ab-angle->ABCF A

Percentage Accurate: 80.3% → 80.2%

Time: 14.6s

Alternatives: 19

Speedup: 1.3×

• Could not converge on a ground truth

  1. a = 1.808092054810868e-247
  2. b = 2.8026197639808644e+134
  3. angle = 1.901445751054359e+34

• 36.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

Initial Program: 80.3% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 80.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \sqrt[3]{\mathsf{PI}\left(\right)}\\ {\left(\cos \left(\left({t\_0}^{2} \cdot t\_0\right) \cdot e^{-\log \left(\frac{180}{angle\_m}\right)}\right) \cdot b\right)}^{2} + {\left(\sin \left(\frac{1}{\frac{\frac{180}{angle\_m}}{\mathsf{PI}\left(\right)}}\right) \cdot a\right)}^{2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (cbrt (PI))))
   (+
    (pow
     (* (cos (* (* (pow t_0 2.0) t_0) (exp (- (log (/ 180.0 angle_m)))))) b)
     2.0)
    (pow (* (sin (/ 1.0 (/ (/ 180.0 angle_m) (PI)))) a) 2.0))))

Derivation

1. Initial program 79.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. clear-num N/A

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

   5. lower-/.f64 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 79.3

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

4. Applied rewrites 79.3%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-/.f64 N/A

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

   4. inv-pow N/A

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

   5. pow-to-exp N/A

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

   6. lower-exp.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-log.f64 39.8

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

6. Applied rewrites 39.8%

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

   1. lift-PI.f64 N/A

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

   2. add-cube-cbrt N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{1}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}\right)\right)}^{2} + {\left(b \cdot \cos \left(e^{\log \left(\frac{180}{angle}\right) \cdot -1} \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right)\right)}^{2} \]

   3. lower-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{1}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}\right)\right)}^{2} + {\left(b \cdot \cos \left(e^{\log \left(\frac{180}{angle}\right) \cdot -1} \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right)\right)}^{2} \]

   4. pow2 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{1}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}\right)\right)}^{2} + {\left(b \cdot \cos \left(e^{\log \left(\frac{180}{angle}\right) \cdot -1} \cdot \left(\color{blue}{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)\right)}^{2} \]

   5. lower-pow.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{1}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}\right)\right)}^{2} + {\left(b \cdot \cos \left(e^{\log \left(\frac{180}{angle}\right) \cdot -1} \cdot \left(\color{blue}{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)\right)}^{2} \]

   6. lift-PI.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{1}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}\right)\right)}^{2} + {\left(b \cdot \cos \left(e^{\log \left(\frac{180}{angle}\right) \cdot -1} \cdot \left({\left(\sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)}}\right)}^{2} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)\right)}^{2} \]

   7. lower-cbrt.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{1}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}\right)\right)}^{2} + {\left(b \cdot \cos \left(e^{\log \left(\frac{180}{angle}\right) \cdot -1} \cdot \left({\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}}^{2} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)\right)}^{2} \]

   8. lift-PI.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{1}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}\right)\right)}^{2} + {\left(b \cdot \cos \left(e^{\log \left(\frac{180}{angle}\right) \cdot -1} \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)}}\right)\right)\right)}^{2} \]

   9. lower-cbrt.f64 39.8

      \[\leadsto {\left(a \cdot \sin \left(\frac{1}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}\right)\right)}^{2} + {\left(b \cdot \cos \left(e^{\log \left(\frac{180}{angle}\right) \cdot -1} \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \color{blue}{\sqrt[3]{\mathsf{PI}\left(\right)}}\right)\right)\right)}^{2} \]

8. Applied rewrites 39.8%

   \[\leadsto {\left(a \cdot \sin \left(\frac{1}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}\right)\right)}^{2} + {\left(b \cdot \cos \left(e^{\log \left(\frac{180}{angle}\right) \cdot -1} \cdot \color{blue}{\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right)\right)}^{2} \]

