 Solution

The first proposed identity can fail even without numerical error because binary fractions have nonunique textual representations. Leading zeroes, optional points and trailing fractional zeroes are normalized differently by a fixed-width output convention. It also fails if the input has more significant bits than Double can preserve, more fractional places than the output buffer can hold, an integer part outside the assumed Int range, or a value that overflows or underflows during decoding. Array identity here must mean equal contents: a newly allocated Scala array is a different software object, so use sameElements rather than reference equality for a content test.

The second identity fails when the stored fractional precision truncates a nonzero bit; with two fractional places, $0.125$ becomes zero. It can also fail outside the finite nonnegative input and integer-conversion assumptions. Within those assumptions, enough fractional positions to encode the finite Double exactly can permit the round trip, but an arbitrary fixed small width does not guarantee it. Every finite Double has a finite binary expansion; saying that a decimal-looking Double necessarily has an infinite binary expansion would confuse the represented number with an exact decimal real.

The source's introductory example is internally inconsistent: the displayed digit string has value $22+1/4+1/32=22.28125$, whereas its shortened character array encodes $22+1/4+1/16=22.3125$. The conversion rules above follow the actual input digits. Round-trip testing needs an independent value oracle as well as the two compositions, because two mutually consistent but incorrect routines can conceal each other's errors.