Average Error: 16.1 → 0.0
Time: 2.9s
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 r548879 = x;
        double r548880 = 1.0;
        double r548881 = r548880 - r548879;
        double r548882 = y;
        double r548883 = r548880 - r548882;
        double r548884 = r548881 * r548883;
        double r548885 = r548879 + r548884;
        return r548885;
}
↓
double f(double x, double y) {
        double r548886 = y;
        double r548887 = x;
        double r548888 = 1.0;
        double r548889 = r548887 - r548888;
        double r548890 = fma(r548886, r548889, r548888);
        return r548890;
}

Error

Bits error versus x

Bits error versus y

Target

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

Derivation

  1. Initial program 16.1

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

  2. Simplified16.1

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

  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 2020064 +o rules:numerics
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.Vectors:renderPlotVectors from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (- (* y x) (- y 1))

  (+ x (* (- 1 x) (- 1 y))))