ab-angle->ABCF A

Percentage Accurate: 79.5% → 79.4%

Time: 16.9s

Alternatives: 17

Speedup: 1.0×

• 36.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

Initial Program: 79.5% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 79.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)}^{2} + {\left(b \cdot \cos \left({\left(\frac{180}{{\mathsf{PI}\left(\right)}^{0.6666666666666666}}\right)}^{-1} \cdot {\left(\frac{\frac{1}{angle}}{\sqrt[3]{\mathsf{PI}\left(\right)}}\right)}^{-1}\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (sin (/ (PI) (/ 180.0 angle)))) 2.0)
  (pow
   (*
    b
    (cos
     (*
      (pow (/ 180.0 (pow (PI) 0.6666666666666666)) -1.0)
      (pow (/ (/ 1.0 angle) (cbrt (PI))) -1.0))))
   2.0)))

Derivation

1. Initial program 77.4%

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

   1. lift-PI.f64 N/A

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

   2. associate-*l/ N/A

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

   3. clear-num N/A

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

   4. associate-/r* N/A

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

   5. clear-num N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 77.4

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

4. Applied rewrites 77.4%

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

   1. clear-num N/A

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

   2. lift-/.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. associate-/r/ N/A

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

   5. inv-pow N/A

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

   6. lift-/.f64 N/A

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

   7. div-inv N/A

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

   8. inv-pow N/A

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

   9. metadata-eval N/A

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

   10. pow-prod-up N/A

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

   11. lift-pow.f64 N/A

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

   12. lift-pow.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. add-cube-cbrt N/A

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

   15. times-frac N/A

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

   16. unpow-prod-down N/A

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

   17. lower-*.f64 N/A

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

6. Applied rewrites 77.5%

   \[\leadsto {\left(a \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left({\left(\frac{180}{{\mathsf{PI}\left(\right)}^{0.6666666666666666}}\right)}^{-1} \cdot {\left(\frac{\frac{1}{angle}}{\sqrt[3]{\mathsf{PI}\left(\right)}}\right)}^{-1}\right)}\right)}^{2} \]
7. Add Preprocessing

Alternative 2: 79.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ {\left(a \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \left({\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot t\_0\right)}^{0.3333333333333333} \cdot \sqrt[3]{t\_0}\right)\right)\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (sqrt (PI))))
   (+
    (pow (* a (sin (/ (PI) (/ 180.0 angle)))) 2.0)
    (pow
     (*
      b
      (cos
       (*
        (/ angle 180.0)
        (* (pow (* (* (PI) (PI)) t_0) 0.3333333333333333) (cbrt t_0)))))
     2.0))))

Derivation

1. Initial program 77.4%

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

   1. lift-PI.f64 N/A

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

   2. associate-*l/ N/A

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

   3. clear-num N/A

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

   4. associate-/r* N/A

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

   5. clear-num N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 77.4

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

4. Applied rewrites 77.4%

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

   1. add-cbrt-cube N/A

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

   2. pow1/3 N/A

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

   3. lift-PI.f64 N/A

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

   4. rem-square-sqrt N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. associate-*r* N/A

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

   8. unpow-prod-down N/A

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

   9. pow1/3 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lift-PI.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-cbrt.f64 77.5

       \[\leadsto {\left(a \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \left({\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}^{0.3333333333333333} \cdot \color{blue}{\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}}\right)\right)\right)}^{2} \]

6. Applied rewrites 77.5%

   \[\leadsto {\left(a \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \color{blue}{\left({\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}\right)\right)}^{2} \]
7. Add Preprocessing

Alternative 3: 79.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 77.4%

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

   1. lift-PI.f64 N/A

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

   2. associate-*l/ N/A

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

   3. clear-num N/A

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

   4. associate-/r* N/A

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

   5. clear-num N/A

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

   6. div-inv N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. div-inv N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 77.5

