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Nimber arithmetic

From Rosetta Code
Nimber arithmetic is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

The nimbers, also known as Grundy numbers, are the values of the heaps in the game of Nim. They have addition and multiplication operations, unrelated to the addition and multiplication of the integers. Both operations are defined recursively:

The nim-sum of two integers m and n, denoted m⊕n is given by

m⊕n=mex(m'⊕n, m⊕ n' : m'<m, n'<n),

where the mex function returns the smallest integer not in the set. More simply: collect all the nim-sums of m and numbers smaller than n, and all nim-sums of n with all numbers less than m and find the smallest number not in that set. Fortunately, this also turns out to be equal to the bitwise xor of the two.

The nim-product is also defined recursively:

m⊗n=mex([m'⊗n]⊕[m⊗n']⊕[m'⊗n'] : m'<m, n'<n)

The product is more complicated and time-consuming to evaluate, but there are a few facts which may help:

  • The operators and are commutative and distributive
  • the nim-product of a Fermat power (22k) and a smaller number is their ordinary product
  • the nim-square of a Fermat power x is the ordinary product 3x/2


Tasks:

  1. Create nimber addition and multiplication tables up to at least 15
  2. Find the nim-sum and nim-product of two five digit integers of your choice

C[edit]

Translation of: FreeBASIC
#include <stdio.h>
#include <stdint.h>
 
// highest power of 2 that divides a given number
uint32_t hpo2(uint32_t n) {
return n & -n;
}
 
// base 2 logarithm of the highest power of 2 dividing a given number
uint32_t lhpo2(uint32_t n) {
uint32_t q = 0, m = hpo2(n);
for (; m % 2 == 0; m >>= 1, ++q) {}
return q;
}
 
// nim-sum of two numbers
uint32_t nimsum(uint32_t x, uint32_t y) {
return x ^ y;
}
 
// nim-product of two numbers
uint32_t nimprod(uint32_t x, uint32_t y) {
if (x < 2 || y < 2)
return x * y;
uint32_t h = hpo2(x);
if (x > h)
return nimprod(h, y) ^ nimprod(x ^ h, y);
if (hpo2(y) < y)
return nimprod(y, x);
uint32_t xp = lhpo2(x), yp = lhpo2(y);
uint32_t comp = xp & yp;
if (comp == 0)
return x * y;
h = hpo2(comp);
return nimprod(nimprod(x >> h, y >> h), 3 << (h - 1));
}
 
void print_table(uint32_t n, char op, uint32_t(*func)(uint32_t, uint32_t)) {
printf(" %c |", op);
for (uint32_t a = 0; a <= n; ++a)
printf("%3d", a);
printf("\n--- -");
for (uint32_t a = 0; a <= n; ++a)
printf("---");
printf("\n");
for (uint32_t b = 0; b <= n; ++b) {
printf("%2d |", b);
for (uint32_t a = 0; a <= n; ++a)
printf("%3d", func(a, b));
printf("\n");
}
}
 
int main() {
print_table(15, '+', nimsum);
printf("\n");
print_table(15, '*', nimprod);
const uint32_t a = 21508, b = 42689;
printf("\n%d + %d = %d\n", a, b, nimsum(a, b));
printf("%d * %d = %d\n", a, b, nimprod(a, b));
return 0;
}
Output:
 + |  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15
--- -------------------------------------------------
 0 |  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15
 1 |  1  0  3  2  5  4  7  6  9  8 11 10 13 12 15 14
 2 |  2  3  0  1  6  7  4  5 10 11  8  9 14 15 12 13
 3 |  3  2  1  0  7  6  5  4 11 10  9  8 15 14 13 12
 4 |  4  5  6  7  0  1  2  3 12 13 14 15  8  9 10 11
 5 |  5  4  7  6  1  0  3  2 13 12 15 14  9  8 11 10
 6 |  6  7  4  5  2  3  0  1 14 15 12 13 10 11  8  9
 7 |  7  6  5  4  3  2  1  0 15 14 13 12 11 10  9  8
 8 |  8  9 10 11 12 13 14 15  0  1  2  3  4  5  6  7
 9 |  9  8 11 10 13 12 15 14  1  0  3  2  5  4  7  6
10 | 10 11  8  9 14 15 12 13  2  3  0  1  6  7  4  5
11 | 11 10  9  8 15 14 13 12  3  2  1  0  7  6  5  4
12 | 12 13 14 15  8  9 10 11  4  5  6  7  0  1  2  3
13 | 13 12 15 14  9  8 11 10  5  4  7  6  1  0  3  2
14 | 14 15 12 13 10 11  8  9  6  7  4  5  2  3  0  1
15 | 15 14 13 12 11 10  9  8  7  6  5  4  3  2  1  0

 * |  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15
--- -------------------------------------------------
 0 |  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
 1 |  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15
 2 |  0  2  3  1  8 10 11  9 12 14 15 13  4  6  7  5
 3 |  0  3  1  2 12 15 13 14  4  7  5  6  8 11  9 10
 4 |  0  4  8 12  6  2 14 10 11 15  3  7 13  9  5  1
 5 |  0  5 10 15  2  7  8 13  3  6  9 12  1  4 11 14
 6 |  0  6 11 13 14  8  5  3  7  1 12 10  9 15  2  4
 7 |  0  7  9 14 10 13  3  4 15  8  6  1  5  2 12 11
 8 |  0  8 12  4 11  3  7 15 13  5  1  9  6 14 10  2
 9 |  0  9 14  7 15  6  1  8  5 12 11  2 10  3  4 13
10 |  0 10 15  5  3  9 12  6  1 11 14  4  2  8 13  7
11 |  0 11 13  6  7 12 10  1  9  2  4 15 14  5  3  8
12 |  0 12  4  8 13  1  9  5  6 10  2 14 11  7 15  3
13 |  0 13  6 11  9  4 15  2 14  3  8  5  7 10  1 12
14 |  0 14  7  9  5 11  2 12 10  4 13  3 15  1  8  6
15 |  0 15  5 10  1 14  4 11  2 13  7  8  3 12  6  9

