ab-angle->ABCF A

Percentage Accurate: 80.5% → 80.5%

Time: 8.5s

Alternatives: 15

Speedup: 1.0×

• Could not converge on a ground truth

  1. a = 2.8077729504057747e-171
  2. b = 5.151736419573034e-91
  3. angle = 1.3281326084223402e+54

• 36.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

Initial Program: 80.5% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 80.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (sqrt (PI))))
   (+
    (pow (* (cos (* (* (/ t_0 180.0) angle) t_0)) b) 2.0)
    (pow (* a (sin (* (PI) (* angle 0.005555555555555556)))) 2.0))))

Derivation

1. Initial program 81.5%

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. add-cube-cbrt N/A

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

   4. associate-*r* N/A

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

   5. add-sqr-sqrt N/A

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

   6. cbrt-prod N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f64 N/A

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

4. Applied rewrites 81.6%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(\left(0.005555555555555556 \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot angle\right)\right) \cdot {\mathsf{PI}\left(\right)}^{0.16666666666666666}\right) \cdot {\mathsf{PI}\left(\right)}^{0.16666666666666666}\right)}\right)}^{2} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(\left(\frac{1}{180} \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot angle\right)\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)}\right)}^{2} \]

   2. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\left(\frac{1}{180} \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot angle\right)\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)}^{2} \]

   3. associate-*l* N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(\frac{1}{180} \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot angle\right)\right) \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)}\right)}^{2} \]

   4. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\frac{1}{180} \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot angle\right)\right)} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)\right)}^{2} \]

   5. lift-pow.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(\frac{1}{180} \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot angle\right)\right) \cdot \left(\color{blue}{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)\right)}^{2} \]

   6. lift-pow.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(\frac{1}{180} \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot angle\right)\right) \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}}\right)\right)\right)}^{2} \]

   7. pow-prod-up N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(\frac{1}{180} \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot angle\right)\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{\left(\frac{1}{6} + \frac{1}{6}\right)}}\right)\right)}^{2} \]

   8. metadata-eval N/A

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

   9. pow1/3 N/A

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

   10. lift-cbrt.f64 N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. associate-*r* N/A

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

   15. lift-pow.f64 N/A

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

   16. pow-plus N/A

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

   17. metadata-eval N/A

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

6. Applied rewrites 81.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 81.7

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

8. Applied rewrites 81.7%

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

9. Final simplification 81.7%

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

Alternative 2: 80.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (sqrt (PI))))
   (+
    (pow (* (cos (* (* (* t_0 angle) 0.005555555555555556) t_0)) b) 2.0)
    (pow (* (sin (* (/ angle 180.0) (PI))) a) 2.0))))

Derivation

1. Initial program 81.5%

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. add-sqr-sqrt N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-/.f64 N/A

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

   8. div-inv N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lift-PI.f64 N/A

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

   15. lower-sqrt.f64 N/A

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

   16. metadata-eval N/A

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

   17. lift-PI.f64 N/A

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

   18. lower-sqrt.f64 81.6

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

4. Applied rewrites 81.6%

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

5. Final simplification 81.6%

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

Alternative 3: 80.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 81.5%

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. add-cube-cbrt N/A

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

   4. associate-*r* N/A

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

   5. add-sqr-sqrt N/A

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

   6. cbrt-prod N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f64 N/A

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

4. Applied rewrites 81.6%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(\left(0.005555555555555556 \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot angle\right)\right) \cdot {\mathsf{PI}\left(\right)}^{0.16666666666666666}\right) \cdot {\mathsf{PI}\left(\right)}^{0.16666666666666666}\right)}\right)}^{2} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(\left(\frac{1}{180} \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot angle\right)\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)}\right)}^{2} \]

   2. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\left(\frac{1}{180} \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot angle\right)\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)}^{2} \]

   3. associate-*l* N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(\frac{1}{180} \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot angle\right)\right) \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)}\right)}^{2} \]

   4. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\frac{1}{180} \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot angle\right)\right)} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)\right)}^{2} \]

   5. lift-pow.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(\frac{1}{180} \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot angle\right)\right) \cdot \left(\color{blue}{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)\right)}^{2} \]

   6. lift-pow.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(\frac{1}{180} \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot angle\right)\right) \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}}\right)\right)\right)}^{2} \]