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

4. Applied rewrites 77.5%

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

5. Final simplification 77.5%

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

Alternative 4: 79.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 77.4%

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

   1. lift-PI.f64 N/A

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

   2. associate-*l/ N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 77.5

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

4. Applied rewrites 77.5%

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

5. Final simplification 77.5%

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

Alternative 5: 79.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 77.4%

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

   1. lift-PI.f64 N/A

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

   2. associate-*l/ N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. div-inv N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 77.5

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

4. Applied rewrites 77.5%

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

5. Final simplification 77.5%

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

Alternative 6: 79.4% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 77.4%

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

   1. lift-PI.f64 N/A

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

   2. associate-*l/ N/A

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

   3. clear-num N/A

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

   4. associate-/r* N/A

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

   5. clear-num N/A

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

   6. div-inv N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. div-inv N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 77.5

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

4. Applied rewrites 77.5%

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

5. Taylor expanded in angle around 0

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

7. Applied rewrites 76.5%

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

8. Final simplification 76.5%

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

Alternative 7: 79.4% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 77.4%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 76.4%

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

6. Final simplification 76.4%

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

Alternative 8: 77.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \left(angle \cdot 0.005555555555555556\right)\\ t_1 := \sin t\_0\\ \mathbf{if}\;\frac{angle}{180} \leq 200000000:\\ \;\;\;\;\mathsf{fma}\left({\left(angle \cdot \left(a \cdot 0.005555555555555556\right)\right)}^{2}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), b \cdot \left(b \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot t\_0\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1 \cdot \left(a \cdot a\right), t\_1, b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (PI) (* angle 0.005555555555555556))) (t_1 (sin t_0)))
   (if (<= (/ angle 180.0) 200000000.0)
     (fma
      (pow (* angle (* a 0.005555555555555556)) 2.0)
      (* (PI) (PI))
      (* b (* b (+ 0.5 (* 0.5 (cos (* 2.0 t_0)))))))
     (fma (* t_1 (* a a)) t_1 (* b b)))))

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2e8

   1. Initial program 87.2%

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

   3. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 84.0

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

   5. Applied rewrites 84.0%

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

   7. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\left(0.005555555555555556 \cdot a\right) \cdot angle\right)}^{2}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), b \cdot \left(b \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\right)} \]

   if 2e8 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 54.8%

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

   4. Applied rewrites 54.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a \cdot a\right), \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \mathsf{PI}\left(\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} \]

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 54.9%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 200000000:\\ \;\;\;\;\mathsf{fma}\left({\left(angle \cdot \left(a \cdot 0.005555555555555556\right)\right)}^{2}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), b \cdot \left(b \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(a \cdot a\right), \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \left(angle \cdot 0.005555555555555556\right)\\ t_1 := 0.005555555555555556 \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\\ t_2 := \sin t\_0\\ \mathbf{if}\;\frac{angle}{180} \leq 200000000:\\ \;\;\;\;\mathsf{fma}\left(t\_1, t\_1, b \cdot \left(b \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot t\_0\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_2 \cdot \left(a \cdot a\right), t\_2, b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (PI) (* angle 0.005555555555555556)))
        (t_1 (* 0.005555555555555556 (* a (* (PI) angle))))
        (t_2 (sin t_0)))
   (if (<= (/ angle 180.0) 200000000.0)
     (fma t_1 t_1 (* b (* b (+ 0.5 (* 0.5 (cos (* 2.0 t_0)))))))
     (fma (* t_2 (* a a)) t_2 (* b b)))))

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2e8

   1. Initial program 87.2%

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

   3. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 84.0

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

   5. Applied rewrites 84.0%

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

   7. Applied rewrites 84.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.005555555555555556 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right), 0.005555555555555556 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right), b \cdot \left(b \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\right)} \]

   if 2e8 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 54.8%

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

   4. Applied rewrites 54.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a \cdot a\right), \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \mathsf{PI}\left(\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} \]

