Average Error: 0.1 → 0.0
Time: 3.7s
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 r654483 = x;
        double r654484 = y;
        double r654485 = z;
        double r654486 = r654484 + r654485;
        double r654487 = r654483 * r654486;
        double r654488 = 5.0;
        double r654489 = r654485 * r654488;
        double r654490 = r654487 + r654489;
        return r654490;
}
↓
double f(double x, double y, double z) {
        double r654491 = x;
        double r654492 = z;
        double r654493 = 5.0;
        double r654494 = y;
        double r654495 = r654491 * r654494;
        double r654496 = fma(r654493, r654492, r654495);
        double r654497 = fma(r654491, r654492, r654496);
        return r654497;
}

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 2020083 +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)))