   7. pow-prod-up N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(\frac{1}{180} \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot angle\right)\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{\left(\frac{1}{6} + \frac{1}{6}\right)}}\right)\right)}^{2} \]

   8. metadata-eval N/A

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

   9. pow1/3 N/A

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

   10. lift-cbrt.f64 N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. associate-*r* N/A

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

   15. lift-pow.f64 N/A

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

   16. pow-plus N/A

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

   17. metadata-eval N/A

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

6. Applied rewrites 81.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 81.7

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

8. Applied rewrites 81.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. lift-sqrt.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. rem-square-sqrt N/A

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

   13. *-commutative N/A

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

   14. rem-square-sqrt N/A

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

   15. lift-sqrt.f64 N/A

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

   16. lift-sqrt.f64 N/A

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

   17. associate-*r* N/A

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

   18. lower-*.f64 N/A

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

   19. lower-*.f64 81.6

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

10. Applied rewrites 81.6%

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

11. Final simplification 81.6%

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

Alternative 4: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\\ \mathsf{fma}\left(t\_0, t\_0, {\left(\cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{-180}\right) \cdot b\right)}^{2}\right) \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* a (sin (* (PI) (* angle 0.005555555555555556))))))
   (fma t_0 t_0 (pow (* (cos (/ (* (PI) angle) -180.0)) b) 2.0))))

Derivation

1. Initial program 81.5%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-fma.f64 81.5

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

4. Applied rewrites 81.6%

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

5. Final simplification 81.6%

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

Alternative 5: 80.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 81.5%

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. add-cube-cbrt N/A

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

   4. associate-*r* N/A

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

   5. add-sqr-sqrt N/A

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

   6. cbrt-prod N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f64 N/A

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

4. Applied rewrites 81.6%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(\left(0.005555555555555556 \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot angle\right)\right) \cdot {\mathsf{PI}\left(\right)}^{0.16666666666666666}\right) \cdot {\mathsf{PI}\left(\right)}^{0.16666666666666666}\right)}\right)}^{2} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(\left(\frac{1}{180} \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot angle\right)\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)}\right)}^{2} \]

   2. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\left(\frac{1}{180} \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot angle\right)\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)}^{2} \]

   3. associate-*l* N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(\frac{1}{180} \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot angle\right)\right) \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)}\right)}^{2} \]

   4. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\frac{1}{180} \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot angle\right)\right)} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)\right)}^{2} \]

   5. lift-pow.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(\frac{1}{180} \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot angle\right)\right) \cdot \left(\color{blue}{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)\right)}^{2} \]

   6. lift-pow.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(\frac{1}{180} \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot angle\right)\right) \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}}\right)\right)\right)}^{2} \]

   7. pow-prod-up N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(\frac{1}{180} \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot angle\right)\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{\left(\frac{1}{6} + \frac{1}{6}\right)}}\right)\right)}^{2} \]

   8. metadata-eval N/A

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

   9. pow1/3 N/A

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

   10. lift-cbrt.f64 N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. associate-*r* N/A

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

   15. lift-pow.f64 N/A

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

   16. pow-plus N/A

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

   17. metadata-eval N/A

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

6. Applied rewrites 81.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 81.7

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

8. Applied rewrites 81.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. lift-sqrt.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. rem-square-sqrt N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 81.6

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

10. Applied rewrites 81.6%

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

11. Final simplification 81.6%

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

Alternative 6: 80.5% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 81.5%

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. add-cube-cbrt N/A

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

   4. associate-*r* N/A

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

   5. add-sqr-sqrt N/A

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

   6. cbrt-prod N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f64 N/A

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

4. Applied rewrites 81.6%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(\left(0.005555555555555556 \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot angle\right)\right) \cdot {\mathsf{PI}\left(\right)}^{0.16666666666666666}\right) \cdot {\mathsf{PI}\left(\right)}^{0.16666666666666666}\right)}\right)}^{2} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(\left(\frac{1}{180} \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot angle\right)\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)}\right)}^{2} \]

   2. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\left(\frac{1}{180} \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot angle\right)\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)}^{2} \]

   3. associate-*l* N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(\frac{1}{180} \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot angle\right)\right) \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)}\right)}^{2} \]

   4. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\frac{1}{180} \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot angle\right)\right)} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)\right)}^{2} \]

   5. lift-pow.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(\frac{1}{180} \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot angle\right)\right) \cdot \left(\color{blue}{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)\right)}^{2} \]

