I'm studying the Hamming's article "On the Distribution of Numbers", but i'm confused in the way he obtain the (cumulative) distribution of the product of two floating point numbers.
Maybe it's only an integration problem but the shaded region of Figure 1 (at page 5) in the article should be the region in which the relation $xy \leq z$ is true for a given $z$, but i can't understand why there are two different and separated shaded region (since $z$ should be continuous in the interval $1/b < z < 1$) and the way in which he obtain the relative equations (i.e. $ y = \frac{z}{x} $, $y = \frac{1}{bx}$, $y=\frac{z}{bx}$, the last relation is the one that hurts me more).
Maybe this is a stupid question, but any help is very appreciated! Thanks in advance!
Perhaps examples will make this clearer.
The areas represent where the product $xy$ has a mantissa between $1/b$ and $z$. Take, for example, $b=10$ and $z=0.3$. Then the shaded regions are where the mantissa of $xy$ is between 0.1 and 0.3. The lower-left shaded region corresponds to where $0.01 \le xy\le z/b = 0.03$. Products like $(0.112)(0.236)=0.026432$ fall into this region.
The other shaded region corresponds to where $1/b \le xy \le z$, i.e., $0.1 \le xy \le 0.3$ (so notice this is "10 times" the other region's inequality: multiplying by 10 doesn't change the mantissa).
Products like $(0.45)(0.51)=0.2295$ fall in this region.
The regions are not contiguous, since products like $(0.04)(0.02)=0.008$ have mantissa greater than $z=0.3$, so fall outside both regions, in the space between the two regions. That is, if $0.03 < xy < 0.1$, then the mantissa of the product is not in the interval $[0.1,0.3]$; so this region is not shaded.