Average Error: 16.5 → 0.0
Time: 23.4s
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 r26888612 = x;
        double r26888613 = 1.0;
        double r26888614 = r26888613 - r26888612;
        double r26888615 = y;
        double r26888616 = r26888613 - r26888615;
        double r26888617 = r26888614 * r26888616;
        double r26888618 = r26888612 + r26888617;
        return r26888618;
}
↓
double f(double x, double y) {
        double r26888619 = y;
        double r26888620 = x;
        double r26888621 = 1.0;
        double r26888622 = r26888620 - r26888621;
        double r26888623 = fma(r26888619, r26888622, r26888621);
        return r26888623;
}

Error

Bits error versus x

Bits error versus y

Target

Original16.5
Target0.0
Herbie0.0
\[y \cdot x - \left(y - 1\right)\]

Derivation

  1. Initial program 16.5

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

  2. Simplified16.5

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

  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 2019168 +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))))