Average Error: 1.3 → 0.3
Time: 8.3s
Precision: binary64
\[[y, z]=\mathsf{sort}([y, z])\]
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \]
↓
\[\frac{\frac{\cos^{-1} \left(0.05555555555555555 \cdot \left(\frac{1}{\frac{y}{\frac{x}{z}}} \cdot \sqrt{t}\right)\right)}{\sqrt[3]{3}}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \]
\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)
↓
\frac{\frac{\cos^{-1} \left(0.05555555555555555 \cdot \left(\frac{1}{\frac{y}{\frac{x}{z}}} \cdot \sqrt{t}\right)\right)}{\sqrt[3]{3}}}{\sqrt[3]{3} \cdot \sqrt[3]{3}}

(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
 (/
  (/
   (acos (* 0.05555555555555555 (* (/ 1.0 (/ y (/ x z))) (sqrt t))))
   (cbrt 3.0))
  (* (cbrt 3.0) (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 (acos(0.05555555555555555 * ((1.0 / (y / (x / z))) * sqrt(t))) / cbrt(3.0)) / (cbrt(3.0) * 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
Herbie0.3
\[\frac{\cos^{-1} \left(\frac{\frac{x}{27}}{y \cdot z} \cdot \frac{\sqrt{t}}{\frac{2}{3}}\right)}{3} \]

Derivation

  1. Initial program 1.3

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

  2. Applied add-cube-cbrt_binary641.3

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

  3. Applied *-un-lft-identity_binary641.3

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

  4. Applied times-frac_binary640.3

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

  5. Applied associate-*l*_binary640.3

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

  6. Simplified0.3

    \[\leadsto \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \color{blue}{\frac{\cos^{-1} \left(\sqrt{t} \cdot \frac{0.1111111111111111 \cdot \frac{x}{y}}{z \cdot 2}\right)}{\sqrt[3]{3}}} \]

  7. Taylor expanded in x around 0 0.3

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

  8. Applied associate-*l/_binary640.3

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

  9. Simplified0.3

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

  10. Applied clear-num_binary640.3

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

  11. Simplified0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2021275 
(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)))))