21508 + 42689 = 62149
21508 * 42689 = 35202

FreeBASIC[edit]

function hpo2( n as uinteger ) as uinteger
'highest power of 2 that divides a given number
return n and -n
end function
 
function lhpo2( n as uinteger ) as uinteger
'base 2 logarithm of the highest power of 2 dividing a given number
dim as uinteger q = 0, m = hpo2( n )
while m mod 2 = 0
m = m shr 1
q += 1
wend
return q
end function
 
function nimsum(x as uinteger, y as uinteger) as uinteger
'nim-sum of two numbers
return x xor y
end function
 
function nimprod(x as uinteger, y as uinteger) as uinteger
'nim-product of two numbers
if x < 2 orelse y < 2 then return x*y
dim as uinteger h = hpo2(x)
if x > h then return nimprod(h, y) xor nimprod(x xor h, y) 'recursively break x into its powers of 2
if hpo2(y) < y then return nimprod(y, x) 'recursively break y into its powers of 2 by flipping the operands
'now both x and y are powers of two
dim as uinteger xp = lhpo2(x), yp = lhpo2(y), comp = xp and yp
if comp = 0 then return x*y 'we have no fermat power in common
h = hpo2(comp)
return nimprod(nimprod(x shr h, y shr h), 3 shl (h - 1)) 'a fermat number square is its sequimultiple
end function
 
'print tables
 
function padto( i as ubyte, j as integer ) as string
return wspace(i-len(str(j)))+str(j)
end function
 
dim as uinteger a, b
dim as string outstr
 
outstr = " + | "
for a = 0 to 15
outstr += padto(2, a)+" "
next a
print outstr
print "--- -------------------------------------------------"
for b = 0 to 15
outstr = padto(2, b)+ " | "
for a = 0 to 15
outstr += padto(2, nimsum(a,b))+" "
next a
print outstr
next b
print
outstr = " * | "
for a = 0 to 15
outstr += padto(2, a)+" "
next a
print outstr
print "--- -------------------------------------------------"
for b = 0 to 15
outstr = padto(2, b)+ " | "
for a = 0 to 15
outstr += padto(2, nimprod(a,b))+" "
next a
print outstr
next b
print
a = 21508
b = 42689
 
print using "##### + ##### = ##########"; a; b; nimsum(a,b)
print using "##### * ##### = ##########"; a; b; nimprod(a,b)
Output:
 + |  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 
--- -------------------------------------------------
 0 |  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 
 1 |  1  0  3  2  5  4  7  6  9  8 11 10 13 12 15 14 
 2 |  2  3  0  1  6  7  4  5 10 11  8  9 14 15 12 13 
 3 |  3  2  1  0  7  6  5  4 11 10  9  8 15 14 13 12 
 4 |  4  5  6  7  0  1  2  3 12 13 14 15  8  9 10 11 
 5 |  5  4  7  6  1  0  3  2 13 12 15 14  9  8 11 10 
 6 |  6  7  4  5  2  3  0  1 14 15 12 13 10 11  8  9 
 7 |  7  6  5  4  3  2  1  0 15 14 13 12 11 10  9  8 
 8 |  8  9 10 11 12 13 14 15  0  1  2  3  4  5  6  7 
 9 |  9  8 11 10 13 12 15 14  1  0  3  2  5  4  7  6 
10 | 10 11  8  9 14 15 12 13  2  3  0  1  6  7  4  5 
11 | 11 10  9  8 15 14 13 12  3  2  1  0  7  6  5  4 
12 | 12 13 14 15  8  9 10 11  4  5  6  7  0  1  2  3 
13 | 13 12 15 14  9  8 11 10  5  4  7  6  1  0  3  2 
14 | 14 15 12 13 10 11  8  9  6  7  4  5  2  3  0  1 
15 | 15 14 13 12 11 10  9  8  7  6  5  4  3  2  1  0 

 * |  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 
--- -------------------------------------------------
 0 |  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 
 1 |  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 
 2 |  0  2  3  1  8 10 11  9 12 14 15 13  4  6  7  5 
 3 |  0  3  1  2 12 15 13 14  4  7  5  6  8 11  9 10 
 4 |  0  4  8 12  6  2 14 10 11 15  3  7 13  9  5  1 
 5 |  0  5 10 15  2  7  8 13  3  6  9 12  1  4 11 14 
 6 |  0  6 11 13 14  8  5  3  7  1 12 10  9 15  2  4 
 7 |  0  7  9 14 10 13  3  4 15  8  6  1  5  2 12 11 
 8 |  0  8 12  4 11  3  7 15 13  5  1  9  6 14 10  2 
 9 |  0  9 14  7 15  6  1  8  5 12 11  2 10  3  4 13 
10 |  0 10 15  5  3  9 12  6  1 11 14  4  2  8 13  7 
11 |  0 11 13  6  7 12 10  1  9  2  4 15 14  5  3  8 
12 |  0 12  4  8 13  1  9  5  6 10  2 14 11  7 15  3 
13 |  0 13  6 11  9  4 15  2 14  3  8  5  7 10  1 12 
14 |  0 14  7  9  5 11  2 12 10  4 13  3 15  1  8  6 
15 |  0 15  5 10  1 14  4 11  2 13  7  8  3 12  6  9 

21508 + 42689 =      62149
21508 * 42689 =      35202

Go[edit]

Translation of: FreeBASIC
package main
 
import (
"fmt"
"strings"
)
 
// Highest power of two that divides a given number.
func hpo2(n uint) uint { return n & (-n) }
 
// Base 2 logarithm of the highest power of 2 dividing a given number.
func lhpo2(n uint) uint {
q := uint(0)
m := hpo2(n)
for m%2 == 0 {
m = m >> 1
q++
}
return q
}
 
// nim-sum of two numbers.
func nimsum(x, y uint) uint { return x ^ y }
 
// nim-product of two numbers.
func nimprod(x, y uint) uint {
if x < 2 || y < 2 {
return x * y
}
h := hpo2(x)
if x > h {
return nimprod(h, y) ^ nimprod(x^h, y) // break x into powers of 2
}
if hpo2(y) < y {
return nimprod(y, x) // break y into powers of 2 by flipping operands
}
xp, yp := lhpo2(x), lhpo2(y)
comp := xp & yp
if comp == 0 {
return x * y // no Fermat power in common
}
h = hpo2(comp)
// a Fermat number square is its sequimultiple
return nimprod(nimprod(x>>h, y>>h), 3<<(h-1))
}
 
type fnop struct {
fn func(x, y uint) uint
op string
}
 
func main() {
for _, f := range []fnop{{nimsum, "+"}, {nimprod, "*"}} {
fmt.Printf(" %s |", f.op)
for i := 0; i <= 15; i++ {
fmt.Printf("%3d", i)
}
fmt.Println("\n--- " + strings.Repeat("-", 48))
for i := uint(0); i <= 15; i++ {
fmt.Printf("%2d |", i)
for j := uint(0); j <= 15; j++ {
fmt.Printf("%3d", f.fn(i, j))
}
fmt.Println()
}
fmt.Println()
}
 
