Max Lst1 0) (need Max Lst2 0)) )
(loop
(while (match '(@A 1 0 0 @B) Lst)
(setq Lst (append @A (0 1 1) @B)) )
(while (match '(@A 1 -1 0 @B) Lst)
(setq Lst (append @A (0 0 1) @B)) )
(while (match '(@A 1 -1 1 @B) Lst)
(setq Lst (append @A (0 0 2) @B)) )
(while (match '(@A 1 0 -1 @B) Lst)
(setq Lst (append @A (0 1 0) @B)) )
(while (match '(@A 2 0 0 @B) Lst)
(setq Lst (append @A (1 1 1) @B)) )
(while (match '(@A 2 -1 0 @B) Lst)
(setq Lst (append @A (1 0 1) @B)) )
(while (match '(@A 2 -1 1 @B) Lst)
(setq Lst (append @A (1 0 2) @B)) )
(while (match '(@A 2 0 -1 @B) Lst)
(setq Lst (append @A (1 1 0) @B)) )
(while (match '(@A 1 -1) Lst)
(setq Lst (append @A (0 1))) )
(while (match '(@A 2 -1) Lst)
(setq Lst (append @A (1 1))) )
(NIL (match '(@A -1 @B) Lst)) )
(reorg (unpad Lst)) ) ) )
(de mulz (Lst1 Lst2)
(let (Sums (list Lst1) Mulz (0))
(mapc
'((X)
(when (= 1 (car X))
(setq Mulz (addz (cdr X) Mulz)) )
Mulz )
(mapcar
'((X)
(cons
X
(push 'Sums (addz (car Sums) (cadr Sums))) ) )
(reverse Lst2) ) ) ) )
(de divz (Lst1 Lst2)
(let Q 0
(while (lez Lst2 Lst1)
(setq Lst1 (subz Lst1 Lst2))
(setq Q (incz Q)) )
(list Q (or Lst1 (0))) ) )
(de reorg (Lst)
(use (@A @B)
(let Lst (reverse Lst)
(loop
(while (match '(@A 1 1 @B) Lst)
(if @B
(inc (nth @B 1))
(setq @B (1)) )
(setq Lst (append @A (0 0) @B) ) )
(while (match '(@A 2 @B) Lst)
(inc
(if (cdr @A)
(tail 2 @A)
@A ) )
(if @B
(inc (nth @B 1))
(setq @B (1)) )
(setq Lst (append @A (0) @B)) )
(NIL
(or
(match '(@A 1 1 @B) Lst)
(match '(@A 2 @B) Lst) ) ) )
(reverse Lst) ) ) )
(de lez (Lst1 Lst2)
(let Max (max (length Lst1) (length Lst2))
(<= (need Max Lst1 0) (need Max Lst2 0)) ) )
(let (X 0 Y 0)
(do 1024
(setq X (rand 1 1024))
(setq Y (rand 1 1024))
(test (numz (+ X Y)) (addz (numz X) (numz Y)))
(test (numz (* X Y)) (mulz (numz X) (numz Y)))
(test (numz (+ X 1)) (incz (numz X))) )
(do 1024
(setq X (rand 129 1024))
(setq Y (rand 1 128))
(test (numz (- X Y)) (subz (numz X) (numz Y)))
(test (numz (/ X Y)) (car (divz (numz X) (numz Y))))
(test (numz (% X Y)) (cadr (divz (numz X) (numz Y))))
(test (numz (- X 1)) (decz (numz X))) ) )
(bye)
Python
import copy
class Zeckendorf:
def __init__(self, x='0'):
q = 1
i = len(x) - 1
self.dLen = int(i / 2)
self.dVal = 0
while i >= 0:
self.dVal = self.dVal + (ord(x[i]) - ord('0')) * q
q = q * 2
i = i -1
def a(self, n):
i = n
while True:
if self.dLen < i:
self.dLen = i
j = (self.dVal >> (i * 2)) & 3
if j == 0 or j == 1:
return
if j == 2:
if (self.dVal >> ((i + 1) * 2) & 1) != 1:
return
self.dVal = self.dVal + (1 << (i * 2 + 1))
return
if j == 3:
temp = 3 << (i * 2)
temp = temp ^ -1
self.dVal = self.dVal & temp
self.b((i + 1) * 2)
i = i + 1
def b(self, pos):
if pos == 0:
self.inc()
return
if (self.dVal >> pos) & 1 == 0:
self.dVal = self.dVal + (1 << pos)
self.a(int(pos / 2))
if pos > 1:
self.a(int(pos / 2) - 1)
else:
temp = 1 << pos
temp = temp ^ -1
self.dVal = self.dVal & temp
self.b(pos + 1)
self.b(pos - (2 if pos > 1 else 1))
def c(self, pos):
if (self.dVal >> pos) & 1 == 1:
temp = 1 << pos
temp = temp ^ -1
self.dVal = self.dVal & temp
return
self.c(pos + 1)
if pos > 0:
self.b(pos - 1)
else:
self.inc()
def inc(self):
self.dVal = self.dVal + 1
self.a(0)
def __add__(self, rhs):
copy = self
rhs_dVal = rhs.dVal
limit = (rhs.dLen + 1) * 2
for gn in range(0, limit):
if ((rhs_dVal >> gn) & 1) == 1:
copy.b(gn)
return copy
def __sub__(self, rhs):
copy = self
rhs_dVal = rhs.dVal
limit = (rhs.dLen + 1) * 2
for gn in range(0, limit):
if (rhs_dVal >> gn) & 1 == 1:
copy.c(gn)
while (((copy.dVal >> ((copy.dLen * 2) & 31)) & 3) == 0) or (copy.dLen == 0):
copy.dLen = copy.dLen - 1
return copy
def __mul__(self, rhs):
na = copy.deepcopy(rhs)
nb = copy.deepcopy(rhs)
nr = Zeckendorf()
dVal = self.dVal
for i in range(0, (self.dLen + 1) * 2):
if ((dVal >> i) & 1) > 0:
nr = nr + nb
nt = copy.deepcopy(nb)
nb = nb + na
na = copy.deepcopy(nt)
return nr
def __str__(self):
dig = ["00", "01", "10"]
dig1 = ["", "1", "10"]
if self.dVal == 0:
return '0'
idx = (self.dVal >> ((self.dLen * 2) & 31)) & 3
sb = dig1[idx]
i = self.dLen - 1
while i >= 0:
idx = (self.dVal >> (i * 2)) & 3
sb = sb + dig[idx]
i = i - 1
return sb
# main
print "Addition:"
g = Zeckendorf("10")
g = g + Zeckendorf("10")
print g
g = g + Zeckendorf("10")
print g
g = g + Zeckendorf("1001")
print g
g = g + Zeckendorf("1000")
print g
g = g + Zeckendorf("10101")
print g
print
print "Subtraction:"
g = Zeckendorf("1000")
g = g - Zeckendorf("101")
print g
g = Zeckendorf("10101010")
g = g - Zeckendorf("1010101")
print g
print
print "Multiplication:"
g = Zeckendorf("1001")
g = g * Zeckendorf("101")
print g
g = Zeckendorf("101010")
g = g + Zeckendorf("101")
print g
- Output:
Addition: 101 1001 10101 100101 1010000 Subtraction: 1 1000000 Multiplication: 1000100 1000100
Quackery
Unsigned (non-negative) Zeckendorf arithmetic.
Implements the required functions; addition, subtraction, multiplication and division, the optional decrement, increment and comparative functions, and additionally double and modulus, since they come for free with addition and division respectively.
The algorithms are of my own devising, without reference to the description in the task or existing research, so are potentially novel, but probably not.
I really should have taken notes as I was going along, so here is the hand-wavy explanation:
Mostly Zeckendorf numbers are represented bitwise to benefit from the inherent parallelism of bitwise logic, but occasionally as nests (the Quackery name for dynamic arrays) of 0s and 1s for ease of coding.
