Can All Integer Values be Perfectly Represented as Doubles?

Are All Integer Values Perfectly Represented as Doubles?

The question arises as to whether all integer values are guaranteed to have a perfect representation as double-precision floating-point numbers. To answer this, we first delve into the representation of doubles.

Doubles are represented as mantissa * 2^exponent, where the mantissa includes fractional digits. This allows for the representation of both integer and decimal values.

Representation of 32-Bit Integers

For 32-bit integers, there are up to 53 bits available for the mantissa. This is sufficient to represent all possible 32-bit integers without fractional loss. Thus, yes, all 32-bit integer values are perfectly represented as doubles.

Representation of 64-Bit Integers

However, for 64-bit integers, the situation is different. IEEE 754 double-precision can guarantee perfect representation for up to 53 bits. Beyond that, there may be rounding errors. Therefore, no, not all 64-bit integer values are perfectly represented as doubles.

Empirically Verifying the Behavior

The following code snippet tests the conversion of integers to doubles:

<code class="cpp">#include <iostream>
#include <limits>
using namespace std;

int main() {
    double test;
    volatile int test_int;
    for (int i = 0; i ::max(); i++) {
        test = i;
        test_int = test;
        if (test_int != i)
            cout <p>Running this code reveals that there are no integer values with fractional conversion errors for 32-bit integers. However, for 64-bit integers, it is possible to find integer values that convert to doubles with fractional differences and round back to the original integer value.</p>
<p><strong>Fractional Differences</strong></p>
<p>Regarding the possibility of fractional differences during conversion, the answer is still no for integers. This is because the step width between double values represented as mantissa * 2^exponent is always a power of two. Therefore, there is never a difference smaller than 2 between two double values, resolving any rounding issues.</p></limits></iostream></code>

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