Average Error: 16.1 → 0.0
Time: 864.0ms
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 r527211 = x;
        double r527212 = 1.0;
        double r527213 = r527212 - r527211;
        double r527214 = y;
        double r527215 = r527212 - r527214;
        double r527216 = r527213 * r527215;
        double r527217 = r527211 + r527216;
        return r527217;
}
↓
double f(double x, double y) {
        double r527218 = y;
        double r527219 = x;
        double r527220 = 1.0;
        double r527221 = r527219 - r527220;
        double r527222 = fma(r527218, r527221, r527220);
        return r527222;
}

Error

Bits error versus x

Bits error versus y

Target

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

Derivation

  1. Initial program 16.1

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

  2. Taylor expanded around 0 0.0

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

  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 2020039 +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))))