Average Error: 1.2 → 0.2
Time: 24.8s
Precision: 64
\[\frac{1.0}{3.0} \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)\]
↓
\[\frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \left(1.0 \cdot \frac{\cos^{-1} \left(\left(\frac{x}{z \cdot y} \cdot \sqrt{t}\right) \cdot 0.05555555555555555\right)}{\sqrt[3]{3.0}}\right)\]
\frac{1.0}{3.0} \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)
↓
\frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \left(1.0 \cdot \frac{\cos^{-1} \left(\left(\frac{x}{z \cdot y} \cdot \sqrt{t}\right) \cdot 0.05555555555555555\right)}{\sqrt[3]{3.0}}\right)
double f(double x, double y, double z, double t) {
        double r33484714 = 1.0;
        double r33484715 = 3.0;
        double r33484716 = r33484714 / r33484715;
        double r33484717 = x;
        double r33484718 = y;
        double r33484719 = 27.0;
        double r33484720 = r33484718 * r33484719;
        double r33484721 = r33484717 / r33484720;
        double r33484722 = r33484715 * r33484721;
        double r33484723 = z;
        double r33484724 = 2.0;
        double r33484725 = r33484723 * r33484724;
        double r33484726 = r33484722 / r33484725;
        double r33484727 = t;
        double r33484728 = sqrt(r33484727);
        double r33484729 = r33484726 * r33484728;
        double r33484730 = acos(r33484729);
        double r33484731 = r33484716 * r33484730;
        return r33484731;
}
↓
double f(double x, double y, double z, double t) {
        double r33484732 = 1.0;
        double r33484733 = 3.0;
        double r33484734 = cbrt(r33484733);
        double r33484735 = r33484734 * r33484734;
        double r33484736 = r33484732 / r33484735;
        double r33484737 = 1.0;
        double r33484738 = x;
        double r33484739 = z;
        double r33484740 = y;
        double r33484741 = r33484739 * r33484740;
        double r33484742 = r33484738 / r33484741;
        double r33484743 = t;
        double r33484744 = sqrt(r33484743);
        double r33484745 = r33484742 * r33484744;
        double r33484746 = 0.05555555555555555;
        double r33484747 = r33484745 * r33484746;
        double r33484748 = acos(r33484747);
        double r33484749 = r33484748 / r33484734;
        double r33484750 = r33484737 * r33484749;
        double r33484751 = r33484736 * r33484750;
        return r33484751;
}

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

Original1.2
Target1.2
Herbie0.2
\[\frac{\cos^{-1} \left(\frac{\frac{x}{27.0}}{y \cdot z} \cdot \frac{\sqrt{t}}{\frac{2.0}{3.0}}\right)}{3.0}\]

Derivation

  1. Initial program 1.2

    \[\frac{1.0}{3.0} \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)\]
  2. Using strategy rm

  3. Applied add-cube-cbrt1.2

    \[\leadsto \frac{1.0}{\color{blue}{\left(\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}\right) \cdot \sqrt[3]{3.0}}} \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)\]

  4. Applied *-un-lft-identity1.2

    \[\leadsto \frac{\color{blue}{1 \cdot 1.0}}{\left(\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}\right) \cdot \sqrt[3]{3.0}} \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)\]

  5. Applied times-frac0.3

    \[\leadsto \color{blue}{\left(\frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \frac{1.0}{\sqrt[3]{3.0}}\right)} \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)\]

  6. Applied associate-*l*0.3

    \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \left(\frac{1.0}{\sqrt[3]{3.0}} \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)\right)}\]

  7. Taylor expanded around 0 0.2

    \[\leadsto \frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \color{blue}{\left(1.0 \cdot \frac{\cos^{-1} \left(0.05555555555555555 \cdot \left(\sqrt{t} \cdot \frac{x}{z \cdot y}\right)\right)}{\sqrt[3]{3.0}}\right)}\]

  8. Final simplification0.2

    \[\leadsto \frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \left(1.0 \cdot \frac{\cos^{-1} \left(\left(\frac{x}{z \cdot y} \cdot \sqrt{t}\right) \cdot 0.05555555555555555\right)}{\sqrt[3]{3.0}}\right)\]

Reproduce

herbie shell --seed 2019162 
(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)))))