Average Error: 16.1 → 0.0
Time: 9.8s
Precision: 64
\[x + \left(1.0 - x\right) \cdot \left(1.0 - y\right)\]
↓
\[\mathsf{fma}\left(y, x - 1.0, 1.0\right)\]
x + \left(1.0 - x\right) \cdot \left(1.0 - y\right)
↓
\mathsf{fma}\left(y, x - 1.0, 1.0\right)
double f(double x, double y) {
        double r27850268 = x;
        double r27850269 = 1.0;
        double r27850270 = r27850269 - r27850268;
        double r27850271 = y;
        double r27850272 = r27850269 - r27850271;
        double r27850273 = r27850270 * r27850272;
        double r27850274 = r27850268 + r27850273;
        return r27850274;
}
↓
double f(double x, double y) {
        double r27850275 = y;
        double r27850276 = x;
        double r27850277 = 1.0;
        double r27850278 = r27850276 - r27850277;
        double r27850279 = fma(r27850275, r27850278, r27850277);
        return r27850279;
}

Error

Bits error versus x

Bits error versus y

Target

Original16.1
Target0.0
Herbie0.0
\[y \cdot x - \left(y - 1.0\right)\]

Derivation

  1. Initial program 16.1

    \[x + \left(1.0 - x\right) \cdot \left(1.0 - y\right)\]

  2. Simplified16.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(1.0 - y, 1.0 - x, x\right)}\]

  3. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{\left(1.0 + x \cdot y\right) - 1.0 \cdot y}\]

  4. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, x - 1.0, 1.0\right)}\]

  5. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(y, x - 1.0, 1.0\right)\]

Reproduce

herbie shell --seed 2019163 +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))))