ab-angle->ABCF C

Percentage Accurate: 79.7% → 79.7%

Time: 11.7s

Alternatives: 14

Speedup: 1.0×

• Could not converge on a ground truth

  1. a = 2.247209502560818e-173
  2. b = 5.504420389618757e-236
  3. angle = 2.0153609070004827e+245

• 35.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

Initial Program: 79.7% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 79.7% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 82.2%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 82.2

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 82.2

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 82.2

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

   10. lift-/.f64 N/A

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

   11. clear-num N/A

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

   12. associate-/r/ N/A

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

   13. lower-*.f64 N/A

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

   14. metadata-eval 82.2

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

4. Applied rewrites 82.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. div-inv N/A

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

   8. lower-/.f64 82.3

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 82.3

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

6. Applied rewrites 82.3%

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

7. Final simplification 82.3%

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

Alternative 2: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 82.2%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 82.2

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 82.2

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 82.2

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

   10. lift-/.f64 N/A

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

   11. clear-num N/A

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

   12. associate-/r/ N/A

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

   13. lower-*.f64 N/A

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

   14. metadata-eval 82.2

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

4. Applied rewrites 82.3%

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

5. Final simplification 82.3%

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

Alternative 3: 79.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* 0.005555555555555556 angle) (PI))))
   (+ (pow (* (cos t_0) a) 2.0) (pow (* (sin t_0) b) 2.0))))

Derivation

1. Initial program 82.2%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 82.2

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 82.2

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 82.2

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

   10. lift-/.f64 N/A

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

   11. clear-num N/A

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

   12. associate-/r/ N/A

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

   13. lower-*.f64 N/A

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

   14. metadata-eval 82.2

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

4. Applied rewrites 82.3%

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

   1. lift-cos.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. metadata-eval N/A

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

   5. distribute-frac-neg N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-*r/ N/A

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

   9. lift-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. cos-neg N/A

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

   12. lower-cos.f64 82.2

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 82.2

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

   16. lift-/.f64 N/A

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

   17. div-inv N/A

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

   18. metadata-eval N/A

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

   19. lower-*.f64 82.3

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

6. Applied rewrites 82.3%

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

7. Final simplification 82.3%

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

Alternative 4: 79.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle (PI)))))
   (+ (pow (* (cos t_0) a) 2.0) (pow (* (sin t_0) b) 2.0))))

Derivation

1. Initial program 82.2%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 82.2

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 82.2

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 82.2

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

   10. lift-/.f64 N/A

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

   11. clear-num N/A

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

   12. associate-/r/ N/A

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

   13. lower-*.f64 N/A

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

   14. metadata-eval 82.2

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

4. Applied rewrites 82.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. div-inv N/A

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

   8. lower-/.f64 82.3

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 82.3

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

6. Applied rewrites 82.3%

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

8. Applied rewrites 82.3%

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

9. Final simplification 82.3%

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

Alternative 5: 79.6% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+ (pow (* 1.0 a) 2.0) (pow (* b (sin (/ (* angle (PI)) 180.0))) 2.0)))

Derivation

1. Initial program 82.2%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 82.2

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 82.2

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 82.2

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

   10. lift-/.f64 N/A

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

   11. clear-num N/A

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

   12. associate-/r/ N/A

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

   13. lower-*.f64 N/A

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

   14. metadata-eval 82.2

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

4. Applied rewrites 82.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. div-inv N/A

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

   8. lower-/.f64 82.3

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 82.3

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

6. Applied rewrites 82.3%

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

7. Taylor expanded in angle around 0

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

9. Applied rewrites 82.1%

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

10. Final simplification 82.1%

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

Alternative 6: 79.6% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 82.2%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 82.0

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

5. Applied rewrites 82.0%

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

6. Final simplification 82.0%

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

Alternative 7: 64.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.7 \cdot 10^{-91}:\\ \;\;\;\;{\left(\cos \left(-0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot a\right)}^{2}\\ \mathbf{elif}\;b \leq 2.15 \cdot 10^{+161}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(b \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), angle \cdot angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left({\left(b \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 4.7e-91)
   (pow (* (cos (* -0.005555555555555556 (* angle (PI)))) a) 2.0)
   (if (<= b 2.15e+161)
     (fma
      (* (* (* b b) 3.08641975308642e-5) (* (PI) (PI)))
      (* angle angle)
      (* a a))
     (* (* (* (pow (* b angle) 2.0) 3.08641975308642e-5) (PI)) (PI)))))

