Implementing equality function with basic arithmetic operations

Question

Given positive-integer inputs x and y, is there a mathematical formula that will return 1 if x==y and 0 otherwise? I am in the unfortunate position of having to use a tool that only allows me to use the following symbols: numerals 0-9; decimal point .; parentheses ( and ); and the four basic arithmetic operations +, -, /, and *.

Currently I am relying on the fact that the tool that evaluates division by zero to be zero. (I can't tell if this is a bug or a feature.) Because of this, I have been able to use ((x-y)/(y-x))+1. Obviously, this is ugly and unideal, especially in the case that it is a bug and they fix it in a future version.

Answer 1

Taking advantage of integer division in C truncates toward 0, the follows works well. No multiplication overflow. Well defined for all "positive-integer inputs x and y".

(x/y) * (y/x)

#include <stdio.h>
#include <limits.h>

void etest(unsigned x, unsigned y) {
  unsigned ref = x == y;
  unsigned z =  (x/y) * (y/x);
  if (ref != z) {
    printf("%u %u %u %u\n", x,y,z,ref);
  }
}

void etests(void) {
  unsigned list[] = { 1,2,3,4,5,6,7,8,9,10,100,1000, UINT_MAX/2 , UINT_MAX - 1, UINT_MAX };
  for (unsigned x = 0; x < sizeof list/sizeof list[0]; x++) {
    for (unsigned y = 0; y < sizeof list/sizeof list[0]; y++) {
      etest(list[x], list[y]);
    }
  }
}

int main(void) {
  etests();
  printf("Done\n");
  return 0;
}

Output (No difference from x == y)

Done

Answer 2

If division is truncating and the numbers are not too big, then:

((x - y) ^ 2 + 2) / ((x - y) ^ 2 + 1) - 1

The division has the value 2 if x = y and otherwise truncates to 1.

(Here x^2 is an abbreviation for x*x.)

This will fail if (x-y)^2 overflows. In that case, you need to independently check x/k = y/k and x%k = y%k where (k-1)*(k-1) doesn't overflow (which will work if k is ceil(sqrt(INT_MAX))). x%k can be computed as x-k*(x/k) and A&&B is simply A*B.

That will work for any x and y in the range [-k*k, k*k].

A slightly incorrect computation, using lots of intermediate values, which assumes that x - y won't overflow (or at least that the overflow won't produce a false 0).

  int delta = x - y;
  int delta_hi = delta / K;
  int delta_lo = delta - K * delta_hi;
  int equal_hi = (delta_hi * delta_hi + 2) / (delta_hi * delta_hi + 1) - 1;
  int equal_lo = (delta_lo * delta_lo + 2) / (delta_lo * delta_lo + 1) - 1;
  int equals = equal_hi * equal_lo;

or written out in full:

((((x-y)/K)*((x-y)/K)+2)/(((x-y)/K)*((x-y)/K)+1)-1)*
((((x-y)-K*((x-y)/K))*((x-y)-K*((x-y)/K))+2)/
 (((x-y)-K*((x-y)/K))*((x-y)-K*((x-y)/K))+1)-1)

(For signed 31-bit integers, use K=46341; for unsigned 32-bit integers, 65536.)

Checked with @chux's test harness, adding the 0 case: live on coliru and with negative values also on coliru.

On a platform where integer subtraction might produce something other than the 2s-complement wraparound, a similar technique could be used, but dividing the numbers into three parts instead of two.

Answer 3

So the problem is that if they fix division by zero, it means that you cannot use any divisor that contains input variables anymore (you'd have to check that the divisor != 0, and implementing that check would solve the original x-y == 0 problem!); hence, division cannot be used at all.

Ergo, only +, -, * and the association operator () can be used. It's not hard to see that with only these operators, the desired behaviour cannot be implemented.