The word canonise puts a Zecekendorf number in canonical form; no two adjacent bits are set to 1, runs of 0s are allowed. The converse operation, defrock puts a number as far from canonical form as possible; no two adjacent bits are set to 0, runs of 1s are allowed. Despite their similarities they are coded quite differently as there was a long gap between coding one and then the other and as noted above, I didn't take notes.
Addition works by isolating the bits in both arguments that are set to 1, removing them from both and then bitwise xoring them together and canonising. After this the isolated bits are doubled and these two numbers (the xored number and the isolated bits number) are added. This is repeated until the xored number is 0. Doubling is achieved by shifting the number an appropriate distance left and right and adding the left shifted and right shifted numbers. zadd and zdouble are mutually recursive.
2blit separates the lowest two bits from a Zeckedorf number so that zdouble can treat them as a special case.
Multiplication is basically the Russian Peasant algorithm with the twist that instead of doubling we start with two instances of one of the multiplicands and repeatedly add them Fibonacci style.
Subtraction is implemented as difference (i.e. abs(a-b) as this is an unsigned implementation.) The process is to reduce both numbers in value until the smaller one equals zero. Continuing the naming theme established by canonise and defrock, the word that removes the bits that are set to 1 in both arguments is called exorcise. The appropriate sequence of exorcisms and defrockings will reduce the smaller argument to zero much of the time.
However, numbers which alternate 1s and 0s (e.g. ...01010101...) are immune to both canonisation and defrocking. When this occurs we add the smaller number to the larger number and double the smaller number and repeat the exorisms and defrockings. Extensive testing leads me to believe with a very high degree of confidence this is sufficient, but I have not proved it in a mathematical sense.
Division is basic binary long division with a twist; instead of multiplying the divisor by 2 until it's large enough, use it to make a fibonacci style sequence, except starting with a couple of copies of the divisor rather than 1s.
The while-again loop computes a nest of all the fibonacci multiples up to the dividend, the witheach loop tries subracting each one from largest to smallest and builds up the result accordingly. The remainder (modulus) comes for free as what is left at the end of all the subtractions.
To demonstrate that these words correctly implement Zeckendorf arithmetic, I have used them to implement Euclid's algorithm for Greatest Common Denominator, and used that to implement Largest Common Multiple. We repeatedly give zlcm two random numbers up to one quintillion (converted to Zeckendorf notation) and print the result (converted back to decimal), next to the same computation made using the conventional representation. zgcd and zlcm exercise the multiplication, division and modulus routines repeatedly, and those exercise the addition, subtraction and comparison routines.
bin is an extension to the Quackery compiler to allow it to understand numbers in binary notation (and hence also Zeckendorf notation).
gcd and lcm are defined at Least common multiple#Quackery, and n->z and z->n are defined at Zeckendorf number representation#Quackery.
[ nextword
dup $ "" = if
[ $ '"bin" needs to be followed by a string.'
message put bail ]
dup
2 base put
$->n
base release
not if
[ drop
$ " is not a binary number."
join message put
bail ]
nip
swap dip join ] builds bin ( [ $ --> [ $ )
[ ^ not ] is zeq ( z z --> b )
[ zeq not ] is zne ( z z --> b )
[ false unrot
[ 2dup zne while
rot drop
dup 1 & unrot
1 >> dip [ 1 >> ]
again ]
2drop ] is zlt ( z z --> b )
[ swap zlt ] is zgt ( z z --> b )
[ zlt not ] is zge ( z z --> b )
[ zgt not ] is zle ( z z --> b )
[ dup 1 << & 0 zeq ] is canonical ( z --> b )
[ [] swap
[ dup 1 & rot join
swap 1 >>
dup not until ]
drop ] is bits ( z --> [ )
[ dup canonical if done
0 0 rot bits
witheach
[ |
[ table
[ 1 << 0 ]
[ 1 << 1 | bin 10 ]
[ 1 << 0 ]
[ 1 >> 1 |
bin 10 << 0 ] ]
do ]
drop again ] is canonise ( z --> z )
[ dup bin -100
& swap bin 11 & ] is 2blit ( z --> z z )
[ 2blit bit | canonise ] is zinc ( z --> z )
[ dup 0 zeq if
[ $ "Cannot zdec zero."
fail ]
1
[ 2dup & if done
1 << again ]
tuck ^
swap 1 <<
[ bin 10 >>
tuck | swap
dup 0 zeq until ]
drop ] is zdec ( z --> z )
forward is zadd ( z --> z )
[ dup 2blit
[ table
0 bin 10 bin 101 ]
unrot bin 10 >>
swap 1 <<
rot | zadd ] is zdouble ( z --> z )
[ 2dup ^ canonise
unrot &
dup 0 zeq iff
drop done
zdouble again ] resolves zadd ( z z --> z )
[ tuck take zadd swap put ] is ztally ( z s --> )
[ 0 temp put
dip dup
[ dup while
dup 1 & if
[ over temp ztally ]
dip [ tuck zadd ]
1 >> again ]
drop 2drop temp take ] is zmult ( z z --> z )
[ 2dup & ~ tuck & dip & ] is exorcise ( z z --> z z )
[ dup
[ 0 ' [ 0 0 0 ] rot 1
[ 2dup > while
1 << again ]
1 <<
[ dup while
2swap 2over & 0 !=
dip
[ dup
' [ 1 0 0 ]
= if
[ drop
' [ 0 1 1 ] ] ]
join
behead
rot 1 << | swap
2swap 1 >> again ]
2drop
witheach
[ dip [ 1 << ] | ]
dup bin 111 &
bin 100 zeq if
[ bin -1000 &
bin 11 | ] ]
2dup zeq iff drop done
nip again ] is defrock ( z --> z )
[ 2dup zlt if swap
dup 0 zeq iff drop done
[ exorcise dup while
dip defrock
exorcise dup while
dup dip zadd
zdouble
again ]
drop canonise ] is zdiff ( z z --> z )
[ dup 0 zeq if
[ $ "Z-division by zero."
fail ]
0 unrot swap
temp put
dup nested
[ dup 0 peek
tuck dip rot zadd
temp share
over zge while
swap join
again ]
drop nip
temp take
swap witheach
[ rot 1 << unrot
2dup zge iff
[ zdiff
dip [ 1 | ] ]
else drop ] ] is zdivmod ( z z --> z z )
[ zdivmod drop ] is zdiv ( z z --> z )
[ zdivmod nip ] is zmod ( z z --> z )
[ [ dup while
tuck zmod again ]
drop ] is zgcd ( z z --> z )
[ 2dup and iff
[ 2dup zgcd
zdiv zmult ]
else and ] is zlcm ( z z --> z )
10 times
[ 10 15 ** random
10 15 ** random
2dup lcm echo cr
n->z dip n->z
zlcm z->n echo cr cr ]- Output:
25624571429859946191396654570 25624571429859946191396654570 24702413608219494319878326100 24702413608219494319878326100 177592191573881063687998734000 177592191573881063687998734000 28221788451919578670971892845 28221788451919578670971892845 99008448632249766843573255321 99008448632249766843573255321 312648960463735816244223692220 312648960463735816244223692220 146093274904252809568841733264 146093274904252809568841733264 169485448104022309641359784180 169485448104022309641359784180 593337022246602746222083444716 593337022246602746222083444716 50904418052185753625716614402 50904418052185753625716614402
Racket
This implementation only handles natural (non-negative numbers). The algorithms for addition and subtraction use the techniques explained in the paper "Efficient algorithms for Zeckendorf arithmetic" (http://arxiv.org/pdf/1207.4497.pdf).
#lang racket (require math)
(define sqrt5 (sqrt 5))
(define phi (* 0.5 (+ 1 sqrt5)))
;; What is the nth fibonnaci number, shifted by 2 so that
;; F(0) = 1, F(1) = 2, ...?
;;
(define (F n)
(fibonacci (+ n 2)))
;; What is the largest n such that F(n) <= m?
