Average Error: 16.6 → 0.0
Time: 15.4s
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 r807565 = x;
        double r807566 = 1.0;
        double r807567 = r807566 - r807565;
        double r807568 = y;
        double r807569 = r807566 - r807568;
        double r807570 = r807567 * r807569;
        double r807571 = r807565 + r807570;
        return r807571;
}
↓
double f(double x, double y) {
        double r807572 = y;
        double r807573 = x;
        double r807574 = 1.0;
        double r807575 = r807573 - r807574;
        double r807576 = fma(r807572, r807575, r807574);
        return r807576;
}

Error

Bits error versus x

Bits error versus y

Target

Original16.6
Target0.0
Herbie0.0
\[y \cdot x - \left(y - 1\right)\]

Derivation

  1. Initial program 16.6

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

  2. Taylor expanded around 0 0.0

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019350 +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))))