Average Error: 16.6 → 0.0
Time: 17.9s
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 r531756 = x;
        double r531757 = 1.0;
        double r531758 = r531757 - r531756;
        double r531759 = y;
        double r531760 = r531757 - r531759;
        double r531761 = r531758 * r531760;
        double r531762 = r531756 + r531761;
        return r531762;
}
↓
double f(double x, double y) {
        double r531763 = y;
        double r531764 = x;
        double r531765 = 1.0;
        double r531766 = r531764 - r531765;
        double r531767 = fma(r531763, r531766, r531765);
        return r531767;
}

Error

Bits error versus x

Bits error versus y

Target

Original16.6
Target0.0
Herbie0.0
\[y \cdot x - \left(y - 1\right)\]

Derivation

  1. Initial program 16.6

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

  2. Simplified16.6

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

  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 2020047 +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))))