;;
(define (F* m)
(let ([n (- (inexact->exact (round (/ (log (* m sqrt5)) (log phi)))) 2)])
(if (<= (F n) m) n (sub1 n))))
(define (zeck->natural z)
(for/sum ([i (reverse z)]
[j (in-naturals)])
(* i (F j))))
(define (natural->zeck n)
(if (zero? n)
null
(for/list ([i (in-range (F* n) -1 -1)])
(let ([f (F i)])
(cond [(>= n f) (set! n (- n f))
1]
[else 0])))))
; Extend list to the right to a length of len with repeated padding elements
;
(define (pad lst len [padding 0])
(append lst (make-list (- len (length lst)) padding)))
; Strip padding elements from the left of the list
;
(define (unpad lst [padding 0])
(cond [(null? lst) lst]
[(equal? (first lst) padding) (unpad (rest lst) padding)]
[else lst]))
;; Run a filter function across a window in a list from left to right
;;
(define (left->right width fn)
(λ (lst)
(let F ([a lst])
(if (< (length a) width)
a
(let ([f (fn (take a width))])
(cons (first f) (F (append (rest f) (drop a width)))))))))
;; Run a function fn across a window in a list from right to left
;;
(define (right->left width fn)
(λ (lst)
(let F ([a lst])
(if (< (length a) width)
a
(let ([f (fn (take-right a width))])
(append (F (append (drop-right a width) (drop-right f 1)))
(list (last f))))))))
;; (a0 a1 a2 ... an) -> (a0 a1 a2 ... (fn ... an))
;;
(define (replace-tail width fn)
(λ (lst)
(append (drop-right lst width) (fn (take-right lst width)))))
(define (rule-a lst)
(match lst
[(list 0 2 0 x) (list 1 0 0 (add1 x))]
[(list 0 3 0 x) (list 1 1 0 (add1 x))]
[(list 0 2 1 x) (list 1 1 0 x)]
[(list 0 1 2 x) (list 1 0 1 x)]
[else lst]))
(define (rule-a-tail lst)
(match lst
[(list x 0 3 0) (list x 1 1 1)]
[(list x 0 2 0) (list x 1 0 1)]
[(list 0 1 2 0) (list 1 0 1 0)]
[(list x y 0 3) (list x y 1 1)]
[(list x y 0 2) (list x y 1 0)]
[(list x 0 1 2) (list x 1 0 0)]
[else lst]))
(define (rule-b lst)
(match lst
[(list 0 1 1) (list 1 0 0)]
[else lst]))
(define (rule-c lst)
(match lst
[(list 1 0 0) (list 0 1 1)]
[(list 1 -1 0) (list 0 0 1)]
[(list 1 -1 1) (list 0 0 2)]
[(list 1 0 -1) (list 0 1 0)]
[(list 2 0 0) (list 1 1 1)]
[(list 2 -1 0) (list 1 0 1)]
[(list 2 -1 1) (list 1 0 2)]
[(list 2 0 -1) (list 1 1 0)]
[else lst]))
(define (zeck-combine op y z [f identity])
(let* ([bits (max (add1 (length y)) (add1 (length z)) 4)]
[f0 (λ (x) (pad (reverse x) bits))]
[f1 (left->right 4 rule-a)]
[f2 (replace-tail 4 rule-a-tail)]
[f3 (right->left 3 rule-b)]
[f4 (left->right 3 rule-b)])
((compose1 unpad f4 f3 f2 f1 f reverse) (map op (f0 y) (f0 z)))))
(define (zeck+ y z)
(zeck-combine + y z))
(define (zeck- y z)
(when (zeck< y z) (error (format "~a" `(zeck-: cannot subtract since ,y < ,z))))
(zeck-combine - y z (left->right 3 rule-c)))
(define (zeck* y z)
(define (M ry Zn Zn_1 [acc null])
(if (null? ry)
acc
(M (rest ry) (zeck+ Zn Zn_1) Zn
(if (zero? (first ry)) acc (zeck+ acc Zn)))))
(cond [(zeck< z y) (zeck* z y)]
[(null? y) null] ; 0 * z -> 0
[else (M (reverse y) z z)]))
(define (zeck-quotient/remainder y z)
(define (M Zn acc)
(if (zeck< y Zn)
(drop-right acc 1)
(M (zeck+ Zn (first acc)) (cons Zn acc))))
(define (D x m [acc null])
(if (null? m)
(values (reverse acc) x)
(let* ([v (first m)]
[smaller (zeck< v x)]
[bit (if smaller 1 0)]
[x_ (if smaller (zeck- x v) x)])
(D x_ (rest m) (cons bit acc)))))
(D y (M z (list z))))
(define (zeck-quotient y z)
(let-values ([(quotient _) (zeck-quotient/remainder y z)])
quotient))
(define (zeck-remainder y z)
(let-values ([(_ remainder) (zeck-quotient/remainder y z)])
remainder))
(define (zeck-add1 z)
(zeck+ z '(1)))
(define (zeck= y z)
(equal? (unpad y) (unpad z)))
(define (zeck< y z)
; Compare equal-length unpadded zecks
(define (LT a b)
(if (null? a)
#f
(let ([a0 (first a)] [b0 (first b)])
(if (= a0 b0)
(LT (rest a) (rest b))
(= a0 0)))))
(let* ([a (unpad y)] [len-a (length a)]
[b (unpad z)] [len-b (length b)])
(cond [(< len-a len-b) #t]
[(> len-a len-b) #f]
[else (LT a b)])))
(define (zeck> y z)
(not (or (zeck= y z) (zeck< y z))))
;; Examples
;;
(define (example op-name op a b)
(let* ([y (natural->zeck a)]
[z (natural->zeck b)]
[x (op y z)]
[c (zeck->natural x)])
(printf "~a ~a ~a = ~a ~a ~a = ~a = ~a\n"
a op-name b y op-name z x c)))
(example '+ zeck+ 888 111)
(example '- zeck- 888 111)
(example '* zeck* 8 111)
(example '/ zeck-quotient 9876 1000)
(example '% zeck-remainder 9876 1000)
- Output:
888 + 111 = (1 0 1 0 0 0 1 0 0 1 0 1 0 0) + (1 0 0 1 0 0 0 0 0 1) = (1 0 0 0 0 0 0 0 0 0 1 0 1 0 1) = 999 888 - 111 = (1 0 1 0 0 0 1 0 0 1 0 1 0 0) - (1 0 0 1 0 0 0 0 0 1) = (1 0 0 1 0 0 0 1 0 0 0 0 1 0) = 777 8 * 111 = (1 0 0 0 0) * (1 0 0 1 0 0 0 0 0 1) = (1 0 1 0 0 0 1 0 0 1 0 1 0 0) = 888 9876 / 1000 = (1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 1) / (1 0 0 0 0 0 0 0 0 1 0 0 0 0 0) = (1 0 0 0 1) = 9 9876 % 1000 = (1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 1) % (1 0 0 0 0 0 0 0 0 1 0 0 0 0 0) = (1 0 1 0 0 0 0 1 0 1 0 1 0 1) = 876
Raku
(formerly Perl 6)
This is a somewhat limited implementation of Zeckendorf arithmetic operators. They only handle positive integer values. There are no actual calculations, everything is done with string manipulations, so it doesn't matter what glyphs you use for 1 and 0.
