Average Error: 16.8 → 0.0
Time: 9.1s
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 r20582723 = x;
        double r20582724 = 1.0;
        double r20582725 = r20582724 - r20582723;
        double r20582726 = y;
        double r20582727 = r20582724 - r20582726;
        double r20582728 = r20582725 * r20582727;
        double r20582729 = r20582723 + r20582728;
        return r20582729;
}
↓
double f(double x, double y) {
        double r20582730 = y;
        double r20582731 = x;
        double r20582732 = 1.0;
        double r20582733 = r20582731 - r20582732;
        double r20582734 = fma(r20582730, r20582733, r20582732);
        return r20582734;
}

Error

Bits error versus x

Bits error versus y

Target

Original16.8
Target0.0
Herbie0.0
\[y \cdot x - \left(y - 1\right)\]

Derivation

  1. Initial program 16.8

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

  2. Simplified16.8

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

  3. Taylor expanded around 0 0.0

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

  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 2019170 +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))))