Average Error: 16.4 → 0.0
Time: 10.4s
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 r28590870 = x;
        double r28590871 = 1.0;
        double r28590872 = r28590871 - r28590870;
        double r28590873 = y;
        double r28590874 = r28590871 - r28590873;
        double r28590875 = r28590872 * r28590874;
        double r28590876 = r28590870 + r28590875;
        return r28590876;
}
↓
double f(double x, double y) {
        double r28590877 = y;
        double r28590878 = x;
        double r28590879 = 1.0;
        double r28590880 = r28590878 - r28590879;
        double r28590881 = fma(r28590877, r28590880, r28590879);
        return r28590881;
}

Error

Bits error versus x

Bits error versus y

Target

Original16.4
Target0.0
Herbie0.0
\[y \cdot x - \left(y - 1\right)\]

Derivation

  1. Initial program 16.4

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

  2. Simplified16.4

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

  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 2019172 +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))))