Average Error: 0.1 → 0.0
Time: 12.8s
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 r22115326 = x;
        double r22115327 = y;
        double r22115328 = z;
        double r22115329 = r22115327 + r22115328;
        double r22115330 = r22115326 * r22115329;
        double r22115331 = 5.0;
        double r22115332 = r22115328 * r22115331;
        double r22115333 = r22115330 + r22115332;
        return r22115333;
}
↓
double f(double x, double y, double z) {
        double r22115334 = 5.0;
        double r22115335 = z;
        double r22115336 = y;
        double r22115337 = r22115335 + r22115336;
        double r22115338 = x;
        double r22115339 = r22115337 * r22115338;
        double r22115340 = fma(r22115334, r22115335, r22115339);
        return r22115340;
}

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 2019169 +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)))