ab-angle->ABCF C

Percentage Accurate: 79.3% → 79.3%

Time: 16.2s

Alternatives: 24

Speedup: 1.0×

• Could not converge on a ground truth

  1. a = 6.591688374111715e-113
  2. b = 7.177232832399209e-279
  3. angle = 1.0749342300752417e+164

• 36.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

Initial Program: 79.3% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 79.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ t_1 := \log t\_0\\ {\left(\sin \left(\frac{angle\_m}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2} + {\left(\cos \left(\frac{\frac{1}{e^{\frac{{t\_1}^{2}}{-\log \left(\frac{t\_0}{angle\_m}\right)}}}}{e^{\frac{{\log angle\_m}^{2}}{t\_1 - \log angle\_m}}} \cdot \frac{t\_0}{180}\right) \cdot a\right)}^{2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (sqrt (PI))) (t_1 (log t_0)))
   (+
    (pow (* (sin (* (/ angle_m 180.0) (PI))) b) 2.0)
    (pow
     (*
      (cos
       (*
        (/
         (/ 1.0 (exp (/ (pow t_1 2.0) (- (log (/ t_0 angle_m))))))
         (exp (/ (pow (log angle_m) 2.0) (- t_1 (log angle_m)))))
        (/ t_0 180.0)))
      a)
     2.0))))

Derivation

1. Initial program 76.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. lift-PI.f64 N/A

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

   6. add-sqr-sqrt N/A

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

   7. div-inv N/A

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

   8. times-frac N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lift-PI.f64 N/A

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

   15. lower-sqrt.f64 N/A

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

   16. inv-pow N/A

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

   17. lower-pow.f64 76.4

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

4. Applied rewrites 76.4%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift-sqrt.f64 N/A

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

   4. pow1/2 N/A

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

   5. pow-to-exp N/A

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

   6. lift-pow.f64 N/A

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

   7. pow-to-exp N/A

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

   8. rec-exp N/A

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

   9. prod-exp N/A

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

   10. lower-exp.f64 N/A

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

   11. rem-log-exp N/A

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

   12. pow-to-exp N/A

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

   13. log-pow N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-log.f64 N/A

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

   16. rem-log-exp N/A

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

   17. rec-exp N/A

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

   18. pow-to-exp N/A

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

   19. unpow-1 N/A

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

   20. remove-double-div N/A

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

   21. lower-log.f64 42.1

       \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{180} \cdot e^{\mathsf{fma}\left(0.5, \log \mathsf{PI}\left(\right), \color{blue}{\log angle}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

6. Applied rewrites 42.1%

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

   1. lift-exp.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. flip-+ N/A

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

   4. div-sub N/A

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

   5. exp-diff N/A

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

   6. lower-/.f64 N/A

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

8. Applied rewrites 42.1%

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

   1. lift-exp.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. distribute-frac-neg N/A

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

   5. exp-neg N/A

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

   6. lower-/.f64 N/A

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

   7. lower-exp.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-neg.f64 42.1

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

   10. lift--.f64 N/A

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

   11. lift-log.f64 N/A

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

   12. lift-log.f64 N/A

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

10. Applied rewrites 42.1%

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

11. Final simplification 42.1%

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

Alternative 2: 79.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ t_1 := \log t\_0\\ {\left(\cos \left(\frac{e^{\frac{{t\_1}^{2}}{t\_1 - \log angle\_m}}}{e^{\frac{{\log angle\_m}^{2}}{\log \left(\frac{t\_0}{angle\_m}\right)}}} \cdot \frac{t\_0}{180}\right) \cdot a\right)}^{2} + {\left(\sin \left(\frac{angle\_m}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (sqrt (PI))) (t_1 (log t_0)))
   (+
    (pow
     (*
      (cos
       (*
        (/
         (exp (/ (pow t_1 2.0) (- t_1 (log angle_m))))
         (exp (/ (pow (log angle_m) 2.0) (log (/ t_0 angle_m)))))
        (/ t_0 180.0)))
      a)
     2.0)
    (pow (* (sin (* (/ angle_m 180.0) (PI))) b) 2.0))))

Derivation

1. Initial program 76.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. lift-PI.f64 N/A

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

   6. add-sqr-sqrt N/A

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

   7. div-inv N/A

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

   8. times-frac N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lift-PI.f64 N/A

