Average Error: 0.2 → 0.1
Time: 14.3s
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 r601100 = x;
        double r601101 = 3.0;
        double r601102 = r601100 * r601101;
        double r601103 = y;
        double r601104 = r601102 * r601103;
        double r601105 = z;
        double r601106 = r601104 - r601105;
        return r601106;
}
↓
double f(double x, double y, double z) {
        double r601107 = x;
        double r601108 = y;
        double r601109 = 3.0;
        double r601110 = r601108 * r601109;
        double r601111 = z;
        double r601112 = -r601111;
        double r601113 = fma(r601107, r601110, r601112);
        return r601113;
}

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))