a := uint(21508)
b := uint(42689)
fmt.Printf("%d + %d = %d\n", a, b, nimsum(a, b))
fmt.Printf("%d * %d = %d\n", a, b, nimprod(a, b))
}
Output:
 + |  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15
--- ------------------------------------------------
 0 |  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15
 1 |  1  0  3  2  5  4  7  6  9  8 11 10 13 12 15 14
 2 |  2  3  0  1  6  7  4  5 10 11  8  9 14 15 12 13
 3 |  3  2  1  0  7  6  5  4 11 10  9  8 15 14 13 12
 4 |  4  5  6  7  0  1  2  3 12 13 14 15  8  9 10 11
 5 |  5  4  7  6  1  0  3  2 13 12 15 14  9  8 11 10
 6 |  6  7  4  5  2  3  0  1 14 15 12 13 10 11  8  9
 7 |  7  6  5  4  3  2  1  0 15 14 13 12 11 10  9  8
 8 |  8  9 10 11 12 13 14 15  0  1  2  3  4  5  6  7
 9 |  9  8 11 10 13 12 15 14  1  0  3  2  5  4  7  6
10 | 10 11  8  9 14 15 12 13  2  3  0  1  6  7  4  5
11 | 11 10  9  8 15 14 13 12  3  2  1  0  7  6  5  4
12 | 12 13 14 15  8  9 10 11  4  5  6  7  0  1  2  3
13 | 13 12 15 14  9  8 11 10  5  4  7  6  1  0  3  2
14 | 14 15 12 13 10 11  8  9  6  7  4  5  2  3  0  1
15 | 15 14 13 12 11 10  9  8  7  6  5  4  3  2  1  0

 * |  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15
--- ------------------------------------------------
 0 |  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
 1 |  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15
 2 |  0  2  3  1  8 10 11  9 12 14 15 13  4  6  7  5
 3 |  0  3  1  2 12 15 13 14  4  7  5  6  8 11  9 10
 4 |  0  4  8 12  6  2 14 10 11 15  3  7 13  9  5  1
 5 |  0  5 10 15  2  7  8 13  3  6  9 12  1  4 11 14
 6 |  0  6 11 13 14  8  5  3  7  1 12 10  9 15  2  4
 7 |  0  7  9 14 10 13  3  4 15  8  6  1  5  2 12 11
 8 |  0  8 12  4 11  3  7 15 13  5  1  9  6 14 10  2
 9 |  0  9 14  7 15  6  1  8  5 12 11  2 10  3  4 13
10 |  0 10 15  5  3  9 12  6  1 11 14  4  2  8 13  7
11 |  0 11 13  6  7 12 10  1  9  2  4 15 14  5  3  8
12 |  0 12  4  8 13  1  9  5  6 10  2 14 11  7 15  3
13 |  0 13  6 11  9  4 15  2 14  3  8  5  7 10  1 12
14 |  0 14  7  9  5 11  2 12 10  4 13  3 15  1  8  6
15 |  0 15  5 10  1 14  4 11  2 13  7  8  3 12  6  9

21508 + 42689 = 62149
21508 * 42689 = 35202

Julia[edit]

Translation of: FreeBASIC
""" highest power of 2 that divides a given number """
hpo2(n) = n & -n
 
""" base 2 logarithm of the highest power of 2 dividing a given number """
lhpo2(n) = begin q, m = 0, hpo2(n); while iseven(m) m >>= 1; q += 1 end; q end
 
""" nim-sum of two numbers """
nimsum(x, y) = x ⊻ y
 
""" nim-product of two numbers """
function nimprod(x, y)
(x < 2 || y < 2) && return x * y
h = hpo2(x)
(x > h) && return nimprod(h, y) ⊻ nimprod(x ⊻ h, y)
(hpo2(y) < y) && return nimprod(y, x)
xp, yp = lhpo2(x), lhpo2(y)
comp = xp & yp
comp == 0 && return x * y
h = hpo2(comp)
return nimprod(nimprod(x >> h, y >> h), 3 << (h - 1))
end
 
""" print a table of nim-sums or nim-products """
function printtable(n, op)
println(" $op |", prod([lpad(i, 3) for i in 0:n]), "\n--- -", "---"^(n + 1))
for j in 0:n
print(lpad(j, 2), " |")
for i in 0:n
print(lpad(op == '⊕' ? nimsum(i, j) : nimprod(i, j), 3))
end
print(j == n ? "\n\n" : "\n")
end
end
 
const a, b = 21508, 42689
 
printtable(15, '⊕')
printtable(15, '⊗')
println("nim-sum: $a ⊕ $b = $(nimsum(a, b))")
println("nim-product: $a ⊗ $b = $(nimprod(a, b))")
 
Output:
 ⊕ |  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15
--- -------------------------------------------------
 0 |  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15
 1 |  1  0  3  2  5  4  7  6  9  8 11 10 13 12 15 14
 2 |  2  3  0  1  6  7  4  5 10 11  8  9 14 15 12 13
 3 |  3  2  1  0  7  6  5  4 11 10  9  8 15 14 13 12
 4 |  4  5  6  7  0  1  2  3 12 13 14 15  8  9 10 11
 5 |  5  4  7  6  1  0  3  2 13 12 15 14  9  8 11 10
 6 |  6  7  4  5  2  3  0  1 14 15 12 13 10 11  8  9
 7 |  7  6  5  4  3  2  1  0 15 14 13 12 11 10  9  8
 8 |  8  9 10 11 12 13 14 15  0  1  2  3  4  5  6  7
 9 |  9  8 11 10 13 12 15 14  1  0  3  2  5  4  7  6
10 | 10 11  8  9 14 15 12 13  2  3  0  1  6  7  4  5
11 | 11 10  9  8 15 14 13 12  3  2  1  0  7  6  5  4
12 | 12 13 14 15  8  9 10 11  4  5  6  7  0  1  2  3
13 | 13 12 15 14  9  8 11 10  5  4  7  6  1  0  3  2
14 | 14 15 12 13 10 11  8  9  6  7  4  5  2  3  0  1
15 | 15 14 13 12 11 10  9  8  7  6  5  4  3  2  1  0

 ⊗ |  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15
--- -------------------------------------------------
 0 |  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
 1 |  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15
 2 |  0  2  3  1  8 10 11  9 12 14 15 13  4  6  7  5
 3 |  0  3  1  2 12 15 13 14  4  7  5  6  8 11  9 10
 4 |  0  4  8 12  6  2 14 10 11 15  3  7 13  9  5  1
 5 |  0  5 10 15  2  7  8 13  3  6  9 12  1  4 11 14
 6 |  0  6 11 13 14  8  5  3  7  1 12 10  9 15  2  4
 7 |  0  7  9 14 10 13  3  4 15  8  6  1  5  2 12 11
 8 |  0  8 12  4 11  3  7 15 13  5  1  9  6 14 10  2
 9 |  0  9 14  7 15  6  1  8  5 12 11  2 10  3  4 13
10 |  0 10 15  5  3  9 12  6  1 11 14  4  2  8 13  7
11 |  0 11 13  6  7 12 10  1  9  2  4 15 14  5  3  8
12 |  0 12  4  8 13  1  9  5  6 10  2 14 11  7 15  3
13 |  0 13  6 11  9  4 15  2 14  3  8  5  7 10  1 12
14 |  0 14  7  9  5 11  2 12 10  4 13  3 15  1  8  6
15 |  0 15  5 10  1 14  4 11  2 13  7  8  3 12  6  9

nim-sum:     21508 ⊕ 42689 = 62149
nim-product: 21508 ⊗ 42689 = 35202


Phix[edit]

