ab-angle->ABCF C

Percentage Accurate: 79.1% → 79.1%

Time: 11.9s

Alternatives: 17

Speedup: 1.3×

• Could not converge on a ground truth

  1. a = 3.4211504114008985e+118
  2. b = 977512175654736000.0
  3. angle = 5.903273066192386e+231

• 35.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \frac{angle}{180}\\ {\left(a \cdot \cos t\_0\right)}^{2} + {\left(b \cdot \sin t\_0\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (PI) (/ angle 180.0))))
   (+ (pow (* a (cos t_0)) 2.0) (pow (* b (sin t_0)) 2.0))))

Initial Program: 79.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \frac{angle}{180}\\ {\left(a \cdot \cos t\_0\right)}^{2} + {\left(b \cdot \sin t\_0\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (PI) (/ angle 180.0))))
   (+ (pow (* a (cos t_0)) 2.0) (pow (* b (sin t_0)) 2.0))))

Alternative 1: 79.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ {\left(a \cdot \cos \left(e^{\log \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot t\_0\right) + \log t\_0}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle\_m}{180}\right)\right)}^{2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (sqrt (PI))))
   (+
    (pow
     (*
      a
      (cos (exp (+ (log (* (* 0.005555555555555556 angle_m) t_0)) (log t_0)))))
     2.0)
    (pow (* b (sin (* (PI) (/ angle_m 180.0)))) 2.0))))

Derivation

1. Initial program 81.5%

   \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. clear-num N/A

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

   5. inv-pow N/A

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

   6. pow-to-exp N/A

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   7. lower-exp.f64 N/A

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   8. lower-*.f64 N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   9. lower-log.f64 N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   10. *-commutative N/A

       \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{180}{\color{blue}{angle \cdot \mathsf{PI}\left(\right)}}\right) \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   11. associate-/r* N/A

       \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   12. lower-/.f64 N/A

       \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   13. lower-/.f64 45.0

       \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{\color{blue}{\frac{180}{angle}}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

4. Applied rewrites 45.0%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
5. Step-by-step derivation

   1. rem-log-exp N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\log \left(e^{\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   2. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(e^{\color{blue}{\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right) \cdot -1}}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   3. lift-log.f64 N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(e^{\color{blue}{\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   4. exp-to-pow N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left({\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)}^{-1}\right)}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   5. unpow-1 N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left(\frac{1}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}\right)}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   6. lift-/.f64 N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{1}{\color{blue}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   7. lift-/.f64 N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{1}{\frac{\color{blue}{\frac{180}{angle}}}{\mathsf{PI}\left(\right)}}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   8. associate-/r* N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{1}{\color{blue}{\frac{180}{angle \cdot \mathsf{PI}\left(\right)}}}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   9. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{1}{\frac{180}{\color{blue}{angle \cdot \mathsf{PI}\left(\right)}}}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   10. clear-num N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*l/ N/A

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

   13. lift-/.f64 N/A

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

   14. lift-PI.f64 N/A

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

   15. add-sqr-sqrt N/A

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

   16. associate-*r* N/A

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

   17. log-prod N/A

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

   18. lower-+.f64 N/A

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

6. Applied rewrites 44.9%

   \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\log \left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) + \log \left(\sqrt{\mathsf{PI}\left(\right)}\right)}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
7. Add Preprocessing

Alternative 2: 79.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(a \cdot \cos \left(e^{\log \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle\_m}{180}\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* a (cos (exp (log (* (* 0.005555555555555556 angle_m) (PI)))))) 2.0)
  (pow (* b (sin (* (PI) (/ angle_m 180.0)))) 2.0)))

Derivation

1. Initial program 81.5%

   \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. clear-num N/A

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

   5. inv-pow N/A

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

   6. pow-to-exp N/A

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   7. lower-exp.f64 N/A

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   8. lower-*.f64 N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   9. lower-log.f64 N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   10. *-commutative N/A

       \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{180}{\color{blue}{angle \cdot \mathsf{PI}\left(\right)}}\right) \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   11. associate-/r* N/A

       \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   12. lower-/.f64 N/A

       \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   13. lower-/.f64 45.0

       \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{\color{blue}{\frac{180}{angle}}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

4. Applied rewrites 45.0%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
5. Step-by-step derivation

   1. rem-log-exp N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\log \left(e^{\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   2. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(e^{\color{blue}{\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right) \cdot -1}}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   3. lift-log.f64 N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(e^{\color{blue}{\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   4. exp-to-pow N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left({\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)}^{-1}\right)}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   5. unpow-1 N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left(\frac{1}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}\right)}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   6. lift-/.f64 N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{1}{\color{blue}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   7. lift-/.f64 N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{1}{\frac{\color{blue}{\frac{180}{angle}}}{\mathsf{PI}\left(\right)}}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   8. associate-/r* N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{1}{\color{blue}{\frac{180}{angle \cdot \mathsf{PI}\left(\right)}}}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   9. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{1}{\frac{180}{\color{blue}{angle \cdot \mathsf{PI}\left(\right)}}}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   10. clear-num N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*l/ N/A

