Average Error: 12.9 → 13.0
Time: 9.1s
Precision: binary64
\[{\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
↓
\[{\left(a \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)}^{2} + {\left(b \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\sqrt{0.005555555555555556} \cdot \left(\left(\pi \cdot angle\right) \cdot \sqrt{0.005555555555555556}\right)\right)\right)\right)\right)}^{2} \]
{\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 \cos \left(\frac{\pi \cdot angle}{180}\right)\right)}^{2} + {\left(b \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\sqrt{0.005555555555555556} \cdot \left(\left(\pi \cdot angle\right) \cdot \sqrt{0.005555555555555556}\right)\right)\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 (cos (/ (* PI angle) 180.0))) 2.0)
  (pow
   (*
    b
    (log1p
     (expm1
      (sin
       (*
        (sqrt 0.005555555555555556)
        (* (* PI angle) (sqrt 0.005555555555555556)))))))
   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 * cos((((double) M_PI) * angle) / 180.0)), 2.0) + pow((b * log1p(expm1(sin(sqrt(0.005555555555555556) * ((((double) M_PI) * angle) * sqrt(0.005555555555555556)))))), 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 12.9

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

  2. Applied associate-*r/_binary6413.0

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

  3. Applied associate-*r/_binary6413.0

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

  4. Simplified13.0

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

  5. Applied log1p-expm1-u_binary6413.0

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

  6. Simplified13.0

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

  7. Applied add-sqr-sqrt_binary6413.0

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

  8. Applied associate-*l*_binary6413.0

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

  9. Final simplification13.0

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

Reproduce

herbie shell --seed 2022067 
(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)))