Average Error: 0.1 → 0.0
Time: 2.9s
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 r531617 = x;
        double r531618 = y;
        double r531619 = z;
        double r531620 = r531618 + r531619;
        double r531621 = r531617 * r531620;
        double r531622 = 5.0;
        double r531623 = r531619 * r531622;
        double r531624 = r531621 + r531623;
        return r531624;
}
↓
double f(double x, double y, double z) {
        double r531625 = x;
        double r531626 = z;
        double r531627 = 5.0;
        double r531628 = y;
        double r531629 = r531625 * r531628;
        double r531630 = fma(r531627, r531626, r531629);
        double r531631 = fma(r531625, r531626, r531630);
        return r531631;
}

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 2020047 +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)))