Average Error: 1.3 → 0.4
Time: 25.4s
Precision: 64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\sqrt[3]{1}}{\frac{\sqrt[3]{3}}{\cos^{-1} \left(\left(\frac{\frac{x}{z}}{y} \cdot 0.05555555555555555247160270937456516548991\right) \cdot \sqrt{t}\right)}}\]
\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{\sqrt[3]{1}}{\frac{\sqrt[3]{3}}{\cos^{-1} \left(\left(\frac{\frac{x}{z}}{y} \cdot 0.05555555555555555247160270937456516548991\right) \cdot \sqrt{t}\right)}}
double f(double x, double y, double z, double t) {
        double r719626 = 1.0;
        double r719627 = 3.0;
        double r719628 = r719626 / r719627;
        double r719629 = x;
        double r719630 = y;
        double r719631 = 27.0;
        double r719632 = r719630 * r719631;
        double r719633 = r719629 / r719632;
        double r719634 = r719627 * r719633;
        double r719635 = z;
        double r719636 = 2.0;
        double r719637 = r719635 * r719636;
        double r719638 = r719634 / r719637;
        double r719639 = t;
        double r719640 = sqrt(r719639);
        double r719641 = r719638 * r719640;
        double r719642 = acos(r719641);
        double r719643 = r719628 * r719642;
        return r719643;
}
↓
double f(double x, double y, double z, double t) {
        double r719644 = 1.0;
        double r719645 = cbrt(r719644);
        double r719646 = r719645 * r719645;
        double r719647 = 3.0;
        double r719648 = cbrt(r719647);
        double r719649 = r719648 * r719648;
        double r719650 = r719646 / r719649;
        double r719651 = x;
        double r719652 = z;
        double r719653 = r719651 / r719652;
        double r719654 = y;
        double r719655 = r719653 / r719654;
        double r719656 = 0.05555555555555555;
        double r719657 = r719655 * r719656;
        double r719658 = t;
        double r719659 = sqrt(r719658);
        double r719660 = r719657 * r719659;
        double r719661 = acos(r719660);
        double r719662 = r719648 / r719661;
        double r719663 = r719645 / r719662;
        double r719664 = r719650 * r719663;
        return r719664;
}

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

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 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-frac0.4

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

  6. Applied associate-*l*0.4

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

  7. Simplified0.4

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

  8. Taylor expanded around 0 0.2

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

  9. Simplified0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019179 
(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)))))