Average Error: 16.5 → 0.0
Time: 1.4s
Precision: 64
\[x + \left(1 - x\right) \cdot \left(1 - y\right)\]
↓
\[\mathsf{fma}\left(y, x, 1 - 1 \cdot y\right)\]
x + \left(1 - x\right) \cdot \left(1 - y\right)
↓
\mathsf{fma}\left(y, x, 1 - 1 \cdot y\right)
double f(double x, double y) {
        double r538591 = x;
        double r538592 = 1.0;
        double r538593 = r538592 - r538591;
        double r538594 = y;
        double r538595 = r538592 - r538594;
        double r538596 = r538593 * r538595;
        double r538597 = r538591 + r538596;
        return r538597;
}
↓
double f(double x, double y) {
        double r538598 = y;
        double r538599 = x;
        double r538600 = 1.0;
        double r538601 = r538600 * r538598;
        double r538602 = r538600 - r538601;
        double r538603 = fma(r538598, r538599, r538602);
        return r538603;
}

Error

Bits error versus x

Bits error versus y

Target

Original16.5
Target0.0
Herbie0.0
\[y \cdot x - \left(y - 1\right)\]

Derivation

  1. Initial program 16.5

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

  2. Taylor expanded around 0 0.0

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2020081 +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))))