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

   13. lift-/.f64 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. lower-log.f64 45.0

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

6. Applied rewrites 45.0%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
7. Add Preprocessing

Alternative 3: 79.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(a \cdot \cos \left(e^{\log \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* a (cos (exp (log (* (* 0.005555555555555556 angle_m) (PI)))))) 2.0)
  (pow (* b (sin (* (PI) (* angle_m 0.005555555555555556)))) 2.0)))

Derivation

1. Initial program 81.5%

   \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. clear-num N/A

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

   5. inv-pow N/A

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

   6. pow-to-exp N/A

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   7. lower-exp.f64 N/A

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   8. lower-*.f64 N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   9. lower-log.f64 N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   10. *-commutative N/A

       \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{180}{\color{blue}{angle \cdot \mathsf{PI}\left(\right)}}\right) \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   11. associate-/r* N/A

       \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   12. lower-/.f64 N/A

       \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   13. lower-/.f64 45.0

       \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{\color{blue}{\frac{180}{angle}}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

4. Applied rewrites 45.0%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
5. Step-by-step derivation

   1. rem-log-exp N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\log \left(e^{\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   2. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(e^{\color{blue}{\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right) \cdot -1}}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   3. lift-log.f64 N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(e^{\color{blue}{\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   4. exp-to-pow N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left({\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)}^{-1}\right)}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   5. unpow-1 N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left(\frac{1}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}\right)}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   6. lift-/.f64 N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{1}{\color{blue}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   7. lift-/.f64 N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{1}{\frac{\color{blue}{\frac{180}{angle}}}{\mathsf{PI}\left(\right)}}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   8. associate-/r* N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{1}{\color{blue}{\frac{180}{angle \cdot \mathsf{PI}\left(\right)}}}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   9. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{1}{\frac{180}{\color{blue}{angle \cdot \mathsf{PI}\left(\right)}}}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   10. clear-num N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*l/ N/A

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

   13. lift-/.f64 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. lower-log.f64 45.0

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

6. Applied rewrites 45.0%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right)}^{2} \]

   2. div-inv N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right)}^{2} \]

   3. metadata-eval N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right)}^{2} \]

   4. lower-*.f64 45.0

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

8. Applied rewrites 45.0%

   \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(angle \cdot 0.005555555555555556\right)}\right)\right)}^{2} \]
9. Add Preprocessing

Alternative 4: 79.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \frac{angle\_m}{180}\\ {\left(a \cdot \cos t\_0\right)}^{2} + {\left(b \cdot \sin t\_0\right)}^{2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (PI) (/ angle_m 180.0))))
   (+ (pow (* a (cos t_0)) 2.0) (pow (* b (sin t_0)) 2.0))))

Derivation

1. Initial program 81.5%

   \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 5: 79.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(a \cdot \cos \left(\left(angle\_m \cdot -0.005555555555555556\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle\_m}{180}\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* a (cos (* (* angle_m -0.005555555555555556) (PI)))) 2.0)
  (pow (* b (sin (* (PI) (/ angle_m 180.0)))) 2.0)))

Derivation

1. Initial program 81.5%

   \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. frac-2neg N/A

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

   6. distribute-frac-neg N/A

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

   7. distribute-lft-neg-out N/A

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

   8. cos-neg N/A

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

   9. lower-cos.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. div-inv N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval 81.5

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

4. Applied rewrites 81.5%

   \[\leadsto {\left(a \cdot \color{blue}{\cos \left(\left(angle \cdot -0.005555555555555556\right) \cdot \mathsf{PI}\left(\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
5. Add Preprocessing

Alternative 6: 79.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \left(angle\_m \cdot 0.005555555555555556\right)\\ {\left(\sin t\_0 \cdot b\right)}^{2} + {\left(\cos t\_0 \cdot a\right)}^{2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (PI) (* angle_m 0.005555555555555556))))
   (+ (pow (* (sin t_0) b) 2.0) (pow (* (cos t_0) a) 2.0))))

Derivation

1. Initial program 81.5%

   \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. clear-num N/A

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

   5. inv-pow N/A

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

   6. pow-to-exp N/A

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   7. lower-exp.f64 N/A

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   8. lower-*.f64 N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   9. lower-log.f64 N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   10. *-commutative N/A

       \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{180}{\color{blue}{angle \cdot \mathsf{PI}\left(\right)}}\right) \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   11. associate-/r* N/A

       \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   12. lower-/.f64 N/A

       \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   13. lower-/.f64 45.0

       \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{\color{blue}{\frac{180}{angle}}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

