Can All Integer Values Be Accurately Represented as Doubles?

Are All Integer Values Accurately Represented as Doubles?

When working with integers and doubles, developers may wonder whether all integer values can be perfectly represented as doubles. This question has sparked discussions, and it's important to explore its implications thoroughly.

The Double Representation

Doubles, as per the IEEE 754 standard, are represented using a mantissa and an exponent. The mantissa holds the fractional part, while the exponent shifts the mantissa to the left or right, representing the integer or decimal value.

Representation of Integers as Doubles

Within the double representation, integers can be represented by using an exponent that effectively removes the decimal part of the number. This allows for the representation of integers using the mantissa's fractional bits.

Limitations for 64-Bit Integers

While this scheme works well for 32-bit integers, it falls short for 64-bit integers. The IEEE 754 double-precision format can only represent integers up to 53 bits (52 significand bits plus an implicit leading 1). This means that 64-bit integers may lose precision when converted to doubles.

Empirical Verification

The following C code empirically tests the conversion of integers to doubles:

<code class="cpp">#include <iostream>
#include <limits>

int main() {
    double test;
    volatile int test_int;
    for(int i=0; i< std::numeric_limits<int>::max(); i++) {
        test = i;
        test_int = test;

        // compare int with int:
        if (test_int != i)
            std::cout << "found integer i=" << i << ", test=" << test << std::endl;
    }
    return 0;
}</code>

Running this code confirms that 64-bit integers do not have perfect double representations.

Conclusion

Therefore, the answer to the question is:

• For 32-bit integers: Yes, they can be perfectly represented as doubles.
• For 64-bit integers: No, perfect double representations are not guaranteed.

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