Average Error: 16.2 → 0.0
Time: 10.1s
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 r431329 = x;
        double r431330 = 1.0;
        double r431331 = r431330 - r431329;
        double r431332 = y;
        double r431333 = r431330 - r431332;
        double r431334 = r431331 * r431333;
        double r431335 = r431329 + r431334;
        return r431335;
}
↓
double f(double x, double y) {
        double r431336 = y;
        double r431337 = x;
        double r431338 = 1.0;
        double r431339 = r431337 - r431338;
        double r431340 = fma(r431336, r431339, r431338);
        return r431340;
}

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))))