Average Error: 0.1 → 0.1
Time: 10.8s
Precision: 64
\[\left(x \cdot 3\right) \cdot y - z\]
↓
\[\left(3 \cdot \left(x \cdot y\right) - z\right) + \left(\left(-z\right) + z\right)\]
\left(x \cdot 3\right) \cdot y - z
↓
\left(3 \cdot \left(x \cdot y\right) - z\right) + \left(\left(-z\right) + z\right)
double f(double x, double y, double z) {
        double r542982 = x;
        double r542983 = 3.0;
        double r542984 = r542982 * r542983;
        double r542985 = y;
        double r542986 = r542984 * r542985;
        double r542987 = z;
        double r542988 = r542986 - r542987;
        return r542988;
}
↓
double f(double x, double y, double z) {
        double r542989 = 3.0;
        double r542990 = x;
        double r542991 = y;
        double r542992 = r542990 * r542991;
        double r542993 = r542989 * r542992;
        double r542994 = z;
        double r542995 = r542993 - r542994;
        double r542996 = -r542994;
        double r542997 = r542996 + r542994;
        double r542998 = r542995 + r542997;
        return r542998;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Target

Original0.1
Target0.2
Herbie0.1
\[x \cdot \left(3 \cdot y\right) - z\]

Derivation

  1. Initial program 0.1

    \[\left(x \cdot 3\right) \cdot y - z\]
  2. Using strategy rm

  3. Applied add-cube-cbrt0.9

    \[\leadsto \left(x \cdot 3\right) \cdot y - \color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}\]

  4. Applied prod-diff0.9

    \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot 3, y, -\sqrt[3]{z} \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{z}, \sqrt[3]{z} \cdot \sqrt[3]{z}, \sqrt[3]{z} \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right)}\]

  5. Simplified0.1

    \[\leadsto \color{blue}{\left(3 \cdot \left(x \cdot y\right) - z\right)} + \mathsf{fma}\left(-\sqrt[3]{z}, \sqrt[3]{z} \cdot \sqrt[3]{z}, \sqrt[3]{z} \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right)\]

  6. Simplified0.1

    \[\leadsto \left(3 \cdot \left(x \cdot y\right) - z\right) + \color{blue}{\left(\left(-z\right) + z\right)}\]

  7. Final simplification0.1

    \[\leadsto \left(3 \cdot \left(x \cdot y\right) - z\right) + \left(\left(-z\right) + z\right)\]

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, B"

  :herbie-target
  (- (* x (* 3.0 y)) z)

  (- (* (* x 3.0) y) z))