Average Error: 16.3 → 0.0
Time: 1.5s
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 r607835 = x;
        double r607836 = 1.0;
        double r607837 = r607836 - r607835;
        double r607838 = y;
        double r607839 = r607836 - r607838;
        double r607840 = r607837 * r607839;
        double r607841 = r607835 + r607840;
        return r607841;
}
↓
double f(double x, double y) {
        double r607842 = y;
        double r607843 = x;
        double r607844 = 1.0;
        double r607845 = r607844 * r607842;
        double r607846 = r607844 - r607845;
        double r607847 = fma(r607842, r607843, r607846);
        return r607847;
}

Error

Bits error versus x

Bits error versus y

Target

Original16.3
Target0.0
Herbie0.0
\[y \cdot x - \left(y - 1\right)\]

Derivation

  1. Initial program 16.3

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

  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 2020036 +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))))