Average Error: 15.8 → 0.0
Time: 1.0s
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 r1300996 = x;
        double r1300997 = 1.0;
        double r1300998 = r1300997 - r1300996;
        double r1300999 = y;
        double r1301000 = r1300997 - r1300999;
        double r1301001 = r1300998 * r1301000;
        double r1301002 = r1300996 + r1301001;
        return r1301002;
}
↓
double f(double x, double y) {
        double r1301003 = y;
        double r1301004 = x;
        double r1301005 = 1.0;
        double r1301006 = r1301004 - r1301005;
        double r1301007 = fma(r1301003, r1301006, r1301005);
        return r1301007;
}

Error

Bits error versus x

Bits error versus y

Target

Original15.8
Target0.0
Herbie0.0
\[y \cdot x - \left(y - 1\right)\]

Derivation

  1. Initial program 15.8

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

  2. Simplified15.8

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

  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 2020034 +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))))