Average Error: 0.1 → 0.1
Time: 10.2s
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 r116842 = x;
        double r116843 = y;
        double r116844 = z;
        double r116845 = r116843 + r116844;
        double r116846 = r116845 + r116844;
        double r116847 = r116846 + r116843;
        double r116848 = t;
        double r116849 = r116847 + r116848;
        double r116850 = r116842 * r116849;
        double r116851 = 5.0;
        double r116852 = r116843 * r116851;
        double r116853 = r116850 + r116852;
        return r116853;
}
↓
double f(double x, double y, double z, double t) {
        double r116854 = x;
        double r116855 = 2.0;
        double r116856 = y;
        double r116857 = z;
        double r116858 = r116856 + r116857;
        double r116859 = t;
        double r116860 = fma(r116855, r116858, r116859);
        double r116861 = 5.0;
        double r116862 = r116856 * r116861;
        double r116863 = fma(r116854, r116860, r116862);
        return r116863;
}

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 2019208 +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)))