Derivation

1. Split input into 3 regimes

2. if b < 4.70000000000000006e-91

   1. Initial program 79.7%

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

      1. unpow1 N/A

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

      2. pow-to-exp N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

      7. exp-1-e N/A

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

      8. lower-E.f64 N/A

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

      9. rem-log-exp N/A

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

      10. pow-to-exp N/A

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

      11. unpow1 N/A

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

      12. lower-log.f64 36.7

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 36.7

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

   4. Applied rewrites 36.3%

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

   5. Taylor expanded in b around 0

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

      1. lower-pow.f64 N/A

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

      2. log-E N/A

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

      3. unpow1 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 61.8

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

   7. Applied rewrites 61.8%

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

   if 4.70000000000000006e-91 < b < 2.15e161

   1. Initial program 75.2%

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

      1. unpow1 N/A

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

      2. pow-to-exp N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

      7. exp-1-e N/A

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

      8. lower-E.f64 N/A

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

      9. rem-log-exp N/A

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

      10. pow-to-exp N/A

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

      11. unpow1 N/A

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

      12. lower-log.f64 29.9

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 29.9

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

   4. Applied rewrites 32.1%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. e-exp-1 N/A

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

      3. log-E N/A

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

      4. exp-prod N/A

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

      5. log-E N/A

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

      6. *-lft-identity N/A

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

      7. rem-exp-log N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 45.6%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 63.5%

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

   if 2.15e161 < b

   1. Initial program 99.6%

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

      1. unpow1 N/A

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

      2. pow-to-exp N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

      7. exp-1-e N/A

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

      8. lower-E.f64 N/A

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

      9. rem-log-exp N/A

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

      10. pow-to-exp N/A

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

      11. unpow1 N/A

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

      12. lower-log.f64 40.8

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 40.8

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

   4. Applied rewrites 43.1%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. e-exp-1 N/A

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

      3. log-E N/A

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

      4. exp-prod N/A

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

      5. log-E N/A

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

      6. *-lft-identity N/A

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

      7. rem-exp-log N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 43.8%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 69.1%

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

   12. Applied rewrites 86.7%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.7 \cdot 10^{-91}:\\ \;\;\;\;{\left(\cos \left(-0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot a\right)}^{2}\\ \mathbf{elif}\;b \leq 2.15 \cdot 10^{+161}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(b \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), angle \cdot angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left({\left(b \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.7 \cdot 10^{-91}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;b \leq 2.15 \cdot 10^{+161}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(b \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), angle \cdot angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left({\left(b \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 4.7e-91)
   (* a a)
   (if (<= b 2.15e+161)
     (fma
      (* (* (* b b) 3.08641975308642e-5) (* (PI) (PI)))
      (* angle angle)
      (* a a))
     (* (* (* (pow (* b angle) 2.0) 3.08641975308642e-5) (PI)) (PI)))))

Derivation

1. Split input into 3 regimes

2. if b < 4.70000000000000006e-91

   1. Initial program 79.7%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.8

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

   5. Applied rewrites 61.8%

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

   if 4.70000000000000006e-91 < b < 2.15e161

   1. Initial program 75.2%

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

      1. unpow1 N/A

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

      2. pow-to-exp N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

      7. exp-1-e N/A

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

      8. lower-E.f64 N/A

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

      9. rem-log-exp N/A

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

      10. pow-to-exp N/A

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

      11. unpow1 N/A

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

      12. lower-log.f64 29.9

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 29.9

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

   4. Applied rewrites 32.1%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. e-exp-1 N/A

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

      3. log-E N/A

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

      4. exp-prod N/A

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

      5. log-E N/A

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

      6. *-lft-identity N/A

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

      7. rem-exp-log N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 45.6%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 63.5%

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

   if 2.15e161 < b

   1. Initial program 99.6%

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

      1. unpow1 N/A

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

      2. pow-to-exp N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

      7. exp-1-e N/A

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

      8. lower-E.f64 N/A

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

      9. rem-log-exp N/A

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

      10. pow-to-exp N/A

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

      11. unpow1 N/A

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

      12. lower-log.f64 40.8

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 40.8

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

   4. Applied rewrites 43.1%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. e-exp-1 N/A

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

      3. log-E N/A

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

      4. exp-prod N/A

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

      5. log-E N/A

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

      6. *-lft-identity N/A

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

      7. rem-exp-log N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 43.8%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 69.1%

