Average Error: 16.1 → 0.0
Time: 10.3s
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 r16150618 = x;
        double r16150619 = 1.0;
        double r16150620 = r16150619 - r16150618;
        double r16150621 = y;
        double r16150622 = r16150619 - r16150621;
        double r16150623 = r16150620 * r16150622;
        double r16150624 = r16150618 + r16150623;
        return r16150624;
}
↓
double f(double x, double y) {
        double r16150625 = y;
        double r16150626 = x;
        double r16150627 = 1.0;
        double r16150628 = r16150626 - r16150627;
        double r16150629 = fma(r16150625, r16150628, r16150627);
        return r16150629;
}

Error

Bits error versus x

Bits error versus y

Target

Original16.1
Target0.0
Herbie0.0
\[y \cdot x - \left(y - 1.0\right)\]

Derivation

  1. Initial program 16.1

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

  2. Simplified16.1

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

  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 2019163 +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))))