4. Applied rewrites 45.0%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
5. Step-by-step derivation

   1. rem-log-exp N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\log \left(e^{\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   2. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(e^{\color{blue}{\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right) \cdot -1}}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   3. lift-log.f64 N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(e^{\color{blue}{\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   4. exp-to-pow N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left({\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)}^{-1}\right)}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   5. unpow-1 N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left(\frac{1}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}\right)}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   6. lift-/.f64 N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{1}{\color{blue}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   7. lift-/.f64 N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{1}{\frac{\color{blue}{\frac{180}{angle}}}{\mathsf{PI}\left(\right)}}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   8. associate-/r* N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{1}{\color{blue}{\frac{180}{angle \cdot \mathsf{PI}\left(\right)}}}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   9. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{1}{\frac{180}{\color{blue}{angle \cdot \mathsf{PI}\left(\right)}}}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   10. clear-num N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*l/ N/A

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

   13. lift-/.f64 N/A

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

   14. lift-PI.f64 N/A

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

   15. add-sqr-sqrt N/A

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

   16. associate-*r* N/A

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

   17. log-prod N/A

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

   18. lower-+.f64 N/A

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

6. Applied rewrites 44.9%

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

8. Applied rewrites 81.5%

   \[\leadsto \color{blue}{{\left(\sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot b\right)}^{2} + {\left(\cos \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot a\right)}^{2}} \]
9. Add Preprocessing

Alternative 7: 79.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle\_m}{180}\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+ (pow (* a 1.0) 2.0) (pow (* b (sin (* (PI) (/ angle_m 180.0)))) 2.0)))

Derivation

1. Initial program 81.5%

   \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
2. Add Preprocessing

3. Taylor expanded in angle around 0

   \[\leadsto {\left(a \cdot \color{blue}{1}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
4. Step-by-step derivation

5. Applied rewrites 81.2%

   \[\leadsto {\left(a \cdot \color{blue}{1}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
6. Add Preprocessing

Alternative 8: 65.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 5.5 \cdot 10^{-89}:\\ \;\;\;\;{\cos \left(\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\_m\right)}^{2} \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot angle\_m\right) \cdot angle\_m, b, {\left(\cos \left(\frac{angle\_m \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot a\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 5.5e-89)
   (* (pow (cos (* (* 0.005555555555555556 (PI)) angle_m)) 2.0) (* a a))
   (fma
    (* (* (* (* (* (PI) (PI)) b) 3.08641975308642e-5) angle_m) angle_m)
    b
    (pow (* (cos (/ (* angle_m (PI)) -180.0)) a) 2.0))))

Derivation

1. Split input into 2 regimes

2. if b < 5.50000000000000012e-89

   1. Initial program 81.5%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. clear-num N/A

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

      5. inv-pow N/A

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

      6. pow-to-exp N/A

         \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      7. lower-exp.f64 N/A

         \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      8. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      9. lower-log.f64 N/A

         \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      10. *-commutative N/A

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{180}{\color{blue}{angle \cdot \mathsf{PI}\left(\right)}}\right) \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      11. associate-/r* N/A

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      12. lower-/.f64 N/A

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      13. lower-/.f64 45.0

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{\color{blue}{\frac{180}{angle}}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   4. Applied rewrites 45.0%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {a}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {a}^{2}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \color{blue}{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}} \cdot {a}^{2} \]

      4. *-commutative N/A

         \[\leadsto {\cos \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}}^{2} \cdot {a}^{2} \]

      5. associate-*r* N/A

         \[\leadsto {\cos \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{180}\right)\right)}}^{2} \cdot {a}^{2} \]

      6. *-commutative N/A

         \[\leadsto {\cos \left(angle \cdot \color{blue}{\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)}^{2} \cdot {a}^{2} \]

      7. lower-cos.f64 N/A

         \[\leadsto {\color{blue}{\cos \left(angle \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}}^{2} \cdot {a}^{2} \]

      8. *-commutative N/A

         \[\leadsto {\cos \color{blue}{\left(\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}}^{2} \cdot {a}^{2} \]

      9. lower-*.f64 N/A

         \[\leadsto {\cos \color{blue}{\left(\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}}^{2} \cdot {a}^{2} \]

      10. lower-*.f64 N/A

          \[\leadsto {\cos \left(\color{blue}{\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)} \cdot angle\right)}^{2} \cdot {a}^{2} \]

      11. lower-PI.f64 N/A

          \[\leadsto {\cos \left(\left(\frac{1}{180} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot angle\right)}^{2} \cdot {a}^{2} \]

      12. unpow2 N/A

          \[\leadsto {\cos \left(\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot \color{blue}{\left(a \cdot a\right)} \]

      13. lower-*.f64 66.7

          \[\leadsto {\cos \left(\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot \color{blue}{\left(a \cdot a\right)} \]

   7. Applied rewrites 66.7%

      \[\leadsto \color{blue}{{\cos \left(\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot \left(a \cdot a\right)} \]

   if 5.50000000000000012e-89 < b

   1. Initial program 81.5%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), b, {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)} \]