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

   12. Applied rewrites 86.7%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.7 \cdot 10^{-91}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;b \leq 2.15 \cdot 10^{+161}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(b \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), angle \cdot angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left({\left(b \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.7 \cdot 10^{-91}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;b \leq 2.15 \cdot 10^{+161}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(b \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), angle \cdot angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(b \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 4.7e-91)
   (* a a)
   (if (<= b 2.15e+161)
     (fma
      (* (* (* b b) 3.08641975308642e-5) (* (PI) (PI)))
      (* angle angle)
      (* a a))
     (* (pow (* (* b angle) (PI)) 2.0) 3.08641975308642e-5))))

Derivation

1. Split input into 3 regimes

2. if b < 4.70000000000000006e-91

   1. Initial program 79.7%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.8

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

   5. Applied rewrites 61.8%

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

   if 4.70000000000000006e-91 < b < 2.15e161

   1. Initial program 75.2%

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

      1. unpow1 N/A

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

      2. pow-to-exp N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

      7. exp-1-e N/A

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

      8. lower-E.f64 N/A

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

      9. rem-log-exp N/A

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

      10. pow-to-exp N/A

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

      11. unpow1 N/A

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

      12. lower-log.f64 29.9

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 29.9

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

   4. Applied rewrites 32.1%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. e-exp-1 N/A

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

      3. log-E N/A

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

      4. exp-prod N/A

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

      5. log-E N/A

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

      6. *-lft-identity N/A

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

      7. rem-exp-log N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 45.6%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 63.5%

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

   if 2.15e161 < b

   1. Initial program 99.6%

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

      1. unpow1 N/A

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

      2. pow-to-exp N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

      7. exp-1-e N/A

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

      8. lower-E.f64 N/A

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

      9. rem-log-exp N/A

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

      10. pow-to-exp N/A

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

      11. unpow1 N/A

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

      12. lower-log.f64 40.8

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 40.8

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

   4. Applied rewrites 43.1%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. e-exp-1 N/A

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

      3. log-E N/A

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

      4. exp-prod N/A

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

      5. log-E N/A

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

      6. *-lft-identity N/A

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

      7. rem-exp-log N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 43.8%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 69.1%

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

   12. Applied rewrites 86.6%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.7 \cdot 10^{-91}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;b \leq 2.15 \cdot 10^{+161}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(b \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), angle \cdot angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(b \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.6% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;b \leq 4.7 \cdot 10^{-91}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;b \leq 2.15 \cdot 10^{+161}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(b \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot t\_0, angle \cdot angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(b \cdot angle\right) \cdot angle\right) \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (PI) (PI))))
   (if (<= b 4.7e-91)
     (* a a)
     (if (<= b 2.15e+161)
       (fma (* (* (* b b) 3.08641975308642e-5) t_0) (* angle angle) (* a a))
       (* (* (* (* (* b angle) angle) b) 3.08641975308642e-5) t_0)))))

Derivation

1. Split input into 3 regimes

2. if b < 4.70000000000000006e-91

   1. Initial program 79.7%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.8

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

   5. Applied rewrites 61.8%

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

   if 4.70000000000000006e-91 < b < 2.15e161

   1. Initial program 75.2%

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

      1. unpow1 N/A

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

      2. pow-to-exp N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

      7. exp-1-e N/A

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

      8. lower-E.f64 N/A

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

      9. rem-log-exp N/A

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

      10. pow-to-exp N/A

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

      11. unpow1 N/A

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

      12. lower-log.f64 29.9

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 29.9

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

   4. Applied rewrites 32.1%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. e-exp-1 N/A

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

      3. log-E N/A

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

      4. exp-prod N/A

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

      5. log-E N/A

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

      6. *-lft-identity N/A

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

      7. rem-exp-log N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 45.6%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 63.5%

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

   if 2.15e161 < b

   1. Initial program 99.6%

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

      1. unpow1 N/A

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

      2. pow-to-exp N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

      7. exp-1-e N/A

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

      8. lower-E.f64 N/A

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

      9. rem-log-exp N/A

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

      10. pow-to-exp N/A

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

      11. unpow1 N/A

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

      12. lower-log.f64 40.8

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 40.8

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

   4. Applied rewrites 43.1%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. e-exp-1 N/A

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

      3. log-E N/A

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

      4. exp-prod N/A

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

      5. log-E N/A

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

      6. *-lft-identity N/A

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

      7. rem-exp-log N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 43.8%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 69.1%

