Average Error: 0.1 → 0.0
Time: 1.1s
Precision: 64
\[x \cdot \left(y + z\right) + z \cdot 5\]
↓
\[\mathsf{fma}\left(x, z, \mathsf{fma}\left(5, z, x \cdot y\right)\right)\]
x \cdot \left(y + z\right) + z \cdot 5
↓
\mathsf{fma}\left(x, z, \mathsf{fma}\left(5, z, x \cdot y\right)\right)
double f(double x, double y, double z) {
        double r500203 = x;
        double r500204 = y;
        double r500205 = z;
        double r500206 = r500204 + r500205;
        double r500207 = r500203 * r500206;
        double r500208 = 5.0;
        double r500209 = r500205 * r500208;
        double r500210 = r500207 + r500209;
        return r500210;
}
↓
double f(double x, double y, double z) {
        double r500211 = x;
        double r500212 = z;
        double r500213 = 5.0;
        double r500214 = y;
        double r500215 = r500211 * r500214;
        double r500216 = fma(r500213, r500212, r500215);
        double r500217 = fma(r500211, r500212, r500216);
        return r500217;
}

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. Simplified0.1

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

  3. Taylor expanded around 0 0.1

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020001 +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)))