9. Final simplification 39.8%

   \[\leadsto {\left(\cos \left(\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot e^{-\log \left(\frac{180}{angle}\right)}\right) \cdot b\right)}^{2} + {\left(\sin \left(\frac{1}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}\right) \cdot a\right)}^{2} \]
10. Add Preprocessing

Alternative 2: 80.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(\cos \left(\frac{-0.005555555555555556}{\frac{-1}{angle\_m}} \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2} + {\left(\sin \left(\frac{1}{\frac{\frac{180}{angle\_m}}{\mathsf{PI}\left(\right)}}\right) \cdot a\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* (cos (* (/ -0.005555555555555556 (/ -1.0 angle_m)) (PI))) b) 2.0)
  (pow (* (sin (/ 1.0 (/ (/ 180.0 angle_m) (PI)))) a) 2.0)))

Derivation

1. Initial program 79.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. clear-num N/A

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

   5. lower-/.f64 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 79.3

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

4. Applied rewrites 79.3%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. div-inv N/A

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

   4. unpow-1 N/A

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

   5. lift-pow.f64 N/A

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

   6. associate-/r* N/A

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

   7. metadata-eval N/A

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

   8. frac-2neg N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f64 N/A

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

   12. metadata-eval N/A

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

   13. lift-pow.f64 N/A

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

   14. unpow-1 N/A

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

   15. distribute-neg-frac N/A

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

   16. metadata-eval N/A

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

   17. lower-/.f64 79.4

       \[\leadsto {\left(a \cdot \sin \left(\frac{1}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{-0.005555555555555556}{\color{blue}{\frac{-1}{angle}}} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

6. Applied rewrites 79.4%

   \[\leadsto {\left(a \cdot \sin \left(\frac{1}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\frac{-0.005555555555555556}{\frac{-1}{angle}}} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

7. Final simplification 79.4%

   \[\leadsto {\left(\cos \left(\frac{-0.005555555555555556}{\frac{-1}{angle}} \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2} + {\left(\sin \left(\frac{1}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}\right) \cdot a\right)}^{2} \]
8. Add Preprocessing

Alternative 3: 80.3% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 79.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. clear-num N/A

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

   5. lower-/.f64 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 79.3

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

4. Applied rewrites 79.3%

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

5. Final simplification 79.3%

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

Alternative 4: 80.2% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 79.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. clear-num N/A

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

   5. lower-/.f64 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 79.3

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

4. Applied rewrites 79.3%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lift-*.f64 N/A

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

   5. frac-2neg N/A

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

   6. metadata-eval N/A

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

   7. div-inv N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. distribute-lft-neg-in N/A

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

   13. lower-*.f64 N/A

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

   14. lower-neg.f64 79.3

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

6. Applied rewrites 79.3%

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

7. Final simplification 79.3%

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

Alternative 5: 80.3% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 79.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. clear-num N/A

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

   5. lower-/.f64 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 79.3

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

4. Applied rewrites 79.3%

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

5. Taylor expanded in angle around 0

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

7. Applied rewrites 79.2%

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

8. Final simplification 79.2%

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

Alternative 6: 79.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)\right) \cdot angle\_m\right) \cdot angle\_m, a, {\left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{-180}{angle\_m}}\right) \cdot b\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot a, a, {\left(1 \cdot b\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= (/ angle_m 180.0) 5e+21)
   (fma
    (* (* (* 3.08641975308642e-5 (* (* (PI) (PI)) a)) angle_m) angle_m)
    a
    (pow (* (cos (/ (PI) (/ -180.0 angle_m))) b) 2.0))
   (fma
    (* (pow (sin (* (* 0.005555555555555556 angle_m) (PI))) 2.0) a)
    a
    (pow (* 1.0 b) 2.0))))

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 5e21

   1. Initial program 86.9%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 84.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. lift-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 84.0