   4. Applied rewrites 75.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot b, b, {\left(\cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot a\right)}^{2}\right)} \]

   5. Taylor expanded in angle around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}, b, {\left(\cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot a\right)}^{2}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{32400} \cdot \color{blue}{\left(\left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)}, b, {\left(\cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot a\right)}^{2}\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{32400} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}}, b, {\left(\cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot a\right)}^{2}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\left(angle \cdot angle\right)}, b, {\left(\cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot a\right)}^{2}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{32400} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot angle\right) \cdot angle}, b, {\left(\cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot a\right)}^{2}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{32400} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot angle\right) \cdot angle}, b, {\left(\cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot a\right)}^{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{32400} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot angle\right)} \cdot angle, b, {\left(\cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot a\right)}^{2}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} \cdot angle\right) \cdot angle, b, {\left(\cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot a\right)}^{2}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} \cdot angle\right) \cdot angle, b, {\left(\cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot a\right)}^{2}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot b\right)} \cdot \frac{1}{32400}\right) \cdot angle\right) \cdot angle, b, {\left(\cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot a\right)}^{2}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot b\right)} \cdot \frac{1}{32400}\right) \cdot angle\right) \cdot angle, b, {\left(\cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot a\right)}^{2}\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot b\right) \cdot \frac{1}{32400}\right) \cdot angle\right) \cdot angle, b, {\left(\cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot a\right)}^{2}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot b\right) \cdot \frac{1}{32400}\right) \cdot angle\right) \cdot angle, b, {\left(\cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot a\right)}^{2}\right) \]

      13. lower-PI.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot b\right) \cdot \frac{1}{32400}\right) \cdot angle\right) \cdot angle, b, {\left(\cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot a\right)}^{2}\right) \]

      14. lower-PI.f64 77.9

          \[\leadsto \mathsf{fma}\left(\left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot angle\right) \cdot angle, b, {\left(\cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot a\right)}^{2}\right) \]

   7. Applied rewrites 77.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot angle\right) \cdot angle}, b, {\left(\cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot a\right)}^{2}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 63.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 5.5 \cdot 10^{-89}:\\ \;\;\;\;{\cos \left(\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\_m\right)}^{2} \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right), angle\_m \cdot angle\_m, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\_m\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 5.5e-89)
   (* (pow (cos (* (* 0.005555555555555556 (PI)) angle_m)) 2.0) (* a a))
   (if (<= b 1.2e+167)
     (fma
      (* (* (PI) (PI)) (* 3.08641975308642e-5 (* b b)))
      (* angle_m angle_m)
      (* a a))
     (* (pow (* (* b (PI)) angle_m) 2.0) 3.08641975308642e-5))))

Derivation

1. Split input into 3 regimes

2. if b < 5.50000000000000012e-89

   1. Initial program 81.5%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. clear-num N/A

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

      5. inv-pow N/A

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

      6. pow-to-exp N/A

         \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      7. lower-exp.f64 N/A

         \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      8. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      9. lower-log.f64 N/A

         \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      10. *-commutative N/A

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{180}{\color{blue}{angle \cdot \mathsf{PI}\left(\right)}}\right) \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      11. associate-/r* N/A

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      12. lower-/.f64 N/A

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      13. lower-/.f64 45.0

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{\color{blue}{\frac{180}{angle}}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   4. Applied rewrites 45.0%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {a}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {a}^{2}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \color{blue}{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}} \cdot {a}^{2} \]

      4. *-commutative N/A

         \[\leadsto {\cos \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}}^{2} \cdot {a}^{2} \]

      5. associate-*r* N/A

         \[\leadsto {\cos \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{180}\right)\right)}}^{2} \cdot {a}^{2} \]

      6. *-commutative N/A

         \[\leadsto {\cos \left(angle \cdot \color{blue}{\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)}^{2} \cdot {a}^{2} \]

      7. lower-cos.f64 N/A

         \[\leadsto {\color{blue}{\cos \left(angle \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}}^{2} \cdot {a}^{2} \]

      8. *-commutative N/A

         \[\leadsto {\cos \color{blue}{\left(\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}}^{2} \cdot {a}^{2} \]

      9. lower-*.f64 N/A

         \[\leadsto {\cos \color{blue}{\left(\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}}^{2} \cdot {a}^{2} \]

      10. lower-*.f64 N/A

          \[\leadsto {\cos \left(\color{blue}{\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)} \cdot angle\right)}^{2} \cdot {a}^{2} \]

      11. lower-PI.f64 N/A

          \[\leadsto {\cos \left(\left(\frac{1}{180} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot angle\right)}^{2} \cdot {a}^{2} \]

      12. unpow2 N/A

          \[\leadsto {\cos \left(\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot \color{blue}{\left(a \cdot a\right)} \]