Implemented arithmetic operators:
+z addition -z subtraction ×z multiplication /z division (more of a divmod really) ++z post increment --z post decrement
Comparison operators:
eqz equal nez not equal gtz greater than ltz less than
my $z1 = '1'; # glyph to use for a '1'
my $z0 = '0'; # glyph to use for a '0'
sub zorder($a) { ($z0 lt $z1) ?? $a !! $a.trans([$z0, $z1] => [$z1, $z0]) }
######## Zeckendorf comparison operators #########
# less than
sub infix:<ltz>($a, $b) { $a.&zorder lt $b.&zorder }
# greater than
sub infix:<gtz>($a, $b) { $a.&zorder gt $b.&zorder }
# equal
sub infix:<eqz>($a, $b) { $a eq $b }
# not equal
sub infix:<nez>($a, $b) { $a ne $b }
######## Operators for Zeckendorf arithmetic ########
# post increment
sub postfix:<++z>($a is rw) {
$a = ("$z0$z0"~$a).subst(/("$z0$z0")($z1+ %% $z0)?$/,
-> $/ { "$z0$z1" ~ ($1 ?? $z0 x $1.chars !! '') });
$a ~~ s/^$z0+//;
$a
}
# post decrement
sub postfix:<--z>($a is rw) {
$a.=subst(/$z1($z0*)$/,
-> $/ {$z0 ~ "$z1$z0" x $0.chars div 2 ~ $z1 x $0.chars mod 2});
$a ~~ s/^$z0+(.+)$/$0/;
$a
}
# addition
sub infix:<+z>($a is copy, $b is copy) { $a++z; $a++z while $b--z nez $z0; $a }
# subtraction
sub infix:<-z>($a is copy, $b is copy) { $a--z; $a--z while $b--z nez $z0; $a }
# multiplication
sub infix:<×z>($a, $b) {
return $z0 if $a eqz $z0 or $b eqz $z0;
return $a if $b eqz $z1;
return $b if $a eqz $z1;
my $c = $a;
my $d = $z1;
repeat {
my $e = $z0;
repeat { $c++z; $e++z } until $e eqz $a;
$d++z;
} until $d eqz $b;
$c
}
# division (really more of a div mod)
sub infix:</z>($a is copy, $b is copy) {
fail "Divide by zero" if $b eqz $z0;
return $a if $a eqz $z0 or $b eqz $z1;
my $c = $z0;
repeat {
my $d = $b +z ($z1 ~ $z0);
$c++z;
$a++z;
$a--z while $d--z nez $z0
} until $a ltz $b;
$c ~= " remainder $a" if $a nez $z0;
$c
}
###################### Testing ######################
# helper sub to translate constants into the particular glyphs you used
sub z($a) { $a.trans([<1 0>] => [$z1, $z0]) }
say "Using the glyph '$z1' for 1 and '$z0' for 0\n";
my $fmt = "%-22s = %15s %s\n";
my $zeck = $z1;
printf( $fmt, "$zeck++z", $zeck++z, '# increment' ) for 1 .. 10;
printf $fmt, "$zeck +z {z('1010')}", $zeck +z= z('1010'), '# addition';
printf $fmt, "$zeck -z {z('100')}", $zeck -z= z('100'), '# subtraction';
printf $fmt, "$zeck ×z {z('100101')}", $zeck ×z= z('100101'), '# multiplication';
printf $fmt, "$zeck /z {z('100')}", $zeck /z= z('100'), '# division';
printf( $fmt, "$zeck--z", $zeck--z, '# decrement' ) for 1 .. 5;
printf $fmt, "$zeck ×z {z('101001')}", $zeck ×z= z('101001'), '# multiplication';
printf $fmt, "$zeck /z {z('100')}", $zeck /z= z('100'), '# division';
Testing Output
Using the glyph '1' for 1 and '0' for 0 1++z = 10 # increment 10++z = 100 # increment 100++z = 101 # increment 101++z = 1000 # increment 1000++z = 1001 # increment 1001++z = 1010 # increment 1010++z = 10000 # increment 10000++z = 10001 # increment 10001++z = 10010 # increment 10010++z = 10100 # increment 10100 +z 1010 = 101000 # addition 101000 -z 100 = 100010 # subtraction 100010 ×z 100101 = 100001000001 # multiplication 100001000001 /z 100 = 101010001 # division 101010001--z = 101010000 # decrement 101010000--z = 101001010 # decrement 101001010--z = 101001001 # decrement 101001001--z = 101001000 # decrement 101001000--z = 101000101 # decrement 101000101 ×z 101001 = 101010000010101 # multiplication 101010000010101 /z 100 = 1001010001001 remainder 10 # division
Using alternate glyphs:
Using the glyph 'X' for 1 and 'O' for 0 X++z = XO # increment XO++z = XOO # increment XOO++z = XOX # increment XOX++z = XOOO # increment XOOO++z = XOOX # increment XOOX++z = XOXO # increment XOXO++z = XOOOO # increment XOOOO++z = XOOOX # increment XOOOX++z = XOOXO # increment XOOXO++z = XOXOO # increment XOXOO +z XOXO = XOXOOO # addition XOXOOO -z XOO = XOOOXO # subtraction XOOOXO ×z XOOXOX = XOOOOXOOOOOX # multiplication XOOOOXOOOOOX /z XOO = XOXOXOOOX # division XOXOXOOOX--z = XOXOXOOOO # decrement XOXOXOOOO--z = XOXOOXOXO # decrement XOXOOXOXO--z = XOXOOXOOX # decrement XOXOOXOOX--z = XOXOOXOOO # decrement XOXOOXOOO--z = XOXOOOXOX # decrement XOXOOOXOX ×z XOXOOX = XOXOXOOOOOXOXOX # multiplication XOXOXOOOOOXOXOX /z XOO = XOOXOXOOOXOOX remainder XO # division
Rust
struct Zeckendorf {
d_val: i32,
d_len: i32,
}
impl Zeckendorf {
fn new(x: &str) -> Zeckendorf {
let mut d_val = 0;
let mut q = 1;
let mut i = x.len() as i32 - 1;
let d_len = i / 2;
while i >= 0 {
d_val += (x.chars().nth(i as usize).unwrap() as i32 - '0' as i32) * q;
q *= 2;
i -= 1;
}
Zeckendorf { d_val, d_len }
}
fn a(&mut self, n: i32) {
let mut i = n;
loop {
if self.d_len < i {
self.d_len = i;
}
let j = (self.d_val >> (i * 2)) & 3;
match j {
0 | 1 => return,
2 => {
if ((self.d_val >> ((i + 1) * 2)) & 1) != 1 {
return;
}
self.d_val += 1 << (i * 2 + 1);
return;
}
3 => {
let temp = 3 << (i * 2);
let temp = !temp;
self.d_val &= temp;
self.b((i + 1) * 2);
}
_ => (),
}
i += 1;
}
}
fn b(&mut self, pos: i32) {
if pos == 0 {
self.inc();
return;
}
if ((self.d_val >> pos) & 1) == 0 {
self.d_val += 1 << pos;
self.a(pos / 2);
if pos > 1 {
self.a(pos / 2 - 1);
}
} else {
let temp = 1 << pos;
let temp = !temp;
self.d_val &= temp;
self.b(pos + 1);
self.b(pos - if pos > 1 { 2 } else { 1 });
}
}
fn c(&mut self, pos: i32) {
if ((self.d_val >> pos) & 1) == 1 {
let temp = 1 << pos;
let temp = !temp;
self.d_val &= temp;
return;
}
self.c(pos + 1);
if pos > 0 {
self.b(pos - 1);
} else {
self.inc();
}
}
fn inc(&mut self) -> &mut Self {
self.d_val += 1;
self.a(0);
self
}
fn copy(&self) -> Zeckendorf {
Zeckendorf {
d_val: self.d_val,
d_len: self.d_len,
}
}
fn plus_assign(&mut self, other: &Zeckendorf) {
for gn in 0..(other.d_len + 1) * 2 {
if ((other.d_val >> gn) & 1) == 1 {
self.b(gn);
}
}
}
fn minus_assign(&mut self, other: &Zeckendorf) {
for gn in 0..(other.d_len + 1) * 2 {
if ((other.d_val >> gn) & 1) == 1 {
self.c(gn);
}
}
while (((self.d_val >> self.d_len * 2) & 3) == 0) || (self.d_len == 0) {
self.d_len -= 1;
}
}
fn times_assign(&mut self, other: &Zeckendorf) {
let mut na = other.copy();
let mut nb = other.copy();
let mut nt;
let mut nr = Zeckendorf::new("0");
for i in 0..(self.d_len + 1) * 2 {
if ((self.d_val >> i) & 1) > 0 {
nr.plus_assign(&nb);
}
nt = nb.copy();
nb.plus_assign(&na);
na = nt.copy(); // `na` is now mutable, so this reassignment is allowed
}
self.d_val = nr.d_val;
self.d_len = nr.d_len;
}
fn to_string(&self) -> String {
if self.d_val == 0 {
return "0".to_string();
}
let dig = ["00", "01", "10"];
let dig1 = ["", "1", "10"];
let idx = (self.d_val >> (self.d_len * 2)) & 3;
let mut sb = String::from(dig1[idx as usize]);
for i in (0..self.d_len).rev() {
let idx = (self.d_val >> (i * 2)) & 3;
sb.push_str(dig[idx as usize]);
}
sb
}
}
fn main() {
println!("Addition:");
let mut g = Zeckendorf::new("10");
g.plus_assign(&Zeckendorf::new("10"));
println!("{}", g.to_string());
g.plus_assign(&Zeckendorf::new("10"));
println!("{}", g.to_string());
g.plus_assign(&Zeckendorf::new("1001"));
println!("{}", g.to_string());
g.plus_assign(&Zeckendorf::new("1000"));
println!("{}", g.to_string());
g.plus_assign(&Zeckendorf::new("10101"));
println!("{}", g.to_string());
println!();
println!("Subtraction:");
g = Zeckendorf::new("1000");
g.minus_assign(&Zeckendorf::new("101"));
println!("{}", g.to_string());
g = Zeckendorf::new("10101010");
g.minus_assign(&Zeckendorf::new("1010101"));
println!("{}", g.to_string());
println!();
println!("Multiplication:");
g = Zeckendorf::new("1001");
g.times_assign(&Zeckendorf::new("101"));
println!("{}", g.to_string());
g = Zeckendorf::new("101010");
g.plus_assign(&Zeckendorf::new("101"));
println!("{}", g.to_string());
}
- Output:
Addition: 101 1001 10101 100101 1010000 Subtraction: 1 1000000 Multiplication: 1000100 1000100
Scala
The addition is an implementation of an algorithm suggested in http[:]//arxiv.org/pdf/1207.4497.pdf: Efficient Algorithms for Zeckendorf Arithmetic.
