Average Error: 15.9 → 0.0
Time: 1.6s
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 r542781 = x;
        double r542782 = 1.0;
        double r542783 = r542782 - r542781;
        double r542784 = y;
        double r542785 = r542782 - r542784;
        double r542786 = r542783 * r542785;
        double r542787 = r542781 + r542786;
        return r542787;
}
↓
double f(double x, double y) {
        double r542788 = y;
        double r542789 = x;
        double r542790 = 1.0;
        double r542791 = r542789 - r542790;
        double r542792 = fma(r542788, r542791, r542790);
        return r542792;
}

Error

Bits error versus x

Bits error versus y

Target

Original15.9
Target0.0
Herbie0.0
\[y \cdot x - \left(y - 1\right)\]

Derivation

  1. Initial program 15.9

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

  2. Simplified15.9

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

  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 2020033 +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))))