Average Error: 1.3 → 0.2
Time: 8.5s
Precision: 64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\cos^{-1} \left(\log \left(e^{0.055555555555555552 \cdot \left(\sqrt{t} \cdot \frac{x}{z \cdot y}\right)}\right)\right) \cdot \sqrt[3]{1}}{\sqrt[3]{3}}\]
\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)
↓
\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\cos^{-1} \left(\log \left(e^{0.055555555555555552 \cdot \left(\sqrt{t} \cdot \frac{x}{z \cdot y}\right)}\right)\right) \cdot \sqrt[3]{1}}{\sqrt[3]{3}}
double f(double x, double y, double z, double t) {
        double r727693 = 1.0;
        double r727694 = 3.0;
        double r727695 = r727693 / r727694;
        double r727696 = x;
        double r727697 = y;
        double r727698 = 27.0;
        double r727699 = r727697 * r727698;
        double r727700 = r727696 / r727699;
        double r727701 = r727694 * r727700;
        double r727702 = z;
        double r727703 = 2.0;
        double r727704 = r727702 * r727703;
        double r727705 = r727701 / r727704;
        double r727706 = t;
        double r727707 = sqrt(r727706);
        double r727708 = r727705 * r727707;
        double r727709 = acos(r727708);
        double r727710 = r727695 * r727709;
        return r727710;
}
↓
double f(double x, double y, double z, double t) {
        double r727711 = 1.0;
        double r727712 = cbrt(r727711);
        double r727713 = r727712 * r727712;
        double r727714 = 3.0;
        double r727715 = cbrt(r727714);
        double r727716 = r727715 * r727715;
        double r727717 = r727713 / r727716;
        double r727718 = 0.05555555555555555;
        double r727719 = t;
        double r727720 = sqrt(r727719);
        double r727721 = x;
        double r727722 = z;
        double r727723 = y;
        double r727724 = r727722 * r727723;
        double r727725 = r727721 / r727724;
        double r727726 = r727720 * r727725;
        double r727727 = r727718 * r727726;
        double r727728 = exp(r727727);
        double r727729 = log(r727728);
        double r727730 = acos(r727729);
        double r727731 = r727730 * r727712;
        double r727732 = r727731 / r727715;
        double r727733 = r727717 * r727732;
        return r727733;
}

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

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-cube-cbrt1.3

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

  5. Applied times-frac0.3

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

  6. Applied associate-*l*0.3

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

  7. Taylor expanded around 0 0.3

    \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \color{blue}{\frac{\cos^{-1} \left(0.055555555555555552 \cdot \left(\sqrt{t} \cdot \frac{x}{z \cdot y}\right)\right) \cdot \sqrt[3]{1}}{\sqrt[3]{3}}}\]
  8. Using strategy rm

  9. Applied add-log-exp0.2

    \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\cos^{-1} \color{blue}{\left(\log \left(e^{0.055555555555555552 \cdot \left(\sqrt{t} \cdot \frac{x}{z \cdot y}\right)}\right)\right)} \cdot \sqrt[3]{1}}{\sqrt[3]{3}}\]

  10. Final simplification0.2

    \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\cos^{-1} \left(\log \left(e^{0.055555555555555552 \cdot \left(\sqrt{t} \cdot \frac{x}{z \cdot y}\right)}\right)\right) \cdot \sqrt[3]{1}}{\sqrt[3]{3}}\]

Reproduce

herbie shell --seed 2020081 +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)))))