Average Error: 16.2 → 0.0
Time: 11.4s
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 r24859503 = x;
        double r24859504 = 1.0;
        double r24859505 = r24859504 - r24859503;
        double r24859506 = y;
        double r24859507 = r24859504 - r24859506;
        double r24859508 = r24859505 * r24859507;
        double r24859509 = r24859503 + r24859508;
        return r24859509;
}
↓
double f(double x, double y) {
        double r24859510 = y;
        double r24859511 = x;
        double r24859512 = 1.0;
        double r24859513 = r24859511 - r24859512;
        double r24859514 = fma(r24859510, r24859513, r24859512);
        return r24859514;
}

Error

Bits error versus x

Bits error versus y

Target

Original16.2
Target0.0
Herbie0.0
\[y \cdot x - \left(y - 1.0\right)\]

Derivation

  1. Initial program 16.2

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

  2. Simplified16.2

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

  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 2019162 +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))))