Crate safeeft

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Safe and branchless error-free transformation algorithms for floating point numbers.

§Abstract

Error-free transformation (EFT) algorithms for floating point numbers are those conserve mathematical equality between input and output. For example, twosum by D. E. Knuth[1] is an EFT algorithm that conserves summation of two floating point numbers with non-overlapping property: (a,b) = twosum(x,y), a+b=x+y, 0.5ulp(a) >= b.

fn twosum(x: f64, y: f64) -> (f64, f64) {
    let sum = x + y;
    let tmp = sum - x;
    (sum, (x - (sum - tmp)) + (y - tmp))

But in practical, especially with IEEE 754 Std. floating point arithmetic[4], there were several cases which broke equality and M. Kashiwagi[3] fixed it using branch.

This crate provides safe and branchless EFT impllementations for twosum, split and twoproduct[2]. The word “safe” means “if output is representable with normal/subnormal floating point number, the algorithm is mathematically correct.”

§Accerelation

With nightly compiler and x86 cpu supporting fma, you can accelerate some algorithms using a use-fma feature gate:

$ RUSTFLAGS='-C target-feature=+fma' cargo build --features use-fma

§References

  1. D. E. Knuth, “The Art of Computer Programming”, vol. 2. Addison-Wesley, Reading, MA, 3rd edition, 1998.
  2. T. J. Dekker, “A Floating-Point Technique for Extending the Available Precision”, Numer. Math. 18(3), 224-242, 1971.
  3. M. Kashiwagi, “Emulation of Rounded Arithmetic in Rounding to Nearest(Japanese only)”, NAS2014, 2014.
  4. American National Standards Institute and Institute of Electrical and Electronic Engineers, “IEEE Standard for Binary Floating-Point Arithmetic”, ANSI/IEEE Standard 754-2008, 2008.

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