Average Error: 20.5 → 20.5
Time: 1.6min
Precision: binary64
Cost: 46016
\[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}\]
↓
\[{b}^{2} + {\left(\left|a \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right|\right)}^{1.5} \cdot \sqrt{\left|a \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right|}\]
{\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}
↓
{b}^{2} + {\left(\left|a \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right|\right)}^{1.5} \cdot \sqrt{\left|a \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right|}
(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
 (+
  (pow b 2.0)
  (*
   (pow (fabs (* a (sin (* angle (* PI 0.005555555555555556))))) 1.5)
   (sqrt (fabs (* a (sin (* angle (* PI 0.005555555555555556)))))))))
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) {
	return pow(b, 2.0) + (pow(fabs(a * sin(angle * (((double) M_PI) * 0.005555555555555556))), 1.5) * sqrt(fabs(a * sin(angle * (((double) M_PI) * 0.005555555555555556)))));
}

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

Alternatives

Alternative 1
Error20.5
Cost19904
\[{b}^{2} + {\left(a \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\]
Alternative 2
Error20.6
Cost19904
\[{b}^{2} + {\left(a \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2}\]
Alternative 3
Error21.9
Cost20104
\[\begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -1.727900170858672 \cdot 10^{+64} \lor \neg \left(\frac{angle}{180} \leq 15310195302.444624\right):\\ \;\;\;\;{b}^{2} + {\log 1}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(b \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \left(a \cdot \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(a \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)\\ \end{array}\]
Alternative 4
Error23.7
Cost14664
\[\begin{array}{l} \mathbf{if}\;a \leq -1.1109038320081141 \cdot 10^{+48} \lor \neg \left(a \leq 5.842377750824772 \cdot 10^{-86}\right):\\ \;\;\;\;{\left(b \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \left(a \cdot \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(a \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\pi \cdot \frac{angle}{180}\right) \cdot \left(a \cdot a\right)\right)\\ \end{array}\]
Alternative 5
Error23.9
Cost13825
\[\begin{array}{l} \mathbf{if}\;a \leq -2.999202869143944 \cdot 10^{+70}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{elif}\;a \leq 8.691200890910609 \cdot 10^{-173}:\\ \;\;\;\;{b}^{2} + \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\pi \cdot \frac{angle}{180}\right) \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(a \cdot \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\\ \end{array}\]
Alternative 6
Error23.9
Cost8194
\[\begin{array}{l} \mathbf{if}\;a \leq -1.701469338032445 \cdot 10^{+73}:\\ \;\;\;\;{b}^{2} + \left(a \cdot \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(\frac{angle}{180} \cdot \left(a \cdot \pi\right)\right)\\ \mathbf{elif}\;a \leq 8.691200890910609 \cdot 10^{-173}:\\ \;\;\;\;{b}^{2} + \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\pi \cdot \frac{angle}{180}\right) \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(a \cdot \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\\ \end{array}\]
Alternative 7
Error23.9
Cost7880
\[\begin{array}{l} \mathbf{if}\;a \leq -4.261887214483194 \cdot 10^{+74} \lor \neg \left(a \leq 8.691200890910609 \cdot 10^{-173}\right):\\ \;\;\;\;{b}^{2} + \left(a \cdot \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\pi \cdot \frac{angle}{180}\right) \cdot \left(a \cdot a\right)\right)\\ \end{array}\]
Alternative 8
Error26.1
Cost7552
\[{b}^{2} + \left(a \cdot \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\]
Alternative 9
Error26.1
Cost7552
\[{b}^{2} + \left(a \cdot \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(a \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)\]
Alternative 10
Error53.1
Cost64
\[0\]

Error

Derivation

  1. Initial program 20.5

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

  2. Taylor expanded around 0 20.6

    \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2}\]
  3. Using strategy rm

  4. Applied div-inv_binary64_7520.5

    \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \pi\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\]

  5. Applied associate-*l*_binary64_1920.5

    \[\leadsto {\left(a \cdot \sin \color{blue}{\left(angle \cdot \left(\frac{1}{180} \cdot \pi\right)\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2}\]

  6. Simplified20.5

    \[\leadsto {\left(a \cdot \sin \left(angle \cdot \color{blue}{\left(\pi \cdot 0.005555555555555556\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\]
  7. Using strategy rm

  8. Applied add-sqr-sqrt_binary64_10020.5

    \[\leadsto \color{blue}{\sqrt{{\left(a \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}} \cdot \sqrt{{\left(a \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}}} + {\left(b \cdot 1\right)}^{2}\]

  9. Simplified20.5

    \[\leadsto \color{blue}{\left|\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot a\right|} \cdot \sqrt{{\left(a \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}} + {\left(b \cdot 1\right)}^{2}\]

  10. Simplified20.5

    \[\leadsto \left|\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot a\right| \cdot \color{blue}{\left|\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot a\right|} + {\left(b \cdot 1\right)}^{2}\]
  11. Using strategy rm

  12. Applied add-sqr-sqrt_binary64_10020.5

    \[\leadsto \left|\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot a\right| \cdot \color{blue}{\left(\sqrt{\left|\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot a\right|} \cdot \sqrt{\left|\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot a\right|}\right)} + {\left(b \cdot 1\right)}^{2}\]

  13. Applied associate-*r*_binary64_1820.5

    \[\leadsto \color{blue}{\left(\left|\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot a\right| \cdot \sqrt{\left|\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot a\right|}\right) \cdot \sqrt{\left|\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot a\right|}} + {\left(b \cdot 1\right)}^{2}\]

  14. Simplified20.5

    \[\leadsto \color{blue}{{\left(\left|\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot a\right|\right)}^{1.5}} \cdot \sqrt{\left|\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot a\right|} + {\left(b \cdot 1\right)}^{2}\]

  15. Simplified20.5

    \[\leadsto \color{blue}{{b}^{2} + {\left(\left|a \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right|\right)}^{1.5} \cdot \sqrt{\left|a \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right|}}\]

  16. Final simplification20.5

    \[\leadsto {b}^{2} + {\left(\left|a \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right|\right)}^{1.5} \cdot \sqrt{\left|a \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right|}\]

Reproduce

herbie shell --seed 2021014 
(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)))