Average Error: 16.3 → 0.0
Time: 8.0s
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 r21375563 = x;
        double r21375564 = 1.0;
        double r21375565 = r21375564 - r21375563;
        double r21375566 = y;
        double r21375567 = r21375564 - r21375566;
        double r21375568 = r21375565 * r21375567;
        double r21375569 = r21375563 + r21375568;
        return r21375569;
}
↓
double f(double x, double y) {
        double r21375570 = y;
        double r21375571 = x;
        double r21375572 = 1.0;
        double r21375573 = r21375571 - r21375572;
        double r21375574 = fma(r21375570, r21375573, r21375572);
        return r21375574;
}

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

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

  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 2019171 +o rules:numerics
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.Vectors:renderPlotVectors from Chart-1.5.3"

  :herbie-target
  (- (* y x) (- y 1.0))

  (+ x (* (- 1.0 x) (- 1.0 y))))