Average Error: 16.2 → 0.0
Time: 14.9s
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 r13896566 = x;
        double r13896567 = 1.0;
        double r13896568 = r13896567 - r13896566;
        double r13896569 = y;
        double r13896570 = r13896567 - r13896569;
        double r13896571 = r13896568 * r13896570;
        double r13896572 = r13896566 + r13896571;
        return r13896572;
}
↓
double f(double x, double y) {
        double r13896573 = y;
        double r13896574 = x;
        double r13896575 = 1.0;
        double r13896576 = r13896574 - r13896575;
        double r13896577 = fma(r13896573, r13896576, r13896575);
        return r13896577;
}

Error

Bits error versus x

Bits error versus y

Target

Original16.2
Target0.0
Herbie0.0
\[y \cdot x - \left(y - 1\right)\]

Derivation

  1. Initial program 16.2

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

  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 2019192 +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))))