Average Error: 1.2 → 0.3
Time: 5.9s
Precision: binary64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\cos^{-1} \left(\left(\frac{3}{z} \cdot \frac{x}{2 \cdot \left(y \cdot 27\right)}\right) \cdot \sqrt{t}\right) \cdot \frac{1}{\sqrt[3]{3}}\right)\]
\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)
↓
\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\cos^{-1} \left(\left(\frac{3}{z} \cdot \frac{x}{2 \cdot \left(y \cdot 27\right)}\right) \cdot \sqrt{t}\right) \cdot \frac{1}{\sqrt[3]{3}}\right)
double code(double x, double y, double z, double t) {
	return ((double) ((1.0 / 3.0) * ((double) acos(((double) ((((double) (3.0 * (x / ((double) (y * 27.0))))) / ((double) (z * 2.0))) * ((double) sqrt(t))))))));
}
↓
double code(double x, double y, double z, double t) {
	return ((double) ((1.0 / ((double) (((double) cbrt(3.0)) * ((double) cbrt(3.0))))) * ((double) (((double) acos(((double) (((double) ((3.0 / z) * (x / ((double) (2.0 * ((double) (y * 27.0))))))) * ((double) sqrt(t)))))) * (1.0 / ((double) cbrt(3.0)))))));
}

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

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-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 *-un-lft-identity1.2

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

  5. Applied times-frac0.3

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

  6. Applied associate-*l*0.3

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

  7. Simplified0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2020196 
(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)))))