Why does Oracle claim java.util.Random.nextFloat() generates 2^24 possibilities and not 2^23?

Question

According to the documentation for java.util.Random, the Java class implements nextFloat by next(24) / ((float)(1 << 24)) (ie. a random non-negative integer with 24 bits divided by 2²⁴). The documentation claims that all 2²⁴ possible values can be returned. They are between 0 (inclusive) and 1 (exclusive). However, I don't think that is true.

First, note that according to the IEC 559 (IEEE 754) standard, float has 23 fraction bits. But there is an implicit leading bit (to the left of the binary point) with value 1, unless the exponent is stored with all zeros. Therefore, it is true that there are a total of 2²⁴ values of type float that are between 0 (inclusive - not counting negative zero) and 1 (exclusive), but exactly half of these numbers are subnormal (all bits in the exponent are 0), which makes them all less than 2^-126. Therefore, none of these numbers can be generated by the implementation. This is because they are all strictly smaller than 2^-24 which is used in the implementation.

The layout of float can be found at Single-precision floating-point format.

So what is it that I am missing?

Answer 1

In the interval [+0, 1), there are 127•2²³ values representable in float, not 2²⁴. There is one for each combination of an exponent encoding field from 0 to 126, inclusive, with each value of 23 bits in the significand encoding field.

Every value in the form m•2⁻²⁴, with 0 ≤ m < 2²⁴, is representable. The smallest non-zero value in this form is 2⁻²⁴, which is represented with an exponent code of 103 and a significand code of 0. The mathematical exponent is the code, 103, minus the bias, 127, which equals −24, and the mathematical significand is 1.

For any such m other than zero, let b be the position number of its leading 1 bit (numbering from 0 for the low bit). Then m•2⁻²⁴ is encoded in float with an exponent code of b+103 and a significand code of m•2^{24− b}−2²⁴. For m = 0, it is encoded with all zero bits.

None of the numbers in this form are subnormal.