Average Error: 0.1 → 0.0
Time: 9.5s
Precision: 64
\[x \cdot \left(y + z\right) + z \cdot 5\]
↓
\[\mathsf{fma}\left(z, 5, x \cdot \left(y + z\right)\right)\]
x \cdot \left(y + z\right) + z \cdot 5
↓
\mathsf{fma}\left(z, 5, x \cdot \left(y + z\right)\right)
double f(double x, double y, double z) {
        double r398710 = x;
        double r398711 = y;
        double r398712 = z;
        double r398713 = r398711 + r398712;
        double r398714 = r398710 * r398713;
        double r398715 = 5.0;
        double r398716 = r398712 * r398715;
        double r398717 = r398714 + r398716;
        return r398717;
}
↓
double f(double x, double y, double z) {
        double r398718 = z;
        double r398719 = 5.0;
        double r398720 = x;
        double r398721 = y;
        double r398722 = r398721 + r398718;
        double r398723 = r398720 * r398722;
        double r398724 = fma(r398718, r398719, r398723);
        return r398724;
}

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. Using strategy rm

  3. Applied distribute-lft-in0.1

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

  4. Taylor expanded around 0 0.1

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019303 +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)))