Average Error: 0.1 → 0.0
Time: 13.9s
Precision: 64
\[x \cdot \left(y + z\right) + z \cdot 5\]
↓
\[\mathsf{fma}\left(5, z, \left(z + y\right) \cdot x\right)\]
x \cdot \left(y + z\right) + z \cdot 5
↓
\mathsf{fma}\left(5, z, \left(z + y\right) \cdot x\right)
double f(double x, double y, double z) {
        double r18496889 = x;
        double r18496890 = y;
        double r18496891 = z;
        double r18496892 = r18496890 + r18496891;
        double r18496893 = r18496889 * r18496892;
        double r18496894 = 5.0;
        double r18496895 = r18496891 * r18496894;
        double r18496896 = r18496893 + r18496895;
        return r18496896;
}
↓
double f(double x, double y, double z) {
        double r18496897 = 5.0;
        double r18496898 = z;
        double r18496899 = y;
        double r18496900 = r18496898 + r18496899;
        double r18496901 = x;
        double r18496902 = r18496900 * r18496901;
        double r18496903 = fma(r18496897, r18496898, r18496902);
        return r18496903;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original0.1
Target0.1
Herbie0.0
\[\left(x + 5\right) \cdot z + x \cdot y\]

Derivation

  1. Initial program 0.1

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

  2. Taylor expanded around 0 0.1

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y z)
  :name "Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, C"

  :herbie-target
  (+ (* (+ x 5.0) z) (* x y))

  (+ (* x (+ y z)) (* z 5.0)))