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

   12. Applied rewrites 82.5%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.7 \cdot 10^{-91}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;b \leq 2.15 \cdot 10^{+161}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(b \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), angle \cdot angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(b \cdot angle\right) \cdot angle\right) \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.6% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.22 \cdot 10^{+166}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(b \cdot angle\right) \cdot angle\right) \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.22e+166)
   (* a a)
   (* (* (* (* (* b angle) angle) b) 3.08641975308642e-5) (* (PI) (PI)))))

Derivation

1. Split input into 2 regimes

2. if b < 1.21999999999999993e166

   1. Initial program 78.7%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 59.2

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

   5. Applied rewrites 59.2%

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

   if 1.21999999999999993e166 < b

   1. Initial program 99.6%

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

      1. unpow1 N/A

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

      2. pow-to-exp N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

      7. exp-1-e N/A

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

      8. lower-E.f64 N/A

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

      9. rem-log-exp N/A

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

      10. pow-to-exp N/A

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

      11. unpow1 N/A

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

      12. lower-log.f64 41.8

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 41.8

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

   4. Applied rewrites 44.1%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. e-exp-1 N/A

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

      3. log-E N/A

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

      4. exp-prod N/A

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

      5. log-E N/A

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

      6. *-lft-identity N/A

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

      7. rem-exp-log N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 44.8%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 70.7%

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

   12. Applied rewrites 84.2%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.22 \cdot 10^{+166}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(b \cdot angle\right) \cdot angle\right) \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.0% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{+172}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(angle \cdot angle\right) \cdot b\right) \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 2.9e+172)
   (* a a)
   (* (* (* (* (* angle angle) b) b) 3.08641975308642e-5) (* (PI) (PI)))))

Derivation

1. Split input into 2 regimes

2. if b < 2.8999999999999999e172

   1. Initial program 78.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 59.2

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

   5. Applied rewrites 59.2%

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

   if 2.8999999999999999e172 < b

   1. Initial program 99.6%

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

      1. unpow1 N/A

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

      2. pow-to-exp N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

      7. exp-1-e N/A

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

      8. lower-E.f64 N/A

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

      9. rem-log-exp N/A

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

      10. pow-to-exp N/A

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

      11. unpow1 N/A

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

      12. lower-log.f64 43.8

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 43.8

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

   4. Applied rewrites 43.8%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. e-exp-1 N/A

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

      3. log-E N/A

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

      4. exp-prod N/A

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

      5. log-E N/A

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

      6. *-lft-identity N/A

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

      7. rem-exp-log N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 44.5%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 71.7%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{+172}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(angle \cdot angle\right) \cdot b\right) \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.0% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{+172}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right) \cdot b\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 2.9e+172)
   (* a a)
   (* (* (* (* (* angle angle) 3.08641975308642e-5) b) b) (* (PI) (PI)))))

Derivation

1. Split input into 2 regimes

2. if b < 2.8999999999999999e172

   1. Initial program 78.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 59.2

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

   5. Applied rewrites 59.2%

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

   if 2.8999999999999999e172 < b

   1. Initial program 99.6%

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

      1. unpow1 N/A

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

      2. pow-to-exp N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

      7. exp-1-e N/A

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

      8. lower-E.f64 N/A

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

      9. rem-log-exp N/A

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

      10. pow-to-exp N/A

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

      11. unpow1 N/A

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

      12. lower-log.f64 43.8

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 43.8

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

   4. Applied rewrites 43.8%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. e-exp-1 N/A

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

      3. log-E N/A

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

      4. exp-prod N/A

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

      5. log-E N/A

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

      6. *-lft-identity N/A

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

      7. rem-exp-log N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 44.5%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 71.7%

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

   11. Taylor expanded in b around 0

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

   13. Applied rewrites 71.7%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{+172}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right) \cdot b\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 56.9% accurate, 74.7× speedup

Math

\[\begin{array}{l} \\ a \cdot a \end{array} \]

FPCore

(FPCore (a b angle) :precision binary64 (* a a))

C

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

Fortran

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

Java

public static double code(double a, double b, double angle) {
	return a * a;
}

Python

def code(a, b, angle):
	return a * a

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_] := N[(a * a), $MachinePrecision]

Derivation

1. Initial program 82.2%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 54.3

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

5. Applied rewrites 54.3%

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

Reproduce

herbie shell --seed 2024270 
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* (PI) (/ angle 180.0)))) 2.0) (pow (* b (sin (* (PI) (/ angle 180.0)))) 2.0)))