import scala.collection.mutable.ListBuffer
object ZeckendorfArithmetic extends App {
val elapsed: (=> Unit) => Long = f => {
val s = System.currentTimeMillis
f
(System.currentTimeMillis - s) / 1000
}
val add: (Z, Z) => Z = (z1, z2) => z1 + z2
val subtract: (Z, Z) => Z = (z1, z2) => z1 - z2
val multiply: (Z, Z) => Z = (z1, z2) => z1 * z2
val divide: (Z, Z) => Option[Z] = (z1, z2) => z1 / z2
val modulo: (Z, Z) => Option[Z] = (z1, z2) => z1 % z2
val ops = Map(("+", add), ("-", subtract), ("*", multiply), ("/", divide), ("%", modulo))
val calcs = List(
(Z("101"), "+", Z("10100"))
, (Z("101"), "-", Z("10100"))
, (Z("101"), "*", Z("10100"))
, (Z("101"), "/", Z("10100"))
, (Z("-1010101"), "+", Z("10100"))
, (Z("-1010101"), "-", Z("10100"))
, (Z("-1010101"), "*", Z("10100"))
, (Z("-1010101"), "/", Z("10100"))
, (Z("1000101010"), "+", Z("10101010"))
, (Z("1000101010"), "-", Z("10101010"))
, (Z("1000101010"), "*", Z("10101010"))
, (Z("1000101010"), "/", Z("10101010"))
, (Z("10100"), "+", Z("1010"))
, (Z("100101"), "-", Z("100"))
, (Z("1010101010101010101"), "+", Z("-1010101010101"))
, (Z("1010101010101010101"), "-", Z("-1010101010101"))
, (Z("1010101010101010101"), "*", Z("-1010101010101"))
, (Z("1010101010101010101"), "/", Z("-1010101010101"))
, (Z("1010101010101010101"), "%", Z("-1010101010101"))
, (Z("1010101010101010101"), "+", Z("101010101010101"))
, (Z("1010101010101010101"), "-", Z("101010101010101"))
, (Z("1010101010101010101"), "*", Z("101010101010101"))
, (Z("1010101010101010101"), "/", Z("101010101010101"))
, (Z("1010101010101010101"), "%", Z("101010101010101"))
, (Z("10101010101010101010"), "+", Z("1010101010101010"))
, (Z("10101010101010101010"), "-", Z("1010101010101010"))
, (Z("10101010101010101010"), "*", Z("1010101010101010"))
, (Z("10101010101010101010"), "/", Z("1010101010101010"))
, (Z("10101010101010101010"), "%", Z("1010101010101010"))
, (Z("1010"), "%", Z("10"))
, (Z("1010"), "%", Z("-10"))
, (Z("-1010"), "%", Z("10"))
, (Z("-1010"), "%", Z("-10"))
, (Z("100"), "/", Z("0"))
, (Z("100"), "%", Z("0"))
)
val iadd: (BigInt, BigInt) => BigInt = (a, b) => a + b
val isub: (BigInt, BigInt) => BigInt = (a, b) => a - b
// just for result checking:
import Z._
val imul: (BigInt, BigInt) => BigInt = (a, b) => a * b
val idiv: (BigInt, BigInt) => Option[BigInt] = (a, b) => if (b == 0) None else Some(a / b)
val imod: (BigInt, BigInt) => Option[BigInt] = (a, b) => if (b == 0) None else Some(a % b)
val iops = Map(("+", iadd), ("-", isub), ("*", imul), ("/", idiv), ("%", imod))
case class Z(var zs: String) {
import Z._
require((zs.toSet -- Set('-', '0', '1') == Set()) && (!zs.contains("11")))
//--- fa(summand1.z,summand2.z) --------------------------
val fa: (BigInt, BigInt) => BigInt = (z1, z2) => {
val v = z1.toString.toCharArray.map(_.asDigit).reverse.padTo(5, 0).zipAll(z2.toString.toCharArray.map(_.asDigit).reverse, 0, 0)
val arr1 = (v.map(p => p._1 + p._2) :+ 0).reverse
(0 to arr1.length - 4) foreach { i => //stage1
val a = arr1.slice(i, i + 4).toList
val b = a.foldRight("")("" + _ + _) dropRight 1
val a1 = b match {
case "020" => List(1, 0, 0, a(3) + 1)
case "030" => List(1, 1, 0, a(3) + 1)
case "021" => List(1, 1, 0, a(3))
case "012" => List(1, 0, 1, a(3))
case _ => a
}
0 to 3 foreach { j => arr1(j + i) = a1(j) }
}
val arr2 = arr1.foldRight("")("" + _ + _)
.replace("0120", "1010").replace("030", "111").replace("003", "100").replace("020", "101")
.replace("003", "100").replace("012", "101").replace("021", "110")
.replace("02", "10").replace("03", "11")
.reverse.toArray
(0 to arr2.length - 3) foreach { i => //stage2, step1
val a = arr2.slice(i, i + 3).toList
val b = a.foldRight("")("" + _ + _)
val a1 = b match {
case "110" => List('0', '0', '1')
case _ => a
}
0 to 2 foreach { j => arr2(j + i) = a1(j) }
}
val arr3 = arr2.foldRight("")("" + _ + _).concat("0").reverse.toArray
(0 to arr3.length - 3) foreach { i => //stage2, step2
val a = arr3.slice(i, i + 3).toList
val b = a.foldRight("")("" + _ + _)
val a1 = b match {
case "011" => List('1', '0', '0')
case _ => a
}
0 to 2 foreach { j => arr3(j + i) = a1(j) }
}
BigInt(arr3.foldRight("")("" + _ + _))
}
//--- fs(minuend.z,subtrahend.z) -------------------------
val fs: (BigInt, BigInt) => BigInt = (min, sub) => {
val zmvr = min.toString.toCharArray.map(_.asDigit).reverse
val zsvr = sub.toString.toCharArray.map(_.asDigit).reverse.padTo(zmvr.length, 0)
val v = zmvr.zipAll(zsvr, 0, 0).reverse
val last = v.length - 1
val zma = zmvr.reverse.toArray
val zsa = zsvr.reverse.toArray
for (i <- (0 to last).reverse) {
val e = zma(i) - zsa(i)
if (e < 0) {
zma(i - 1) = zma(i - 1) - 1
zma(i) = 0
val part = Z(((i to last).map(zma(_))).foldRight("")("" + _ + _))
val carry = Z("1".padTo(last - i, "0").foldRight("")("" + _ + _))
val sum = part + carry
val sums = sum.z.toString
(1 to sum.size) foreach { j => zma(last - sum.size + j) = sums(j - 1).asDigit }
if (zma(i - 1) < 0) {
for (j <- (0 until i).reverse) {
if (zma(j) < 0) {
zma(j - 1) = zma(j - 1) - 1
zma(j) = 0
val part = Z(((j to last).map(zma(_))).foldRight("")("" + _ + _))
val carry = Z("1".padTo(last - j, "0").foldRight("")("" + _ + _))
val sum = part + carry
val sums = sum.z.toString
(1 to sum.size) foreach { k => zma(last - sum.size + k) = sums(k - 1).asDigit }
}
}
}
}
else zma(i) = e
zsa(i) = 0