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 54.9%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 200000000:\\ \;\;\;\;\mathsf{fma}\left(0.005555555555555556 \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right), 0.005555555555555556 \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right), b \cdot \left(b \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(a \cdot a\right), \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\\ \mathbf{if}\;a \leq 3.5 \cdot 10^{-15}:\\ \;\;\;\;\left(b \cdot b\right) \cdot {\cos \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_0, b \cdot \left(b \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* a (* (PI) angle)))))
   (if (<= a 3.5e-15)
     (* (* b b) (pow (cos (* angle (* (PI) 0.005555555555555556))) 2.0))
     (fma
      t_0
      t_0
      (*
       b
       (*
        b
        (+
         0.5
         (* 0.5 (cos (* 2.0 (* (PI) (* angle 0.005555555555555556))))))))))))

Derivation

1. Split input into 2 regimes

2. if a < 3.5000000000000001e-15

   1. Initial program 78.8%

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

      1. lift-PI.f64 N/A

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

      2. associate-*l/ N/A

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

      3. clear-num N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. div-inv N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 78.8

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

   4. Applied rewrites 78.8%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. lift-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. frac-2neg N/A

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

      10. metadata-eval N/A

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

      11. frac-times N/A

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

      12. neg-mul-1 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. distribute-neg-frac N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 78.8

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

   6. Applied rewrites 78.8%

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

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-PI.f64 62.9

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

   9. Applied rewrites 62.9%

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

   if 3.5000000000000001e-15 < a

   1. Initial program 74.0%

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

   3. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 70.4

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

   5. Applied rewrites 70.4%

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

   7. Applied rewrites 70.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.005555555555555556 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right), 0.005555555555555556 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right), b \cdot \left(b \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.5 \cdot 10^{-15}:\\ \;\;\;\;\left(b \cdot b\right) \cdot {\cos \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.005555555555555556 \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right), 0.005555555555555556 \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right), b \cdot \left(b \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot angle\\ t_1 := 0.005555555555555556 \cdot \left(a \cdot t\_0\right)\\ \mathbf{if}\;a \leq 3.5 \cdot 10^{-15}:\\ \;\;\;\;\left(b \cdot b\right) \cdot {\cos \left(0.005555555555555556 \cdot t\_0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, t\_1, b \cdot \left(b \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (PI) angle)) (t_1 (* 0.005555555555555556 (* a t_0))))
   (if (<= a 3.5e-15)
     (* (* b b) (pow (cos (* 0.005555555555555556 t_0)) 2.0))
     (fma
      t_1
      t_1
      (*
       b
       (*
        b
        (+
         0.5
         (* 0.5 (cos (* 2.0 (* (PI) (* angle 0.005555555555555556))))))))))))

Derivation

1. Split input into 2 regimes

2. if a < 3.5000000000000001e-15

   1. Initial program 78.8%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-cos.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-PI.f64 62.9

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

   5. Applied rewrites 62.9%

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

   if 3.5000000000000001e-15 < a

   1. Initial program 74.0%

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

   3. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 70.4

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

   5. Applied rewrites 70.4%

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

   7. Applied rewrites 70.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.005555555555555556 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right), 0.005555555555555556 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right), b \cdot \left(b \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.5 \cdot 10^{-15}:\\ \;\;\;\;\left(b \cdot b\right) \cdot {\cos \left(0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.005555555555555556 \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right), 0.005555555555555556 \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right), b \cdot \left(b \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\\ \mathbf{if}\;a \leq 3.5 \cdot 10^{-15}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot 0.011111111111111112\right)\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_0, b \cdot \left(b \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* a (* (PI) angle)))))
   (if (<= a 3.5e-15)
     (* (* b b) (fma 0.5 (cos (* angle (* (PI) 0.011111111111111112))) 0.5))
     (fma
      t_0
      t_0
      (*
       b
       (*
        b
        (+
         0.5
         (* 0.5 (cos (* 2.0 (* (PI) (* angle 0.005555555555555556))))))))))))

