ab-angle->ABCF C

Percentage Accurate: 80.3% → 80.3%

Time: 12.7s

Alternatives: 11

Speedup: 1.0×

• Could not converge on a ground truth

  1. a = 4.0520040055795565e-42
  2. b = 1.911266064355088e-249
  3. angle = 5.838059763809319e+35

• 36.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

Initial Program: 80.3% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 80.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\mathsf{PI}\left(\right)}\\ {\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2} + {\left(\cos \left(\left(0.005555555555555556 \cdot t\_0\right) \cdot \left(angle \cdot {t\_0}^{2}\right)\right) \cdot a\right)}^{2} \end{array} \end{array} \]

FPCore

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

Derivation

1. Initial program 81.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. lift-PI.f64 N/A

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

   6. add-cube-cbrt N/A

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

   7. associate-*l* N/A

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

   8. div-inv N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. lower-cbrt.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. pow2 N/A

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

   16. lower-pow.f64 N/A

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

   17. lift-PI.f64 N/A

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

   18. lower-cbrt.f64 N/A

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

   19. inv-pow N/A

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

   20. lower-pow.f64 81.2

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

4. Applied rewrites 81.2%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. neg-mul-1 N/A

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

   4. lift-pow.f64 N/A

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

   5. sqr-pow N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-pow.f64 N/A

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

   15. metadata-eval 41.6

       \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\mathsf{PI}\left(\right)}}{180} \cdot \left(\frac{-1}{{angle}^{-0.5}} \cdot \frac{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}}{-{angle}^{\color{blue}{-0.5}}}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

6. Applied rewrites 41.6%

   \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\mathsf{PI}\left(\right)}}{180} \cdot \color{blue}{\left(\frac{-1}{{angle}^{-0.5}} \cdot \frac{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}}{-{angle}^{-0.5}}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. metadata-eval N/A

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

   4. lower-*.f64 41.6

      \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\mathsf{PI}\left(\right)}}{180} \cdot \left(\frac{-1}{{angle}^{-0.5}} \cdot \frac{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}}{-{angle}^{-0.5}}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(angle \cdot 0.005555555555555556\right)}\right)\right)}^{2} \]

8. Applied rewrites 41.6%

   \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\mathsf{PI}\left(\right)}}{180} \cdot \left(\frac{-1}{{angle}^{-0.5}} \cdot \frac{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}}{-{angle}^{-0.5}}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(angle \cdot 0.005555555555555556\right)}\right)\right)}^{2} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. frac-times N/A

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

   7. lift-/.f64 N/A

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

   8. frac-times N/A

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

   9. neg-mul-1 N/A

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

10. Applied rewrites 81.2%

    \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot angle\right) \cdot \left(0.005555555555555556 \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

11. Final simplification 81.2%

    \[\leadsto {\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2} + {\left(\cos \left(\left(0.005555555555555556 \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(angle \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}\right)\right) \cdot a\right)}^{2} \]
12. Add Preprocessing

Alternative 2: 80.3% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 81.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 81.2

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 81.2

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

   7. lift-/.f64 N/A

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

   8. clear-num N/A

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

   9. associate-/r/ N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval 81.2

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

4. Applied rewrites 81.2%

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

5. Final simplification 81.2%

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

Alternative 3: 80.3% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 81.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. lift-PI.f64 N/A

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

   6. add-cube-cbrt N/A

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

   7. associate-*l* N/A

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

   8. div-inv N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. lower-cbrt.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. pow2 N/A

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

   16. lower-pow.f64 N/A

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

   17. lift-PI.f64 N/A

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

   18. lower-cbrt.f64 N/A

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

   19. inv-pow N/A

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

   20. lower-pow.f64 81.2

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

4. Applied rewrites 81.2%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. neg-mul-1 N/A

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

   4. lift-pow.f64 N/A

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

   5. sqr-pow N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-pow.f64 N/A

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

   15. metadata-eval 41.6

       \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\mathsf{PI}\left(\right)}}{180} \cdot \left(\frac{-1}{{angle}^{-0.5}} \cdot \frac{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}}{-{angle}^{\color{blue}{-0.5}}}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

6. Applied rewrites 41.6%

   \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\mathsf{PI}\left(\right)}}{180} \cdot \color{blue}{\left(\frac{-1}{{angle}^{-0.5}} \cdot \frac{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}}{-{angle}^{-0.5}}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. metadata-eval N/A

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

   4. lower-*.f64 41.6

      \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\mathsf{PI}\left(\right)}}{180} \cdot \left(\frac{-1}{{angle}^{-0.5}} \cdot \frac{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}}{-{angle}^{-0.5}}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(angle \cdot 0.005555555555555556\right)}\right)\right)}^{2} \]

