Average Error: 1.4 → 0.3
Time: 26.5s
Precision: 64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\sqrt{\cos^{-1} \left(\left(\sqrt{t} \cdot \frac{3}{2}\right) \cdot \frac{\frac{x}{z}}{y \cdot 27}\right) \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{3}}} \cdot \sqrt{\cos^{-1} \left(\left(\sqrt{t} \cdot \frac{3}{2}\right) \cdot \frac{\frac{x}{z}}{y \cdot 27}\right) \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{3}}}\right)\]
\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)
↓
\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\sqrt{\cos^{-1} \left(\left(\sqrt{t} \cdot \frac{3}{2}\right) \cdot \frac{\frac{x}{z}}{y \cdot 27}\right) \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{3}}} \cdot \sqrt{\cos^{-1} \left(\left(\sqrt{t} \cdot \frac{3}{2}\right) \cdot \frac{\frac{x}{z}}{y \cdot 27}\right) \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{3}}}\right)
double f(double x, double y, double z, double t) {
        double r681833 = 1.0;
        double r681834 = 3.0;
        double r681835 = r681833 / r681834;
        double r681836 = x;
        double r681837 = y;
        double r681838 = 27.0;
        double r681839 = r681837 * r681838;
        double r681840 = r681836 / r681839;
        double r681841 = r681834 * r681840;
        double r681842 = z;
        double r681843 = 2.0;
        double r681844 = r681842 * r681843;
        double r681845 = r681841 / r681844;
        double r681846 = t;
        double r681847 = sqrt(r681846);
        double r681848 = r681845 * r681847;
        double r681849 = acos(r681848);
        double r681850 = r681835 * r681849;
        return r681850;
}
↓
double f(double x, double y, double z, double t) {
        double r681851 = 1.0;
        double r681852 = cbrt(r681851);
        double r681853 = r681852 * r681852;
        double r681854 = 3.0;
        double r681855 = cbrt(r681854);
        double r681856 = r681855 * r681855;
        double r681857 = r681853 / r681856;
        double r681858 = t;
        double r681859 = sqrt(r681858);
        double r681860 = 2.0;
        double r681861 = r681854 / r681860;
        double r681862 = r681859 * r681861;
        double r681863 = x;
        double r681864 = z;
        double r681865 = r681863 / r681864;
        double r681866 = y;
        double r681867 = 27.0;
        double r681868 = r681866 * r681867;
        double r681869 = r681865 / r681868;
        double r681870 = r681862 * r681869;
        double r681871 = acos(r681870);
        double r681872 = r681852 / r681855;
        double r681873 = r681871 * r681872;
        double r681874 = sqrt(r681873);
        double r681875 = r681874 * r681874;
        double r681876 = r681857 * r681875;
        return r681876;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original1.4
Target1.3
Herbie0.3
\[\frac{\cos^{-1} \left(\frac{\frac{x}{27}}{y \cdot z} \cdot \frac{\sqrt{t}}{\frac{2}{3}}\right)}{3}\]

Derivation

  1. Initial program 1.4

    \[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
  2. Using strategy rm

  3. Applied add-cube-cbrt1.4

    \[\leadsto \frac{1}{\color{blue}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}}} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]

  4. Applied add-cube-cbrt1.4

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]

  5. Applied times-frac0.4

    \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{3}}\right)} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]

  6. Applied associate-*l*0.4

    \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{\sqrt[3]{1}}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\right)}\]

  7. Simplified0.4

    \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \color{blue}{\left(\frac{\sqrt[3]{1}}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\frac{\frac{x}{27 \cdot y} \cdot 3}{z \cdot 2} \cdot \sqrt{t}\right)\right)}\]
  8. Using strategy rm

  9. Applied times-frac0.4

    \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{\sqrt[3]{1}}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\color{blue}{\left(\frac{\frac{x}{27 \cdot y}}{z} \cdot \frac{3}{2}\right)} \cdot \sqrt{t}\right)\right)\]

  10. Simplified0.3

    \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{\sqrt[3]{1}}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\left(\color{blue}{\frac{\frac{x}{z}}{y \cdot 27}} \cdot \frac{3}{2}\right) \cdot \sqrt{t}\right)\right)\]
  11. Using strategy rm

  12. Applied add-sqr-sqrt0.3

    \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \color{blue}{\left(\sqrt{\frac{\sqrt[3]{1}}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\left(\frac{\frac{x}{z}}{y \cdot 27} \cdot \frac{3}{2}\right) \cdot \sqrt{t}\right)} \cdot \sqrt{\frac{\sqrt[3]{1}}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\left(\frac{\frac{x}{z}}{y \cdot 27} \cdot \frac{3}{2}\right) \cdot \sqrt{t}\right)}\right)}\]

  13. Simplified0.3

    \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\color{blue}{\sqrt{\cos^{-1} \left(\left(\sqrt{t} \cdot \frac{3}{2}\right) \cdot \frac{\frac{x}{z}}{y \cdot 27}\right) \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{3}}}} \cdot \sqrt{\frac{\sqrt[3]{1}}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\left(\frac{\frac{x}{z}}{y \cdot 27} \cdot \frac{3}{2}\right) \cdot \sqrt{t}\right)}\right)\]

  14. Simplified0.3

    \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\sqrt{\cos^{-1} \left(\left(\sqrt{t} \cdot \frac{3}{2}\right) \cdot \frac{\frac{x}{z}}{y \cdot 27}\right) \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{3}}} \cdot \color{blue}{\sqrt{\cos^{-1} \left(\left(\sqrt{t} \cdot \frac{3}{2}\right) \cdot \frac{\frac{x}{z}}{y \cdot 27}\right) \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{3}}}}\right)\]

  15. Final simplification0.3

    \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\sqrt{\cos^{-1} \left(\left(\sqrt{t} \cdot \frac{3}{2}\right) \cdot \frac{\frac{x}{z}}{y \cdot 27}\right) \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{3}}} \cdot \sqrt{\cos^{-1} \left(\left(\sqrt{t} \cdot \frac{3}{2}\right) \cdot \frac{\frac{x}{z}}{y \cdot 27}\right) \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{3}}}\right)\]

Reproduce

herbie shell --seed 2019194 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, D"

  :herbie-target
  (/ (acos (* (/ (/ x 27.0) (* y z)) (/ (sqrt t) (/ 2.0 3.0)))) 3.0)

  (* (/ 1.0 3.0) (acos (* (/ (* 3.0 (/ x (* y 27.0))) (* z 2.0)) (sqrt t)))))