Average Error: 17.0 → 0.0
Time: 7.2s
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 r26907540 = x;
        double r26907541 = 1.0;
        double r26907542 = r26907541 - r26907540;
        double r26907543 = y;
        double r26907544 = r26907541 - r26907543;
        double r26907545 = r26907542 * r26907544;
        double r26907546 = r26907540 + r26907545;
        return r26907546;
}
↓
double f(double x, double y) {
        double r26907547 = y;
        double r26907548 = x;
        double r26907549 = 1.0;
        double r26907550 = r26907548 - r26907549;
        double r26907551 = fma(r26907547, r26907550, r26907549);
        return r26907551;
}

Error

Bits error versus x

Bits error versus y

Target

Original17.0
Target0.0
Herbie0.0
\[y \cdot x - \left(y - 1.0\right)\]

Derivation

  1. Initial program 17.0

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

  2. Simplified17.0

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

  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 2019164 +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))))