Average Error: 1.4 → 0.4
Time: 19.9s
Precision: 64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\frac{1 \cdot \cos^{-1} \left(\frac{\frac{\sqrt{t} \cdot \left(3 \cdot \frac{\frac{x}{y}}{27}\right)}{2}}{z}\right)}{\sqrt[3]{3}} \cdot \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\]
\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)
↓
\frac{1 \cdot \cos^{-1} \left(\frac{\frac{\sqrt{t} \cdot \left(3 \cdot \frac{\frac{x}{y}}{27}\right)}{2}}{z}\right)}{\sqrt[3]{3}} \cdot \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}
double f(double x, double y, double z, double t) {
        double r653874 = 1.0;
        double r653875 = 3.0;
        double r653876 = r653874 / r653875;
        double r653877 = x;
        double r653878 = y;
        double r653879 = 27.0;
        double r653880 = r653878 * r653879;
        double r653881 = r653877 / r653880;
        double r653882 = r653875 * r653881;
        double r653883 = z;
        double r653884 = 2.0;
        double r653885 = r653883 * r653884;
        double r653886 = r653882 / r653885;
        double r653887 = t;
        double r653888 = sqrt(r653887);
        double r653889 = r653886 * r653888;
        double r653890 = acos(r653889);
        double r653891 = r653876 * r653890;
        return r653891;
}
↓
double f(double x, double y, double z, double t) {
        double r653892 = 1.0;
        double r653893 = t;
        double r653894 = sqrt(r653893);
        double r653895 = 3.0;
        double r653896 = x;
        double r653897 = y;
        double r653898 = r653896 / r653897;
        double r653899 = 27.0;
        double r653900 = r653898 / r653899;
        double r653901 = r653895 * r653900;
        double r653902 = r653894 * r653901;
        double r653903 = 2.0;
        double r653904 = r653902 / r653903;
        double r653905 = z;
        double r653906 = r653904 / r653905;
        double r653907 = acos(r653906);
        double r653908 = r653892 * r653907;
        double r653909 = cbrt(r653895);
        double r653910 = r653908 / r653909;
        double r653911 = 1.0;
        double r653912 = r653909 * r653909;
        double r653913 = r653911 / r653912;
        double r653914 = r653910 * r653913;
        return r653914;
}

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

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 *-un-lft-identity1.4

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

  5. Applied times-frac0.4

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

  6. Applied associate-*l*0.4

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

  7. Simplified0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2019196 
(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)))))