   6. lift-pow.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(\frac{1}{180} \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot angle\right)\right) \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}}\right)\right)\right)}^{2} \]

   7. pow-prod-up N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(\frac{1}{180} \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot angle\right)\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{\left(\frac{1}{6} + \frac{1}{6}\right)}}\right)\right)}^{2} \]

   8. metadata-eval N/A

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

   9. pow1/3 N/A

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

   10. lift-cbrt.f64 N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. associate-*r* N/A

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

   15. lift-pow.f64 N/A

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

   16. pow-plus N/A

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

   17. metadata-eval N/A

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

6. Applied rewrites 81.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 81.7

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

8. Applied rewrites 81.7%

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

9. Taylor expanded in angle around 0

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

11. Applied rewrites 81.4%

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

12. Final simplification 81.4%

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

Alternative 7: 65.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.5 \cdot 10^{-16}:\\ \;\;\;\;\left(b \cdot b\right) \cdot {\cos \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2} + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(\left(\left(angle \cdot angle\right) \cdot a\right) \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 3.5e-16)
   (* (* b b) (pow (cos (* (* (PI) 0.005555555555555556) angle)) 2.0))
   (+
    (pow (* (cos (* (/ angle 180.0) (PI))) b) 2.0)
    (* (* (PI) (PI)) (* (* (* (* angle angle) a) a) 3.08641975308642e-5)))))

Derivation

1. Split input into 2 regimes

2. if a < 3.50000000000000017e-16

   1. Initial program 80.3%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 64.1

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

   5. Applied rewrites 64.1%

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

   if 3.50000000000000017e-16 < a

   1. Initial program 85.3%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. lower-PI.f64 74.4

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

   5. Applied rewrites 74.4%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.5 \cdot 10^{-16}:\\ \;\;\;\;\left(b \cdot b\right) \cdot {\cos \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2} + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(\left(\left(angle \cdot angle\right) \cdot a\right) \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.5 \cdot 10^{-16}:\\ \;\;\;\;\left(b \cdot b\right) \cdot {\cos \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{2}\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), angle \cdot angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(a \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 3.5e-16)
   (* (* b b) (pow (cos (* (* (PI) 0.005555555555555556) angle)) 2.0))
   (if (<= a 7.5e+153)
     (fma
      (* (* (* (* a a) 3.08641975308642e-5) (PI)) (PI))
      (* angle angle)
      (* b b))
     (* (pow (* (* a (PI)) angle) 2.0) 3.08641975308642e-5))))

Derivation

1. Split input into 3 regimes

2. if a < 3.50000000000000017e-16

   1. Initial program 80.3%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 64.1

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

   5. Applied rewrites 64.1%

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

   if 3.50000000000000017e-16 < a < 7.50000000000000065e153

   1. Initial program 73.8%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 51.4%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 64.4%

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

   if 7.50000000000000065e153 < a

   1. Initial program 97.1%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 33.9%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 50.0%

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

   10. Applied rewrites 75.3%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.5 \cdot 10^{-16}:\\ \;\;\;\;\left(b \cdot b\right) \cdot {\cos \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{2}\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), angle \cdot angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(a \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.5 \cdot 10^{-16}:\\ \;\;\;\;{\cos \left(-0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}^{2} \cdot \left(b \cdot b\right)\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), angle \cdot angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(a \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 3.5e-16)
   (* (pow (cos (* -0.005555555555555556 (* (PI) angle))) 2.0) (* b b))
   (if (<= a 7.5e+153)
     (fma
      (* (* (* (* a a) 3.08641975308642e-5) (PI)) (PI))
      (* angle angle)
      (* b b))
     (* (pow (* (* a (PI)) angle) 2.0) 3.08641975308642e-5))))

Derivation

1. Split input into 3 regimes

2. if a < 3.50000000000000017e-16

   1. Initial program 80.3%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. unpow-prod-down N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 73.8%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 64.0

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

   7. Applied rewrites 64.0%

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

   if 3.50000000000000017e-16 < a < 7.50000000000000065e153

   1. Initial program 73.8%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 51.4%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 64.4%

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

   if 7.50000000000000065e153 < a

   1. Initial program 97.1%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 33.9%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 50.0%

