Average Error: 15.6 → 0.0
Time: 7.4s
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 r446648 = x;
        double r446649 = 1.0;
        double r446650 = r446649 - r446648;
        double r446651 = y;
        double r446652 = r446649 - r446651;
        double r446653 = r446650 * r446652;
        double r446654 = r446648 + r446653;
        return r446654;
}
↓
double f(double x, double y) {
        double r446655 = y;
        double r446656 = x;
        double r446657 = 1.0;
        double r446658 = r446656 - r446657;
        double r446659 = fma(r446655, r446658, r446657);
        return r446659;
}

Error

Bits error versus x

Bits error versus y

Target

Original15.6
Target0.0
Herbie0.0
\[y \cdot x - \left(y - 1\right)\]

Derivation

  1. Initial program 15.6

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

  2. Simplified15.6

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

  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 2019198 +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))))