Average Error: 0.1 → 0.0
Time: 13.6s
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 r22103729 = x;
        double r22103730 = y;
        double r22103731 = z;
        double r22103732 = r22103730 + r22103731;
        double r22103733 = r22103729 * r22103732;
        double r22103734 = 5.0;
        double r22103735 = r22103731 * r22103734;
        double r22103736 = r22103733 + r22103735;
        return r22103736;
}
↓
double f(double x, double y, double z) {
        double r22103737 = 5.0;
        double r22103738 = z;
        double r22103739 = y;
        double r22103740 = r22103738 + r22103739;
        double r22103741 = x;
        double r22103742 = r22103740 * r22103741;
        double r22103743 = fma(r22103737, r22103738, r22103742);
        return r22103743;
}

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 2019172 +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)))