Average Error: 1.4 → 0.4
Time: 23.8s
Precision: 64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\left(\left(\frac{x}{y} \cdot \frac{\frac{3}{27}}{2}\right) \cdot \frac{\sqrt{t}}{\sqrt[3]{z}}\right) \cdot \frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right)\right)\]
\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(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\left(\left(\frac{x}{y} \cdot \frac{\frac{3}{27}}{2}\right) \cdot \frac{\sqrt{t}}{\sqrt[3]{z}}\right) \cdot \frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right)\right)
double f(double x, double y, double z, double t) {
        double r710958 = 1.0;
        double r710959 = 3.0;
        double r710960 = r710958 / r710959;
        double r710961 = x;
        double r710962 = y;
        double r710963 = 27.0;
        double r710964 = r710962 * r710963;
        double r710965 = r710961 / r710964;
        double r710966 = r710959 * r710965;
        double r710967 = z;
        double r710968 = 2.0;
        double r710969 = r710967 * r710968;
        double r710970 = r710966 / r710969;
        double r710971 = t;
        double r710972 = sqrt(r710971);
        double r710973 = r710970 * r710972;
        double r710974 = acos(r710973);
        double r710975 = r710960 * r710974;
        return r710975;
}
↓
double f(double x, double y, double z, double t) {
        double r710976 = 1.0;
        double r710977 = 3.0;
        double r710978 = cbrt(r710977);
        double r710979 = r710978 * r710978;
        double r710980 = r710976 / r710979;
        double r710981 = 1.0;
        double r710982 = r710981 / r710978;
        double r710983 = x;
        double r710984 = y;
        double r710985 = r710983 / r710984;
        double r710986 = 27.0;
        double r710987 = r710977 / r710986;
        double r710988 = 2.0;
        double r710989 = r710987 / r710988;
        double r710990 = r710985 * r710989;
        double r710991 = t;
        double r710992 = sqrt(r710991);
        double r710993 = z;
        double r710994 = cbrt(r710993);
        double r710995 = r710992 / r710994;
        double r710996 = r710990 * r710995;
        double r710997 = r710994 * r710994;
        double r710998 = r710976 / r710997;
        double r710999 = r710996 * r710998;
        double r711000 = acos(r710999);
        double r711001 = r710982 * r711000;
        double r711002 = r710980 * r711001;
        return r711002;
}

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

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. Simplified1.2

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

  9. Applied add-cube-cbrt1.2

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

  10. Applied *-un-lft-identity1.2

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

  11. Applied sqrt-prod1.2

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

  12. Applied times-frac1.2

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

  13. Applied associate-*l*0.4

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

  14. Simplified0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2019196 +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)))))