Average Error: 16.2 → 0.0
Time: 10.3s
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 r23831411 = x;
        double r23831412 = 1.0;
        double r23831413 = r23831412 - r23831411;
        double r23831414 = y;
        double r23831415 = r23831412 - r23831414;
        double r23831416 = r23831413 * r23831415;
        double r23831417 = r23831411 + r23831416;
        return r23831417;
}
↓
double f(double x, double y) {
        double r23831418 = y;
        double r23831419 = x;
        double r23831420 = 1.0;
        double r23831421 = r23831419 - r23831420;
        double r23831422 = fma(r23831418, r23831421, r23831420);
        return r23831422;
}

Error

Bits error versus x

Bits error versus y

Target

Original16.2
Target0.0
Herbie0.0
\[y \cdot x - \left(y - 1\right)\]

Derivation

  1. Initial program 16.2

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

  2. Simplified16.2

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

  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 2019179 +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))))