Average Error: 0.1 → 0.0
Time: 8.9s
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 r20963252 = x;
        double r20963253 = y;
        double r20963254 = z;
        double r20963255 = r20963253 + r20963254;
        double r20963256 = r20963252 * r20963255;
        double r20963257 = 5.0;
        double r20963258 = r20963254 * r20963257;
        double r20963259 = r20963256 + r20963258;
        return r20963259;
}
↓
double f(double x, double y, double z) {
        double r20963260 = 5.0;
        double r20963261 = z;
        double r20963262 = y;
        double r20963263 = r20963261 + r20963262;
        double r20963264 = x;
        double r20963265 = r20963263 * r20963264;
        double r20963266 = fma(r20963260, r20963261, r20963265);
        return r20963266;
}

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)))