Average Error: 1.3 → 0.3
Time: 17.4s
Precision: 64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\cos^{-1} \left(0.05555555555555555247160270937456516548991 \cdot \left(\sqrt{t} \cdot \frac{x}{z \cdot y}\right)\right)}{\sqrt[3]{3}}\]
\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)
↓
\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\cos^{-1} \left(0.05555555555555555247160270937456516548991 \cdot \left(\sqrt{t} \cdot \frac{x}{z \cdot y}\right)\right)}{\sqrt[3]{3}}
double f(double x, double y, double z, double t) {
        double r570868 = 1.0;
        double r570869 = 3.0;
        double r570870 = r570868 / r570869;
        double r570871 = x;
        double r570872 = y;
        double r570873 = 27.0;
        double r570874 = r570872 * r570873;
        double r570875 = r570871 / r570874;
        double r570876 = r570869 * r570875;
        double r570877 = z;
        double r570878 = 2.0;
        double r570879 = r570877 * r570878;
        double r570880 = r570876 / r570879;
        double r570881 = t;
        double r570882 = sqrt(r570881);
        double r570883 = r570880 * r570882;
        double r570884 = acos(r570883);
        double r570885 = r570870 * r570884;
        return r570885;
}
↓
double f(double x, double y, double z, double t) {
        double r570886 = 1.0;
        double r570887 = 3.0;
        double r570888 = cbrt(r570887);
        double r570889 = r570888 * r570888;
        double r570890 = r570886 / r570889;
        double r570891 = 0.05555555555555555;
        double r570892 = t;
        double r570893 = sqrt(r570892);
        double r570894 = x;
        double r570895 = z;
        double r570896 = y;
        double r570897 = r570895 * r570896;
        double r570898 = r570894 / r570897;
        double r570899 = r570893 * r570898;
        double r570900 = r570891 * r570899;
        double r570901 = acos(r570900);
        double r570902 = r570901 / r570888;
        double r570903 = r570890 * r570902;
        return r570903;
}

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-frac0.4

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

  6. Applied associate-*l*0.4

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

  7. Taylor expanded around 0 0.3

    \[\leadsto \frac{\sqrt{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \color{blue}{\frac{\cos^{-1} \left(0.05555555555555555247160270937456516548991 \cdot \left(\sqrt{t} \cdot \frac{x}{z \cdot y}\right)\right) \cdot \sqrt{1}}{\sqrt[3]{3}}}\]
  8. Using strategy rm

  9. Applied add-cube-cbrt0.3

    \[\leadsto \frac{\sqrt{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\cos^{-1} \left(0.05555555555555555247160270937456516548991 \cdot \left(\sqrt{t} \cdot \frac{x}{z \cdot y}\right)\right) \cdot \sqrt{1}}{\color{blue}{\left(\sqrt[3]{\sqrt[3]{3}} \cdot \sqrt[3]{\sqrt[3]{3}}\right) \cdot \sqrt[3]{\sqrt[3]{3}}}}\]

  10. Applied associate-/r*0.3

    \[\leadsto \frac{\sqrt{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \color{blue}{\frac{\frac{\cos^{-1} \left(0.05555555555555555247160270937456516548991 \cdot \left(\sqrt{t} \cdot \frac{x}{z \cdot y}\right)\right) \cdot \sqrt{1}}{\sqrt[3]{\sqrt[3]{3}} \cdot \sqrt[3]{\sqrt[3]{3}}}}{\sqrt[3]{\sqrt[3]{3}}}}\]

  11. Final simplification0.3

    \[\leadsto \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\cos^{-1} \left(0.05555555555555555247160270937456516548991 \cdot \left(\sqrt{t} \cdot \frac{x}{z \cdot y}\right)\right)}{\sqrt[3]{3}}\]

Reproduce

herbie shell --seed 2019297 
(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)))))