Average Error: 0.1 → 0.1
Time: 22.2s
Precision: 64
\[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5\]
↓
\[\mathsf{fma}\left(x, \mathsf{fma}\left(2, y + z, t\right), y \cdot 5\right)\]
x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5
↓
\mathsf{fma}\left(x, \mathsf{fma}\left(2, y + z, t\right), y \cdot 5\right)
double f(double x, double y, double z, double t) {
        double r162838 = x;
        double r162839 = y;
        double r162840 = z;
        double r162841 = r162839 + r162840;
        double r162842 = r162841 + r162840;
        double r162843 = r162842 + r162839;
        double r162844 = t;
        double r162845 = r162843 + r162844;
        double r162846 = r162838 * r162845;
        double r162847 = 5.0;
        double r162848 = r162839 * r162847;
        double r162849 = r162846 + r162848;
        return r162849;
}
↓
double f(double x, double y, double z, double t) {
        double r162850 = x;
        double r162851 = 2.0;
        double r162852 = y;
        double r162853 = z;
        double r162854 = r162852 + r162853;
        double r162855 = t;
        double r162856 = fma(r162851, r162854, r162855);
        double r162857 = 5.0;
        double r162858 = r162852 * r162857;
        double r162859 = fma(r162850, r162856, r162858);
        return r162859;
}

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

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(2, y + z, t\right), y \cdot 5\right)}\]

  3. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(2, y + z, t\right), y \cdot 5\right)\]

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, B"
  (+ (* x (+ (+ (+ (+ y z) z) y) t)) (* y 5.0)))