Average Error: 0.1 → 0.1
Time: 3.2s
Precision: 64
\[x \cdot \left(1 - x \cdot y\right)\]
↓
\[\left(x \cdot 1 + x \cdot \left(-x \cdot y\right)\right) + x \cdot \mathsf{fma}\left(-y, x, y \cdot x\right)\]
x \cdot \left(1 - x \cdot y\right)
↓
\left(x \cdot 1 + x \cdot \left(-x \cdot y\right)\right) + x \cdot \mathsf{fma}\left(-y, x, y \cdot x\right)
double f(double x, double y) {
        double r89848 = x;
        double r89849 = 1.0;
        double r89850 = y;
        double r89851 = r89848 * r89850;
        double r89852 = r89849 - r89851;
        double r89853 = r89848 * r89852;
        return r89853;
}
↓
double f(double x, double y) {
        double r89854 = x;
        double r89855 = 1.0;
        double r89856 = r89854 * r89855;
        double r89857 = y;
        double r89858 = r89854 * r89857;
        double r89859 = -r89858;
        double r89860 = r89854 * r89859;
        double r89861 = r89856 + r89860;
        double r89862 = -r89857;
        double r89863 = r89857 * r89854;
        double r89864 = fma(r89862, r89854, r89863);
        double r89865 = r89854 * r89864;
        double r89866 = r89861 + r89865;
        return r89866;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.1

    \[x \cdot \left(1 - x \cdot y\right)\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt0.1

    \[\leadsto x \cdot \left(\color{blue}{\sqrt{1} \cdot \sqrt{1}} - x \cdot y\right)\]

  4. Applied prod-diff0.1

    \[\leadsto x \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt{1}, \sqrt{1}, -y \cdot x\right) + \mathsf{fma}\left(-y, x, y \cdot x\right)\right)}\]

  5. Applied distribute-lft-in0.1

    \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(\sqrt{1}, \sqrt{1}, -y \cdot x\right) + x \cdot \mathsf{fma}\left(-y, x, y \cdot x\right)}\]

  6. Simplified0.1

    \[\leadsto \color{blue}{x \cdot \left(1 - x \cdot y\right)} + x \cdot \mathsf{fma}\left(-y, x, y \cdot x\right)\]
  7. Using strategy rm

  8. Applied sub-neg0.1

    \[\leadsto x \cdot \color{blue}{\left(1 + \left(-x \cdot y\right)\right)} + x \cdot \mathsf{fma}\left(-y, x, y \cdot x\right)\]

  9. Applied distribute-lft-in0.1

    \[\leadsto \color{blue}{\left(x \cdot 1 + x \cdot \left(-x \cdot y\right)\right)} + x \cdot \mathsf{fma}\left(-y, x, y \cdot x\right)\]

  10. Final simplification0.1

    \[\leadsto \left(x \cdot 1 + x \cdot \left(-x \cdot y\right)\right) + x \cdot \mathsf{fma}\left(-y, x, y \cdot x\right)\]

Reproduce

herbie shell --seed 2019354 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, A"
  :precision binary64
  (* x (- 1 (* x y))))