Translation of: FreeBASIC
function hpo2(integer n)
-- highest power of 2 that divides a given number
return and_bits(n,-n)
end function
 
function lhpo2(integer n)
-- base 2 logarithm of the highest power of 2 dividing a given number
integer q = 0, m = hpo2(n)
while remainder(m,2)=0 do
m = floor(m/2)
q += 1
end while
return q
end function
 
function nimsum(integer x, y)
-- nim-sum of two numbers
return xor_bits(x,y)
end function
 
function nimprod(integer x, y)
-- nim-product of two numbers
if x < 2 or y < 2 then return x*y end if
integer h = hpo2(x)
if x > h then
return xor_bits(nimprod(h, y),nimprod(xor_bits(x,h), y)) -- recursively break x into its powers of 2
elsif hpo2(y) < y then
return nimprod(y, x) -- recursively break y into its powers of 2 by flipping the operands
end if
-- now both x and y are powers of two
integer xp = lhpo2(x), yp = lhpo2(y), comp = and_bits(xp,yp)
if comp = 0 then return x*y end if -- we have no fermat power in common
h = hpo2(comp)
return nimprod(nimprod(floor(x/power(2,h)), floor(y/power(2,h))), 3*power(2,h-1)) -- a fermat number square is its sequimultiple
end function
 
procedure print_table(integer n, op)
-- print a table of nim-sums or nim-products
printf(1," %c | "&join(repeat("%3d",n+1))&"\n",op&tagset(n,0))
printf(1,"---+%s\n",repeat('-',(n+1)*4))
for j=0 to n do
printf(1,"%2d |",j)
for i=0 to n do
printf(1,"%4d",iff(op='+' ? nimsum(i, j) : nimprod(i, j)))
end for
printf(1,"\n")
end for
printf(1,"\n")
end procedure
 
print_table(25, '+')
print_table(25, '*')
constant a = 21508, b = 42689
printf(1,"%5d + %5d = %5d\n",{a,b,nimsum(a,b)})
printf(1,"%5d * %5d = %5d\n",{a,b,nimprod(a,b)})
Output:
 + |   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25
---+--------------------------------------------------------------------------------------------------------
 0 |   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25
 1 |   1   0   3   2   5   4   7   6   9   8  11  10  13  12  15  14  17  16  19  18  21  20  23  22  25  24
 2 |   2   3   0   1   6   7   4   5  10  11   8   9  14  15  12  13  18  19  16  17  22  23  20  21  26  27
 3 |   3   2   1   0   7   6   5   4  11  10   9   8  15  14  13  12  19  18  17  16  23  22  21  20  27  26
 4 |   4   5   6   7   0   1   2   3  12  13  14  15   8   9  10  11  20  21  22  23  16  17  18  19  28  29
 5 |   5   4   7   6   1   0   3   2  13  12  15  14   9   8  11  10  21  20  23  22  17  16  19  18  29  28
 6 |   6   7   4   5   2   3   0   1  14  15  12  13  10  11   8   9  22  23  20  21  18  19  16  17  30  31
 7 |   7   6   5   4   3   2   1   0  15  14  13  12  11  10   9   8  23  22  21  20  19  18  17  16  31  30
 8 |   8   9  10  11  12  13  14  15   0   1   2   3   4   5   6   7  24  25  26  27  28  29  30  31  16  17
 9 |   9   8  11  10  13  12  15  14   1   0   3   2   5   4   7   6  25  24  27  26  29  28  31  30  17  16
10 |  10  11   8   9  14  15  12  13   2   3   0   1   6   7   4   5  26  27  24  25  30  31  28  29  18  19
11 |  11  10   9   8  15  14  13  12   3   2   1   0   7   6   5   4  27  26  25  24  31  30  29  28  19  18
12 |  12  13  14  15   8   9  10  11   4   5   6   7   0   1   2   3  28  29  30  31  24  25  26  27  20  21
13 |  13  12  15  14   9   8  11  10   5   4   7   6   1   0   3   2  29  28  31  30  25  24  27  26  21  20
14 |  14  15  12  13  10  11   8   9   6   7   4   5   2   3   0   1  30  31  28  29  26  27  24  25  22  23
15 |  15  14  13  12  11  10   9   8   7   6   5   4   3   2   1   0  31  30  29  28  27  26  25  24  23  22
16 |  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31   0   1   2   3   4   5   6   7   8   9
17 |  17  16  19  18  21  20  23  22  25  24  27  26  29  28  31  30   1   0   3   2   5   4   7   6   9   8
18 |  18  19  16  17  22  23  20  21  26  27  24  25  30  31  28  29   2   3   0   1   6   7   4   5  10  11
19 |  19  18  17  16  23  22  21  20  27  26  25  24  31  30  29  28   3   2   1   0   7   6   5   4  11  10
20 |  20  21  22  23  16  17  18  19  28  29  30  31  24  25  26  27   4   5   6   7   0   1   2   3  12  13
21 |  21  20  23  22  17  16  19  18  29  28  31  30  25  24  27  26   5   4   7   6   1   0   3   2  13  12
22 |  22  23  20  21  18  19  16  17  30  31  28  29  26  27  24  25   6   7   4   5   2   3   0   1  14  15
23 |  23  22  21  20  19  18  17  16  31  30  29  28  27  26  25  24   7   6   5   4   3   2   1   0  15  14
24 |  24  25  26  27  28  29  30  31  16  17  18  19  20  21  22  23   8   9  10  11  12  13  14  15   0   1
25 |  25  24  27  26  29  28  31  30  17  16  19  18  21  20  23  22   9   8  11  10  13  12  15  14   1   0

