Average Error: 1.3 → 0.3
Time: 25.8s
Precision: 64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\frac{\sqrt{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\cos^{-1} \left(0.055555555555555552 \cdot \left(\sqrt{t} \cdot \frac{x}{z \cdot y}\right)\right) \cdot \sqrt{1}}{\sqrt[3]{3}}\]
\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)
↓
\frac{\sqrt{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\cos^{-1} \left(0.055555555555555552 \cdot \left(\sqrt{t} \cdot \frac{x}{z \cdot y}\right)\right) \cdot \sqrt{1}}{\sqrt[3]{3}}
double f(double x, double y, double z, double t) {
        double r535956 = 1.0;
        double r535957 = 3.0;
        double r535958 = r535956 / r535957;
        double r535959 = x;
        double r535960 = y;
        double r535961 = 27.0;
        double r535962 = r535960 * r535961;
        double r535963 = r535959 / r535962;
        double r535964 = r535957 * r535963;
        double r535965 = z;
        double r535966 = 2.0;
        double r535967 = r535965 * r535966;
        double r535968 = r535964 / r535967;
        double r535969 = t;
        double r535970 = sqrt(r535969);
        double r535971 = r535968 * r535970;
        double r535972 = acos(r535971);
        double r535973 = r535958 * r535972;
        return r535973;
}
↓
double f(double x, double y, double z, double t) {
        double r535974 = 1.0;
        double r535975 = sqrt(r535974);
        double r535976 = 3.0;
        double r535977 = cbrt(r535976);
        double r535978 = r535977 * r535977;
        double r535979 = r535975 / r535978;
        double r535980 = 0.05555555555555555;
        double r535981 = t;
        double r535982 = sqrt(r535981);
        double r535983 = x;
        double r535984 = z;
        double r535985 = y;
        double r535986 = r535984 * r535985;
        double r535987 = r535983 / r535986;
        double r535988 = r535982 * r535987;
        double r535989 = r535980 * r535988;
        double r535990 = acos(r535989);
        double r535991 = r535990 * r535975;
        double r535992 = r535991 / r535977;
        double r535993 = r535979 * r535992;
        return r535993;
}

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
Target1.2
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 add-sqr-sqrt1.3

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

  5. Applied times-frac0.3

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

  6. Applied associate-*l*0.3

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

  7. Taylor expanded around 0 0.3

    \[\leadsto \frac{\sqrt{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \color{blue}{\frac{\cos^{-1} \left(0.055555555555555552 \cdot \left(\sqrt{t} \cdot \frac{x}{z \cdot y}\right)\right) \cdot \sqrt{1}}{\sqrt[3]{3}}}\]

  8. Final simplification0.3

    \[\leadsto \frac{\sqrt{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\cos^{-1} \left(0.055555555555555552 \cdot \left(\sqrt{t} \cdot \frac{x}{z \cdot y}\right)\right) \cdot \sqrt{1}}{\sqrt[3]{3}}\]

Reproduce

herbie shell --seed 2019198 +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)))))