      13. lower-*.f64 66.7

          \[\leadsto {\cos \left(\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot \color{blue}{\left(a \cdot a\right)} \]

   7. Applied rewrites 66.7%

      \[\leadsto \color{blue}{{\cos \left(\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot \left(a \cdot a\right)} \]

   if 5.50000000000000012e-89 < b < 1.19999999999999999e167

   1. Initial program 72.9%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. clear-num N/A

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

      5. inv-pow N/A

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

      6. pow-to-exp N/A

         \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      7. lower-exp.f64 N/A

         \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      8. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      9. lower-log.f64 N/A

         \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      10. *-commutative N/A

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{180}{\color{blue}{angle \cdot \mathsf{PI}\left(\right)}}\right) \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      11. associate-/r* N/A

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      12. lower-/.f64 N/A

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      13. lower-/.f64 44.3

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{\color{blue}{\frac{180}{angle}}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   4. Applied rewrites 44.3%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}} + {a}^{2} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), {angle}^{2}, {a}^{2}\right)} \]

   7. Applied rewrites 37.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5} \cdot b, b, -3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right), angle \cdot angle, a \cdot a\right)} \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{32400} \cdot {b}^{2}\right), angle \cdot angle, a \cdot a\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 67.4%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right), angle \cdot angle, a \cdot a\right) \]

   if 1.19999999999999999e167 < b

   1. Initial program 99.8%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. clear-num N/A

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

      5. inv-pow N/A

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

      6. pow-to-exp N/A

         \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      7. lower-exp.f64 N/A

         \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      8. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      9. lower-log.f64 N/A

         \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      10. *-commutative N/A

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{180}{\color{blue}{angle \cdot \mathsf{PI}\left(\right)}}\right) \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      11. associate-/r* N/A

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      12. lower-/.f64 N/A

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      13. lower-/.f64 46.1

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{\color{blue}{\frac{180}{angle}}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   4. Applied rewrites 46.1%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}} + {a}^{2} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), {angle}^{2}, {a}^{2}\right)} \]

   7. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5} \cdot b, b, -3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right), angle \cdot angle, a \cdot a\right)} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{1}{32400} \cdot \color{blue}{\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 69.2%

       \[\leadsto \left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \color{blue}{\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right) \cdot b\right)} \]

   12. Applied rewrites 85.2%

       \[\leadsto \color{blue}{{\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.5 \cdot 10^{-89}:\\ \;\;\;\;{\cos \left(\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right), angle \cdot angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 5.5 \cdot 10^{-89}:\\ \;\;\;\;{\cos \left(-0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\_m\right)\right)}^{2} \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right), angle\_m \cdot angle\_m, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\_m\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 5.5e-89)
   (* (pow (cos (* -0.005555555555555556 (* (PI) angle_m))) 2.0) (* a a))
   (if (<= b 1.2e+167)
     (fma
      (* (* (PI) (PI)) (* 3.08641975308642e-5 (* b b)))
      (* angle_m angle_m)
      (* a a))
     (* (pow (* (* b (PI)) angle_m) 2.0) 3.08641975308642e-5))))

Derivation

1. Split input into 3 regimes

2. if b < 5.50000000000000012e-89

   1. Initial program 81.5%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), b, {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)} \]

   4. Applied rewrites 78.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot b, b, {\left(\cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot a\right)}^{2}\right)} \]

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{2} \cdot {\cos \left(\frac{-1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{{\cos \left(\frac{-1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {a}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{{\cos \left(\frac{-1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {a}^{2}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \color{blue}{{\cos \left(\frac{-1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}} \cdot {a}^{2} \]

      4. lower-cos.f64 N/A

         \[\leadsto {\color{blue}{\cos \left(\frac{-1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}^{2} \cdot {a}^{2} \]

      5. lower-*.f64 N/A

         \[\leadsto {\cos \color{blue}{\left(\frac{-1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}^{2} \cdot {a}^{2} \]

      6. *-commutative N/A

         \[\leadsto {\cos \left(\frac{-1}{180} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right)}^{2} \cdot {a}^{2} \]

      7. lower-*.f64 N/A

         \[\leadsto {\cos \left(\frac{-1}{180} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right)}^{2} \cdot {a}^{2} \]

      8. lower-PI.f64 N/A

         \[\leadsto {\cos \left(\frac{-1}{180} \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot angle\right)\right)}^{2} \cdot {a}^{2} \]

      9. unpow2 N/A

         \[\leadsto {\cos \left(\frac{-1}{180} \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}^{2} \cdot \color{blue}{\left(a \cdot a\right)} \]

      10. lower-*.f64 66.6

          \[\leadsto {\cos \left(-0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}^{2} \cdot \color{blue}{\left(a \cdot a\right)} \]

