Average Error: 1.4 → 0.4
Time: 6.2s
Precision: 64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\left(\sqrt[3]{\frac{1}{\sqrt[3]{3}}} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{3}}}\right) \cdot \left(\sqrt[3]{\frac{1}{\sqrt[3]{3}}} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\right)\right)\]
\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 \left(\left(\sqrt[3]{\frac{1}{\sqrt[3]{3}}} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{3}}}\right) \cdot \left(\sqrt[3]{\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)\right)
double f(double x, double y, double z, double t) {
        double r1033904 = 1.0;
        double r1033905 = 3.0;
        double r1033906 = r1033904 / r1033905;
        double r1033907 = x;
        double r1033908 = y;
        double r1033909 = 27.0;
        double r1033910 = r1033908 * r1033909;
        double r1033911 = r1033907 / r1033910;
        double r1033912 = r1033905 * r1033911;
        double r1033913 = z;
        double r1033914 = 2.0;
        double r1033915 = r1033913 * r1033914;
        double r1033916 = r1033912 / r1033915;
        double r1033917 = t;
        double r1033918 = sqrt(r1033917);
        double r1033919 = r1033916 * r1033918;
        double r1033920 = acos(r1033919);
        double r1033921 = r1033906 * r1033920;
        return r1033921;
}
↓
double f(double x, double y, double z, double t) {
        double r1033922 = 1.0;
        double r1033923 = 3.0;
        double r1033924 = cbrt(r1033923);
        double r1033925 = r1033924 * r1033924;
        double r1033926 = r1033922 / r1033925;
        double r1033927 = 1.0;
        double r1033928 = r1033927 / r1033924;
        double r1033929 = cbrt(r1033928);
        double r1033930 = r1033929 * r1033929;
        double r1033931 = x;
        double r1033932 = y;
        double r1033933 = 27.0;
        double r1033934 = r1033932 * r1033933;
        double r1033935 = r1033931 / r1033934;
        double r1033936 = r1033923 * r1033935;
        double r1033937 = z;
        double r1033938 = 2.0;
        double r1033939 = r1033937 * r1033938;
        double r1033940 = r1033936 / r1033939;
        double r1033941 = t;
        double r1033942 = sqrt(r1033941);
        double r1033943 = r1033940 * r1033942;
        double r1033944 = acos(r1033943);
        double r1033945 = r1033929 * r1033944;
        double r1033946 = r1033930 * r1033945;
        double r1033947 = r1033926 * r1033946;
        return r1033947;
}

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. Using strategy rm

  8. Applied add-cube-cbrt1.4

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

  9. Applied associate-*l*0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019362 +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)))))