Average Error: 20.6 → 20.6
Time: 14.1s
Precision: binary64
\[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
↓
\[\begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \sqrt[3]{t_0}\\ t_2 := \sqrt[3]{t_1}\\ {\left(a \cdot \sin t_0\right)}^{2} + {\left(b \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(t_1 \cdot \left(t_1 \cdot \sqrt[3]{t_1 \cdot \left(t_1 \cdot \left(t_2 \cdot \left(t_2 \cdot t_2\right)\right)\right)}\right)\right)\right)\right)\right)}^{2} \end{array} \]
{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}
↓
\begin{array}{l}
t_0 := \frac{angle}{180} \cdot \pi\\
t_1 := \sqrt[3]{t_0}\\
t_2 := \sqrt[3]{t_1}\\
{\left(a \cdot \sin t_0\right)}^{2} + {\left(b \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(t_1 \cdot \left(t_1 \cdot \sqrt[3]{t_1 \cdot \left(t_1 \cdot \left(t_2 \cdot \left(t_2 \cdot t_2\right)\right)\right)}\right)\right)\right)\right)\right)}^{2}
\end{array}

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (sin (* (/ angle 180.0) PI))) 2.0)
  (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))
↓
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI)) (t_1 (cbrt t_0)) (t_2 (cbrt t_1)))
   (+
    (pow (* a (sin t_0)) 2.0)
    (pow
     (*
      b
      (log1p
       (expm1
        (cos (* t_1 (* t_1 (cbrt (* t_1 (* t_1 (* t_2 (* t_2 t_2)))))))))))
     2.0))))
double code(double a, double b, double angle) {
	return pow((a * sin((angle / 180.0) * ((double) M_PI))), 2.0) + pow((b * cos((angle / 180.0) * ((double) M_PI))), 2.0);
}
↓
double code(double a, double b, double angle) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = cbrt(t_0);
	double t_2 = cbrt(t_1);
	return pow((a * sin(t_0)), 2.0) + pow((b * log1p(expm1(cos(t_1 * (t_1 * cbrt(t_1 * (t_1 * (t_2 * (t_2 * t_2))))))))), 2.0);
}

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 20.6

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

  2. Applied add-cube-cbrt_binary6420.6

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

  3. Simplified20.6

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

  4. Simplified20.6

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

  5. Applied add-cbrt-cube_binary6420.6

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

  6. Applied add-cube-cbrt_binary6420.6

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

  7. Applied log1p-expm1-u_binary6420.6

    \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\left(\sqrt[3]{\left(\sqrt[3]{\pi \cdot \frac{angle}{180}} \cdot \left(\left(\sqrt[3]{\sqrt[3]{\pi \cdot \frac{angle}{180}}} \cdot \sqrt[3]{\sqrt[3]{\pi \cdot \frac{angle}{180}}}\right) \cdot \sqrt[3]{\sqrt[3]{\pi \cdot \frac{angle}{180}}}\right)\right) \cdot \sqrt[3]{\pi \cdot \frac{angle}{180}}} \cdot \sqrt[3]{\pi \cdot \frac{angle}{180}}\right) \cdot \sqrt[3]{\pi \cdot \frac{angle}{180}}\right)\right)\right)}\right)}^{2} \]

  8. Final simplification20.6

    \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\sqrt[3]{\frac{angle}{180} \cdot \pi} \cdot \left(\sqrt[3]{\frac{angle}{180} \cdot \pi} \cdot \sqrt[3]{\sqrt[3]{\frac{angle}{180} \cdot \pi} \cdot \left(\sqrt[3]{\frac{angle}{180} \cdot \pi} \cdot \left(\sqrt[3]{\sqrt[3]{\frac{angle}{180} \cdot \pi}} \cdot \left(\sqrt[3]{\sqrt[3]{\frac{angle}{180} \cdot \pi}} \cdot \sqrt[3]{\sqrt[3]{\frac{angle}{180} \cdot \pi}}\right)\right)\right)}\right)\right)\right)\right)\right)}^{2} \]

Reproduce

herbie shell --seed 2021224 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))