8. Applied rewrites 41.6%

   \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\mathsf{PI}\left(\right)}}{180} \cdot \left(\frac{-1}{{angle}^{-0.5}} \cdot \frac{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}}{-{angle}^{-0.5}}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(angle \cdot 0.005555555555555556\right)}\right)\right)}^{2} \]

10. Applied rewrites 81.2%

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

Alternative 4: 80.3% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 81.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. lift-PI.f64 N/A

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

   6. add-cube-cbrt N/A

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

   7. associate-*l* N/A

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

   8. div-inv N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. lower-cbrt.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. pow2 N/A

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

   16. lower-pow.f64 N/A

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

   17. lift-PI.f64 N/A

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

   18. lower-cbrt.f64 N/A

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

   19. inv-pow N/A

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

   20. lower-pow.f64 81.2

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

4. Applied rewrites 81.2%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. neg-mul-1 N/A

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

   4. lift-pow.f64 N/A

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

   5. sqr-pow N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-pow.f64 N/A

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

   15. metadata-eval 41.6

       \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\mathsf{PI}\left(\right)}}{180} \cdot \left(\frac{-1}{{angle}^{-0.5}} \cdot \frac{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}}{-{angle}^{\color{blue}{-0.5}}}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

6. Applied rewrites 41.6%

   \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\mathsf{PI}\left(\right)}}{180} \cdot \color{blue}{\left(\frac{-1}{{angle}^{-0.5}} \cdot \frac{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}}{-{angle}^{-0.5}}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. metadata-eval N/A

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

   4. lower-*.f64 41.6

      \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\mathsf{PI}\left(\right)}}{180} \cdot \left(\frac{-1}{{angle}^{-0.5}} \cdot \frac{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}}{-{angle}^{-0.5}}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(angle \cdot 0.005555555555555556\right)}\right)\right)}^{2} \]

8. Applied rewrites 41.6%

   \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\mathsf{PI}\left(\right)}}{180} \cdot \left(\frac{-1}{{angle}^{-0.5}} \cdot \frac{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}}{-{angle}^{-0.5}}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(angle \cdot 0.005555555555555556\right)}\right)\right)}^{2} \]

10. Applied rewrites 81.2%

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

11. Final simplification 81.2%

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

Alternative 5: 80.3% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 81.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. lift-PI.f64 N/A

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

   6. add-cube-cbrt N/A

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

   7. associate-*l* N/A

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

   8. div-inv N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. lower-cbrt.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. pow2 N/A

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

   16. lower-pow.f64 N/A

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

   17. lift-PI.f64 N/A

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

   18. lower-cbrt.f64 N/A

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

   19. inv-pow N/A

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

   20. lower-pow.f64 81.2

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

4. Applied rewrites 81.2%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. neg-mul-1 N/A

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

   4. lift-pow.f64 N/A

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

   5. sqr-pow N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-pow.f64 N/A

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

   15. metadata-eval 41.6

       \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\mathsf{PI}\left(\right)}}{180} \cdot \left(\frac{-1}{{angle}^{-0.5}} \cdot \frac{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}}{-{angle}^{\color{blue}{-0.5}}}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

6. Applied rewrites 41.6%

   \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\mathsf{PI}\left(\right)}}{180} \cdot \color{blue}{\left(\frac{-1}{{angle}^{-0.5}} \cdot \frac{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}}{-{angle}^{-0.5}}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. metadata-eval N/A

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

   4. lower-*.f64 41.6

      \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\mathsf{PI}\left(\right)}}{180} \cdot \left(\frac{-1}{{angle}^{-0.5}} \cdot \frac{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}}{-{angle}^{-0.5}}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(angle \cdot 0.005555555555555556\right)}\right)\right)}^{2} \]

8. Applied rewrites 41.6%

   \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\mathsf{PI}\left(\right)}}{180} \cdot \left(\frac{-1}{{angle}^{-0.5}} \cdot \frac{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}}{-{angle}^{-0.5}}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(angle \cdot 0.005555555555555556\right)}\right)\right)}^{2} \]

9. Taylor expanded in angle around 0

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

11. Applied rewrites 81.1%

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

12. Final simplification 81.1%

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

Alternative 6: 80.2% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 81.2%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 81.1

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

5. Applied rewrites 81.1%

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

6. Final simplification 81.1%

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

Alternative 7: 65.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{-124}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(b \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), angle \cdot angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.15e-124)
   (* a a)
   (if (<= b 6.8e+155)
     (fma
      (* (* (* (* b b) 3.08641975308642e-5) (PI)) (PI))
      (* angle angle)
      (* a a))
     (* (pow (* (* b (PI)) angle) 2.0) 3.08641975308642e-5))))

