Average Error: 31.4 → 31.3
Time: 14.9s
Precision: binary64
\[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
↓
\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \sqrt[3]{\cos t_0}\\ \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t_0\right) \cdot \left(t_1 \cdot \left(\sqrt[3]{\cos \left(\pi \cdot \frac{angle}{180}\right)} \cdot t_1\right)\right) \end{array} \]
\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)
↓
\begin{array}{l}
t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\
t_1 := \sqrt[3]{\cos t_0}\\
\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t_0\right) \cdot \left(t_1 \cdot \left(\sqrt[3]{\cos \left(\pi \cdot \frac{angle}{180}\right)} \cdot t_1\right)\right)
\end{array}

(FPCore (a b angle)
 :precision binary64
 (*
  (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* PI (/ angle 180.0))))
  (cos (* PI (/ angle 180.0)))))
↓
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))) (t_1 (cbrt (cos t_0))))
   (*
    (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin t_0))
    (* t_1 (* (cbrt (cos (* PI (/ angle 180.0)))) t_1)))))
double code(double a, double b, double angle) {
	return ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(((double) M_PI) * (angle / 180.0))) * cos(((double) M_PI) * (angle / 180.0));
}
↓
double code(double a, double b, double angle) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = cbrt(cos(t_0));
	return ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(t_0)) * (t_1 * (cbrt(cos(((double) M_PI) * (angle / 180.0))) * t_1));
}

Error

Bits error versus a

Bits error versus b

Bits error versus angle

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.4

    \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

  2. Applied add-cube-cbrt_binary6431.4

    \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\cos \left(\pi \cdot \frac{angle}{180}\right)} \cdot \sqrt[3]{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \sqrt[3]{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right)} \]

  3. Taylor expanded in angle around inf 31.3

    \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(\left(\sqrt[3]{\cos \left(\pi \cdot \frac{angle}{180}\right)} \cdot \sqrt[3]{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \sqrt[3]{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right) \]

  4. Taylor expanded in angle around inf 31.4

    \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(\left(\sqrt[3]{\cos \left(\pi \cdot \frac{angle}{180}\right)} \cdot \sqrt[3]{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right) \cdot \sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right) \]

  5. Taylor expanded in angle around inf 31.3

    \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \left(\left(\sqrt[3]{\cos \left(\pi \cdot \frac{angle}{180}\right)} \cdot \sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right) \]

  6. Final simplification31.3

    \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \left(\sqrt[3]{\cos \left(\pi \cdot \frac{angle}{180}\right)} \cdot \sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right) \]

Reproduce

herbie shell --seed 2022081 
(FPCore (a b angle)
  :name "ab-angle->ABCF B"
  :precision binary64
  (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* PI (/ angle 180.0)))) (cos (* PI (/ angle 180.0)))))