Average Error: 15.8 → 0.0
Time: 1.4s
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 r703593 = x;
        double r703594 = 1.0;
        double r703595 = r703594 - r703593;
        double r703596 = y;
        double r703597 = r703594 - r703596;
        double r703598 = r703595 * r703597;
        double r703599 = r703593 + r703598;
        return r703599;
}
↓
double f(double x, double y) {
        double r703600 = y;
        double r703601 = x;
        double r703602 = 1.0;
        double r703603 = r703601 - r703602;
        double r703604 = fma(r703600, r703603, r703602);
        return r703604;
}

Error

Bits error versus x

Bits error versus y

Target

Original15.8
Target0.0
Herbie0.0
\[y \cdot x - \left(y - 1\right)\]

Derivation

  1. Initial program 15.8

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

  2. Simplified15.8

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

  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 2019354 +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))))