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

   10. Applied rewrites 75.3%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.5 \cdot 10^{-16}:\\ \;\;\;\;{\cos \left(-0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}^{2} \cdot \left(b \cdot b\right)\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), angle \cdot angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(a \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.5 \cdot 10^{-16}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), angle \cdot angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(a \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 3.5e-16)
   (* b b)
   (if (<= a 7.5e+153)
     (fma
      (* (* (* (* a a) 3.08641975308642e-5) (PI)) (PI))
      (* angle angle)
      (* b b))
     (* (pow (* (* a (PI)) angle) 2.0) 3.08641975308642e-5))))

Derivation

1. Split input into 3 regimes

2. if a < 3.50000000000000017e-16

   1. Initial program 80.3%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 63.9

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

   5. Applied rewrites 63.9%

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

   if 3.50000000000000017e-16 < a < 7.50000000000000065e153

   1. Initial program 73.8%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 51.4%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 64.4%

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

   if 7.50000000000000065e153 < a

   1. Initial program 97.1%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 33.9%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 50.0%

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

   10. Applied rewrites 75.3%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.5 \cdot 10^{-16}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), angle \cdot angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(a \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.8% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.85 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5} \cdot a, a, -3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right), angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 3.85e+87)
   (fma
    (*
     (* (* (PI) (PI)) angle)
     (fma (* 3.08641975308642e-5 a) a (* -3.08641975308642e-5 (* b b))))
    angle
    (* b b))
   (* b b)))

Derivation

1. Split input into 2 regimes

2. if b < 3.85000000000000015e87

   1. Initial program 78.9%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 50.2%

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

   7. Applied rewrites 55.0%

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

   if 3.85000000000000015e87 < b

   1. Initial program 94.4%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 92.3

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

   5. Applied rewrites 92.3%

      \[\leadsto \color{blue}{b \cdot b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.85 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5} \cdot a, a, -3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right), angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.8% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.5 \cdot 10^{-16}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), angle \cdot angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \cdot \left(\left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 3.5e-16)
   (* b b)
   (if (<= a 1.35e+154)
     (fma
      (* (* (* (* a a) 3.08641975308642e-5) (PI)) (PI))
      (* angle angle)
      (* b b))
     (* (* (* (PI) (PI)) a) (* (* (* angle angle) 3.08641975308642e-5) a)))))

Derivation

1. Split input into 3 regimes

2. if a < 3.50000000000000017e-16

   1. Initial program 80.3%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 63.9

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

   5. Applied rewrites 63.9%

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

   if 3.50000000000000017e-16 < a < 1.35000000000000003e154

   1. Initial program 72.2%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 50.4%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 63.0%

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

   if 1.35000000000000003e154 < a

   1. Initial program 99.6%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 34.4%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 51.1%

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

   10. Applied rewrites 64.2%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.5 \cdot 10^{-16}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), angle \cdot angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \cdot \left(\left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.0% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 4.5 \cdot 10^{+125}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \cdot \left(\left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 4.5e+125)
   (* b b)
   (* (* (* (PI) (PI)) a) (* (* (* angle angle) 3.08641975308642e-5) a))))

Derivation

1. Split input into 2 regimes

2. if a < 4.5e125

   1. Initial program 79.3%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.6

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

   5. Applied rewrites 61.6%

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

   if 4.5e125 < a

   1. Initial program 95.2%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 37.7%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 49.0%

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

   10. Applied rewrites 60.0%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.5 \cdot 10^{+125}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \cdot \left(\left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 60.5% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.9 \cdot 10^{+139}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \cdot a\right) \cdot \left(\left(3.08641975308642 \cdot 10^{-5} \cdot angle\right) \cdot angle\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 3.9e+139)
   (* b b)
   (* (* (* (* (PI) (PI)) a) a) (* (* 3.08641975308642e-5 angle) angle))))

Derivation

1. Split input into 2 regimes

2. if a < 3.90000000000000006e139

   1. Initial program 79.6%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.6

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

   5. Applied rewrites 61.6%

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

   if 3.90000000000000006e139 < a

   1. Initial program 94.7%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 35.1%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 47.4%

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

   10. Applied rewrites 47.4%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.9 \cdot 10^{+139}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \cdot a\right) \cdot \left(\left(3.08641975308642 \cdot 10^{-5} \cdot angle\right) \cdot angle\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 57.4% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.5%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 58.6

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

5. Applied rewrites 58.6%

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

Reproduce

herbie shell --seed 2024304 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) (PI)))) 2.0) (pow (* b (cos (* (/ angle 180.0) (PI)))) 2.0)))