An iterator over floating-point numbers which are hard to convert from/to decimal form.
Converting a number x from/to another base (usually 2 or 10), as done in
atof or ftoa functions, involves multiplying
x × bⁿ = digits + ε
where |ε| < 1/2 and digits is the rounded result, used for display.
A conversion is hard is ε is very close to ±1/2, meaning that any lack of precision might make rounding go the wrong side.
Usually x itself is expressed in some base B, meaning that the
computation is:
mantissa × Bᵖ × bⁿ = digits + ε
where mantissa and digits have fixed predefined precision,
and the exponents n and p can be computed from each other.
Thus the problem can be expressed as enumerating all fractions
mantissa / digits, within the range of fixed precision integers,
which are very close to Bᵖ × bⁿ up to some precision.
An easy way to perform this enueration is to walk the Stern-Brocot tree, which arranges rational numbers in an (unbounded depth) binary search tree, such that children of a node always have larger denominator than their parent node.
Manipulating the Stern-Brocot tree is easiest where elements are written in continued fraction form.
The algorithm is implemented in Python 3 and Go.
For each pair of interesting exponents, a interval of fractions
around 2ᵖ × 10ⁿ is computed, such that lower and upper bounds
have numerators and denominators, bounded by a max integer M.
This can easily by precomputed as a small table for languages where arbitrary-precision arithmetic ("big integers") is hard or inconvenient.
Then the algorithm traverses entirely the tree between these 2 bounds,
stopping at the depth where integers would overflow the bound M.
This step can be done using finite precision arithmetic only.
The Python script takes about 1 minute to enumerate double-precision floats (float64) such that formatting in decimal form is hard even using 96-bit arithmetic (about 1 million values).
It takes less than a second to enumerate the single-precision (float32) values that are hard to format when using 52-bit arithmetic (about 280 values).
The Go implementation can enumerate an interval of 1 million rational numbers in a few milliseconds.
The following torture tests are implemented in Go:
-
TestCarry64, TestCarry128: check wide-precision powers of ten for exceptional carries. This is used in the proof of Ryū-like algorithms.
-
TestTortureFixed32/64: check edge cases for fixed-precission decimal formatting. The iterators provide the expected answer so it is checked exactly without depending on strconv correctness.
-
TestTortureShortest32/64: check edge cases for shortest floating-point formatting. Currently exact verification is not done, instead the test verifies that selected numbers are formatted shorter than their neighbours. The edge cases make it hard to find the correct rounding direction.
-
TestTortureAtof32/64: check edge cases for parsing floats. The inputs are decimal representations whch are very close to midpoints between consecutive float32/float64s. The iterator gives the expected answer without using strconv functions.
Exact midpoints (commonly found when using small exponents) are not tested. Small exponents are not tested (|exp| < 55 for float64, |exp| < 10 for float32)