Average Error: 16.8 → 0.0
Time: 1.4s
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 r523515 = x;
        double r523516 = 1.0;
        double r523517 = r523516 - r523515;
        double r523518 = y;
        double r523519 = r523516 - r523518;
        double r523520 = r523517 * r523519;
        double r523521 = r523515 + r523520;
        return r523521;
}
↓
double f(double x, double y) {
        double r523522 = y;
        double r523523 = x;
        double r523524 = 1.0;
        double r523525 = r523523 - r523524;
        double r523526 = fma(r523522, r523525, r523524);
        return r523526;
}

Error

Bits error versus x

Bits error versus y

Target

Original16.8
Target0.0
Herbie0.0
\[y \cdot x - \left(y - 1\right)\]

Derivation

  1. Initial program 16.8

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

  2. Simplified16.8

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

  3. Taylor expanded around 0 0.0

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

  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 2020035 +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))))