Largest odd integer that can be represented as a float

Question

I was wondering what largest odd integer that can be represented exactly as a float? And why there is a difference between the largest even integer represented as a float in this case.

I believe it would have to do with the base 2 exponents 2^n-1, however I am not familiar enough with data representation in C to see the distinction.

Answer 1

For IEEE-754 basic 32-bit binary floating-point, the largest representable odd integer is 2 ²⁴ −1.

For IEEE-754 basic 64-bit binary floating-point, the largest representable odd integer is 2 ⁵³ −1.

This is due to the fact that the formats have 24-bit and 53-bit significands. (The significand is the fraction part of a floating-point number.)

The values represented by the bits in the significand are scaled according to the exponent of the floating-point number. In order to represent an odd number, the floating-point number must have a bit in the significand that represents 2 ⁰ . With a 24-bit significand, if the lowest bit represents 2 ⁰ , then the highest bit represents 2 ²³ . The largest value is obtained when all the bits are on, which makes the value 2 ⁰ + 2 ¹ + 2 ² + … 2 ²³ , which equals 2 ²⁴ −1.

More generally, the largest representable odd integer is normally scalbnf(1, FLT_MANT_DIG) - 1 . This can also be computed as (2 - FLT_EPSILON) / FLT_EPSILON . (This assumes a normal case in which FLT_RADIX is even and FLT_MANT_DIG <= FLT_MAX_EXP . Note that if FLT_MANT_DIG == FLT_MAX_EXP , the latter expression, with FLT_EPSILON , should be used, because the former overflows.)

The abnormal cases, just for completeness:

• If FLT_RADIX is odd and FLT_MANT_DIG <= FLT_MAX_EXP , the largest representable odd integer is FLT_MAX if FLT_MANT_DIG is odd and FLT_MAX - scalbnf(FLT_EPSILON, FLT_MAX_EXP+1) otherwise.
• If FLT_RADIX is even and FLT_MANT_DIG > FLT_MAX_EXP , then: If FLT_MAX_EXP > 0 , the largest representable odd integer is floorf(FLT_MAX) . Otherwise, no odd integers are representable.
• If FLT_RADIX is odd and FLT_MANT_DIG > FLT_MAX_EXP , then: If FLT_MAX_EXP > 0 , the largest representable odd integer is floorf(FLT_MAX) if FLT_MANT_DIG - FLT_MAX_EXP is odd or floorf(FLT_MAX)-1 otherwise. Otherwise, no odd integers are representable.

Answer 2

The largest odd integer representable as a 32-bit float is 2^24 - 1.

More specifically, a 32-bit float can exactly represent the following integers:

• All integers up to 2^24
• All even integers up to 2^25
• All multiples of 4 up to 2^26
• All multiples of 8 up to 2^27
• etc.

In other words, here are the integers exactly representable:

0
1
2
...
16777215
16777216 (= 2^24)
16777218
16777220
...
33554430
33554432 (= 2^25)
33554436
33554440
...
67108860
67108864 (= 2^26)
67108872
67108880
...

Note that this applies both for positive and negative integers, for example, all negative integers down to -2^24 can be exactly represented.

Below is some C++ code doing integers-to-float conversions for integers around 2^24, around 2^25, and around 2^26, to see this in practice.

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#include <iostream>
#include <iomanip>
#include <vector>

int main()
{
    int _2pow24 = 1 << 24;
    int _2pow25 = 1 << 25;
    int _2pow26 = 1 << 26;
    std::cout << "2^24 = " << _2pow24 << std::endl;
    std::cout << "2^25 = " << _2pow25 << std::endl;
    std::cout << "2^26 = " << _2pow26 << std::endl;
    std::vector<int> v;
    for (int i = -4; i < 4; ++i) v.push_back(_2pow24 + i);
    for (int i = -8; i < 8; ++i) v.push_back(_2pow25 + i);
    for (int i = -16; i < 16; ++i) v.push_back(_2pow26 + i);
    for (int i : v) {
        std::cout << i << " -> "
                  << std::fixed << std::setprecision(1)
                  << static_cast<float>(i)
                  << std::endl;
    }
    return 0;
}

Output:

2^24 = 16777216
2^25 = 33554432
2^26 = 67108864
16777212 -> 16777212.0
16777213 -> 16777213.0
16777214 -> 16777214.0
16777215 -> 16777215.0
16777216 -> 16777216.0
16777217 -> 16777216.0
16777218 -> 16777218.0
16777219 -> 16777220.0
33554424 -> 33554424.0
33554425 -> 33554424.0
33554426 -> 33554426.0
33554427 -> 33554428.0
33554428 -> 33554428.0
33554429 -> 33554428.0
33554430 -> 33554430.0
33554431 -> 33554432.0
33554432 -> 33554432.0
33554433 -> 33554432.0
33554434 -> 33554432.0
33554435 -> 33554436.0
33554436 -> 33554436.0
33554437 -> 33554436.0
33554438 -> 33554440.0
33554439 -> 33554440.0
67108848 -> 67108848.0
67108849 -> 67108848.0
67108850 -> 67108848.0
67108851 -> 67108852.0
67108852 -> 67108852.0
67108853 -> 67108852.0
67108854 -> 67108856.0
67108855 -> 67108856.0
67108856 -> 67108856.0
67108857 -> 67108856.0
67108858 -> 67108856.0
67108859 -> 67108860.0
67108860 -> 67108860.0
67108861 -> 67108860.0
67108862 -> 67108864.0
67108863 -> 67108864.0
67108864 -> 67108864.0
67108865 -> 67108864.0
67108866 -> 67108864.0
67108867 -> 67108864.0
67108868 -> 67108864.0
67108869 -> 67108872.0
67108870 -> 67108872.0
67108871 -> 67108872.0
67108872 -> 67108872.0
67108873 -> 67108872.0
67108874 -> 67108872.0
67108875 -> 67108872.0
67108876 -> 67108880.0
67108877 -> 67108880.0
67108878 -> 67108880.0
67108879 -> 67108880.0