Average Error: 1.3 → 0.3
Time: 6.2s
Precision: 64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\frac{\sqrt{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{\sqrt{1}}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\left(0.055555555555555552 \cdot \frac{x}{z \cdot y}\right) \cdot \sqrt{t}\right)\right)\]
\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)
↓
\frac{\sqrt{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{\sqrt{1}}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\left(0.055555555555555552 \cdot \frac{x}{z \cdot y}\right) \cdot \sqrt{t}\right)\right)
double f(double x, double y, double z, double t) {
        double r599742 = 1.0;
        double r599743 = 3.0;
        double r599744 = r599742 / r599743;
        double r599745 = x;
        double r599746 = y;
        double r599747 = 27.0;
        double r599748 = r599746 * r599747;
        double r599749 = r599745 / r599748;
        double r599750 = r599743 * r599749;
        double r599751 = z;
        double r599752 = 2.0;
        double r599753 = r599751 * r599752;
        double r599754 = r599750 / r599753;
        double r599755 = t;
        double r599756 = sqrt(r599755);
        double r599757 = r599754 * r599756;
        double r599758 = acos(r599757);
        double r599759 = r599744 * r599758;
        return r599759;
}
↓
double f(double x, double y, double z, double t) {
        double r599760 = 1.0;
        double r599761 = sqrt(r599760);
        double r599762 = 3.0;
        double r599763 = cbrt(r599762);
        double r599764 = r599763 * r599763;
        double r599765 = r599761 / r599764;
        double r599766 = r599761 / r599763;
        double r599767 = 0.05555555555555555;
        double r599768 = x;
        double r599769 = z;
        double r599770 = y;
        double r599771 = r599769 * r599770;
        double r599772 = r599768 / r599771;
        double r599773 = r599767 * r599772;
        double r599774 = t;
        double r599775 = sqrt(r599774);
        double r599776 = r599773 * r599775;
        double r599777 = acos(r599776);
        double r599778 = r599766 * r599777;
        double r599779 = r599765 * r599778;
        return r599779;
}

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*0.3

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

  7. Taylor expanded around 0 0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2020046 
(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)))))