Average Error: 16.4 → 0.0
Time: 20.1s
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 r20802 = x;
        double r20803 = 1.0;
        double r20804 = r20803 - r20802;
        double r20805 = y;
        double r20806 = r20803 - r20805;
        double r20807 = r20804 * r20806;
        double r20808 = r20802 + r20807;
        return r20808;
}
↓
double f(double x, double y) {
        double r20809 = y;
        double r20810 = x;
        double r20811 = 1.0;
        double r20812 = r20810 - r20811;
        double r20813 = fma(r20809, r20812, r20811);
        return r20813;
}

Error

Bits error versus x

Bits error versus y

Target

Original16.4
Target0.0
Herbie0.0
\[y \cdot x - \left(y - 1\right)\]

Derivation

  1. Initial program 16.4

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

  2. Simplified16.4

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

  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 2019315 +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))))