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

   6. Applied rewrites 84.0%

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

   7. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. lower-PI.f64 84.4

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

   9. Applied rewrites 84.4%

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

   if 5e21 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 59.3%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 59.2%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 60.4%

      \[\leadsto \mathsf{fma}\left({\sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{2} \cdot a, a, {\left(\color{blue}{1} \cdot b\right)}^{2}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)\right) \cdot angle\right) \cdot angle, a, {\left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{-180}{angle}}\right) \cdot b\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot a, a, {\left(1 \cdot b\right)}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.4% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 79.2%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 79.1%

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

6. Final simplification 79.1%

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

Alternative 8: 73.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;a \leq 5 \cdot 10^{+134}:\\ \;\;\;\;{\left(\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\_m\right) \cdot 0.005555555555555556\right) \cdot b\right)}^{2} + \left(\left(\left(\left(a \cdot a\right) \cdot angle\_m\right) \cdot angle\_m\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(t\_0 \cdot a\right)\right) \cdot angle\_m\right) \cdot angle\_m, a, {\left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{-180}{angle\_m}}\right) \cdot b\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (PI) (PI))))
   (if (<= a 5e+134)
     (+
      (pow (* (cos (* (* (PI) angle_m) 0.005555555555555556)) b) 2.0)
      (* (* (* (* (* a a) angle_m) angle_m) 3.08641975308642e-5) t_0))
     (fma
      (* (* (* 3.08641975308642e-5 (* t_0 a)) angle_m) angle_m)
      a
      (pow (* (cos (/ (PI) (/ -180.0 angle_m))) b) 2.0)))))

Derivation

1. Split input into 2 regimes

2. if a < 4.99999999999999981e134

   1. Initial program 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-*l/ N/A

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

      5. div-inv N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 76.9

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \frac{\mathsf{PI}\left(\right)}{\color{blue}{{angle}^{-1}}}\right)\right)}^{2} \]

   4. Applied rewrites 76.9%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(0.005555555555555556 \cdot \frac{\mathsf{PI}\left(\right)}{{angle}^{-1}}\right)}\right)}^{2} \]

   5. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. lower-PI.f64 70.8

          \[\leadsto \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \frac{\mathsf{PI}\left(\right)}{{angle}^{-1}}\right)\right)}^{2} \]

   7. Applied rewrites 70.8%

      \[\leadsto \color{blue}{\left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \frac{\mathsf{PI}\left(\right)}{{angle}^{-1}}\right)\right)}^{2} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-1 N/A

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

      5. remove-double-div N/A

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

      6. lower-*.f64 70.8

         \[\leadsto \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right)\right)}^{2} \]

   9. Applied rewrites 70.8%

      \[\leadsto \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right)\right)}^{2} \]

   if 4.99999999999999981e134 < a

   1. Initial program 97.1%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 90.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. lift-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 90.6

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

   6. Applied rewrites 90.6%

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

   7. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. lower-PI.f64 97.1

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

   9. Applied rewrites 97.1%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5 \cdot 10^{+134}:\\ \;\;\;\;{\left(\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b\right)}^{2} + \left(\left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)\right) \cdot angle\right) \cdot angle, a, {\left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{-180}{angle}}\right) \cdot b\right)}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\\ t_1 := \mathsf{PI}\left(\right) \cdot angle\_m\\ \mathbf{if}\;a \leq 10^{+60}:\\ \;\;\;\;{\left(\cos \left(t\_1 \cdot 0.005555555555555556\right) \cdot b\right)}^{2} + \left(\left(\left(\left(a \cdot a\right) \cdot angle\_m\right) \cdot angle\_m\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(angle\_m \cdot angle\_m\right) \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot t\_0, a, {\left(\cos \left(\frac{t\_1}{-180}\right) \cdot b\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (PI) (PI))) (t_1 (* (PI) angle_m)))
   (if (<= a 1e+60)
     (+
      (pow (* (cos (* t_1 0.005555555555555556)) b) 2.0)
      (* (* (* (* (* a a) angle_m) angle_m) 3.08641975308642e-5) t_0))
     (fma
      (* (* (* (* angle_m angle_m) a) 3.08641975308642e-5) t_0)
      a
      (pow (* (cos (/ t_1 -180.0)) b) 2.0)))))

