Division with numerator 2^64

Question

How to to divide the constant 2^64 (i.e. ULLONG_MAX + 1) by uint64 larger than 2, without using unit128?

In other words, given x such as 2 <= x <= 2^64-1, how to obtain the quotient 2^64 / x, using just uint64?

The problem is that I cannot represent 2^64, let alone to divide it so I was hoping there is a trick which would simulate the result.

Given your constants, may I assume that you are limiting this question to the C#, C++, F#, and Visual Basic languages? This is trivial in many other languages, such as in Python. — Rory Daulton, Oct 31 '17 at 18:22
@RoryDaulton Basically any language without `int128_t` or bignums, yes. — Ecir Hana, Oct 31 '17 at 18:23
Perhaps just do `(2^63)/x` and then multiply that result by `2`? — ImaginaryHuman072889, Nov 01 '17 at 10:48
@ImaginaryHuman072889 then the result is necessarily even, but that's not right in general — harold, Nov 02 '17 at 19:18

chux - Reinstate Monica · Answer 1 · 2017-11-03T22:05:24.487

2

How to to divide the constant 2^64 (i.e. ULLONG_MAX + 1) by uint64 larger than 2

a/b --> (a-b)/b + 1

First subtract x from (max _value + 1), then divide by x, add 1.

// C solution:
uint64_t foo(uint64_t x) {
  return (0u - x)/x + 1; // max_value + 1 is 0 and unsigned subtraction wraps around.
}

Of course division by 0 is a no-no. Code works for x >= 2, but not x == 1 as the quotient is also not representable.

edited Nov 03 '17 at 22:05

answered Nov 03 '17 at 17:26

chux - Reinstate Monica

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score 1 · Accepted Answer · answered Oct 31 '17 at 18:26

Take ULLONG_MAX / denom, and add 1 if denom is a power of 2. In pseudocode:

if (denom == 0) {
    throw ZeroDivisionException;
} else if (denom == 1) {
    throw OverflowException;
} else {
    return ULLONG_MAX / denom + (denom & (denom-1) == 0);
}

Alternatively, take ULLONG_MAX / denom for odd denom, and take 2^63 / (denom / 2) for even denom:

if (denom == 0) {
    throw ZeroDivisionException;
} else if (denom == 1) {
    throw OverflowException;
} else if (denom & 1) {
    return ULLONG_MAX / denom;
} else {
    return (1ULL << 63) / (denom >> 1);
}

Division with numerator 2^64

2 Answers2