Average Error: 1.3 → 0.4
Time: 12.4s
Precision: binary64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\cos^{-1} \left(\left(1.5 \cdot \left(\frac{1}{y} \cdot \frac{\frac{x}{27}}{z}\right)\right) \cdot \sqrt{t}\right) \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{3}}\right)\]
\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(\cos^{-1} \left(\left(1.5 \cdot \left(\frac{1}{y} \cdot \frac{\frac{x}{27}}{z}\right)\right) \cdot \sqrt{t}\right) \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{3}}\right)
(FPCore (x y z t)
 :precision binary64
 (* (/ 1.0 3.0) (acos (* (/ (* 3.0 (/ x (* y 27.0))) (* z 2.0)) (sqrt t)))))
↓
(FPCore (x y z t)
 :precision binary64
 (*
  (/ (* (cbrt 1.0) (cbrt 1.0)) (* (cbrt 3.0) (cbrt 3.0)))
  (*
   (acos (* (* 1.5 (* (/ 1.0 y) (/ (/ x 27.0) z))) (sqrt t)))
   (/ (cbrt 1.0) (cbrt 3.0)))))
double code(double x, double y, double z, double t) {
	return (1.0 / 3.0) * acos(((3.0 * (x / (y * 27.0))) / (z * 2.0)) * sqrt(t));
}
↓
double code(double x, double y, double z, double t) {
	return ((cbrt(1.0) * cbrt(1.0)) / (cbrt(3.0) * cbrt(3.0))) * (acos((1.5 * ((1.0 / y) * ((x / 27.0) / z))) * sqrt(t)) * (cbrt(1.0) / cbrt(3.0)));
}

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

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-cbrt_binary64_120081.3

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

  4. Applied add-cube-cbrt_binary64_120081.3

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

  5. Applied times-frac_binary64_120320.3

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

  6. Applied associate-*l*_binary64_120970.3

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

  7. Simplified0.3

    \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \color{blue}{\left(\cos^{-1} \left(\left(1.5 \cdot \frac{\frac{x}{y \cdot 27}}{z}\right) \cdot \sqrt{t}\right) \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{3}}\right)}\]
  8. Using strategy rm

  9. Applied *-un-lft-identity_binary64_120370.3

    \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\cos^{-1} \left(\left(1.5 \cdot \frac{\frac{x}{y \cdot 27}}{\color{blue}{1 \cdot z}}\right) \cdot \sqrt{t}\right) \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{3}}\right)\]

  10. Applied *-un-lft-identity_binary64_120370.3

    \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\cos^{-1} \left(\left(1.5 \cdot \frac{\frac{\color{blue}{1 \cdot x}}{y \cdot 27}}{1 \cdot z}\right) \cdot \sqrt{t}\right) \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{3}}\right)\]

  11. Applied times-frac_binary64_120320.3

    \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\cos^{-1} \left(\left(1.5 \cdot \frac{\color{blue}{\frac{1}{y} \cdot \frac{x}{27}}}{1 \cdot z}\right) \cdot \sqrt{t}\right) \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{3}}\right)\]

  12. Applied times-frac_binary64_120320.4

    \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\cos^{-1} \left(\left(1.5 \cdot \color{blue}{\left(\frac{\frac{1}{y}}{1} \cdot \frac{\frac{x}{27}}{z}\right)}\right) \cdot \sqrt{t}\right) \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{3}}\right)\]

  13. Simplified0.4

    \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\cos^{-1} \left(\left(1.5 \cdot \left(\color{blue}{\frac{1}{y}} \cdot \frac{\frac{x}{27}}{z}\right)\right) \cdot \sqrt{t}\right) \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{3}}\right)\]

  14. Final simplification0.4

    \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\cos^{-1} \left(\left(1.5 \cdot \left(\frac{1}{y} \cdot \frac{\frac{x}{27}}{z}\right)\right) \cdot \sqrt{t}\right) \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{3}}\right)\]

Reproduce

herbie shell --seed 2020268 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, D"
  :precision binary64

  :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)))))