Average Error: 20.1 → 20.1
Time: 13.9s
Precision: binary64
\[{\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
↓
\[{\left(a \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(\log \left(e^{\mathsf{expm1}\left(\cos \left(\pi \cdot \frac{angle}{180}\right)\right)}\right)\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}^{2} \]
{\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2}
↓
{\left(a \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(\log \left(e^{\mathsf{expm1}\left(\cos \left(\pi \cdot \frac{angle}{180}\right)\right)}\right)\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}^{2}

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (cos (* PI (/ angle 180.0)))) 2.0)
  (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))
↓
(FPCore (a b angle)
 :precision binary64
 (+
  (pow
   (*
    a
    (expm1 (log1p (log1p (log (exp (expm1 (cos (* PI (/ angle 180.0))))))))))
   2.0)
  (pow (* b (sin (* 0.005555555555555556 (* PI angle)))) 2.0)))
double code(double a, double b, double angle) {
	return pow((a * cos(((double) M_PI) * (angle / 180.0))), 2.0) + pow((b * sin(((double) M_PI) * (angle / 180.0))), 2.0);
}
↓
double code(double a, double b, double angle) {
	return pow((a * expm1(log1p(log1p(log(exp(expm1(cos(((double) M_PI) * (angle / 180.0))))))))), 2.0) + pow((b * sin(0.005555555555555556 * (((double) M_PI) * angle))), 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 20.1

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

  2. Applied expm1-log1p-u_binary6420.1

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

  3. Applied log1p-expm1-u_binary6420.1

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

  4. Applied add-log-exp_binary6420.1

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

  5. Taylor expanded in angle around inf 20.1

    \[\leadsto {\left(a \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(\log \left(e^{\mathsf{expm1}\left(\cos \left(\pi \cdot \frac{angle}{180}\right)\right)}\right)\right)\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}^{2} \]

  6. Final simplification20.1

    \[\leadsto {\left(a \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(\log \left(e^{\mathsf{expm1}\left(\cos \left(\pi \cdot \frac{angle}{180}\right)\right)}\right)\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}^{2} \]

Reproduce

herbie shell --seed 2022101 
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* PI (/ angle 180.0)))) 2.0) (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))