Average Error: 1.4 → 0.5
Time: 23.2s
Precision: 64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\frac{\sqrt{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\sqrt{\frac{\sqrt{1}}{\sqrt[3]{3}}} \cdot \left(\sqrt{\frac{\sqrt{1}}{\sqrt[3]{3}}} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\right)\right)\]
\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(\sqrt{\frac{\sqrt{1}}{\sqrt[3]{3}}} \cdot \left(\sqrt{\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)\right)
double f(double x, double y, double z, double t) {
        double r493736 = 1.0;
        double r493737 = 3.0;
        double r493738 = r493736 / r493737;
        double r493739 = x;
        double r493740 = y;
        double r493741 = 27.0;
        double r493742 = r493740 * r493741;
        double r493743 = r493739 / r493742;
        double r493744 = r493737 * r493743;
        double r493745 = z;
        double r493746 = 2.0;
        double r493747 = r493745 * r493746;
        double r493748 = r493744 / r493747;
        double r493749 = t;
        double r493750 = sqrt(r493749);
        double r493751 = r493748 * r493750;
        double r493752 = acos(r493751);
        double r493753 = r493738 * r493752;
        return r493753;
}
↓
double f(double x, double y, double z, double t) {
        double r493754 = 1.0;
        double r493755 = sqrt(r493754);
        double r493756 = 3.0;
        double r493757 = cbrt(r493756);
        double r493758 = r493757 * r493757;
        double r493759 = r493755 / r493758;
        double r493760 = r493755 / r493757;
        double r493761 = sqrt(r493760);
        double r493762 = x;
        double r493763 = y;
        double r493764 = 27.0;
        double r493765 = r493763 * r493764;
        double r493766 = r493762 / r493765;
        double r493767 = r493756 * r493766;
        double r493768 = z;
        double r493769 = 2.0;
        double r493770 = r493768 * r493769;
        double r493771 = r493767 / r493770;
        double r493772 = t;
        double r493773 = sqrt(r493772);
        double r493774 = r493771 * r493773;
        double r493775 = acos(r493774);
        double r493776 = r493761 * r493775;
        double r493777 = r493761 * r493776;
        double r493778 = r493759 * r493777;
        return r493778;
}

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

Original1.4
Target1.2
Herbie0.5
\[\frac{\cos^{-1} \left(\frac{\frac{x}{27}}{y \cdot z} \cdot \frac{\sqrt{t}}{\frac{2}{3}}\right)}{3}\]

Derivation

  1. Initial program 1.4

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

  3. Applied add-cube-cbrt1.4

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

  4. Applied add-sqr-sqrt1.4

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

  5. Applied times-frac0.5

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

  6. Applied associate-*l*0.4

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

  8. Applied add-sqr-sqrt1.4

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

  9. Applied associate-*l*0.5

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

  10. Final simplification0.5

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

Reproduce

herbie shell --seed 2019323 +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)))))