   7. Applied rewrites 66.6%

      \[\leadsto \color{blue}{{\cos \left(-0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}^{2} \cdot \left(a \cdot a\right)} \]

   if 5.50000000000000012e-89 < b < 1.19999999999999999e167

   1. Initial program 72.9%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. clear-num N/A

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

      5. inv-pow N/A

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

      6. pow-to-exp N/A

         \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      7. lower-exp.f64 N/A

         \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      8. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      9. lower-log.f64 N/A

         \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      10. *-commutative N/A

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{180}{\color{blue}{angle \cdot \mathsf{PI}\left(\right)}}\right) \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      11. associate-/r* N/A

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      12. lower-/.f64 N/A

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      13. lower-/.f64 44.3

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{\color{blue}{\frac{180}{angle}}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   4. Applied rewrites 44.3%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}} + {a}^{2} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), {angle}^{2}, {a}^{2}\right)} \]

   7. Applied rewrites 37.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5} \cdot b, b, -3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right), angle \cdot angle, a \cdot a\right)} \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{32400} \cdot {b}^{2}\right), angle \cdot angle, a \cdot a\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 67.4%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right), angle \cdot angle, a \cdot a\right) \]

   if 1.19999999999999999e167 < b

   1. Initial program 99.8%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. clear-num N/A

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

      5. inv-pow N/A

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

      6. pow-to-exp N/A

         \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      7. lower-exp.f64 N/A

         \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      8. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      9. lower-log.f64 N/A

         \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      10. *-commutative N/A

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{180}{\color{blue}{angle \cdot \mathsf{PI}\left(\right)}}\right) \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      11. associate-/r* N/A

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      12. lower-/.f64 N/A

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      13. lower-/.f64 46.1

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{\color{blue}{\frac{180}{angle}}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   4. Applied rewrites 46.1%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}} + {a}^{2} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), {angle}^{2}, {a}^{2}\right)} \]

   7. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5} \cdot b, b, -3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right), angle \cdot angle, a \cdot a\right)} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{1}{32400} \cdot \color{blue}{\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 69.2%

       \[\leadsto \left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \color{blue}{\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right) \cdot b\right)} \]

   12. Applied rewrites 85.2%

       \[\leadsto \color{blue}{{\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.5 \cdot 10^{-89}:\\ \;\;\;\;{\cos \left(-0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}^{2} \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right), angle \cdot angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.7% accurate, 3.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{-32}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right), angle\_m \cdot angle\_m, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\_m\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 2.9e-32)
   (* a a)
   (if (<= b 1.2e+167)
     (fma
      (* (* (PI) (PI)) (* 3.08641975308642e-5 (* b b)))
      (* angle_m angle_m)
      (* a a))
     (* (pow (* (* b (PI)) angle_m) 2.0) 3.08641975308642e-5))))

Derivation

1. Split input into 3 regimes

2. if b < 2.89999999999999996e-32

   1. Initial program 80.1%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{a}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \color{blue}{a \cdot a} \]

      2. lower-*.f64 65.8

         \[\leadsto \color{blue}{a \cdot a} \]

   5. Applied rewrites 65.8%

      \[\leadsto \color{blue}{a \cdot a} \]

   if 2.89999999999999996e-32 < b < 1.19999999999999999e167

   1. Initial program 76.3%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. clear-num N/A

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

      5. inv-pow N/A

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

      6. pow-to-exp N/A

         \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      7. lower-exp.f64 N/A

         \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      8. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      9. lower-log.f64 N/A

         \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      10. *-commutative N/A

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{180}{\color{blue}{angle \cdot \mathsf{PI}\left(\right)}}\right) \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      11. associate-/r* N/A

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      12. lower-/.f64 N/A

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      13. lower-/.f64 42.3

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{\color{blue}{\frac{180}{angle}}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   4. Applied rewrites 42.3%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}} + {a}^{2} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), {angle}^{2}, {a}^{2}\right)} \]

   7. Applied rewrites 41.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5} \cdot b, b, -3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right), angle \cdot angle, a \cdot a\right)} \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{32400} \cdot {b}^{2}\right), angle \cdot angle, a \cdot a\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 70.7%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right), angle \cdot angle, a \cdot a\right) \]

   if 1.19999999999999999e167 < b

   1. Initial program 99.8%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. clear-num N/A

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

      5. inv-pow N/A

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

      6. pow-to-exp N/A

         \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      7. lower-exp.f64 N/A

         \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      8. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      9. lower-log.f64 N/A

         \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      10. *-commutative N/A

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{180}{\color{blue}{angle \cdot \mathsf{PI}\left(\right)}}\right) \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      11. associate-/r* N/A

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      12. lower-/.f64 N/A

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      13. lower-/.f64 46.1

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{\color{blue}{\frac{180}{angle}}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   4. Applied rewrites 46.1%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}} + {a}^{2} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), {angle}^{2}, {a}^{2}\right)} \]