 * |   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25
---+--------------------------------------------------------------------------------------------------------
 0 |   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0
 1 |   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25
 2 |   0   2   3   1   8  10  11   9  12  14  15  13   4   6   7   5  32  34  35  33  40  42  43  41  44  46
 3 |   0   3   1   2  12  15  13  14   4   7   5   6   8  11   9  10  48  51  49  50  60  63  61  62  52  55
 4 |   0   4   8  12   6   2  14  10  11  15   3   7  13   9   5   1  64  68  72  76  70  66  78  74  75  79
 5 |   0   5  10  15   2   7   8  13   3   6   9  12   1   4  11  14  80  85  90  95  82  87  88  93  83  86
 6 |   0   6  11  13  14   8   5   3   7   1  12  10   9  15   2   4  96 102 107 109 110 104 101  99 103  97
 7 |   0   7   9  14  10  13   3   4  15   8   6   1   5   2  12  11 112 119 121 126 122 125 115 116 127 120
 8 |   0   8  12   4  11   3   7  15  13   5   1   9   6  14  10   2 128 136 140 132 139 131 135 143 141 133
 9 |   0   9  14   7  15   6   1   8   5  12  11   2  10   3   4  13 144 153 158 151 159 150 145 152 149 156
10 |   0  10  15   5   3   9  12   6   1  11  14   4   2   8  13   7 160 170 175 165 163 169 172 166 161 171
11 |   0  11  13   6   7  12  10   1   9   2   4  15  14   5   3   8 176 187 189 182 183 188 186 177 185 178
12 |   0  12   4   8  13   1   9   5   6  10   2  14  11   7  15   3 192 204 196 200 205 193 201 197 198 202
13 |   0  13   6  11   9   4  15   2  14   3   8   5   7  10   1  12 208 221 214 219 217 212 223 210 222 211
14 |   0  14   7   9   5  11   2  12  10   4  13   3  15   1   8   6 224 238 231 233 229 235 226 236 234 228
15 |   0  15   5  10   1  14   4  11   2  13   7   8   3  12   6   9 240 255 245 250 241 254 244 251 242 253
16 |   0  16  32  48  64  80  96 112 128 144 160 176 192 208 224 240  24   8  56  40  88  72 120 104 152 136
17 |   0  17  34  51  68  85 102 119 136 153 170 187 204 221 238 255   8  25  42  59  76  93 110 127 128 145
18 |   0  18  35  49  72  90 107 121 140 158 175 189 196 214 231 245  56  42  27   9 112  98  83  65 180 166
19 |   0  19  33  50  76  95 109 126 132 151 165 182 200 219 233 250  40  59   9  26 100 119  69  86 172 191
20 |   0  20  40  60  70  82 110 122 139 159 163 183 205 217 229 241  88  76 112 100  30  10  54  34 211 199
21 |   0  21  42  63  66  87 104 125 131 150 169 188 193 212 235 254  72  93  98 119  10  31  32  53 203 222
22 |   0  22  43  61  78  88 101 115 135 145 172 186 201 223 226 244 120 110  83  69  54  32  29  11 255 233
23 |   0  23  41  62  74  93  99 116 143 152 166 177 197 210 236 251 104 127  65  86  34  53  11  28 231 240
24 |   0  24  44  52  75  83 103 127 141 149 161 185 198 222 234 242 152 128 180 172 211 203 255 231  21  13
25 |   0  25  46  55  79  86  97 120 133 156 171 178 202 211 228 253 136 145 166 191 199 222 233 240  13  20

21508 + 42689 = 62149
21508 * 42689 = 35202

Prolog[edit]

Translation of: FreeBASIC
Works with: SWI Prolog
% highest power of 2 that divides a given number
hpo2(N, P):-
P is N /\ -N.
 
% base 2 logarithm of the highest power of 2 dividing a given number
lhpo2(N, Q):-
hpo2(N, M),
lhpo2_(M, 0, Q).
 
lhpo2_(M, Q, Q):-
1 is M mod 2,
!.
lhpo2_(M, Q1, Q):-
M1 is M >> 1,
Q2 is Q1 + 1,
lhpo2_(M1, Q2, Q).
 
% nim-sum of two numbers
nimsum(X, Y, Sum):-
Sum is X xor Y.
 
% nim-product of twp numbers
nimprod(X, Y, Product):-
(X < 2 ; Y < 2),
!,
Product is X * Y.
nimprod(X, Y, Product):-
hpo2(X, H),
X > H,
!,
nimprod(H, Y, P1),
X1 is X xor H,
nimprod(X1, Y, P2),
Product is P1 xor P2.
nimprod(X, Y, Product):-
hpo2(Y, H),
H < Y,
!,
nimprod(Y, X, Product).
nimprod(X, Y, Product):-
lhpo2(X, Xp),
lhpo2(Y, Yp),
Comp is Xp /\ Yp,
(Comp == 0 ->
Product is X * Y
;
hpo2(Comp, H),
X1 is X >> H,
Y1 is Y >> H,
Z is 3 << (H - 1),
nimprod(X1, Y1, P),
nimprod(P, Z, Product)
).
 
print_row(N, B, Function):-
writef('%3r |', [B]),
Goal =.. [Function, A, B, C],
forall(between(0, N, A), (Goal, writef('%3r', [C]))),
nl.
 
print_table(N, Operator, Function):-
writef('  %w |', [Operator]),
forall(between(0, N, A), writef('%3r', [A])),
writef('\n --- -', []),
forall(between(0, N, _), writef('---', [])),
nl,
forall(between(0, N, A), print_row(N, A, Function)).
 
main:-
print_table(15, '+', nimsum),
nl,
print_table(15, '*', nimprod),
nl,
A = 21508, B = 42689,
nimsum(A, B, Sum),
nimprod(A, B, Product),
writef('%w + %w = %w\n', [A, B, Sum]),
writef('%w * %w = %w\n', [A, B, Product]).
Output:
  + |  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15
 --- -------------------------------------------------
  0 |  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15
  1 |  1  0  3  2  5  4  7  6  9  8 11 10 13 12 15 14
  2 |  2  3  0  1  6  7  4  5 10 11  8  9 14 15 12 13
  3 |  3  2  1  0  7  6  5  4 11 10  9  8 15 14 13 12
  4 |  4  5  6  7  0  1  2  3 12 13 14 15  8  9 10 11
  5 |  5  4  7  6  1  0  3  2 13 12 15 14  9  8 11 10
  6 |  6  7  4  5  2  3  0  1 14 15 12 13 10 11  8  9
  7 |  7  6  5  4  3  2  1  0 15 14 13 12 11 10  9  8
  8 |  8  9 10 11 12 13 14 15  0  1  2  3  4  5  6  7
  9 |  9  8 11 10 13 12 15 14  1  0  3  2  5  4  7  6
 10 | 10 11  8  9 14 15 12 13  2  3  0  1  6  7  4  5
 11 | 11 10  9  8 15 14 13 12  3  2  1  0  7  6  5  4
 12 | 12 13 14 15  8  9 10 11  4  5  6  7  0  1  2  3
 13 | 13 12 15 14  9  8 11 10  5  4  7  6  1  0  3  2
 14 | 14 15 12 13 10 11  8  9  6  7  4  5  2  3  0  1
 15 | 15 14 13 12 11 10  9  8  7  6  5  4  3  2  1  0

