Average Error: 1.3 → 0.3
Time: 27.7s
Precision: 64
\[\frac{1.0}{3.0} \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)\]
↓
\[\frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \sqrt[3]{\frac{1.0 \cdot \left(1.0 \cdot 1.0\right)}{3.0} \cdot \left(\left(\cos^{-1} \left(\sqrt{t} \cdot \frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0}\right) \cdot \cos^{-1} \left(\sqrt{t} \cdot \frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0}\right)\right) \cdot \cos^{-1} \left(\sqrt{t} \cdot \frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0}\right)\right)}\]
\frac{1.0}{3.0} \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)
↓
\frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \sqrt[3]{\frac{1.0 \cdot \left(1.0 \cdot 1.0\right)}{3.0} \cdot \left(\left(\cos^{-1} \left(\sqrt{t} \cdot \frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0}\right) \cdot \cos^{-1} \left(\sqrt{t} \cdot \frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0}\right)\right) \cdot \cos^{-1} \left(\sqrt{t} \cdot \frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0}\right)\right)}
double f(double x, double y, double z, double t) {
        double r35180792 = 1.0;
        double r35180793 = 3.0;
        double r35180794 = r35180792 / r35180793;
        double r35180795 = x;
        double r35180796 = y;
        double r35180797 = 27.0;
        double r35180798 = r35180796 * r35180797;
        double r35180799 = r35180795 / r35180798;
        double r35180800 = r35180793 * r35180799;
        double r35180801 = z;
        double r35180802 = 2.0;
        double r35180803 = r35180801 * r35180802;
        double r35180804 = r35180800 / r35180803;
        double r35180805 = t;
        double r35180806 = sqrt(r35180805);
        double r35180807 = r35180804 * r35180806;
        double r35180808 = acos(r35180807);
        double r35180809 = r35180794 * r35180808;
        return r35180809;
}
↓
double f(double x, double y, double z, double t) {
        double r35180810 = 1.0;
        double r35180811 = 3.0;
        double r35180812 = cbrt(r35180811);
        double r35180813 = r35180812 * r35180812;
        double r35180814 = r35180810 / r35180813;
        double r35180815 = 1.0;
        double r35180816 = r35180815 * r35180815;
        double r35180817 = r35180815 * r35180816;
        double r35180818 = r35180817 / r35180811;
        double r35180819 = t;
        double r35180820 = sqrt(r35180819);
        double r35180821 = x;
        double r35180822 = y;
        double r35180823 = 27.0;
        double r35180824 = r35180822 * r35180823;
        double r35180825 = r35180821 / r35180824;
        double r35180826 = r35180811 * r35180825;
        double r35180827 = z;
        double r35180828 = 2.0;
        double r35180829 = r35180827 * r35180828;
        double r35180830 = r35180826 / r35180829;
        double r35180831 = r35180820 * r35180830;
        double r35180832 = acos(r35180831);
        double r35180833 = r35180832 * r35180832;
        double r35180834 = r35180833 * r35180832;
        double r35180835 = r35180818 * r35180834;
        double r35180836 = cbrt(r35180835);
        double r35180837 = r35180814 * r35180836;
        return r35180837;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original1.3
Target1.3
Herbie0.3
\[\frac{\cos^{-1} \left(\frac{\frac{x}{27.0}}{y \cdot z} \cdot \frac{\sqrt{t}}{\frac{2.0}{3.0}}\right)}{3.0}\]

Derivation

  1. Initial program 1.3

    \[\frac{1.0}{3.0} \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)\]
  2. Using strategy rm

  3. Applied add-cube-cbrt1.3

    \[\leadsto \frac{1.0}{\color{blue}{\left(\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}\right) \cdot \sqrt[3]{3.0}}} \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)\]

  4. Applied *-un-lft-identity1.3

    \[\leadsto \frac{\color{blue}{1 \cdot 1.0}}{\left(\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}\right) \cdot \sqrt[3]{3.0}} \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)\]

  5. Applied times-frac0.3

    \[\leadsto \color{blue}{\left(\frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \frac{1.0}{\sqrt[3]{3.0}}\right)} \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)\]

  6. Applied associate-*l*0.3

    \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \left(\frac{1.0}{\sqrt[3]{3.0}} \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)\right)}\]
  7. Using strategy rm

  8. Applied add-cbrt-cube1.3

    \[\leadsto \frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \left(\frac{1.0}{\sqrt[3]{3.0}} \cdot \color{blue}{\sqrt[3]{\left(\cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right) \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)\right) \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)}}\right)\]

  9. Applied add-cbrt-cube1.3

    \[\leadsto \frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \left(\frac{\color{blue}{\sqrt[3]{\left(1.0 \cdot 1.0\right) \cdot 1.0}}}{\sqrt[3]{3.0}} \cdot \sqrt[3]{\left(\cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right) \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)\right) \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)}\right)\]

  10. Applied cbrt-undiv0.3

    \[\leadsto \frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \left(\color{blue}{\sqrt[3]{\frac{\left(1.0 \cdot 1.0\right) \cdot 1.0}{3.0}}} \cdot \sqrt[3]{\left(\cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right) \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)\right) \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)}\right)\]

  11. Applied cbrt-unprod0.3

    \[\leadsto \frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \color{blue}{\sqrt[3]{\frac{\left(1.0 \cdot 1.0\right) \cdot 1.0}{3.0} \cdot \left(\left(\cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right) \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)\right) \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)\right)}}\]

  12. Final simplification0.3

    \[\leadsto \frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \sqrt[3]{\frac{1.0 \cdot \left(1.0 \cdot 1.0\right)}{3.0} \cdot \left(\left(\cos^{-1} \left(\sqrt{t} \cdot \frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0}\right) \cdot \cos^{-1} \left(\sqrt{t} \cdot \frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0}\right)\right) \cdot \cos^{-1} \left(\sqrt{t} \cdot \frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0}\right)\right)}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(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)))))