Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, D

Average Error: 1.2 → 0.3

Time: 5.8s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r736759 = 1.0;
        double r736760 = 3.0;
        double r736761 = r736759 / r736760;
        double r736762 = x;
        double r736763 = y;
        double r736764 = 27.0;
        double r736765 = r736763 * r736764;
        double r736766 = r736762 / r736765;
        double r736767 = r736760 * r736766;
        double r736768 = z;
        double r736769 = 2.0;
        double r736770 = r736768 * r736769;
        double r736771 = r736767 / r736770;
        double r736772 = t;
        double r736773 = sqrt(r736772);
        double r736774 = r736771 * r736773;
        double r736775 = acos(r736774);
        double r736776 = r736761 * r736775;
        return r736776;
}

↓

double f(double x, double y, double z, double t) {
        double r736777 = 1.0;
        double r736778 = cbrt(r736777);
        double r736779 = r736778 * r736778;
        double r736780 = 3.0;
        double r736781 = cbrt(r736780);
        double r736782 = r736781 * r736781;
        double r736783 = r736779 / r736782;
        double r736784 = r736778 / r736781;
        double r736785 = sqrt(r736784);
        double r736786 = x;
        double r736787 = y;
        double r736788 = 27.0;
        double r736789 = r736787 * r736788;
        double r736790 = r736786 / r736789;
        double r736791 = r736780 * r736790;
        double r736792 = z;
        double r736793 = 2.0;
        double r736794 = r736792 * r736793;
        double r736795 = r736791 / r736794;
        double r736796 = t;
        double r736797 = sqrt(r736796);
        double r736798 = r736795 * r736797;
        double r736799 = acos(r736798);
        double r736800 = sqrt(r736799);
        double r736801 = r736785 * r736800;
        double r736802 = r736801 * r736801;
        double r736803 = r736783 * r736802;
        return r736803;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	1.2
Target	1.2
Herbie	0.3

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

Derivation

1. Initial program 1.2

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

3. Applied add-cube-cbrt 1.2

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

4. Applied add-cube-cbrt 1.2

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

5. Applied times-frac 0.3

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

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. Using strategy rm

8. Applied add-sqr-sqrt 1.2

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

9. Applied add-sqr-sqrt 0.3

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

10. Applied unswap-sqr 0.3

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

11. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020018 
(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)))))