Average Error: 16.5 → 0.0
Time: 13.3s
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 r26390983 = x;
        double r26390984 = 1.0;
        double r26390985 = r26390984 - r26390983;
        double r26390986 = y;
        double r26390987 = r26390984 - r26390986;
        double r26390988 = r26390985 * r26390987;
        double r26390989 = r26390983 + r26390988;
        return r26390989;
}
↓
double f(double x, double y) {
        double r26390990 = y;
        double r26390991 = x;
        double r26390992 = 1.0;
        double r26390993 = r26390991 - r26390992;
        double r26390994 = fma(r26390990, r26390993, r26390992);
        return r26390994;
}

Error

Bits error versus x

Bits error versus y

Target

Original16.5
Target0.0
Herbie0.0
\[y \cdot x - \left(y - 1.0\right)\]

Derivation

  1. Initial program 16.5

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

  2. Simplified16.5

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

  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 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))))