Average Error: 0.1 → 0.0
Time: 3.9s
Precision: 64
\[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5\]
↓
\[\mathsf{fma}\left(y + z, x + x, \mathsf{fma}\left(5, y, t \cdot x\right)\right)\]
x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5
↓
\mathsf{fma}\left(y + z, x + x, \mathsf{fma}\left(5, y, t \cdot x\right)\right)
double f(double x, double y, double z, double t) {
        double r167861 = x;
        double r167862 = y;
        double r167863 = z;
        double r167864 = r167862 + r167863;
        double r167865 = r167864 + r167863;
        double r167866 = r167865 + r167862;
        double r167867 = t;
        double r167868 = r167866 + r167867;
        double r167869 = r167861 * r167868;
        double r167870 = 5.0;
        double r167871 = r167862 * r167870;
        double r167872 = r167869 + r167871;
        return r167872;
}
↓
double f(double x, double y, double z, double t) {
        double r167873 = y;
        double r167874 = z;
        double r167875 = r167873 + r167874;
        double r167876 = x;
        double r167877 = r167876 + r167876;
        double r167878 = 5.0;
        double r167879 = t;
        double r167880 = r167879 * r167876;
        double r167881 = fma(r167878, r167873, r167880);
        double r167882 = fma(r167875, r167877, r167881);
        return r167882;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 0.1

    \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(y + z, x + x, \mathsf{fma}\left(t, x, y \cdot 5\right)\right)}\]

  3. Taylor expanded around 0 0.1

    \[\leadsto \mathsf{fma}\left(y + z, x + x, \color{blue}{5 \cdot y + t \cdot x}\right)\]

  4. Simplified0.0

    \[\leadsto \mathsf{fma}\left(y + z, x + x, \color{blue}{\mathsf{fma}\left(5, y, t \cdot x\right)}\right)\]

  5. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(y + z, x + x, \mathsf{fma}\left(5, y, t \cdot x\right)\right)\]

Reproduce

herbie shell --seed 2020021 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, B"
  :precision binary64
  (+ (* x (+ (+ (+ (+ y z) z) y) t)) (* y 5)))