  * |  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15
 --- -------------------------------------------------
  0 |  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
  1 |  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15
  2 |  0  2  3  1  8 10 11  9 12 14 15 13  4  6  7  5
  3 |  0  3  1  2 12 15 13 14  4  7  5  6  8 11  9 10
  4 |  0  4  8 12  6  2 14 10 11 15  3  7 13  9  5  1
  5 |  0  5 10 15  2  7  8 13  3  6  9 12  1  4 11 14
  6 |  0  6 11 13 14  8  5  3  7  1 12 10  9 15  2  4
  7 |  0  7  9 14 10 13  3  4 15  8  6  1  5  2 12 11
  8 |  0  8 12  4 11  3  7 15 13  5  1  9  6 14 10  2
  9 |  0  9 14  7 15  6  1  8  5 12 11  2 10  3  4 13
 10 |  0 10 15  5  3  9 12  6  1 11 14  4  2  8 13  7
 11 |  0 11 13  6  7 12 10  1  9  2  4 15 14  5  3  8
 12 |  0 12  4  8 13  1  9  5  6 10  2 14 11  7 15  3
 13 |  0 13  6 11  9  4 15  2 14  3  8  5  7 10  1 12
 14 |  0 14  7  9  5 11  2 12 10  4 13  3 15  1  8  6
 15 |  0 15  5 10  1 14  4 11  2 13  7  8  3 12  6  9

21508 + 42689 = 62149
21508 * 42689 = 35202

Raku[edit]

Works with: Rakudo version 2020.05
Translation of: FreeBasic

(or at least, heavily inspired by FreeBasic)

Not limited by integer size. Doesn't rely on twos complement bitwise and.

sub infix:<> (Int $x, Int $y) { $x +^ $y }
 
sub infix:<> (Int $x, Int $y) {
return $x × $y if so $x|$y < 2;
my $h = exp $x.lsb, 2;
return ($h$y)(($x$h)$y) if $x > $h;
return ($y$x) if $y.lsb < $y.msb;
return $x × $y unless my $comp = $x.lsb +& $y.lsb;
$h = exp $comp.lsb, 2;
(($x +> $h)($y +> $h))(3 +< ($h - 1))
}
 
# TESTING
 
my $upto = 26;
 
for <>, &infix:<>,
<>, &infix:<>
-> $op, &f {
 
put " $op │", ^$upto .fmt('%3s'), "\n───┼", '────' x $upto;
-> $r { put $r.fmt('%2s'), ' │', ^$upto .map: { &f($r, $_).fmt('%3s')} } for ^$upto;
put "\n";
}
 
put "21508 ⊕ 42689 = ", 2150842689;
put "21508 ⊗ 42689 = ", 2150842689;
 
put "2150821508215082150821508 ⊕ 4268942689426894268942689 = ", 21508215082150821508215084268942689426894268942689;
put "2150821508215082150821508 ⊗ 4268942689426894268942689 = ", 21508215082150821508215084268942689426894268942689;
Output:
 ⊕ │  0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25
───┼────────────────────────────────────────────────────────────────────────────────────────────────────────
 0 │  0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25
 1 │  1   0   3   2   5   4   7   6   9   8  11  10  13  12  15  14  17  16  19  18  21  20  23  22  25  24
 2 │  2   3   0   1   6   7   4   5  10  11   8   9  14  15  12  13  18  19  16  17  22  23  20  21  26  27
 3 │  3   2   1   0   7   6   5   4  11  10   9   8  15  14  13  12  19  18  17  16  23  22  21  20  27  26
 4 │  4   5   6   7   0   1   2   3  12  13  14  15   8   9  10  11  20  21  22  23  16  17  18  19  28  29
 5 │  5   4   7   6   1   0   3   2  13  12  15  14   9   8  11  10  21  20  23  22  17  16  19  18  29  28
 6 │  6   7   4   5   2   3   0   1  14  15  12  13  10  11   8   9  22  23  20  21  18  19  16  17  30  31
 7 │  7   6   5   4   3   2   1   0  15  14  13  12  11  10   9   8  23  22  21  20  19  18  17  16  31  30
 8 │  8   9  10  11  12  13  14  15   0   1   2   3   4   5   6   7  24  25  26  27  28  29  30  31  16  17
 9 │  9   8  11  10  13  12  15  14   1   0   3   2   5   4   7   6  25  24  27  26  29  28  31  30  17  16
10 │ 10  11   8   9  14  15  12  13   2   3   0   1   6   7   4   5  26  27  24  25  30  31  28  29  18  19
11 │ 11  10   9   8  15  14  13  12   3   2   1   0   7   6   5   4  27  26  25  24  31  30  29  28  19  18
12 │ 12  13  14  15   8   9  10  11   4   5   6   7   0   1   2   3  28  29  30  31  24  25  26  27  20  21
13 │ 13  12  15  14   9   8  11  10   5   4   7   6   1   0   3   2  29  28  31  30  25  24  27  26  21  20
14 │ 14  15  12  13  10  11   8   9   6   7   4   5   2   3   0   1  30  31  28  29  26  27  24  25  22  23
15 │ 15  14  13  12  11  10   9   8   7   6   5   4   3   2   1   0  31  30  29  28  27  26  25  24  23  22
16 │ 16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31   0   1   2   3   4   5   6   7   8   9
17 │ 17  16  19  18  21  20  23  22  25  24  27  26  29  28  31  30   1   0   3   2   5   4   7   6   9   8
18 │ 18  19  16  17  22  23  20  21  26  27  24  25  30  31  28  29   2   3   0   1   6   7   4   5  10  11
19 │ 19  18  17  16  23  22  21  20  27  26  25  24  31  30  29  28   3   2   1   0   7   6   5   4  11  10
20 │ 20  21  22  23  16  17  18  19  28  29  30  31  24  25  26  27   4   5   6   7   0   1   2   3  12  13
21 │ 21  20  23  22  17  16  19  18  29  28  31  30  25  24  27  26   5   4   7   6   1   0   3   2  13  12
22 │ 22  23  20  21  18  19  16  17  30  31  28  29  26  27  24  25   6   7   4   5   2   3   0   1  14  15
23 │ 23  22  21  20  19  18  17  16  31  30  29  28  27  26  25  24   7   6   5   4   3   2   1   0  15  14
24 │ 24  25  26  27  28  29  30  31  16  17  18  19  20  21  22  23   8   9  10  11  12  13  14  15   0   1
25 │ 25  24  27  26  29  28  31  30  17  16  19  18  21  20  23  22   9   8  11  10  13  12  15  14   1   0