Derivation

1. Split input into 2 regimes

2. if a < 9.9999999999999995e59

   1. Initial program 77.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-*l/ N/A

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

      5. div-inv N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 77.5

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \frac{\mathsf{PI}\left(\right)}{\color{blue}{{angle}^{-1}}}\right)\right)}^{2} \]

   4. Applied rewrites 77.5%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(0.005555555555555556 \cdot \frac{\mathsf{PI}\left(\right)}{{angle}^{-1}}\right)}\right)}^{2} \]

   5. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. lower-PI.f64 71.4

          \[\leadsto \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \frac{\mathsf{PI}\left(\right)}{{angle}^{-1}}\right)\right)}^{2} \]

   7. Applied rewrites 71.4%

      \[\leadsto \color{blue}{\left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \frac{\mathsf{PI}\left(\right)}{{angle}^{-1}}\right)\right)}^{2} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-1 N/A

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

      5. remove-double-div N/A

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

      6. lower-*.f64 71.4

         \[\leadsto \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right)\right)}^{2} \]

   9. Applied rewrites 71.4%

      \[\leadsto \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right)\right)}^{2} \]

   if 9.9999999999999995e59 < a

   1. Initial program 88.4%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 83.6%

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

   5. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 N/A

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

      12. lower-PI.f64 81.8

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

   7. Applied rewrites 81.8%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 10^{+60}:\\ \;\;\;\;{\left(\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b\right)}^{2} + \left(\left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(angle \cdot angle\right) \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), a, {\left(\cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{-180}\right) \cdot b\right)}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\_m\right) \cdot 0.005555555555555556\right) \cdot b\right)}^{2} + \left(\left(\left(\left(a \cdot a\right) \cdot angle\_m\right) \cdot angle\_m\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* (cos (* (* (PI) angle_m) 0.005555555555555556)) b) 2.0)
  (* (* (* (* (* a a) angle_m) angle_m) 3.08641975308642e-5) (* (PI) (PI)))))

Derivation

1. Initial program 79.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. associate-*l/ N/A

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

   5. div-inv N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. inv-pow N/A

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

   11. lower-pow.f64 79.2

       \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \frac{\mathsf{PI}\left(\right)}{\color{blue}{{angle}^{-1}}}\right)\right)}^{2} \]

4. Applied rewrites 79.2%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(0.005555555555555556 \cdot \frac{\mathsf{PI}\left(\right)}{{angle}^{-1}}\right)}\right)}^{2} \]

5. Taylor expanded in angle around 0

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-PI.f64 N/A

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

   15. lower-PI.f64 73.0

       \[\leadsto \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \frac{\mathsf{PI}\left(\right)}{{angle}^{-1}}\right)\right)}^{2} \]

7. Applied rewrites 73.0%

   \[\leadsto \color{blue}{\left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \frac{\mathsf{PI}\left(\right)}{{angle}^{-1}}\right)\right)}^{2} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow-1 N/A

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

   5. remove-double-div N/A

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

   6. lower-*.f64 73.1

      \[\leadsto \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right)\right)}^{2} \]

9. Applied rewrites 73.1%

   \[\leadsto \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right)\right)}^{2} \]

10. Final simplification 73.1%

    \[\leadsto {\left(\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b\right)}^{2} + \left(\left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \]
11. Add Preprocessing

