ab-angle->ABCF C

Percentage Accurate: 79.7% → 79.6%

Time: 14.0s

Alternatives: 14

Speedup: 1.3×

• Could not converge on a ground truth

  1. a = 3.09847315372408e-119
  2. b = 5.440014231388029e+136
  3. angle = 4.217362169441825e+211

• 36.4% 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: 79.6% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow
   (* (sin (* (/ 0.005555555555555556 (/ -1.0 angle)) (/ (PI) -1.0))) b)
   2.0)
  (pow (* (cos (* (- 0.005555555555555556) (/ (PI) (/ -1.0 angle)))) a) 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

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. associate-*r/ N/A

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

   5. *-commutative N/A

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

   6. div-inv N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f64 N/A

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

   11. inv-pow N/A

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

   12. lower-pow.f64 82.3

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

4. Applied rewrites 82.3%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow-1 N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 82.3

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

6. Applied rewrites 82.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. remove-double-neg N/A

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

   6. lift-neg.f64 N/A

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

   7. neg-mul-1 N/A

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

   8. div-inv N/A

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

   9. times-frac N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. times-frac N/A

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

   13. div-inv N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*r/ N/A

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

   16. metadata-eval N/A

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

   17. times-frac N/A

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

   18. *-commutative N/A

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

   19. times-frac N/A

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

   20. lift-neg.f64 N/A

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

   21. metadata-eval N/A

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

   22. frac-2neg N/A

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

   23. lower-*.f64 N/A

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

   24. lower-/.f64 N/A

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

   25. lower-/.f64 82.4

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

8. Applied rewrites 82.4%

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

9. Final simplification 82.4%

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

Alternative 2: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. associate-*r/ N/A

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

   5. *-commutative N/A

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

   6. div-inv N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f64 N/A

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

   11. inv-pow N/A

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

   12. lower-pow.f64 82.3

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

4. Applied rewrites 82.3%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow-1 N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 82.3

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

6. Applied rewrites 82.3%

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

7. Final simplification 82.3%

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

Alternative 3: 79.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* (sin (* (* (PI) 0.005555555555555556) angle)) b) 2.0)
  (pow (* (cos (* (- 0.005555555555555556) (/ (PI) (/ -1.0 angle)))) a) 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

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. associate-*r/ N/A

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

   5. *-commutative N/A

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

   6. div-inv N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f64 N/A

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

   11. inv-pow N/A

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

   12. lower-pow.f64 82.3

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

4. Applied rewrites 82.3%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow-1 N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 82.3

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

6. Applied rewrites 82.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. div-inv N/A

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

   4. metadata-eval N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 82.3

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

8. Applied rewrites 82.3%

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

9. Final simplification 82.3%

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

Alternative 4: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. div-inv N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 82.3

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

4. Applied rewrites 82.3%

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

5. Final simplification 82.3%

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

Alternative 5: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. associate-*r/ N/A

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

   5. *-commutative N/A

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

   6. div-inv N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f64 N/A

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

   11. inv-pow N/A

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

   12. lower-pow.f64 82.3

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

4. Applied rewrites 82.3%

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

6. Applied rewrites 82.3%

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

7. Final simplification 82.3%

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

Alternative 6: 79.7% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* 1.0 a) 2.0)
  (pow
   (* (sin (* (/ 0.005555555555555556 (/ -1.0 angle)) (/ (PI) -1.0))) 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

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. associate-*r/ N/A

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

   5. *-commutative N/A

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

   6. div-inv N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f64 N/A

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

   11. inv-pow N/A

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

   12. lower-pow.f64 82.3

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

4. Applied rewrites 82.3%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow-1 N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 82.3

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

6. Applied rewrites 82.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. remove-double-neg N/A

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

   6. lift-neg.f64 N/A

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

   7. neg-mul-1 N/A

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

   8. div-inv N/A

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

   9. times-frac N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. times-frac N/A

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

   13. div-inv N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*r/ N/A

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

   16. metadata-eval N/A

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

   17. times-frac N/A

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

   18. *-commutative N/A

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

   19. times-frac N/A

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

   20. lift-neg.f64 N/A

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

   21. metadata-eval N/A

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

   22. frac-2neg N/A

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

   23. lower-*.f64 N/A

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

   24. lower-/.f64 N/A

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

   25. lower-/.f64 82.4

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

8. Applied rewrites 82.4%

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

9. Taylor expanded in angle around 0

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

11. Applied rewrites 82.1%

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

12. Final simplification 82.1%

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

Alternative 7: 79.6% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+ (* a a) (pow (* (sin (* (* angle (PI)) 0.005555555555555556)) 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-pow.f64 N/A

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

   2. unpow1 N/A

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

   3. pow-to-exp N/A

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

   4. pow-exp N/A

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

   5. *-commutative N/A

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

   6. pow-exp N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. rem-log-exp N/A

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

   10. pow-to-exp N/A

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

   11. unpow1 N/A

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

   12. lower-log.f64 46.2

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 46.2

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

4. Applied rewrites 47.3%

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

5. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 47.4

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

7. Applied rewrites 47.4%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

9. Applied rewrites 82.1%

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

10. Final simplification 82.1%

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

Alternative 8: 56.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot 0.005555555555555556\\ t_1 := t\_0 \cdot angle\\ \mathbf{if}\;a \leq 6 \cdot 10^{-168}:\\ \;\;\;\;{\left(\sin t\_1 \cdot b\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, \left(\left(b \cdot b\right) \cdot angle\right) \cdot t\_0, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (PI) 0.005555555555555556)) (t_1 (* t_0 angle)))
   (if (<= a 6e-168)
     (pow (* (sin t_1) b) 2.0)
     (fma t_1 (* (* (* b b) angle) t_0) (* a a)))))

