Average Error: 1.2 → 0.2
Time: 3.8m
Precision: 64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{\sqrt[3]{1}}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}{z \cdot \left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right)} \cdot \left(\frac{\frac{x}{\sqrt[3]{y} \cdot 27}}{\sqrt[3]{2}} \cdot \sqrt{t}\right)\right)\right)\]
\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)
↓
\frac{\sqrt[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{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}{z \cdot \left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right)} \cdot \left(\frac{\frac{x}{\sqrt[3]{y} \cdot 27}}{\sqrt[3]{2}} \cdot \sqrt{t}\right)\right)\right)
double code(double x, double y, double z, double t) {
	return ((double) (((double) (1.0 / 3.0)) * ((double) acos(((double) (((double) (((double) (3.0 * ((double) (x / ((double) (y * 27.0)))))) / ((double) (z * 2.0)))) * ((double) sqrt(t))))))));
}
↓
double code(double x, double y, double z, double t) {
	return ((double) (((double) (((double) (((double) cbrt(1.0)) * ((double) cbrt(1.0)))) / ((double) (((double) cbrt(3.0)) * ((double) cbrt(3.0)))))) * ((double) (((double) (((double) cbrt(1.0)) / ((double) cbrt(3.0)))) * ((double) acos(((double) (((double) (((double) (3.0 * ((double) (1.0 / ((double) (((double) cbrt(y)) * ((double) cbrt(y)))))))) / ((double) (z * ((double) (((double) cbrt(2.0)) * ((double) cbrt(2.0)))))))) * ((double) (((double) (((double) (x / ((double) (((double) cbrt(y)) * 27.0)))) / ((double) cbrt(2.0)))) * ((double) sqrt(t))))))))))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

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Your Program's Arguments

Results

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Target

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

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-cbrt1.2

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

  4. Applied add-cube-cbrt1.2

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

  5. Applied times-frac0.3

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

  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-cube-cbrt0.3

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

  9. Applied associate-*r*0.3

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

  10. Applied add-cube-cbrt0.3

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

  11. Applied associate-*l*0.3

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

  12. Applied *-un-lft-identity0.3

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

  13. Applied times-frac0.2

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

  14. Applied associate-*r*0.2

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

  15. Applied times-frac0.1

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

  16. Applied associate-*l*0.2

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

  17. Final simplification0.2

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

Reproduce

herbie shell --seed 2020113 +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)))))