Average Error: 17.1 → 0.0
Time: 10.8s
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 r351044 = x;
        double r351045 = 1.0;
        double r351046 = r351045 - r351044;
        double r351047 = y;
        double r351048 = r351045 - r351047;
        double r351049 = r351046 * r351048;
        double r351050 = r351044 + r351049;
        return r351050;
}
↓
double f(double x, double y) {
        double r351051 = y;
        double r351052 = x;
        double r351053 = 1.0;
        double r351054 = r351052 - r351053;
        double r351055 = fma(r351051, r351054, r351053);
        return r351055;
}

Error

Bits error versus x

Bits error versus y

Target

Original17.1
Target0.0
Herbie0.0
\[y \cdot x - \left(y - 1\right)\]

Derivation

  1. Initial program 17.1

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

  2. Simplified17.1

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

  3. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{\left(x \cdot y + 1\right) - 1 \cdot y}\]

  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 2019212 +o rules:numerics
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.Vectors:renderPlotVectors from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (- (* y x) (- y 1))

  (+ x (* (- 1 x) (- 1 y))))