Average Error: 1.3 → 0.4
Time: 22.2s
Precision: 64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\frac{\frac{x}{27 \cdot y} \cdot 3}{2 \cdot z} \cdot \sqrt{t}\right)\right) \cdot \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\]
\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)
↓
\left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\frac{\frac{x}{27 \cdot y} \cdot 3}{2 \cdot z} \cdot \sqrt{t}\right)\right) \cdot \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}
double f(double x, double y, double z, double t) {
        double r38127942 = 1.0;
        double r38127943 = 3.0;
        double r38127944 = r38127942 / r38127943;
        double r38127945 = x;
        double r38127946 = y;
        double r38127947 = 27.0;
        double r38127948 = r38127946 * r38127947;
        double r38127949 = r38127945 / r38127948;
        double r38127950 = r38127943 * r38127949;
        double r38127951 = z;
        double r38127952 = 2.0;
        double r38127953 = r38127951 * r38127952;
        double r38127954 = r38127950 / r38127953;
        double r38127955 = t;
        double r38127956 = sqrt(r38127955);
        double r38127957 = r38127954 * r38127956;
        double r38127958 = acos(r38127957);
        double r38127959 = r38127944 * r38127958;
        return r38127959;
}
↓
double f(double x, double y, double z, double t) {
        double r38127960 = 1.0;
        double r38127961 = 3.0;
        double r38127962 = cbrt(r38127961);
        double r38127963 = r38127960 / r38127962;
        double r38127964 = x;
        double r38127965 = 27.0;
        double r38127966 = y;
        double r38127967 = r38127965 * r38127966;
        double r38127968 = r38127964 / r38127967;
        double r38127969 = r38127968 * r38127961;
        double r38127970 = 2.0;
        double r38127971 = z;
        double r38127972 = r38127970 * r38127971;
        double r38127973 = r38127969 / r38127972;
        double r38127974 = t;
        double r38127975 = sqrt(r38127974);
        double r38127976 = r38127973 * r38127975;
        double r38127977 = acos(r38127976);
        double r38127978 = r38127963 * r38127977;
        double r38127979 = 1.0;
        double r38127980 = r38127962 * r38127962;
        double r38127981 = r38127979 / r38127980;
        double r38127982 = r38127978 * r38127981;
        return r38127982;
}

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
Target1.3
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 1.3

    \[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
  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-frac0.4

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

  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. Final simplification0.4

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

Reproduce

herbie shell --seed 2019169 +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)))))