   7. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5} \cdot b, b, -3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right), angle \cdot angle, a \cdot a\right)} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{1}{32400} \cdot \color{blue}{\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 69.2%

       \[\leadsto \left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \color{blue}{\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right) \cdot b\right)} \]

   12. Applied rewrites 85.2%

       \[\leadsto \color{blue}{{\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{-32}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right), angle \cdot angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.7% accurate, 8.3× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 6 \cdot 10^{-173}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right), angle\_m \cdot angle\_m, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= (/ angle_m 180.0) 6e-173)
   (* a a)
   (fma
    (* (* (PI) (PI)) (* 3.08641975308642e-5 (* b b)))
    (* angle_m angle_m)
    (* a a))))

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 6.0000000000000002e-173

   1. Initial program 85.2%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{a}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \color{blue}{a \cdot a} \]

      2. lower-*.f64 67.5

         \[\leadsto \color{blue}{a \cdot a} \]

   5. Applied rewrites 67.5%

      \[\leadsto \color{blue}{a \cdot a} \]

   if 6.0000000000000002e-173 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 75.6%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. clear-num N/A

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

      5. inv-pow N/A

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

      6. pow-to-exp N/A

         \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      7. lower-exp.f64 N/A

         \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      8. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      9. lower-log.f64 N/A

         \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      10. *-commutative N/A

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{180}{\color{blue}{angle \cdot \mathsf{PI}\left(\right)}}\right) \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      11. associate-/r* N/A

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      12. lower-/.f64 N/A

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      13. lower-/.f64 75.7

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{\color{blue}{\frac{180}{angle}}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   4. Applied rewrites 75.7%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}} + {a}^{2} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), {angle}^{2}, {a}^{2}\right)} \]

   7. Applied rewrites 39.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5} \cdot b, b, -3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right), angle \cdot angle, a \cdot a\right)} \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{32400} \cdot {b}^{2}\right), angle \cdot angle, a \cdot a\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 65.5%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right), angle \cdot angle, a \cdot a\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 6 \cdot 10^{-173}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right), angle \cdot angle, a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 66.7% accurate, 8.3× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 6 \cdot 10^{-173}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right) \cdot b, angle\_m \cdot angle\_m, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= (/ angle_m 180.0) 6e-173)
   (* a a)
   (fma
    (* (* (* (* (PI) (PI)) 3.08641975308642e-5) b) b)
    (* angle_m angle_m)
    (* a a))))

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 6.0000000000000002e-173

   1. Initial program 85.2%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{a}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \color{blue}{a \cdot a} \]

      2. lower-*.f64 67.5

         \[\leadsto \color{blue}{a \cdot a} \]

   5. Applied rewrites 67.5%

      \[\leadsto \color{blue}{a \cdot a} \]

   if 6.0000000000000002e-173 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 75.6%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. clear-num N/A

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

      5. inv-pow N/A

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

      6. pow-to-exp N/A

         \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      7. lower-exp.f64 N/A

         \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      8. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      9. lower-log.f64 N/A

         \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      10. *-commutative N/A

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{180}{\color{blue}{angle \cdot \mathsf{PI}\left(\right)}}\right) \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      11. associate-/r* N/A

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      12. lower-/.f64 N/A

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      13. lower-/.f64 75.7

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{\color{blue}{\frac{180}{angle}}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   4. Applied rewrites 75.7%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}} + {a}^{2} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), {angle}^{2}, {a}^{2}\right)} \]

   7. Applied rewrites 39.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5} \cdot b, b, -3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right), angle \cdot angle, a \cdot a\right)} \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \color{blue}{angle} \cdot angle, a \cdot a\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 65.5%

       \[\leadsto \mathsf{fma}\left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right) \cdot b, \color{blue}{angle} \cdot angle, a \cdot a\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 6 \cdot 10^{-173}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right) \cdot b, angle \cdot angle, a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 60.6% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 5.4 \cdot 10^{+131}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(angle\_m \cdot angle\_m\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 5.4e+131)
   (* a a)
   (* (* (* (* (* angle_m angle_m) 3.08641975308642e-5) b) (* (PI) (PI))) b)))

Derivation

1. Split input into 2 regimes

2. if b < 5.40000000000000008e131

   1. Initial program 79.5%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{a}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \color{blue}{a \cdot a} \]

      2. lower-*.f64 64.7

         \[\leadsto \color{blue}{a \cdot a} \]

   5. Applied rewrites 64.7%

      \[\leadsto \color{blue}{a \cdot a} \]

   if 5.40000000000000008e131 < b

   1. Initial program 97.0%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. clear-num N/A

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

      5. inv-pow N/A

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

      6. pow-to-exp N/A

         \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      7. lower-exp.f64 N/A

         \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      8. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      9. lower-log.f64 N/A

         \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      10. *-commutative N/A