 ⊗ │  0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25
───┼────────────────────────────────────────────────────────────────────────────────────────────────────────
 0 │  0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0
 1 │  0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25
 2 │  0   2   3   1   8  10  11   9  12  14  15  13   4   6   7   5  32  34  35  33  40  42  43  41  44  46
 3 │  0   3   1   2  12  15  13  14   4   7   5   6   8  11   9  10  48  51  49  50  60  63  61  62  52  55
 4 │  0   4   8  12   6   2  14  10  11  15   3   7  13   9   5   1  64  68  72  76  70  66  78  74  75  79
 5 │  0   5  10  15   2   7   8  13   3   6   9  12   1   4  11  14  80  85  90  95  82  87  88  93  83  86
 6 │  0   6  11  13  14   8   5   3   7   1  12  10   9  15   2   4  96 102 107 109 110 104 101  99 103  97
 7 │  0   7   9  14  10  13   3   4  15   8   6   1   5   2  12  11 112 119 121 126 122 125 115 116 127 120
 8 │  0   8  12   4  11   3   7  15  13   5   1   9   6  14  10   2 128 136 140 132 139 131 135 143 141 133
 9 │  0   9  14   7  15   6   1   8   5  12  11   2  10   3   4  13 144 153 158 151 159 150 145 152 149 156
10 │  0  10  15   5   3   9  12   6   1  11  14   4   2   8  13   7 160 170 175 165 163 169 172 166 161 171
11 │  0  11  13   6   7  12  10   1   9   2   4  15  14   5   3   8 176 187 189 182 183 188 186 177 185 178
12 │  0  12   4   8  13   1   9   5   6  10   2  14  11   7  15   3 192 204 196 200 205 193 201 197 198 202
13 │  0  13   6  11   9   4  15   2  14   3   8   5   7  10   1  12 208 221 214 219 217 212 223 210 222 211
14 │  0  14   7   9   5  11   2  12  10   4  13   3  15   1   8   6 224 238 231 233 229 235 226 236 234 228
15 │  0  15   5  10   1  14   4  11   2  13   7   8   3  12   6   9 240 255 245 250 241 254 244 251 242 253
16 │  0  16  32  48  64  80  96 112 128 144 160 176 192 208 224 240  24   8  56  40  88  72 120 104 152 136
17 │  0  17  34  51  68  85 102 119 136 153 170 187 204 221 238 255   8  25  42  59  76  93 110 127 128 145
18 │  0  18  35  49  72  90 107 121 140 158 175 189 196 214 231 245  56  42  27   9 112  98  83  65 180 166
19 │  0  19  33  50  76  95 109 126 132 151 165 182 200 219 233 250  40  59   9  26 100 119  69  86 172 191
20 │  0  20  40  60  70  82 110 122 139 159 163 183 205 217 229 241  88  76 112 100  30  10  54  34 211 199
21 │  0  21  42  63  66  87 104 125 131 150 169 188 193 212 235 254  72  93  98 119  10  31  32  53 203 222
22 │  0  22  43  61  78  88 101 115 135 145 172 186 201 223 226 244 120 110  83  69  54  32  29  11 255 233
23 │  0  23  41  62  74  93  99 116 143 152 166 177 197 210 236 251 104 127  65  86  34  53  11  28 231 240
24 │  0  24  44  52  75  83 103 127 141 149 161 185 198 222 234 242 152 128 180 172 211 203 255 231  21  13
25 │  0  25  46  55  79  86  97 120 133 156 171 178 202 211 228 253 136 145 166 191 199 222 233 240  13  20


21508 ⊕ 42689 = 62149
21508 ⊗ 42689 = 35202
2150821508215082150821508 ⊕ 4268942689426894268942689 = 2722732241575131661744101
2150821508215082150821508 ⊗ 4268942689426894268942689 = 221974472829844568827862736061997038065

Rust[edit]

Translation of: FreeBASIC
// highest power of 2 that divides a given number
fn hpo2(n: u32) -> u32 {
n & (0xFFFFFFFF - n + 1)
}
 
// base 2 logarithm of the highest power of 2 dividing a given number
fn lhpo2(n: u32) -> u32 {
let mut q: u32 = 0;
let mut m: u32 = hpo2(n);
while m % 2 == 0 {
m >>= 1;
q += 1;
}
q
}
 
// nim-sum of two numbers
fn nimsum(x: u32, y: u32) -> u32 {
x ^ y
}
 
// nim-product of two numbers
fn nimprod(x: u32, y: u32) -> u32 {
if x < 2 || y < 2 {
return x * y;
}
let mut h: u32 = hpo2(x);
if x > h {
return nimprod(h, y) ^ nimprod(x ^ h, y);
}
if hpo2(y) < y {
return nimprod(y, x);
}
let xp: u32 = lhpo2(x);
let yp: u32 = lhpo2(y);
let comp: u32 = xp & yp;
if comp == 0 {
return x * y;
}
h = hpo2(comp);
nimprod(nimprod(x >> h, y >> h), 3 << (h - 1))
}
 
fn print_table(n: u32, op: char, func: fn(u32, u32) -> u32) {
print!(" {} |", op);
for a in 0..=n {
print!("{:3}", a);
}
print!("\n--- -");
for _ in 0..=n {
print!("---");
}
println!();
for b in 0..=n {
print!("{:2} |", b);
for a in 0..=n {
print!("{:3}", func(a, b));
}
println!();
}
}
 
fn main() {
print_table(15, '+', nimsum);
println!();
print_table(15, '*', nimprod);
let a: u32 = 21508;
let b: u32 = 42689;
println!("\n{} + {} = {}", a, b, nimsum(a, b));
println!("{} * {} = {}", a, b, nimprod(a, b));
}
Output:
 + |  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15
--- -------------------------------------------------
 0 |  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15
 1 |  1  0  3  2  5  4  7  6  9  8 11 10 13 12 15 14
 2 |  2  3  0  1  6  7  4  5 10 11  8  9 14 15 12 13
 3 |  3  2  1  0  7  6  5  4 11 10  9  8 15 14 13 12
 4 |  4  5  6  7  0  1  2  3 12 13 14 15  8  9 10 11
 5 |  5  4  7  6  1  0  3  2 13 12 15 14  9  8 11 10
 6 |  6  7  4  5  2  3  0  1 14 15 12 13 10 11  8  9
 7 |  7  6  5  4  3  2  1  0 15 14 13 12 11 10  9  8
 8 |  8  9 10 11 12 13 14 15  0  1  2  3  4  5  6  7
 9 |  9  8 11 10 13 12 15 14  1  0  3  2  5  4  7  6
10 | 10 11  8  9 14 15 12 13  2  3  0  1  6  7  4  5
11 | 11 10  9  8 15 14 13 12  3  2  1  0  7  6  5  4
12 | 12 13 14 15  8  9 10 11  4  5  6  7  0  1  2  3
13 | 13 12 15 14  9  8 11 10  5  4  7  6  1  0  3  2
14 | 14 15 12 13 10 11  8  9  6  7  4  5  2  3  0  1
15 | 15 14 13 12 11 10  9  8  7  6  5  4  3  2  1  0

