Average Error: 0.2 → 0.2
Time: 13.3s
Precision: 64
\[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right)\]
↓
\[\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{3}} \cdot \sqrt[3]{\sqrt[3]{3}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{3}} \cdot {x}^{2}\right)\right) - 2 \cdot {x}^{3}\]
\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right)
↓
\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{3}} \cdot \sqrt[3]{\sqrt[3]{3}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{3}} \cdot {x}^{2}\right)\right) - 2 \cdot {x}^{3}
double f(double x) {
        double r501991 = x;
        double r501992 = r501991 * r501991;
        double r501993 = 3.0;
        double r501994 = 2.0;
        double r501995 = r501991 * r501994;
        double r501996 = r501993 - r501995;
        double r501997 = r501992 * r501996;
        return r501997;
}
↓
double f(double x) {
        double r501998 = 3.0;
        double r501999 = cbrt(r501998);
        double r502000 = r501999 * r501999;
        double r502001 = cbrt(r501999);
        double r502002 = r502001 * r502001;
        double r502003 = x;
        double r502004 = 2.0;
        double r502005 = pow(r502003, r502004);
        double r502006 = r502001 * r502005;
        double r502007 = r502002 * r502006;
        double r502008 = r502000 * r502007;
        double r502009 = 2.0;
        double r502010 = 3.0;
        double r502011 = pow(r502003, r502010);
        double r502012 = r502009 * r502011;
        double r502013 = r502008 - r502012;
        return r502013;
}

Error

Bits error versus x

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Results

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Target

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

Derivation

  1. Initial program 0.2

    \[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right)\]

  2. Taylor expanded around 0 0.1

    \[\leadsto \color{blue}{3 \cdot {x}^{2} - 2 \cdot {x}^{3}}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt0.1

    \[\leadsto \color{blue}{\left(\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}\right)} \cdot {x}^{2} - 2 \cdot {x}^{3}\]

  5. Applied associate-*l*0.2

    \[\leadsto \color{blue}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \left(\sqrt[3]{3} \cdot {x}^{2}\right)} - 2 \cdot {x}^{3}\]
  6. Using strategy rm

  7. Applied add-cube-cbrt0.2

    \[\leadsto \left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{3}} \cdot \sqrt[3]{\sqrt[3]{3}}\right) \cdot \sqrt[3]{\sqrt[3]{3}}\right)} \cdot {x}^{2}\right) - 2 \cdot {x}^{3}\]

  8. Applied associate-*l*0.2

    \[\leadsto \left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{3}} \cdot \sqrt[3]{\sqrt[3]{3}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{3}} \cdot {x}^{2}\right)\right)} - 2 \cdot {x}^{3}\]

  9. Final simplification0.2

    \[\leadsto \left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{3}} \cdot \sqrt[3]{\sqrt[3]{3}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{3}} \cdot {x}^{2}\right)\right) - 2 \cdot {x}^{3}\]

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (x)
  :name "Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2"
  :precision binary64

  :herbie-target
  (* x (* x (- 3 (* x 2))))

  (* (* x x) (- 3 (* x 2))))