Average Error: 16.3 → 0.0
Time: 19.9s
Precision: 64
\[x + \left(1 - x\right) \cdot \left(1 - y\right)\]
↓
\[\mathsf{fma}\left(y, x - 1, 1\right)\]
x + \left(1 - x\right) \cdot \left(1 - y\right)
↓
\mathsf{fma}\left(y, x - 1, 1\right)
double f(double x, double y) {
        double r19224872 = x;
        double r19224873 = 1.0;
        double r19224874 = r19224873 - r19224872;
        double r19224875 = y;
        double r19224876 = r19224873 - r19224875;
        double r19224877 = r19224874 * r19224876;
        double r19224878 = r19224872 + r19224877;
        return r19224878;
}
↓
double f(double x, double y) {
        double r19224879 = y;
        double r19224880 = x;
        double r19224881 = 1.0;
        double r19224882 = r19224880 - r19224881;
        double r19224883 = fma(r19224879, r19224882, r19224881);
        return r19224883;
}

Error

Bits error versus x

Bits error versus y

Target

Original16.3
Target0.0
Herbie0.0
\[y \cdot x - \left(y - 1\right)\]

Derivation

  1. Initial program 16.3

    \[x + \left(1 - x\right) \cdot \left(1 - y\right)\]

  2. Simplified16.3

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

  3. Taylor expanded around 0 0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.Vectors:renderPlotVectors from Chart-1.5.3"

  :herbie-target
  (- (* y x) (- y 1.0))

  (+ x (* (- 1.0 x) (- 1.0 y))))