}
BigInt(zma.foldRight("")("" + _ + _))
}
//--- fm(multiplicand.z,multplier.z) ---------------------
val fm: (BigInt, BigInt) => BigInt = (mc, mp) => {
val mct = mt(Z(mc.toString))
val mpxi = mp.toString.reverse.toCharArray.map(_.asDigit).zipWithIndex.filter(_._1 != 0).map(_._2)
mpxi.foldRight(Z("0"))((fi, sum) => sum + mct(fi)).z
}
//--- fd(dividend.z,divisor.z) ---------------------------
val fd: (BigInt, BigInt) => BigInt = (dd, ds) => {
val dst = dt(Z(dd.toString), Z(ds.toString)).reverse
var diff = Z(dd.toString)
val zd = ListBuffer[String]()
0 until dst.length foreach { i =>
if (dst(i) > diff) zd += "0" else {
diff = diff - dst(i)
zd += "1"
}
}
BigInt(zd.mkString)
}
val fasig: (Z, Z) => Int = (z1, z2) => if (z1.z.abs > z2.z.abs) z1.z.signum else z2.z.signum
val fssig: (Z, Z) => Int = (z1, z2) =>
if ((z1.z.abs > z2.z.abs && z1.z.signum > 0) || (z1.z.abs < z2.z.abs && z1.z.signum < 0)) 1 else -1
var z: BigInt = BigInt(zs)
override def toString: String = "" + z + "Z(i:" + z2i(this) + ")"
def size: Int = z.abs.toString.length
def ++ : Z = {
val za = this + Z("1")
this.zs = za.zs
this.z = za.z
this
}
def +(that: Z): Z =
if (this == Z("0")) that
else if (that == Z("0")) this
else if (this.z.signum == that.z.signum) Z((fa(this.z.abs.max(that.z.abs), this.z.abs.min(that.z.abs)) * this.z.signum).toString)
else if (this.z.abs == that.z.abs) Z("0")
else Z((fs(this.z.abs.max(that.z.abs), this.z.abs.min(that.z.abs)) * fasig(this, that)).toString)
def -- : Z = {
val zs = this - Z("1")
this.zs = zs.zs
this.z = zs.z
this
}
def -(that: Z): Z =
if (this == Z("0")) Z((that.z * (-1)).toString)
else if (that == Z("0")) this
else if (this.z.signum != that.z.signum) Z((fa(this.z.abs.max(that.z.abs), this.z.abs.min(that.z.abs)) * this.z.signum).toString)
else if (this.z.abs == that.z.abs) Z("0")
else Z((fs(this.z.abs.max(that.z.abs), this.z.abs.min(that.z.abs)) * fssig(this, that)).toString)
def %(that: Z): Option[Z] =
if (that == Z("0")) None
else if (this == Z("0")) Some(Z("0"))
else if (that == Z("1")) Some(Z("0"))
else if (this.z.abs < that.z.abs) Some(this)
else if (this.z == that.z) Some(Z("0"))
else this / that match {
case None => None
case Some(z) => Some(this - z * that)
}
def *(that: Z): Z =
if (this == Z("0") || that == Z("0")) Z("0")
else if (this == Z("1")) that
else if (that == Z("1")) this
else Z((fm(this.z.abs.max(that.z.abs), this.z.abs.min(that.z.abs)) * this.z.signum * that.z.signum).toString)
def /(that: Z): Option[Z] =
if (that == Z("0")) None
else if (this == Z("0")) Some(Z("0"))
else if (that == Z("1")) Some(Z("1"))
else if (this.z.abs < that.z.abs) Some(Z("0"))
else if (this.z == that.z) Some(Z("1"))
else Some(Z((fd(this.z.abs.max(that.z.abs), this.z.abs.min(that.z.abs)) * this.z.signum * that.z.signum).toString))
def <(that: Z): Boolean = this.z < that.z
def <=(that: Z): Boolean = this.z <= that.z
def >(that: Z): Boolean = this.z > that.z
def >=(that: Z): Boolean = this.z >= that.z
}
object Z {
// only for comfort and result checking:
val fibs: LazyList[BigInt] = {
def series(i: BigInt, j: BigInt): LazyList[BigInt] = i #:: series(j, i + j)
series(1, 0).tail.tail.tail
}
val z2i: Z => BigInt = z => z.z.abs.toString.toCharArray.map(_.asDigit).reverse.zipWithIndex.map { case (v, i) => v * fibs(i) }.foldRight(BigInt(0))(_ + _) * z.z.signum
var fmts: Map[Z, List[Z]] = Map(Z("0") -> List[Z](Z("0"))) //map of Fibonacci multiples table of divisors
// get division table (division weight vector)
def dt(dd: Z, ds: Z): List[Z] = {
val wv = new ListBuffer[Z]
wv ++= mt(ds)
var zs = ds.z.abs.toString
val upper = dd.z.abs.toString
while ((zs.length < upper.length)) {
wv += (wv.toList.last + wv.toList.reverse.tail.head)
zs = "1" + zs
}
wv.toList
}
// get multiply table from fmts
def mt(z: Z): List[Z] = {
fmts.getOrElse(z, Nil) match {
case Nil =>
val e = mwv(z)
fmts = fmts + (z -> e)
e
case l => l
}
}
// multiply weight vector
def mwv(z: Z): List[Z] = {
val wv = new ListBuffer[Z]
wv += z
wv += (z + z)
var zs = "11"
val upper = z.z.abs.toString
while ((zs.length < upper.length)) {
wv += (wv.toList.last + wv.toList.reverse.tail.head)
zs = "1" + zs
}
wv.toList
}
}
println("elapsed time: " + elapsed {
calcs foreach { case (op1, op, op2) => println("" + op1 + " " + op + " " + op2 + " = "
+ {
(ops(op)) (op1, op2) match {
case None => None
case Some(z) => z
case z => z
}
}
.ensuring { x =>
(iops(op)) (z2i(op1), z2i(op2)) match {
case None => None == x
case Some(i) => i == z2i(x.asInstanceOf[Z])
case i => i == z2i(x.asInstanceOf[Z])
}
})
}
} + " sec"
)
}
Output:
