Average Error: 16.7 → 0.0
Time: 13.6s
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 r25225377 = x;
        double r25225378 = 1.0;
        double r25225379 = r25225378 - r25225377;
        double r25225380 = y;
        double r25225381 = r25225378 - r25225380;
        double r25225382 = r25225379 * r25225381;
        double r25225383 = r25225377 + r25225382;
        return r25225383;
}
↓
double f(double x, double y) {
        double r25225384 = y;
        double r25225385 = x;
        double r25225386 = 1.0;
        double r25225387 = r25225385 - r25225386;
        double r25225388 = fma(r25225384, r25225387, r25225386);
        return r25225388;
}

Error

Bits error versus x

Bits error versus y

Target

Original16.7
Target0.0
Herbie0.0
\[y \cdot x - \left(y - 1\right)\]

Derivation

  1. Initial program 16.7

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

  2. Simplified16.7

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

  3. Taylor expanded around 0 0.0

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

  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 2019174 +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))))