Alternative 11: 64.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;a \leq 1.3 \cdot 10^{-84}:\\ \;\;\;\;\left(b \cdot b\right) \cdot {\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\_m\right) \cdot -0.005555555555555556\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle\_m \cdot angle\_m\right), t\_0, 1\right) \cdot b\right)}^{2} + \left(\left(\left(\left(a \cdot a\right) \cdot angle\_m\right) \cdot angle\_m\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (PI) (PI))))
   (if (<= a 1.3e-84)
     (* (* b b) (pow (cos (* (* (PI) angle_m) -0.005555555555555556)) 2.0))
     (+
      (pow
       (* (fma (* -1.54320987654321e-5 (* angle_m angle_m)) t_0 1.0) b)
       2.0)
      (* (* (* (* (* a a) angle_m) angle_m) 3.08641975308642e-5) t_0)))))

Derivation

1. Split input into 2 regimes

2. if a < 1.3e-84

   1. Initial program 78.6%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 76.6%

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

   5. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 66.9

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

   7. Applied rewrites 66.9%

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

   if 1.3e-84 < a

   1. Initial program 80.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-*l/ N/A

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

      5. div-inv N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 80.4

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \frac{\mathsf{PI}\left(\right)}{\color{blue}{{angle}^{-1}}}\right)\right)}^{2} \]

   4. Applied rewrites 80.4%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(0.005555555555555556 \cdot \frac{\mathsf{PI}\left(\right)}{{angle}^{-1}}\right)}\right)}^{2} \]

   5. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. lower-PI.f64 74.6

          \[\leadsto \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \frac{\mathsf{PI}\left(\right)}{{angle}^{-1}}\right)\right)}^{2} \]

   7. Applied rewrites 74.6%

      \[\leadsto \color{blue}{\left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \frac{\mathsf{PI}\left(\right)}{{angle}^{-1}}\right)\right)}^{2} \]

   8. Taylor expanded in angle around 0

      \[\leadsto \left(\frac{1}{32400} \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) + {\left(b \cdot \color{blue}{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)}^{2} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{32400} \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) + {\left(b \cdot \color{blue}{\left(\frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)}\right)}^{2} \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{1}{32400} \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) + {\left(b \cdot \left(\color{blue}{\left(\frac{-1}{64800} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}} + 1\right)\right)}^{2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{32400} \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) + {\left(b \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{64800} \cdot {angle}^{2}, {\mathsf{PI}\left(\right)}^{2}, 1\right)}\right)}^{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{32400} \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) + {\left(b \cdot \mathsf{fma}\left(\color{blue}{\frac{-1}{64800} \cdot {angle}^{2}}, {\mathsf{PI}\left(\right)}^{2}, 1\right)\right)}^{2} \]

      5. unpow2 N/A

         \[\leadsto \left(\frac{1}{32400} \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) + {\left(b \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{2}, 1\right)\right)}^{2} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{32400} \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) + {\left(b \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{2}, 1\right)\right)}^{2} \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{32400} \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) + {\left(b \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right)\right)}^{2} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{32400} \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) + {\left(b \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right)\right)}^{2} \]

      9. lower-PI.f64 N/A

         \[\leadsto \left(\frac{1}{32400} \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) + {\left(b \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right), 1\right)\right)}^{2} \]

      10. lower-PI.f64 74.2

          \[\leadsto \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) + {\left(b \cdot \mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}, 1\right)\right)}^{2} \]

   10. Applied rewrites 74.2%

       \[\leadsto \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) + {\left(b \cdot \color{blue}{\mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)}\right)}^{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.3 \cdot 10^{-84}:\\ \;\;\;\;\left(b \cdot b\right) \cdot {\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot -0.005555555555555556\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot b\right)}^{2} + \left(\left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;a \leq 5 \cdot 10^{-85}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle\_m \cdot angle\_m\right), t\_0, 1\right) \cdot b\right)}^{2} + \left(\left(\left(\left(a \cdot a\right) \cdot angle\_m\right) \cdot angle\_m\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (PI) (PI))))
   (if (<= a 5e-85)
     (* b b)
     (+
      (pow
       (* (fma (* -1.54320987654321e-5 (* angle_m angle_m)) t_0 1.0) b)
       2.0)
      (* (* (* (* (* a a) angle_m) angle_m) 3.08641975308642e-5) t_0)))))

