RoundingFiasco: rounding variants floor, ceil and truncate for floating point operations +-*/√…

There is an exact definition for +-*/√ over the real numbers in mathematics. However, for performant, flexible, and ergonomic numerical computations one ought to restrict oneself to only using a finite subset of rational numbers. The most common data types for such use cases are the single and double floating point formats.

Not every floating point value is a real number. There are NaNs, positive and negative infinity, and the negative zero. Combining two real floating point numbers by an operator yields a mathematical and uncontroversially defined result. This exact result might not be representable as a floating point number. Therefore, rounding is necessary.

The most common way of rounding is to select the nearest representable float (round-to-nearest-even-on-tie). This rounding variant minimizes the accumulation of rounding errors over several floating point operations.

Other rounding variants (floor, ceil and truncate) are useful for computing error bounds of chained floating point instructions. Floor chooses the next lower representable value relative to the exact result. Ceil chooses the greater neighbor. Truncate chooses the value that is closer to zero.

This library implements the floating-point-instructions-like operations in pure Haskell. The implementation avoids C++ functions like fegetround, for example. That way they can be used in the WebAssembly backend of GHC since WebAssembly supports neither rounding variants nor fegetround.

This module aims to expose the fastest possible pure interface of rounding variants. Should there ever be some compiler intrinsics for rounding variants then these intrinsics shall be used in a future version.

Every combination of number type (Float, Double) and operator (+,-,*,/,√,id) is exported separately. The exported functions are designed as a basis for interval arithmetic.

Internally, the module extensively uses the Rational data type. First, the operation's result is calculated twice. Once done exactly using Rational. Then a round-to-nearest-even-on-tie result is also calculated. After that, both numbers are compared to determine whether the round-to-nearest-even-on-tie result happens to be rounded in the correct direction. Should that not be the case, the other neighbor is determined and returned.

• Rounding