Derivation

1. Split input into 2 regimes

2. if a < 5.99999999999999983e-168

   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

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 36.8%

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

   6. Taylor expanded in b around inf

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

      1. exp-to-pow N/A

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

      2. *-commutative N/A

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

      3. exp-to-pow N/A

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

      4. *-commutative N/A

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

      5. exp-sum N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. exp-prod N/A

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

      9. exp-sum N/A

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

      10. rem-exp-log N/A

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

      11. rem-exp-log N/A

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

      12. lower-pow.f64 N/A

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

   8. Applied rewrites 41.5%

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

   if 5.99999999999999983e-168 < a

   1. Initial program 86.7%

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

      1. 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

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

   4. Applied rewrites 58.1%

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

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. log-prod N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. rem-log-exp N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-log.f64 27.6

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

   6. Applied rewrites 27.6%

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

   7. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

   9. Applied rewrites 79.3%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 6 \cdot 10^{-168}:\\ \;\;\;\;{\left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right) \cdot b\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle, \left(\left(b \cdot b\right) \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right), a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot 0.005555555555555556\\ \mathbf{if}\;b \leq 8.4 \cdot 10^{+126}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot angle, \left(\left(b \cdot b\right) \cdot angle\right) \cdot t\_0, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (PI) 0.005555555555555556)))
   (if (<= b 8.4e+126)
     (fma (* t_0 angle) (* (* (* b b) angle) t_0) (* a a))
     (* (pow (* (* b (PI)) angle) 2.0) 3.08641975308642e-5))))

Derivation

1. Split input into 2 regimes

2. if b < 8.3999999999999997e126

   1. Initial program 80.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   4. Applied rewrites 56.0%

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

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. log-prod N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. rem-log-exp N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-log.f64 18.1

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

   6. Applied rewrites 18.1%

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

   7. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

   9. Applied rewrites 75.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle, \left(angle \cdot \left(b \cdot b\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right), a \cdot a\right)} \]

   if 8.3999999999999997e126 < b

   1. Initial program 92.7%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 29.2%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 45.9%

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

   10. Applied rewrites 73.5%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.4 \cdot 10^{+126}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle, \left(\left(b \cdot b\right) \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.8% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-167}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;\frac{angle}{180} \leq 4 \cdot 10^{+198}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right) \cdot b, angle \cdot angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) 5e-167)
   (* a a)
   (if (<= (/ angle 180.0) 4e+198)
     (fma
      (* (* (* (* (PI) (PI)) 3.08641975308642e-5) b) b)
      (* angle angle)
      (* a a))
     (* a a))))

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 5.0000000000000002e-167 or 4.00000000000000007e198 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 83.3%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 64.5

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

   5. Applied rewrites 64.5%

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

   if 5.0000000000000002e-167 < (/.f64 angle #s(literal 180 binary64)) < 4.00000000000000007e198

   1. Initial program 79.8%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 42.3%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 71.9%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-167}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;\frac{angle}{180} \leq 4 \cdot 10^{+198}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right) \cdot b, angle \cdot angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.1% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot 0.005555555555555556\\ \mathsf{fma}\left(t\_0 \cdot angle, \left(\left(b \cdot b\right) \cdot angle\right) \cdot t\_0, a \cdot a\right) \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (PI) 0.005555555555555556)))
   (fma (* t_0 angle) (* (* (* b b) angle) t_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. 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

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

4. Applied rewrites 52.6%

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

   1. lift-log.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. log-prod N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. pow-to-exp N/A

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

   7. rem-log-exp N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-log.f64 N/A

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

   10. lower-log.f64 21.7

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

6. Applied rewrites 21.7%

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

7. Taylor expanded in angle around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

9. Applied rewrites 72.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle, \left(angle \cdot \left(b \cdot b\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right), a \cdot a\right)} \]

10. Final simplification 72.8%

    \[\leadsto \mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle, \left(\left(b \cdot b\right) \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right), a \cdot a\right) \]
11. Add Preprocessing

Alternative 12: 61.6% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{+128}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right) \cdot \left(\left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.8e+128)
   (* a a)
   (* (* (* (PI) (PI)) b) (* (* (* angle angle) 3.08641975308642e-5) b))))

Derivation

1. Split input into 2 regimes

2. if b < 1.80000000000000014e128

   1. Initial program 80.6%

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

   3. 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 63.0

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

   5. Applied rewrites 63.0%

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

   if 1.80000000000000014e128 < b

   1. Initial program 92.5%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 29.9%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 47.1%

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

   10. Applied rewrites 53.4%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{+128}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right) \cdot \left(\left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.5% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.5 \cdot 10^{+187}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right) \cdot b\right) \cdot \left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 5.5e+187)
   (* a a)
   (* (* (* (* (PI) (PI)) b) b) (* (* angle angle) 3.08641975308642e-5))))

Derivation

1. Split input into 2 regimes

2. if b < 5.49999999999999997e187

   1. Initial program 80.3%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.5

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

   5. Applied rewrites 61.5%

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

   if 5.49999999999999997e187 < 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. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 39.0%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 58.2%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.5 \cdot 10^{+187}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right) \cdot b\right) \cdot \left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\\ \end{array} \]
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 59.4

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

5. Applied rewrites 59.4%

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

Reproduce

herbie shell --seed 2024249 
(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)))