101Z(i:4) + 10100Z(i:11) = 100010Z(i:15) 101Z(i:4) - 10100Z(i:11) = -1010Z(i:-7) 101Z(i:4) * 10100Z(i:11) = 10010010Z(i:44) 101Z(i:4) / 10100Z(i:11) = 0Z(i:0) -1010101Z(i:-33) + 10100Z(i:11) = -1000001Z(i:-22) -1010101Z(i:-33) - 10100Z(i:11) = -10010010Z(i:-44) -1010101Z(i:-33) * 10100Z(i:11) = -101010001010Z(i:-363) -1010101Z(i:-33) / 10100Z(i:11) = -100Z(i:-3) 1000101010Z(i:109) + 10101010Z(i:54) = 10000101001Z(i:163) 1000101010Z(i:109) - 10101010Z(i:54) = 100000000Z(i:55) 1000101010Z(i:109) * 10101010Z(i:54) = 101000001000101001Z(i:5886) 1000101010Z(i:109) / 10101010Z(i:54) = 10Z(i:2) 10100Z(i:11) + 1010Z(i:7) = 101000Z(i:18) 100101Z(i:17) - 100Z(i:3) = 100001Z(i:14) 1010101010101010101Z(i:10945) + -1010101010101Z(i:-609) = 1010100000000000000Z(i:10336) 1010101010101010101Z(i:10945) - -1010101010101Z(i:-609) = 10000001010101010100Z(i:11554) 1010101010101010101Z(i:10945) * -1010101010101Z(i:-609) = -100010001000001001010010100100001Z(i:-6665505) 1010101010101010101Z(i:10945) / -1010101010101Z(i:-609) = -100101Z(i:-17) 1010101010101010101Z(i:10945) % -1010101010101Z(i:-609) = 1010100100100Z(i:592) 1010101010101010101Z(i:10945) + 101010101010101Z(i:1596) = 10000101010101010100Z(i:12541) 1010101010101010101Z(i:10945) - 101010101010101Z(i:1596) = 1010000000000000000Z(i:9349) 1010101010101010101Z(i:10945) * 101010101010101Z(i:1596) = 10001000100001010001001010001001001Z(i:17468220) 1010101010101010101Z(i:10945) / 101010101010101Z(i:1596) = 1001Z(i:6) 1010101010101010101Z(i:10945) % 101010101010101Z(i:1596) = 101000000001000Z(i:1369) 10101010101010101010Z(i:17710) + 1010101010101010Z(i:2583) = 100001010101010101001Z(i:20293) 10101010101010101010Z(i:17710) - 1010101010101010Z(i:2583) = 10100000000000000000Z(i:15127) 10101010101010101010Z(i:17710) * 1010101010101010Z(i:2583) = 1000100010001000000000001000100010001Z(i:45744930) 10101010101010101010Z(i:17710) / 1010101010101010Z(i:2583) = 1001Z(i:6) 10101010101010101010Z(i:17710) % 1010101010101010Z(i:2583) = 1010000000001000Z(i:2212) 1010Z(i:7) % 10Z(i:2) = 1Z(i:1) 1010Z(i:7) % -10Z(i:-2) = 1Z(i:1) -1010Z(i:-7) % 10Z(i:2) = -1Z(i:-1) -1010Z(i:-7) % -10Z(i:-2) = -1Z(i:-1) 100Z(i:3) / 0Z(i:0) = None 100Z(i:3) % 0Z(i:0) = None elapsed time: 1 sec
Tcl
namespace eval zeckendorf {
# Want to use alternate symbols? Change these
variable zero "0"
variable one "1"
# Base operations: increment and decrement
proc zincr var {
upvar 1 $var a
namespace upvar [namespace current] zero 0 one 1
if {![regsub "$0$" $a $1$0 a]} {append a $1}
while {[regsub "$0$1$1" $a "$1$0$0" a]
|| [regsub "^$1$1" $a "$1$0$0" a]} {}
regsub ".$" $a "" a
return $a
}
proc zdecr var {
upvar 1 $var a
namespace upvar [namespace current] zero 0 one 1
regsub "^$0+(.+)$" [subst [regsub "${1}($0*)$" $a "$0\[
string repeat {$1$0} \[regsub -all .. {\\1} {} x]]\[
string repeat {$1} \[expr {\$x ne {}}]]"]
] {\1} a
return $a
}
# Exported operations
proc eq {a b} {
expr {$a eq $b}
}
proc add {a b} {
variable zero
while {![eq $b $zero]} {
zincr a
zdecr b
}
return $a
}
proc sub {a b} {
variable zero
while {![eq $b $zero]} {
zdecr a
zdecr b
}
return $a
}
proc mul {a b} {
variable zero
variable one
if {[eq $a $zero] || [eq $b $zero]} {return $zero}
if {[eq $a $one]} {return $b}
if {[eq $b $one]} {return $a}
set c $a
while {![eq [zdecr b] $zero]} {
set c [add $c $a]
}
return $c
}
proc div {a b} {
variable zero
variable one
if {[eq $b $zero]} {error "div zero"}
if {[eq $a $zero] || [eq $b $one]} {return $a}
set r $zero
while {![eq $a $zero]} {
if {![eq $a [add [set a [sub $a $b]] $b]]} break
zincr r
}
return $r
}
# Note that there aren't any ordering operations in this version
# Assemble into a coherent API
namespace export \[a-y\]*
namespace ensemble create
}
Demonstrating:
puts [zeckendorf add "10100" "1010"]
puts [zeckendorf sub "10100" "1010"]
puts [zeckendorf mul "10100" "1010"]
puts [zeckendorf div "10100" "1010"]
puts [zeckendorf div [zeckendorf mul "10100" "1010"] "1010"]
- Output:
101000 101 101000001 1 10100
Visual Basic .NET
Imports System.Text
Module Module1
Class Zeckendorf
Implements IComparable(Of Zeckendorf)
Private Shared ReadOnly dig As String() = {"00", "01", "10"}
Private Shared ReadOnly dig1 As String() = {"", "1", "10"}
Private dVal As Integer = 0
Private dLen As Integer = 0
Public Sub New(Optional x As String = "0")
Dim q = 1
Dim i = x.Length - 1
dLen = i \ 2
Dim z = Asc("0")
While i >= 0
Dim a = Asc(x(i))
dVal += (a - z) * q
q *= 2
i -= 1
End While
End Sub
Private Sub A(n As Integer)
Dim i = n
While True
If dLen < i Then
dLen = i
End If
Dim j = (dVal >> (i * 2)) And 3
If j = 0 OrElse j = 1 Then
Return
ElseIf j = 2 Then
If ((dVal >> ((i + 1) * 2)) And 1) <> 1 Then
Return
End If
dVal += 1 << (i * 2 + 1)
Return
ElseIf j = 3 Then
Dim temp = 3 << (i * 2)
temp = temp Xor -1
dVal = dVal And temp
B((i + 1) * 2)
End If
i += 1
End While
End Sub
Private Sub B(pos As Integer)
If pos = 0 Then
Inc()
Return
End If
If ((dVal >> pos) And 1) = 0 Then
dVal += 1 << pos
A(pos \ 2)
If pos > 1 Then
A(pos \ 2 - 1)
End If
Else
Dim temp = 1 << pos
temp = temp Xor -1
dVal = dVal And temp
B(pos + 1)
B(pos - If(pos > 1, 2, 1))
End If
End Sub
Private Sub C(pos As Integer)
If ((dVal >> pos) And 1) = 1 Then
Dim temp = 1 << pos
temp = temp Xor -1
dVal = dVal And temp
Return
End If
C(pos + 1)
If pos > 0 Then
B(pos - 1)
Else
Inc()
End If
End Sub
Public Function Inc() As Zeckendorf
dVal += 1
A(0)
Return Me
End Function
Public Function Copy() As Zeckendorf
Dim z As New Zeckendorf With {
.dVal = dVal,
.dLen = dLen
}
Return z
End Function
Public Sub PlusAssign(other As Zeckendorf)
Dim gn = 0
While gn < (other.dLen + 1) * 2
If ((other.dVal >> gn) And 1) = 1 Then
B(gn)
End If
gn += 1
End While
End Sub
Public Sub MinusAssign(other As Zeckendorf)
Dim gn = 0
While gn < (other.dLen + 1) * 2
If ((other.dVal >> gn) And 1) = 1 Then
C(gn)
End If
gn += 1
End While
While (((dVal >> dLen * 2) And 3) = 0) OrElse dLen = 0
dLen -= 1
End While
End Sub
Public Sub TimesAssign(other As Zeckendorf)
Dim na = other.Copy
Dim nb = other.Copy
Dim nt As Zeckendorf
