Adding Floating-Point Decimals for Fun and Profit
2 days ago
- Decimal calculations using floating-point numbers introduce rounding errors, as shown by 0.1 + 0.2 not equaling 0.3.
- The pattern of errors when summing multiples of 0.01 up to 1.00 is visualized, with errors appearing as too large, too small, or correct.
- IEEE double-precision floats have 1 sign bit, 11 exponent bits, and 52 fraction bits, with the value determined by the exponent and significand.
- Floating-point addition involves rounding the exact sum to the nearest representable number, which can lead to ties being resolved by rounding to an even significand.
- Printing floating-point numbers follows criteria: round-trip accuracy, shortest representation, and closest possible value, resulting in outputs like 0.30000000000000004.
- The difference between computed and exact sums is at most 1 ulp (unit in the last place), as derived from error bounds.
- The error function shows periodic patterns due to modular arithmetic with ulp, with the number of steps before wrapping depending on the range of values.
- Error-free transformations like 2Sum, Veltkamp splitting, and Dekker product enable exact error computation within floating-point arithmetic.