Derivation

1. Split input into 2 regimes

2. if a < 5.0000000000000002e-85

   1. Initial program 78.6%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 66.7

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

   5. Applied rewrites 66.7%

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

   if 5.0000000000000002e-85 < a

   1. Initial program 80.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-*l/ N/A

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

      5. div-inv N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 80.4

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \frac{\mathsf{PI}\left(\right)}{\color{blue}{{angle}^{-1}}}\right)\right)}^{2} \]

   4. Applied rewrites 80.4%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(0.005555555555555556 \cdot \frac{\mathsf{PI}\left(\right)}{{angle}^{-1}}\right)}\right)}^{2} \]

   5. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. lower-PI.f64 74.6

          \[\leadsto \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \frac{\mathsf{PI}\left(\right)}{{angle}^{-1}}\right)\right)}^{2} \]

   7. Applied rewrites 74.6%

      \[\leadsto \color{blue}{\left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \frac{\mathsf{PI}\left(\right)}{{angle}^{-1}}\right)\right)}^{2} \]

   8. Taylor expanded in angle around 0

      \[\leadsto \left(\frac{1}{32400} \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) + {\left(b \cdot \color{blue}{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)}^{2} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{32400} \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) + {\left(b \cdot \color{blue}{\left(\frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)}\right)}^{2} \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{1}{32400} \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) + {\left(b \cdot \left(\color{blue}{\left(\frac{-1}{64800} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}} + 1\right)\right)}^{2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{32400} \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) + {\left(b \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{64800} \cdot {angle}^{2}, {\mathsf{PI}\left(\right)}^{2}, 1\right)}\right)}^{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{32400} \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) + {\left(b \cdot \mathsf{fma}\left(\color{blue}{\frac{-1}{64800} \cdot {angle}^{2}}, {\mathsf{PI}\left(\right)}^{2}, 1\right)\right)}^{2} \]

      5. unpow2 N/A

         \[\leadsto \left(\frac{1}{32400} \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) + {\left(b \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{2}, 1\right)\right)}^{2} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{32400} \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) + {\left(b \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{2}, 1\right)\right)}^{2} \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{32400} \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) + {\left(b \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right)\right)}^{2} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{32400} \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) + {\left(b \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right)\right)}^{2} \]

      9. lower-PI.f64 N/A

         \[\leadsto \left(\frac{1}{32400} \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) + {\left(b \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right), 1\right)\right)}^{2} \]

      10. lower-PI.f64 74.2

          \[\leadsto \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) + {\left(b \cdot \mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}, 1\right)\right)}^{2} \]

   10. Applied rewrites 74.2%

       \[\leadsto \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) + {\left(b \cdot \color{blue}{\mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)}\right)}^{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5 \cdot 10^{-85}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot b\right)}^{2} + \left(\left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.2% accurate, 10.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 5 \cdot 10^{-85}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), angle\_m \cdot angle\_m, b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 5e-85)
   (* b b)
   (fma
    (* (* (* a a) 3.08641975308642e-5) (* (PI) (PI)))
    (* angle_m angle_m)
    (* b b))))

Derivation

1. Split input into 2 regimes

2. if a < 5.0000000000000002e-85

   1. Initial program 78.6%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 66.7

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

   5. Applied rewrites 66.7%

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

   if 5.0000000000000002e-85 < a

   1. Initial program 80.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 80.6

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

   4. Applied rewrites 80.6%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 42.1%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 74.6%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5 \cdot 10^{-85}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), angle \cdot angle, b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 63.2% accurate, 10.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 5 \cdot 10^{-85}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), angle\_m \cdot angle\_m, b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 5e-85)
   (* b b)
   (fma
    (* (* (* (* a a) 3.08641975308642e-5) (PI)) (PI))
    (* angle_m angle_m)
    (* b b))))

