Average Error: 0.1 → 0.0
Time: 3.0s
Precision: 64
\[x \cdot \left(y + z\right) + z \cdot 5\]
↓
\[\mathsf{fma}\left(x, z, \mathsf{fma}\left(5, z, x \cdot y\right)\right)\]
x \cdot \left(y + z\right) + z \cdot 5
↓
\mathsf{fma}\left(x, z, \mathsf{fma}\left(5, z, x \cdot y\right)\right)
double f(double x, double y, double z) {
        double r541062 = x;
        double r541063 = y;
        double r541064 = z;
        double r541065 = r541063 + r541064;
        double r541066 = r541062 * r541065;
        double r541067 = 5.0;
        double r541068 = r541064 * r541067;
        double r541069 = r541066 + r541068;
        return r541069;
}
↓
double f(double x, double y, double z) {
        double r541070 = x;
        double r541071 = z;
        double r541072 = 5.0;
        double r541073 = y;
        double r541074 = r541070 * r541073;
        double r541075 = fma(r541072, r541071, r541074);
        double r541076 = fma(r541070, r541071, r541075);
        return r541076;
}

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. Simplified0.1

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

  3. Taylor expanded around 0 0.1

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(FPCore (x y z)
  :name "Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, C"
  :precision binary64

  :herbie-target
  (+ (* (+ x 5) z) (* x y))

  (+ (* x (+ y z)) (* z 5)))