Average Error: 16.5 → 0.0
Time: 22.5s
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 r370857 = x;
        double r370858 = 1.0;
        double r370859 = r370858 - r370857;
        double r370860 = y;
        double r370861 = r370858 - r370860;
        double r370862 = r370859 * r370861;
        double r370863 = r370857 + r370862;
        return r370863;
}
↓
double f(double x, double y) {
        double r370864 = y;
        double r370865 = x;
        double r370866 = 1.0;
        double r370867 = r370865 - r370866;
        double r370868 = fma(r370864, r370867, r370866);
        return r370868;
}

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

    \[\leadsto \color{blue}{\left(x \cdot y + 1\right) - 1 \cdot y}\]

  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 2020043 +o rules:numerics
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.Vectors:renderPlotVectors from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (- (* y x) (- y 1))

  (+ x (* (- 1 x) (- 1 y))))