Finding Floating Number Operations

Purpose: To find parsimonious arithmetic expressions that explain "surprising" floating-point values.

People are often startled to see values like 0.05000000000001137 where they expected 0.05. This tool finds a short sequence of elementary arithmetic operations on common literals that could have produced the unexpected value.

How it works

Search uses bottom-up dynamic programming keyed on literal count (the number of numeric constants in the expression):

• Level 1 — atomic literals: 0.1, 0.2, …, 1, 2, …, 20, 100, 200, …
• Level k — all (level-i op level-(k-i)) pairs under +, −, ×, ÷

Float values are stored by exact IEEE-754 bit pattern, so each reachable float is represented once by its simplest known expression. The search terminates as soon as the target bit pattern is matched.

Template exploration runs after a match is found. Each distinct literal in the expression is treated as a parameter and swept over ~12 000 candidates (integers 1–10 000, powers of 2 up to 2³⁰, common decimal fractions, negatives). Same-valued literals are varied together — they are the same "variable". The output shows which values produce the same surprising result, revealing the underlying pattern.

Example

Target: 0.05000000000001137

Expression : (200 - (200 - 0.05))
Literals   : 3  (operations: 2)

Template : (N0 - (N0 - N1))
  N0 = 200.0,  N1 = 0.05

Varying N0 — positive values that work (384):
  129.0  130.0  …  512.0          ← exactly two IEEE-754 binades (2⁷ to 2⁹)

Varying N1 — values that work (1):
  0.05                             ← the error is specific to this literal

The 384-integer range [129, 512] is two complete binary binade intervals (2⁷–2⁸ and 2⁸–2⁹). Rounding behaviour changes at powers of two, which is exactly what IEEE-754 arithmetic predicts.

Usage

# Default demo target
python3 find_float.py

# Provide your own target
python3 find_float.py 0.30000000000000004
python3 find_float.py 0.6000000000000001

# Tune the search
python3 find_float.py 0.05000000000001137 --max-literals 6 --time-limit 120

# Skip template exploration
python3 find_float.py 0.30000000000000004 --no-explore

Options

Flag	Default	Description
target	0.05000000000001137	Float value to explain
-m, --max-literals	7	Maximum literals in the expression
-t, --time-limit	60	Wall-clock search limit (seconds)
--no-explore	off	Skip template exploration
-q, --quiet	off	Suppress progress output

Requirements

Python 3.10+, no dependencies beyond the standard library.

Credits

Code generated by Claude Code with Sonnet 4.6.