Average Error: 1.3 → 0.6
Time: 10.4s
Precision: binary64
\[[y, z]=\mathsf{sort}([y, z])\]
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \]
↓
\[\begin{array}{l} t_1 := \sqrt{\cos^{-1} \left(x \cdot \frac{\frac{\sqrt{t}}{z}}{y \cdot 18}\right)}\\ t_2 := \sqrt[3]{t_1}\\ \left(0.3333333333333333 \cdot t_1\right) \cdot \left(t_2 \cdot \left(t_2 \cdot t_2\right)\right) \end{array} \]
\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)
↓
\begin{array}{l}
t_1 := \sqrt{\cos^{-1} \left(x \cdot \frac{\frac{\sqrt{t}}{z}}{y \cdot 18}\right)}\\
t_2 := \sqrt[3]{t_1}\\
\left(0.3333333333333333 \cdot t_1\right) \cdot \left(t_2 \cdot \left(t_2 \cdot t_2\right)\right)
\end{array}

(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
 (let* ((t_1 (sqrt (acos (* x (/ (/ (sqrt t) z) (* y 18.0))))))
        (t_2 (cbrt t_1)))
   (* (* 0.3333333333333333 t_1) (* t_2 (* t_2 t_2)))))
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) {
	double t_1 = sqrt(acos(x * ((sqrt(t) / z) / (y * 18.0))));
	double t_2 = cbrt(t_1);
	return (0.3333333333333333 * t_1) * (t_2 * (t_2 * t_2));
}

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
Target1.3
Herbie0.6
\[\frac{\cos^{-1} \left(\frac{\frac{x}{27}}{y \cdot z} \cdot \frac{\sqrt{t}}{\frac{2}{3}}\right)}{3} \]

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. Simplified1.5

    \[\leadsto \color{blue}{0.3333333333333333 \cdot \cos^{-1} \left(x \cdot \frac{\frac{\sqrt{t}}{z}}{y \cdot 18}\right)} \]

  3. Applied add-sqr-sqrt_binary642.1

    \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(\sqrt{\cos^{-1} \left(x \cdot \frac{\frac{\sqrt{t}}{z}}{y \cdot 18}\right)} \cdot \sqrt{\cos^{-1} \left(x \cdot \frac{\frac{\sqrt{t}}{z}}{y \cdot 18}\right)}\right)} \]

  4. Applied associate-*r*_binary641.5

    \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \sqrt{\cos^{-1} \left(x \cdot \frac{\frac{\sqrt{t}}{z}}{y \cdot 18}\right)}\right) \cdot \sqrt{\cos^{-1} \left(x \cdot \frac{\frac{\sqrt{t}}{z}}{y \cdot 18}\right)}} \]

  5. Applied add-cube-cbrt_binary640.6

    \[\leadsto \left(0.3333333333333333 \cdot \sqrt{\cos^{-1} \left(x \cdot \frac{\frac{\sqrt{t}}{z}}{y \cdot 18}\right)}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt{\cos^{-1} \left(x \cdot \frac{\frac{\sqrt{t}}{z}}{y \cdot 18}\right)}} \cdot \sqrt[3]{\sqrt{\cos^{-1} \left(x \cdot \frac{\frac{\sqrt{t}}{z}}{y \cdot 18}\right)}}\right) \cdot \sqrt[3]{\sqrt{\cos^{-1} \left(x \cdot \frac{\frac{\sqrt{t}}{z}}{y \cdot 18}\right)}}\right)} \]

  6. Final simplification0.6

    \[\leadsto \left(0.3333333333333333 \cdot \sqrt{\cos^{-1} \left(x \cdot \frac{\frac{\sqrt{t}}{z}}{y \cdot 18}\right)}\right) \cdot \left(\sqrt[3]{\sqrt{\cos^{-1} \left(x \cdot \frac{\frac{\sqrt{t}}{z}}{y \cdot 18}\right)}} \cdot \left(\sqrt[3]{\sqrt{\cos^{-1} \left(x \cdot \frac{\frac{\sqrt{t}}{z}}{y \cdot 18}\right)}} \cdot \sqrt[3]{\sqrt{\cos^{-1} \left(x \cdot \frac{\frac{\sqrt{t}}{z}}{y \cdot 18}\right)}}\right)\right) \]

Reproduce

herbie shell --seed 2021280 
(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)))))