Derivation

1. Split input into 2 regimes

2. if a < 3.5000000000000001e-15

   1. Initial program 78.8%

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

   4. Applied rewrites 73.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a \cdot a\right), \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \mathsf{PI}\left(\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(b \cdot b\right)} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(b \cdot b\right)} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{2}\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{2}\right)} \]

      6. lower-cos.f64 N/A

         \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}, \frac{1}{2}\right) \]

      7. *-commutative N/A

         \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{90}\right)}, \frac{1}{2}\right) \]

      8. associate-*l* N/A

         \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right)}, \frac{1}{2}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right)}, \frac{1}{2}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)}\right), \frac{1}{2}\right) \]

      11. lower-PI.f64 62.9

          \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(angle \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 0.011111111111111112\right)\right), 0.5\right) \]

   7. Applied rewrites 62.9%

      \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot 0.011111111111111112\right)\right), 0.5\right)} \]

   if 3.5000000000000001e-15 < a

   1. Initial program 74.0%

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

   3. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 70.4

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

   5. Applied rewrites 70.4%

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

   7. Applied rewrites 70.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.005555555555555556 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right), 0.005555555555555556 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right), b \cdot \left(b \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.5 \cdot 10^{-15}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot 0.011111111111111112\right)\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.005555555555555556 \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right), 0.005555555555555556 \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right), b \cdot \left(b \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 67.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.5 \cdot 10^{-15}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot 0.011111111111111112\right)\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\left(angle \cdot \left(a \cdot 0.005555555555555556\right)\right)}^{2}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 3.5e-15)
   (* (* b b) (fma 0.5 (cos (* angle (* (PI) 0.011111111111111112))) 0.5))
   (fma (pow (* angle (* a 0.005555555555555556)) 2.0) (* (PI) (PI)) (* b b))))

Derivation

1. Split input into 2 regimes

2. if a < 3.5000000000000001e-15

   1. Initial program 78.8%

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

   4. Applied rewrites 73.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a \cdot a\right), \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \mathsf{PI}\left(\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(b \cdot b\right)} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(b \cdot b\right)} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{2}\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{2}\right)} \]

      6. lower-cos.f64 N/A

         \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}, \frac{1}{2}\right) \]

      7. *-commutative N/A

         \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{90}\right)}, \frac{1}{2}\right) \]

      8. associate-*l* N/A

         \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right)}, \frac{1}{2}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right)}, \frac{1}{2}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)}\right), \frac{1}{2}\right) \]

      11. lower-PI.f64 62.9

          \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(angle \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 0.011111111111111112\right)\right), 0.5\right) \]

   7. Applied rewrites 62.9%

      \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot 0.011111111111111112\right)\right), 0.5\right)} \]

   if 3.5000000000000001e-15 < a

   1. Initial program 74.0%

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

   3. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 70.4

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

   5. Applied rewrites 70.4%

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

   7. Applied rewrites 70.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\left(0.005555555555555556 \cdot a\right) \cdot angle\right)}^{2}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), b \cdot \left(b \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\right)} \]

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 69.9%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.5 \cdot 10^{-15}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot 0.011111111111111112\right)\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\left(angle \cdot \left(a \cdot 0.005555555555555556\right)\right)}^{2}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 63.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := angle \cdot \left(a \cdot 0.005555555555555556\right)\\ \mathbf{if}\;a \leq 2.6 \cdot 10^{+127}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot 0.011111111111111112\right)\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* angle (* a 0.005555555555555556))))
   (if (<= a 2.6e+127)
     (* (* b b) (fma 0.5 (cos (* angle (* (PI) 0.011111111111111112))) 0.5))
     (* t_0 (* (* (PI) (PI)) t_0)))))

Derivation

1. Split input into 2 regimes

2. if a < 2.6000000000000002e127

   1. Initial program 75.8%

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

   4. Applied rewrites 71.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a \cdot a\right), \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \mathsf{PI}\left(\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(b \cdot b\right)} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(b \cdot b\right)} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{2}\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{2}\right)} \]

