Average Error: 15.9 → 0.0
Time: 23.3s
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 r389012 = x;
        double r389013 = 1.0;
        double r389014 = r389013 - r389012;
        double r389015 = y;
        double r389016 = r389013 - r389015;
        double r389017 = r389014 * r389016;
        double r389018 = r389012 + r389017;
        return r389018;
}
↓
double f(double x, double y) {
        double r389019 = y;
        double r389020 = x;
        double r389021 = 1.0;
        double r389022 = r389020 - r389021;
        double r389023 = fma(r389019, r389022, r389021);
        return r389023;
}

Error

Bits error versus x

Bits error versus y

Target

Original15.9
Target0.0
Herbie0.0
\[y \cdot x - \left(y - 1\right)\]

Derivation

  1. Initial program 15.9

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

  2. Simplified15.9

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

  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 2019306 +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))))