 * |  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15
--- -------------------------------------------------
 0 |  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
 1 |  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15
 2 |  0  2  3  1  8 10 11  9 12 14 15 13  4  6  7  5
 3 |  0  3  1  2 12 15 13 14  4  7  5  6  8 11  9 10
 4 |  0  4  8 12  6  2 14 10 11 15  3  7 13  9  5  1
 5 |  0  5 10 15  2  7  8 13  3  6  9 12  1  4 11 14
 6 |  0  6 11 13 14  8  5  3  7  1 12 10  9 15  2  4
 7 |  0  7  9 14 10 13  3  4 15  8  6  1  5  2 12 11
 8 |  0  8 12  4 11  3  7 15 13  5  1  9  6 14 10  2
 9 |  0  9 14  7 15  6  1  8  5 12 11  2 10  3  4 13
10 |  0 10 15  5  3  9 12  6  1 11 14  4  2  8 13  7
11 |  0 11 13  6  7 12 10  1  9  2  4 15 14  5  3  8
12 |  0 12  4  8 13  1  9  5  6 10  2 14 11  7 15  3
13 |  0 13  6 11  9  4 15  2 14  3  8  5  7 10  1 12
14 |  0 14  7  9  5 11  2 12 10  4 13  3 15  1  8  6
15 |  0 15  5 10  1 14  4 11  2 13  7  8  3 12  6  9

21508 + 42689 = 62149
21508 * 42689 = 35202

Wren[edit]

Translation of: FreeBASIC
Library: Wren-fmt
import "/fmt" for Fmt
 
// Highest power of two that divides a given number.
var hpo2 = Fn.new { |n| n & (-n) }
 
// Base 2 logarithm of the highest power of 2 dividing a given number.
var lhpo2 = Fn.new { |n|
var q = 0
var m = hpo2.call(n)
while (m%2 == 0) {
m = m >> 1
q = q + 1
}
return q
}
 
// nim-sum of two numbers.
var nimsum = Fn.new { |x, y| x ^ y }
 
// nim-product of two numbers.
var nimprod // recursive
nimprod = Fn.new { |x, y|
if (x < 2 || y < 2) return x * y
var h = hpo2.call(x)
System.write("") // fixes VM recursion bug
if (x > h) return nimprod.call(h, y) ^ nimprod.call(x ^ h, y) // break x into powers of 2
if (hpo2.call(y) < y) return nimprod.call(y, x) // break y into powers of 2
var xp = lhpo2.call(x)
var yp = lhpo2.call(y)
var comp = xp & yp
if (comp == 0) return x * y // no Fermat power in common
h = hpo2.call(comp)
// a Fermat number square is its sequimultiple
return nimprod.call(nimprod.call(x >> h, y >> h), 3 << (h-1))
}
 
var fns = [[nimsum, "⊕"], [nimprod, "⊗"]]
for (fn in fns) {
System.write(" %(fn[1]) |")
for (i in 0..15) System.write(Fmt.d(3, i))
System.print("\n---+%("-" * 48)")
for (i in 0..15) {
System.write("%(Fmt.d(2, i)) |")
for (j in 0..15) System.write(Fmt.d(3, fn[0].call(i, j)))
System.print()
}
System.print()
}
var a = 21508
var b = 42689
System.print("%(a) + %(b) = %(nimsum.call(a, b))")
System.print("%(a) * %(b) = %(nimprod.call(a, b))")
Output:
 ⊕ |  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15
---+------------------------------------------------
 0 |  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15
 1 |  1  0  3  2  5  4  7  6  9  8 11 10 13 12 15 14
 2 |  2  3  0  1  6  7  4  5 10 11  8  9 14 15 12 13
 3 |  3  2  1  0  7  6  5  4 11 10  9  8 15 14 13 12
 4 |  4  5  6  7  0  1  2  3 12 13 14 15  8  9 10 11
 5 |  5  4  7  6  1  0  3  2 13 12 15 14  9  8 11 10
 6 |  6  7  4  5  2  3  0  1 14 15 12 13 10 11  8  9
 7 |  7  6  5  4  3  2  1  0 15 14 13 12 11 10  9  8
 8 |  8  9 10 11 12 13 14 15  0  1  2  3  4  5  6  7
 9 |  9  8 11 10 13 12 15 14  1  0  3  2  5  4  7  6
10 | 10 11  8  9 14 15 12 13  2  3  0  1  6  7  4  5
11 | 11 10  9  8 15 14 13 12  3  2  1  0  7  6  5  4
12 | 12 13 14 15  8  9 10 11  4  5  6  7  0  1  2  3
13 | 13 12 15 14  9  8 11 10  5  4  7  6  1  0  3  2
14 | 14 15 12 13 10 11  8  9  6  7  4  5  2  3  0  1
15 | 15 14 13 12 11 10  9  8  7  6  5  4  3  2  1  0

 ⊗ |  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15
---+------------------------------------------------
 0 |  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
 1 |  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15
 2 |  0  2  3  1  8 10 11  9 12 14 15 13  4  6  7  5
 3 |  0  3  1  2 12 15 13 14  4  7  5  6  8 11  9 10
 4 |  0  4  8 12  6  2 14 10 11 15  3  7 13  9  5  1
 5 |  0  5 10 15  2  7  8 13  3  6  9 12  1  4 11 14
 6 |  0  6 11 13 14  8  5  3  7  1 12 10  9 15  2  4
 7 |  0  7  9 14 10 13  3  4 15  8  6  1  5  2 12 11
 8 |  0  8 12  4 11  3  7 15 13  5  1  9  6 14 10  2
 9 |  0  9 14  7 15  6  1  8  5 12 11  2 10  3  4 13
10 |  0 10 15  5  3  9 12  6  1 11 14  4  2  8 13  7
11 |  0 11 13  6  7 12 10  1  9  2  4 15 14  5  3  8
12 |  0 12  4  8 13  1  9  5  6 10  2 14 11  7 15  3
13 |  0 13  6 11  9  4 15  2 14  3  8  5  7 10  1 12
14 |  0 14  7  9  5 11  2 12 10  4 13  3 15  1  8  6
15 |  0 15  5 10  1 14  4 11  2 13  7  8  3 12  6  9

21508 + 42689 = 62149
21508 * 42689 = 35202