Derivation

1. Split input into 2 regimes

2. if a < 5.0000000000000002e-85

   1. Initial program 78.6%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 66.7

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

   5. Applied rewrites 66.7%

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

   if 5.0000000000000002e-85 < a

   1. Initial program 80.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 80.6

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

   4. Applied rewrites 80.6%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 42.1%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 74.6%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5 \cdot 10^{-85}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), angle \cdot angle, b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 62.5% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 3 \cdot 10^{+134}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\_m\right) \cdot \left(\left(\left(a \cdot a\right) \cdot angle\_m\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 3e+134)
   (* b b)
   (* (* (* (PI) (PI)) angle_m) (* (* (* a a) angle_m) 3.08641975308642e-5))))

Derivation

1. Split input into 2 regimes

2. if a < 2.99999999999999997e134

   1. Initial program 76.8%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 65.9

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

   5. Applied rewrites 65.9%

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

   if 2.99999999999999997e134 < a

   1. Initial program 97.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 97.8

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

   4. Applied rewrites 97.8%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 67.2%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 84.5%

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

   12. Applied rewrites 90.8%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3 \cdot 10^{+134}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 63.8% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 3 \cdot 10^{+134}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(angle\_m \cdot a\right) \cdot \left(\left(3.08641975308642 \cdot 10^{-5} \cdot a\right) \cdot angle\_m\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 3e+134)
   (* b b)
   (* (* (* angle_m a) (* (* 3.08641975308642e-5 a) angle_m)) (* (PI) (PI)))))

Derivation

1. Split input into 2 regimes

2. if a < 2.99999999999999997e134

   1. Initial program 76.8%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 65.9

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

   5. Applied rewrites 65.9%

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

   if 2.99999999999999997e134 < a

   1. Initial program 97.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 97.8

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

   4. Applied rewrites 97.8%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 67.2%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 84.5%

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

   12. Applied rewrites 90.9%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3 \cdot 10^{+134}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(angle \cdot a\right) \cdot \left(\left(3.08641975308642 \cdot 10^{-5} \cdot a\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 62.2% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 3 \cdot 10^{+134}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3.08641975308642 \cdot 10^{-5} \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(angle\_m \cdot angle\_m\right) \cdot a\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 3e+134)
   (* b b)
   (* (* (* 3.08641975308642e-5 a) (* (PI) (PI))) (* (* angle_m angle_m) a))))

Derivation

1. Split input into 2 regimes

2. if a < 2.99999999999999997e134

   1. Initial program 76.8%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 65.9

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

   5. Applied rewrites 65.9%

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

   if 2.99999999999999997e134 < a

   1. Initial program 97.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 97.8

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

   4. Applied rewrites 97.8%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 67.2%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 84.5%

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

   12. Applied rewrites 84.5%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3 \cdot 10^{+134}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3.08641975308642 \cdot 10^{-5} \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 62.2% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 3 \cdot 10^{+134}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(angle\_m \cdot angle\_m\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot a\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 3e+134)
   (* b b)
   (* (* (* (* angle_m angle_m) a) (* (PI) (PI))) (* 3.08641975308642e-5 a))))

Derivation

1. Split input into 2 regimes

2. if a < 2.99999999999999997e134

   1. Initial program 76.8%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 65.9

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

   5. Applied rewrites 65.9%

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

   if 2.99999999999999997e134 < a

   1. Initial program 97.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 97.8

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

   4. Applied rewrites 97.8%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 67.2%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 84.5%

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

   12. Applied rewrites 84.5%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3 \cdot 10^{+134}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(angle \cdot angle\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 57.4% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

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 = b * b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 61.2

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

5. Applied rewrites 61.2%

   \[\leadsto \color{blue}{b \cdot b} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024268 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) (PI)))) 2.0) (pow (* b (cos (* (/ angle 180.0) (PI)))) 2.0)))