Dim nr As New Zeckendorf
Dim i = 0
While i < (dLen + 1) * 2
If ((dVal >> i) And 1) > 0 Then
nr.PlusAssign(nb)
End If
nt = nb.Copy
nb.PlusAssign(na)
na = nt.Copy
i += 1
End While
dVal = nr.dVal
dLen = nr.dLen
End Sub
Public Function CompareTo(other As Zeckendorf) As Integer Implements IComparable(Of Zeckendorf).CompareTo
Return dVal.CompareTo(other.dVal)
End Function
Public Overrides Function ToString() As String
If dVal = 0 Then
Return "0"
End If
Dim idx = (dVal >> (dLen * 2)) And 3
Dim sb As New StringBuilder(dig1(idx))
Dim i = dLen - 1
While i >= 0
idx = (dVal >> (i * 2)) And 3
sb.Append(dig(idx))
i -= 1
End While
Return sb.ToString
End Function
End Class
Sub Main()
Console.WriteLine("Addition:")
Dim g As New Zeckendorf("10")
g.PlusAssign(New Zeckendorf("10"))
Console.WriteLine(g)
g.PlusAssign(New Zeckendorf("10"))
Console.WriteLine(g)
g.PlusAssign(New Zeckendorf("1001"))
Console.WriteLine(g)
g.PlusAssign(New Zeckendorf("1000"))
Console.WriteLine(g)
g.PlusAssign(New Zeckendorf("10101"))
Console.WriteLine(g)
Console.WriteLine()
Console.WriteLine("Subtraction:")
g = New Zeckendorf("1000")
g.MinusAssign(New Zeckendorf("101"))
Console.WriteLine(g)
g = New Zeckendorf("10101010")
g.MinusAssign(New Zeckendorf("1010101"))
Console.WriteLine(g)
Console.WriteLine()
Console.WriteLine("Multiplication:")
g = New Zeckendorf("1001")
g.TimesAssign(New Zeckendorf("101"))
Console.WriteLine(g)
g = New Zeckendorf("101010")
g.PlusAssign(New Zeckendorf("101"))
Console.WriteLine(g)
End Sub
End Module
- Output:
Addition: 101 1001 10101 100101 1010000 Subtraction: 1 1000000 Multiplication: 1000100 1000100
V (Vlang)
import strings
const (
dig = ["00", "01", "10"]
dig1 = ["", "1", "10"]
)
struct Zeckendorf {
mut:
d_val int
d_len int
}
fn new_zeck(xx string) Zeckendorf {
mut z := Zeckendorf{}
mut x := xx
if x == "" {
x = "0"
}
mut q := 1
mut i := x.len - 1
z.d_len = i / 2
for ; i >= 0; i-- {
z.d_val += int(x[i]-'0'[0]) * q
q *= 2
}
return z
}
fn (mut z Zeckendorf) a(ii int) {
mut i:=ii
for ; ; i++ {
if z.d_len < i {
z.d_len = i
}
j := (z.d_val >> u32(i*2)) & 3
if j in [0, 1] {
return
} else if j==2 {
if ((z.d_val >> (u32(i+1) * 2)) & 1) != 1 {
return
}
z.d_val += 1 << u32(i*2+1)
return
} else {// 3
z.d_val &= ~(3 << u32(i*2))
z.b((i + 1) * 2)
}
}
}
fn (mut z Zeckendorf) b(p int) {
mut pos := p
if pos == 0 {
z.inc()
return
}
if ((z.d_val >> u32(pos)) & 1) == 0 {
z.d_val += 1 << u32(pos)
z.a(pos / 2)
if pos > 1 {
z.a(pos/2 - 1)
}
} else {
z.d_val &= ~(1 << u32(pos))
z.b(pos + 1)
mut temp := 1
if pos > 1 {
temp = 2
}
z.b(pos - temp)
}
}
fn (mut z Zeckendorf) c(p int) {
mut pos := p
if ((z.d_val >> u32(pos)) & 1) == 1 {
z.d_val &= ~(1 << u32(pos))
return
}
z.c(pos + 1)
if pos > 0 {
z.b(pos - 1)
} else {
z.inc()
}
}
fn (mut z Zeckendorf) inc() {
z.d_val++
z.a(0)
}
fn (mut z1 Zeckendorf) plus_assign(z2 Zeckendorf) {
for gn := 0; gn < (z2.d_len+1)*2; gn++ {
if ((z2.d_val >> u32(gn)) & 1) == 1 {
z1.b(gn)
}
}
}
fn (mut z1 Zeckendorf) minus_assign(z2 Zeckendorf) {
for gn := 0; gn < (z2.d_len+1)*2; gn++ {
if ((z2.d_val >> u32(gn)) & 1) == 1 {
z1.c(gn)
}
}
for z1.d_len > 0 && ((z1.d_val>>u32(z1.d_len*2))&3) == 0 {
z1.d_len--
}
}
fn (mut z1 Zeckendorf) times_assign(z2 Zeckendorf) {
mut na := z2.copy()
mut nb := z2.copy()
mut nr := Zeckendorf{}
for i := 0; i <= (z1.d_len+1)*2; i++ {
if ((z1.d_val >> u32(i)) & 1) > 0 {
nr.plus_assign(nb)
}
nt := nb.copy()
nb.plus_assign(na)
na = nt.copy()
}
z1.d_val = nr.d_val
z1.d_len = nr.d_len
}
fn (z Zeckendorf) copy() Zeckendorf {
return Zeckendorf{z.d_val, z.d_len}
}
fn (z1 Zeckendorf) compare(z2 Zeckendorf) int {
if z1.d_val < z2.d_val {
return -1
} else if z1.d_val > z2.d_val {
return 1
} else {
return 0
}
}
fn (z Zeckendorf) str() string {
if z.d_val == 0 {
return "0"
}
mut sb := strings.new_builder(128)
sb.write_string(dig1[(z.d_val>>u32(z.d_len*2))&3])
for i := z.d_len - 1; i >= 0; i-- {
sb.write_string(dig[(z.d_val>>u32(i*2))&3])
}
return sb.str()
}
fn main() {
println("Addition:")
mut g := new_zeck("10")
g.plus_assign(new_zeck("10"))
println(g)
g.plus_assign(new_zeck("10"))
println(g)
g.plus_assign(new_zeck("1001"))
println(g)
g.plus_assign(new_zeck("1000"))
println(g)
g.plus_assign(new_zeck("10101"))
println(g)
println("\nSubtraction:")
g = new_zeck("1000")
g.minus_assign(new_zeck("101"))
println(g)
g = new_zeck("10101010")
g.minus_assign(new_zeck("1010101"))
println(g)
println("\nMultiplication:")
g = new_zeck("1001")
g.times_assign(new_zeck("101"))
println(g)
g = new_zeck("101010")
g.plus_assign(new_zeck("101"))
println(g)
}- Output:
Addition: 101 1001 10101 100101 1010000 Subtraction: 1 1000000 Multiplication: 1000100 1000100
Wren
import "./trait" for Comparable
class Zeckendorf is Comparable {
static dig { ["00", "01", "10"] }
static dig1 { ["", "1", "10"] }
construct new(x) {
var q = 1
var i = x.count - 1
_dLen = (i / 2).floor
_dVal = 0
while (i >= 0) {
_dVal = _dVal + (x[i].bytes[0] - 48) * q
q = q * 2
i = i - 1
}
}
dLen { _dLen }
dVal { _dVal }
dLen=(v) { _dLen = v }
dVal=(v) { _dVal = v }
a(n) {
var i = n
while (true) {
if (_dLen < i) _dLen = i
var j = (_dVal >> (i * 2)) & 3
if (j == 0 || j == 1) return
if (j == 2) {
if (((_dVal >> ((i + 1) * 2)) & 1) != 1) return
_dVal = _dVal + (1 << (i * 2 + 1))
return
}
if (j == 3) {
_dVal = _dVal & ~(3 << (i * 2))
b((i + 1) * 2)
}
i = i + 1