Average Error: 15.9 → 0.0
Time: 16.7s
Precision: 64
\[x + \left(1.0 - x\right) \cdot \left(1.0 - y\right)\]
↓
\[\mathsf{fma}\left(y, x - 1.0, 1.0\right)\]
x + \left(1.0 - x\right) \cdot \left(1.0 - y\right)
↓
\mathsf{fma}\left(y, x - 1.0, 1.0\right)
double f(double x, double y) {
        double r23159716 = x;
        double r23159717 = 1.0;
        double r23159718 = r23159717 - r23159716;
        double r23159719 = y;
        double r23159720 = r23159717 - r23159719;
        double r23159721 = r23159718 * r23159720;
        double r23159722 = r23159716 + r23159721;
        return r23159722;
}
↓
double f(double x, double y) {
        double r23159723 = y;
        double r23159724 = x;
        double r23159725 = 1.0;
        double r23159726 = r23159724 - r23159725;
        double r23159727 = fma(r23159723, r23159726, r23159725);
        return r23159727;
}

Error

Bits error versus x

Bits error versus y

Target

Original15.9
Target0.0
Herbie0.0
\[y \cdot x - \left(y - 1.0\right)\]

Derivation

  1. Initial program 15.9

    \[x + \left(1.0 - x\right) \cdot \left(1.0 - y\right)\]

  2. Simplified15.9

    \[\leadsto \color{blue}{\mathsf{fma}\left(1.0 - y, 1.0 - x, x\right)}\]

  3. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{\left(1.0 + x \cdot y\right) - 1.0 \cdot y}\]

  4. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, x - 1.0, 1.0\right)}\]

  5. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(y, x - 1.0, 1.0\right)\]

Reproduce

herbie shell --seed 2019165 +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))))