Average Error: 20.0 → 20.0
Time: 11.0s
Precision: binary64
\[{\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2}\]
↓
\[{\left(a \cdot \cos \left(\sqrt{0.005555555555555556} \cdot \left(\left(\pi \cdot angle\right) \cdot \sqrt{0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi \cdot angle}{180}\right)\right)}^{2}\]
{\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(\sqrt{0.005555555555555556} \cdot \left(\left(\pi \cdot angle\right) \cdot \sqrt{0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi \cdot angle}{180}\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
     (*
      (sqrt 0.005555555555555556)
      (* (* PI angle) (sqrt 0.005555555555555556)))))
   2.0)
  (pow (* b (sin (/ (* 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 * sin(((double) M_PI) * (angle / 180.0))), 2.0);
}
↓
double code(double a, double b, double angle) {
	return pow((a * cos(sqrt(0.005555555555555556) * ((((double) M_PI) * angle) * sqrt(0.005555555555555556)))), 2.0) + pow((b * sin((((double) M_PI) * angle) / 180.0)), 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 20.0

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

  3. Applied associate-*r/_binary64_2020.0

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

  4. Taylor expanded around 0 20.0

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

  5. Simplified20.0

    \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi \cdot angle}{180}\right)\right)}^{2}\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt_binary64_10020.0

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

  8. Applied associate-*r*_binary64_1820.0

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

  9. Final simplification20.0

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

Reproduce

herbie shell --seed 2020353 
(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)))