Derivation

1. Split input into 3 regimes

2. if b < 1.15000000000000006e-124

   1. Initial program 80.3%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 66.1

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

   5. Applied rewrites 66.1%

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

   if 1.15000000000000006e-124 < b < 6.8000000000000002e155

   1. Initial program 73.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-PI.f64 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*l* N/A

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

      8. div-inv N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lower-cbrt.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. lower-pow.f64 N/A

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

      17. lift-PI.f64 N/A

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

      18. lower-cbrt.f64 N/A

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

      19. inv-pow N/A

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

      20. lower-pow.f64 73.3

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

   4. Applied rewrites 73.3%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 42.2%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 63.2%

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

   if 6.8000000000000002e155 < b

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-PI.f64 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*l* N/A

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

      8. div-inv N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lower-cbrt.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. lower-pow.f64 N/A

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

      17. lift-PI.f64 N/A

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

      18. lower-cbrt.f64 N/A

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

      19. inv-pow N/A

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

      20. lower-pow.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 50.9%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 68.5%

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

   12. Applied rewrites 94.3%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{-124}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(b \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), angle \cdot angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.0% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{-124}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;b \leq 9 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(b \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), angle \cdot angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right) \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.15e-124)
   (* a a)
   (if (<= b 9e+155)
     (fma
      (* (* (* (* b b) 3.08641975308642e-5) (PI)) (PI))
      (* angle angle)
      (* a a))
     (* (* (* (* (* angle angle) 3.08641975308642e-5) b) (* b (PI))) (PI)))))

Derivation

1. Split input into 3 regimes

2. if b < 1.15000000000000006e-124

   1. Initial program 80.3%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 66.1

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

   5. Applied rewrites 66.1%

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

   if 1.15000000000000006e-124 < b < 8.99999999999999947e155

   1. Initial program 73.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-PI.f64 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*l* N/A

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

      8. div-inv N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lower-cbrt.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. lower-pow.f64 N/A

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

      17. lift-PI.f64 N/A

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

      18. lower-cbrt.f64 N/A

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

      19. inv-pow N/A

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

      20. lower-pow.f64 73.3

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

   4. Applied rewrites 73.3%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 42.2%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 63.2%

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

   if 8.99999999999999947e155 < b

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-PI.f64 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*l* N/A

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

      8. div-inv N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lower-cbrt.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. lower-pow.f64 N/A

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

      17. lift-PI.f64 N/A

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

      18. lower-cbrt.f64 N/A

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

      19. inv-pow N/A

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

      20. lower-pow.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 50.9%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 68.5%

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

   12. Applied rewrites 78.4%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{-124}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;b \leq 9 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(b \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), angle \cdot angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right) \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.0% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.5 \cdot 10^{+161}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right) \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 2.5e+161)
   (* a a)
   (* (* (* (* (* angle angle) 3.08641975308642e-5) b) (* b (PI))) (PI))))

Derivation

1. Split input into 2 regimes

2. if b < 2.4999999999999998e161

   1. Initial program 78.4%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.7

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

   5. Applied rewrites 61.7%

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

   if 2.4999999999999998e161 < b

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-PI.f64 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*l* N/A

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

      8. div-inv N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lower-cbrt.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. lower-pow.f64 N/A

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

      17. lift-PI.f64 N/A

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

      18. lower-cbrt.f64 N/A

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

      19. inv-pow N/A

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

      20. lower-pow.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 52.3%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 70.4%

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

   12. Applied rewrites 77.8%

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

Alternative 10: 61.5% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{+161}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(b \cdot b\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(3.08641975308642 \cdot 10^{-5} \cdot angle\right) \cdot angle\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 4.2e+161)
   (* a a)
   (* (* (* (* b b) (PI)) (PI)) (* (* 3.08641975308642e-5 angle) angle))))

Derivation

1. Split input into 2 regimes

2. if b < 4.2e161

   1. Initial program 78.4%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.7

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

   5. Applied rewrites 61.7%

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

   if 4.2e161 < b

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-PI.f64 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*l* N/A

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

      8. div-inv N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lower-cbrt.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. lower-pow.f64 N/A

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

      17. lift-PI.f64 N/A

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

      18. lower-cbrt.f64 N/A

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

      19. inv-pow N/A

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

      20. lower-pow.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 52.3%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 70.4%

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

   12. Applied rewrites 70.4%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{+161}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(b \cdot b\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(3.08641975308642 \cdot 10^{-5} \cdot angle\right) \cdot angle\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.8% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.2%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 56.9

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

5. Applied rewrites 56.9%

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

Reproduce

herbie shell --seed 2024283 
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* (PI) (/ angle 180.0)))) 2.0) (pow (* b (sin (* (PI) (/ angle 180.0)))) 2.0)))