Average Error: 1.3 → 0.3
Time: 27.6s
Precision: 64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(1 \cdot \left(\frac{\frac{\pi}{2}}{\sqrt[3]{3}} - \frac{\sin^{-1} \left(\frac{\frac{\frac{x \cdot 3}{27 \cdot y}}{z} \cdot \sqrt{t}}{2}\right)}{\sqrt[3]{3}}\right)\right)\]
\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(1 \cdot \left(\frac{\frac{\pi}{2}}{\sqrt[3]{3}} - \frac{\sin^{-1} \left(\frac{\frac{\frac{x \cdot 3}{27 \cdot y}}{z} \cdot \sqrt{t}}{2}\right)}{\sqrt[3]{3}}\right)\right)
double f(double x, double y, double z, double t) {
        double r32851894 = 1.0;
        double r32851895 = 3.0;
        double r32851896 = r32851894 / r32851895;
        double r32851897 = x;
        double r32851898 = y;
        double r32851899 = 27.0;
        double r32851900 = r32851898 * r32851899;
        double r32851901 = r32851897 / r32851900;
        double r32851902 = r32851895 * r32851901;
        double r32851903 = z;
        double r32851904 = 2.0;
        double r32851905 = r32851903 * r32851904;
        double r32851906 = r32851902 / r32851905;
        double r32851907 = t;
        double r32851908 = sqrt(r32851907);
        double r32851909 = r32851906 * r32851908;
        double r32851910 = acos(r32851909);
        double r32851911 = r32851896 * r32851910;
        return r32851911;
}
↓
double f(double x, double y, double z, double t) {
        double r32851912 = 1.0;
        double r32851913 = 3.0;
        double r32851914 = cbrt(r32851913);
        double r32851915 = r32851914 * r32851914;
        double r32851916 = r32851912 / r32851915;
        double r32851917 = 1.0;
        double r32851918 = atan2(1.0, 0.0);
        double r32851919 = 2.0;
        double r32851920 = r32851918 / r32851919;
        double r32851921 = r32851920 / r32851914;
        double r32851922 = x;
        double r32851923 = r32851922 * r32851913;
        double r32851924 = 27.0;
        double r32851925 = y;
        double r32851926 = r32851924 * r32851925;
        double r32851927 = r32851923 / r32851926;
        double r32851928 = z;
        double r32851929 = r32851927 / r32851928;
        double r32851930 = t;
        double r32851931 = sqrt(r32851930);
        double r32851932 = r32851929 * r32851931;
        double r32851933 = 2.0;
        double r32851934 = r32851932 / r32851933;
        double r32851935 = asin(r32851934);
        double r32851936 = r32851935 / r32851914;
        double r32851937 = r32851921 - r32851936;
        double r32851938 = r32851917 * r32851937;
        double r32851939 = r32851916 * r32851938;
        return r32851939;
}

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 *-un-lft-identity1.3

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

  5. Applied times-frac0.3

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

  6. Applied associate-*l*0.3

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

  8. Applied div-inv0.3

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

  9. Applied associate-*l*0.3

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

  10. Simplified0.3

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

  12. Applied acos-asin0.3

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

  13. Applied div-sub0.3

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

  14. Final simplification0.3

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

Reproduce

herbie shell --seed 2019172 +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)))))