Average Error: 1.4 → 0.2
Time: 24.8s
Precision: 64
\[\frac{1.0}{3.0} \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)\]
↓
\[\frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \left(\frac{1.0}{\sqrt[3]{3.0}} \cdot \cos^{-1} \left(\left(\frac{x}{z \cdot y} \cdot 0.05555555555555555\right) \cdot \sqrt{t}\right)\right)\]
\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 \left(\frac{1.0}{\sqrt[3]{3.0}} \cdot \cos^{-1} \left(\left(\frac{x}{z \cdot y} \cdot 0.05555555555555555\right) \cdot \sqrt{t}\right)\right)
double f(double x, double y, double z, double t) {
        double r36886902 = 1.0;
        double r36886903 = 3.0;
        double r36886904 = r36886902 / r36886903;
        double r36886905 = x;
        double r36886906 = y;
        double r36886907 = 27.0;
        double r36886908 = r36886906 * r36886907;
        double r36886909 = r36886905 / r36886908;
        double r36886910 = r36886903 * r36886909;
        double r36886911 = z;
        double r36886912 = 2.0;
        double r36886913 = r36886911 * r36886912;
        double r36886914 = r36886910 / r36886913;
        double r36886915 = t;
        double r36886916 = sqrt(r36886915);
        double r36886917 = r36886914 * r36886916;
        double r36886918 = acos(r36886917);
        double r36886919 = r36886904 * r36886918;
        return r36886919;
}
↓
double f(double x, double y, double z, double t) {
        double r36886920 = 1.0;
        double r36886921 = 3.0;
        double r36886922 = cbrt(r36886921);
        double r36886923 = r36886922 * r36886922;
        double r36886924 = r36886920 / r36886923;
        double r36886925 = 1.0;
        double r36886926 = r36886925 / r36886922;
        double r36886927 = x;
        double r36886928 = z;
        double r36886929 = y;
        double r36886930 = r36886928 * r36886929;
        double r36886931 = r36886927 / r36886930;
        double r36886932 = 0.05555555555555555;
        double r36886933 = r36886931 * r36886932;
        double r36886934 = t;
        double r36886935 = sqrt(r36886934);
        double r36886936 = r36886933 * r36886935;
        double r36886937 = acos(r36886936);
        double r36886938 = r36886926 * r36886937;
        double r36886939 = r36886924 * r36886938;
        return r36886939;
}

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
Herbie0.2
\[\frac{\cos^{-1} \left(\frac{\frac{x}{27.0}}{y \cdot z} \cdot \frac{\sqrt{t}}{\frac{2.0}{3.0}}\right)}{3.0}\]

Derivation

  1. Initial program 1.4

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

  3. Applied add-cube-cbrt1.4

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

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

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

  5. Applied times-frac0.4

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

  6. Applied associate-*l*0.4

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

  7. Taylor expanded around 0 0.2

    \[\leadsto \frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \left(\frac{1.0}{\sqrt[3]{3.0}} \cdot \cos^{-1} \left(\color{blue}{\left(0.05555555555555555 \cdot \frac{x}{z \cdot y}\right)} \cdot \sqrt{t}\right)\right)\]

  8. Final simplification0.2

    \[\leadsto \frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \left(\frac{1.0}{\sqrt[3]{3.0}} \cdot \cos^{-1} \left(\left(\frac{x}{z \cdot y} \cdot 0.05555555555555555\right) \cdot \sqrt{t}\right)\right)\]

Reproduce

herbie shell --seed 2019164 +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)))))