Average Error: 1.4 → 0.2
Time: 12.2s
Precision: binary64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \]
↓
\[e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(\left(\frac{x}{y \cdot z} \cdot \sqrt{t}\right) \cdot 0.05555555555555555\right)\right)} - 1 \]
\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)
↓
e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(\left(\frac{x}{y \cdot z} \cdot \sqrt{t}\right) \cdot 0.05555555555555555\right)\right)} - 1

(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
 (-
  (exp
   (log1p
    (*
     0.3333333333333333
     (acos (* (* (/ x (* y z)) (sqrt t)) 0.05555555555555555)))))
  1.0))
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 exp(log1p((0.3333333333333333 * acos((((x / (y * z)) * sqrt(t)) * 0.05555555555555555))))) - 1.0;
}

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

Original1.4
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 1.4

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

  2. Taylor expanded in x around 0 1.2

    \[\leadsto \color{blue}{0.3333333333333333 \cdot \cos^{-1} \left(0.05555555555555555 \cdot \left(\frac{x}{y \cdot z} \cdot \sqrt{t}\right)\right)} \]

  3. Applied egg-rr0.2

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(\left(\frac{x}{y \cdot z} \cdot \sqrt{t}\right) \cdot 0.05555555555555555\right)\right)} - 1} \]

  4. Final simplification0.2

    \[\leadsto e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(\left(\frac{x}{y \cdot z} \cdot \sqrt{t}\right) \cdot 0.05555555555555555\right)\right)} - 1 \]

Reproduce

herbie shell --seed 2022130 
(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)))))