      6. lower-cos.f64 N/A

         \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}, \frac{1}{2}\right) \]

      7. *-commutative N/A

         \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{90}\right)}, \frac{1}{2}\right) \]

      8. associate-*l* N/A

         \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right)}, \frac{1}{2}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right)}, \frac{1}{2}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)}\right), \frac{1}{2}\right) \]

      11. lower-PI.f64 60.5

          \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(angle \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 0.011111111111111112\right)\right), 0.5\right) \]

   7. Applied rewrites 60.5%

      \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot 0.011111111111111112\right)\right), 0.5\right)} \]

   if 2.6000000000000002e127 < a

   1. Initial program 87.6%

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

      1. lift-PI.f64 N/A

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

      2. associate-*l/ N/A

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

      3. clear-num N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. div-inv N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 87.7

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

   4. Applied rewrites 87.7%

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

   5. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   7. Applied rewrites 44.8%

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

   8. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-PI.f64 N/A

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

      19. lower-PI.f64 64.8

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

   10. Applied rewrites 64.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-PI.f64 N/A

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

       4. lift-PI.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

   12. Applied rewrites 75.1%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.6 \cdot 10^{+127}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot 0.011111111111111112\right)\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot \left(a \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(angle \cdot \left(a \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 63.0% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := angle \cdot \left(a \cdot 0.005555555555555556\right)\\ \mathbf{if}\;a \leq 2.6 \cdot 10^{+127}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* angle (* a 0.005555555555555556))))
   (if (<= a 2.6e+127) (* b b) (* t_0 (* (* (PI) (PI)) t_0)))))

Derivation

1. Split input into 2 regimes

2. if a < 2.6000000000000002e127

   1. Initial program 75.8%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 60.2

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

   5. Applied rewrites 60.2%

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

   if 2.6000000000000002e127 < a

   1. Initial program 87.6%

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

      1. lift-PI.f64 N/A

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

      2. associate-*l/ N/A

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

      3. clear-num N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. div-inv N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 87.7

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

   4. Applied rewrites 87.7%

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

   5. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   7. Applied rewrites 44.8%

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

   8. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-PI.f64 N/A

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

      19. lower-PI.f64 64.8

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

   10. Applied rewrites 64.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-PI.f64 N/A

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

       4. lift-PI.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

   12. Applied rewrites 75.1%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.6 \cdot 10^{+127}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot \left(a \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(angle \cdot \left(a \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 63.0% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\\ \mathbf{if}\;a \leq 3 \cdot 10^{+127}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;3.08641975308642 \cdot 10^{-5} \cdot \left(t\_0 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* a (* (PI) angle))))
   (if (<= a 3e+127) (* b b) (* 3.08641975308642e-5 (* t_0 t_0)))))

Derivation

1. Split input into 2 regimes

2. if a < 3.0000000000000002e127

   1. Initial program 75.8%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 60.2

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

   5. Applied rewrites 60.2%

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

   if 3.0000000000000002e127 < a

   1. Initial program 87.6%

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

      1. lift-PI.f64 N/A

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

      2. associate-*l/ N/A

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

      3. clear-num N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. div-inv N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 87.7

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

   4. Applied rewrites 87.7%

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

   5. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   7. Applied rewrites 44.8%

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

   8. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-PI.f64 N/A

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

      19. lower-PI.f64 64.8

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

   10. Applied rewrites 64.8%

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

   11. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. unpow2 N/A

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

       4. unpow2 N/A

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

       5. unswap-sqr N/A

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

       6. unswap-sqr N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-PI.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 N/A

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

       13. lower-PI.f64 75.0

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

   13. Applied rewrites 75.0%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3 \cdot 10^{+127}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;3.08641975308642 \cdot 10^{-5} \cdot \left(\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 57.4% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 56.6

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

5. Applied rewrites 56.6%

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

Reproduce

herbie shell --seed 2024216 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) (PI)))) 2.0) (pow (* b (cos (* (/ angle 180.0) (PI)))) 2.0)))