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{180}{\color{blue}{angle \cdot \mathsf{PI}\left(\right)}}\right) \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      11. associate-/r* N/A

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      12. lower-/.f64 N/A

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      13. lower-/.f64 45.4

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{\color{blue}{\frac{180}{angle}}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   4. Applied rewrites 45.4%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}} + {a}^{2} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), {angle}^{2}, {a}^{2}\right)} \]

   7. Applied rewrites 48.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5} \cdot b, b, -3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right), angle \cdot angle, a \cdot a\right)} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{1}{32400} \cdot \color{blue}{\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 69.2%

       \[\leadsto \left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \color{blue}{\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right) \cdot b\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 70.0%

       \[\leadsto \left(\left(\left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.4 \cdot 10^{+131}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 60.6% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 5.4 \cdot 10^{+131}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(angle\_m \cdot angle\_m\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 5.4e+131)
   (* a a)
   (* (* (* (* angle_m angle_m) 3.08641975308642e-5) b) (* (* (PI) (PI)) b))))

Derivation

1. Split input into 2 regimes

2. if b < 5.40000000000000008e131

   1. Initial program 79.5%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{a}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \color{blue}{a \cdot a} \]

      2. lower-*.f64 64.7

         \[\leadsto \color{blue}{a \cdot a} \]

   5. Applied rewrites 64.7%

      \[\leadsto \color{blue}{a \cdot a} \]

   if 5.40000000000000008e131 < b

   1. Initial program 97.0%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. clear-num N/A

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

      5. inv-pow N/A

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

      6. pow-to-exp N/A

         \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      7. lower-exp.f64 N/A

         \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      8. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      9. lower-log.f64 N/A

         \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      10. *-commutative N/A

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{180}{\color{blue}{angle \cdot \mathsf{PI}\left(\right)}}\right) \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      11. associate-/r* N/A

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      12. lower-/.f64 N/A

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      13. lower-/.f64 45.4

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{\color{blue}{\frac{180}{angle}}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   4. Applied rewrites 45.4%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}} + {a}^{2} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), {angle}^{2}, {a}^{2}\right)} \]

   7. Applied rewrites 48.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5} \cdot b, b, -3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right), angle \cdot angle, a \cdot a\right)} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{1}{32400} \cdot \color{blue}{\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 69.2%

       \[\leadsto \left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \color{blue}{\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right) \cdot b\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 70.0%

       \[\leadsto \left(\left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{b}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.4 \cdot 10^{+131}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 59.9% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 5.4 \cdot 10^{+131}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(angle\_m \cdot angle\_m\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right) \cdot b\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 5.4e+131)
   (* a a)
   (* (* (* angle_m angle_m) 3.08641975308642e-5) (* (* (* (PI) (PI)) b) b))))

Derivation

1. Split input into 2 regimes

2. if b < 5.40000000000000008e131

   1. Initial program 79.5%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{a}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \color{blue}{a \cdot a} \]

      2. lower-*.f64 64.7

         \[\leadsto \color{blue}{a \cdot a} \]

   5. Applied rewrites 64.7%

      \[\leadsto \color{blue}{a \cdot a} \]

   if 5.40000000000000008e131 < b

   1. Initial program 97.0%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. clear-num N/A

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

      5. inv-pow N/A

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

      6. pow-to-exp N/A

         \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      7. lower-exp.f64 N/A

         \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      8. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      9. lower-log.f64 N/A

         \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      10. *-commutative N/A

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{180}{\color{blue}{angle \cdot \mathsf{PI}\left(\right)}}\right) \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      11. associate-/r* N/A

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      12. lower-/.f64 N/A

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

      13. lower-/.f64 45.4

          \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\frac{\color{blue}{\frac{180}{angle}}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   4. Applied rewrites 45.4%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}} + {a}^{2} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), {angle}^{2}, {a}^{2}\right)} \]

   7. Applied rewrites 48.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5} \cdot b, b, -3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right), angle \cdot angle, a \cdot a\right)} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{1}{32400} \cdot \color{blue}{\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 69.2%

       \[\leadsto \left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \color{blue}{\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right) \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.4 \cdot 10^{+131}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right) \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 56.1% accurate, 74.7× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ a \cdot a \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m) :precision binary64 (* a a))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return a * a;
}

Fortran

angle_m = abs(angle)
real(8) function code(a, b, angle_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle_m
    code = a * a
end function

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return a * a;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return a * a

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64(a * a)
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = a * a;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(a * a), $MachinePrecision]

Derivation

1. Initial program 81.5%

   \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
2. Add Preprocessing

3. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{{a}^{2}} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \color{blue}{a \cdot a} \]

   2. lower-*.f64 61.5

      \[\leadsto \color{blue}{a \cdot a} \]

5. Applied rewrites 61.5%

   \[\leadsto \color{blue}{a \cdot a} \]

6. Final simplification 61.5%

   \[\leadsto a \cdot a \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024298 
(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)))