Average Error: 1.4 → 0.5
Time: 22.7s
Precision: 64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\frac{\sqrt{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\sqrt{\frac{\sqrt{1}}{\sqrt[3]{3}}} \cdot \left(\sqrt{\frac{\sqrt{1}}{\sqrt[3]{3}}} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\right)\right)\]
\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)
↓
\frac{\sqrt{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\sqrt{\frac{\sqrt{1}}{\sqrt[3]{3}}} \cdot \left(\sqrt{\frac{\sqrt{1}}{\sqrt[3]{3}}} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\right)\right)
double f(double x, double y, double z, double t) {
        double r493841 = 1.0;
        double r493842 = 3.0;
        double r493843 = r493841 / r493842;
        double r493844 = x;
        double r493845 = y;
        double r493846 = 27.0;
        double r493847 = r493845 * r493846;
        double r493848 = r493844 / r493847;
        double r493849 = r493842 * r493848;
        double r493850 = z;
        double r493851 = 2.0;
        double r493852 = r493850 * r493851;
        double r493853 = r493849 / r493852;
        double r493854 = t;
        double r493855 = sqrt(r493854);
        double r493856 = r493853 * r493855;
        double r493857 = acos(r493856);
        double r493858 = r493843 * r493857;
        return r493858;
}
↓
double f(double x, double y, double z, double t) {
        double r493859 = 1.0;
        double r493860 = sqrt(r493859);
        double r493861 = 3.0;
        double r493862 = cbrt(r493861);
        double r493863 = r493862 * r493862;
        double r493864 = r493860 / r493863;
        double r493865 = r493860 / r493862;
        double r493866 = sqrt(r493865);
        double r493867 = x;
        double r493868 = y;
        double r493869 = 27.0;
        double r493870 = r493868 * r493869;
        double r493871 = r493867 / r493870;
        double r493872 = r493861 * r493871;
        double r493873 = z;
        double r493874 = 2.0;
        double r493875 = r493873 * r493874;
        double r493876 = r493872 / r493875;
        double r493877 = t;
        double r493878 = sqrt(r493877);
        double r493879 = r493876 * r493878;
        double r493880 = acos(r493879);
        double r493881 = r493866 * r493880;
        double r493882 = r493866 * r493881;
        double r493883 = r493864 * r493882;
        return r493883;
}

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

Original1.4
Target1.2
Herbie0.5
\[\frac{\cos^{-1} \left(\frac{\frac{x}{27}}{y \cdot z} \cdot \frac{\sqrt{t}}{\frac{2}{3}}\right)}{3}\]

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-sqr-sqrt1.4

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

  5. Applied times-frac0.5

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

  6. Applied associate-*l*0.4

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

  8. Applied add-sqr-sqrt1.4

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

  9. Applied associate-*l*0.5

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

  10. Final simplification0.5

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, D"
  :precision binary64

  :herbie-target
  (/ (acos (* (/ (/ x 27) (* y z)) (/ (sqrt t) (/ 2 3)))) 3)

  (* (/ 1 3) (acos (* (/ (* 3 (/ x (* y 27))) (* z 2)) (sqrt t)))))