Average Error: 0.1 → 0.0
Time: 8.1s
Precision: 64
\[x \cdot \left(y + z\right) + z \cdot 5\]
↓
\[\mathsf{fma}\left(z, 5, \left(z + y\right) \cdot x\right)\]
x \cdot \left(y + z\right) + z \cdot 5
↓
\mathsf{fma}\left(z, 5, \left(z + y\right) \cdot x\right)
double f(double x, double y, double z) {
        double r478184 = x;
        double r478185 = y;
        double r478186 = z;
        double r478187 = r478185 + r478186;
        double r478188 = r478184 * r478187;
        double r478189 = 5.0;
        double r478190 = r478186 * r478189;
        double r478191 = r478188 + r478190;
        return r478191;
}
↓
double f(double x, double y, double z) {
        double r478192 = z;
        double r478193 = 5.0;
        double r478194 = y;
        double r478195 = r478192 + r478194;
        double r478196 = x;
        double r478197 = r478195 * r478196;
        double r478198 = fma(r478192, r478193, r478197);
        return r478198;
}

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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019194 +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)))