Average Error: 1.3 → 0.3
Time: 50.7s
Precision: 64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \sqrt[3]{\frac{1 \cdot \left(1 \cdot 1\right)}{3} \cdot \left(\left(\cos^{-1} \left(\sqrt{t} \cdot \frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2}\right) \cdot \cos^{-1} \left(\sqrt{t} \cdot \frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2}\right)\right) \cdot \cos^{-1} \left(\sqrt{t} \cdot \frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2}\right)\right)}\]
\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)
↓
\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \sqrt[3]{\frac{1 \cdot \left(1 \cdot 1\right)}{3} \cdot \left(\left(\cos^{-1} \left(\sqrt{t} \cdot \frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2}\right) \cdot \cos^{-1} \left(\sqrt{t} \cdot \frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2}\right)\right) \cdot \cos^{-1} \left(\sqrt{t} \cdot \frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2}\right)\right)}
double f(double x, double y, double z, double t) {
        double r23363728 = 1.0;
        double r23363729 = 3.0;
        double r23363730 = r23363728 / r23363729;
        double r23363731 = x;
        double r23363732 = y;
        double r23363733 = 27.0;
        double r23363734 = r23363732 * r23363733;
        double r23363735 = r23363731 / r23363734;
        double r23363736 = r23363729 * r23363735;
        double r23363737 = z;
        double r23363738 = 2.0;
        double r23363739 = r23363737 * r23363738;
        double r23363740 = r23363736 / r23363739;
        double r23363741 = t;
        double r23363742 = sqrt(r23363741);
        double r23363743 = r23363740 * r23363742;
        double r23363744 = acos(r23363743);
        double r23363745 = r23363730 * r23363744;
        return r23363745;
}
↓
double f(double x, double y, double z, double t) {
        double r23363746 = 1.0;
        double r23363747 = 3.0;
        double r23363748 = cbrt(r23363747);
        double r23363749 = r23363748 * r23363748;
        double r23363750 = r23363746 / r23363749;
        double r23363751 = 1.0;
        double r23363752 = r23363751 * r23363751;
        double r23363753 = r23363751 * r23363752;
        double r23363754 = r23363753 / r23363747;
        double r23363755 = t;
        double r23363756 = sqrt(r23363755);
        double r23363757 = x;
        double r23363758 = y;
        double r23363759 = 27.0;
        double r23363760 = r23363758 * r23363759;
        double r23363761 = r23363757 / r23363760;
        double r23363762 = r23363747 * r23363761;
        double r23363763 = z;
        double r23363764 = 2.0;
        double r23363765 = r23363763 * r23363764;
        double r23363766 = r23363762 / r23363765;
        double r23363767 = r23363756 * r23363766;
        double r23363768 = acos(r23363767);
        double r23363769 = r23363768 * r23363768;
        double r23363770 = r23363769 * r23363768;
        double r23363771 = r23363754 * r23363770;
        double r23363772 = cbrt(r23363771);
        double r23363773 = r23363750 * r23363772;
        return r23363773;
}

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

Original1.3
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 1.3

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

  3. Applied add-cube-cbrt1.3

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

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

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

  5. Applied times-frac0.3

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

  6. Applied associate-*l*0.3

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

  8. Applied add-cbrt-cube1.3

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

  9. Applied add-cbrt-cube1.3

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

  10. Applied cbrt-undiv0.3

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

  11. Applied cbrt-unprod0.3

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

  12. Final simplification0.3

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

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)))))