Average Error: 17.0 → 0.0
Time: 975.0ms
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 r580074 = x;
        double r580075 = 1.0;
        double r580076 = r580075 - r580074;
        double r580077 = y;
        double r580078 = r580075 - r580077;
        double r580079 = r580076 * r580078;
        double r580080 = r580074 + r580079;
        return r580080;
}
↓
double f(double x, double y) {
        double r580081 = y;
        double r580082 = x;
        double r580083 = 1.0;
        double r580084 = r580082 - r580083;
        double r580085 = fma(r580081, r580084, r580083);
        return r580085;
}

Error

Bits error versus x

Bits error versus y

Target

Original17.0
Target0.0
Herbie0.0
\[y \cdot x - \left(y - 1\right)\]

Derivation

  1. Initial program 17.0

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

  2. Simplified17.0

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

  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 2019353 +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))))