Average Error: 0.2 → 0.1
Time: 17.5s
Precision: 64
\[\left(x \cdot 3\right) \cdot y - z\]
↓
\[\mathsf{fma}\left(x, y \cdot 3, -z\right)\]
\left(x \cdot 3\right) \cdot y - z
↓
\mathsf{fma}\left(x, y \cdot 3, -z\right)
double f(double x, double y, double z) {
        double r495798 = x;
        double r495799 = 3.0;
        double r495800 = r495798 * r495799;
        double r495801 = y;
        double r495802 = r495800 * r495801;
        double r495803 = z;
        double r495804 = r495802 - r495803;
        return r495804;
}
↓
double f(double x, double y, double z) {
        double r495805 = x;
        double r495806 = y;
        double r495807 = 3.0;
        double r495808 = r495806 * r495807;
        double r495809 = z;
        double r495810 = -r495809;
        double r495811 = fma(r495805, r495808, r495810);
        return r495811;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

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

Derivation

  1. Initial program 0.2

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

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

  6. Simplified0.1

    \[\leadsto \mathsf{fma}\left(x, y \cdot 3, -z\right) + \color{blue}{0}\]

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (- (* x (* 3 y)) z)

  (- (* (* x 3) y) z))