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

   15. lower-sqrt.f64 N/A

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

   16. inv-pow N/A

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

   17. lower-pow.f64 76.4

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

4. Applied rewrites 76.4%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift-sqrt.f64 N/A

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

   4. pow1/2 N/A

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

   5. pow-to-exp N/A

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

   6. lift-pow.f64 N/A

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

   7. pow-to-exp N/A

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

   8. rec-exp N/A

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

   9. prod-exp N/A

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

   10. lower-exp.f64 N/A

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

   11. rem-log-exp N/A

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

   12. pow-to-exp N/A

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

   13. log-pow N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-log.f64 N/A

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

   16. rem-log-exp N/A

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

   17. rec-exp N/A

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

   18. pow-to-exp N/A

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

   19. unpow-1 N/A

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

   20. remove-double-div N/A

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

   21. lower-log.f64 42.1

       \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{180} \cdot e^{\mathsf{fma}\left(0.5, \log \mathsf{PI}\left(\right), \color{blue}{\log angle}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

6. Applied rewrites 42.1%

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

   1. lift-exp.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. flip-+ N/A

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

   4. div-sub N/A

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

   5. exp-diff N/A

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

   6. lower-/.f64 N/A

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

8. Applied rewrites 42.1%

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

   1. lift--.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. lift-log.f64 N/A

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

   4. diff-log N/A

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

   5. lower-log.f64 N/A

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

   6. lower-/.f64 42.1

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

10. Applied rewrites 42.1%

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

11. Final simplification 42.1%

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

Alternative 3: 79.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ t_1 := \log t\_0\\ {\left(\cos \left(\frac{{t\_0}^{\left(\frac{t\_1}{\log \left(\frac{t\_0}{angle\_m}\right)}\right)}}{e^{\frac{{\log angle\_m}^{2}}{t\_1 - \log angle\_m}}} \cdot \frac{t\_0}{180}\right) \cdot a\right)}^{2} + {\left(\sin \left(\frac{angle\_m}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (sqrt (PI))) (t_1 (log t_0)))
   (+
    (pow
     (*
      (cos
       (*
        (/
         (pow t_0 (/ t_1 (log (/ t_0 angle_m))))
         (exp (/ (pow (log angle_m) 2.0) (- t_1 (log angle_m)))))
        (/ t_0 180.0)))
      a)
     2.0)
    (pow (* (sin (* (/ angle_m 180.0) (PI))) b) 2.0))))

Derivation

1. Initial program 76.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. lift-PI.f64 N/A

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

   6. add-sqr-sqrt N/A

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

   7. div-inv N/A

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

   8. times-frac N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lift-PI.f64 N/A

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

   15. lower-sqrt.f64 N/A

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

   16. inv-pow N/A

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

   17. lower-pow.f64 76.4

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

4. Applied rewrites 76.4%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift-sqrt.f64 N/A

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

   4. pow1/2 N/A

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

   5. pow-to-exp N/A

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

   6. lift-pow.f64 N/A

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

   7. pow-to-exp N/A

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

   8. rec-exp N/A

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

   9. prod-exp N/A

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

   10. lower-exp.f64 N/A

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

   11. rem-log-exp N/A

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

   12. pow-to-exp N/A

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

   13. log-pow N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-log.f64 N/A

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

   16. rem-log-exp N/A

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

   17. rec-exp N/A

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

   18. pow-to-exp N/A

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

   19. unpow-1 N/A

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

   20. remove-double-div N/A

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

   21. lower-log.f64 42.1

       \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{180} \cdot e^{\mathsf{fma}\left(0.5, \log \mathsf{PI}\left(\right), \color{blue}{\log angle}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

6. Applied rewrites 42.1%

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

   1. lift-exp.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. flip-+ N/A

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

   4. div-sub N/A

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

   5. exp-diff N/A

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

   6. lower-/.f64 N/A

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

8. Applied rewrites 42.1%

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

   1. lift-exp.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. associate-/l* N/A

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

   6. lift-log.f64 N/A

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

   7. exp-to-pow N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-/.f64 42.1

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

   10. lift--.f64 N/A

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

   11. lift-log.f64 N/A

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

   12. lift-log.f64 N/A

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

   13. diff-log N/A

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

10. Applied rewrites 42.1%

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

11. Final simplification 42.1%

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

Alternative 4: 79.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(\cos \left(e^{\mathsf{fma}\left(0.5, \log \mathsf{PI}\left(\right), \log angle\_m\right)} \cdot \frac{\sqrt{\mathsf{PI}\left(\right)}}{180}\right) \cdot a\right)}^{2} + {\left(\sin \left(\frac{angle\_m}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow
   (*
    (cos (* (exp (fma 0.5 (log (PI)) (log angle_m))) (/ (sqrt (PI)) 180.0)))
    a)
   2.0)
  (pow (* (sin (* (/ angle_m 180.0) (PI))) b) 2.0)))

Derivation

1. Initial program 76.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. lift-PI.f64 N/A

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

   6. add-sqr-sqrt N/A

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

   7. div-inv N/A

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

   8. times-frac N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lift-PI.f64 N/A

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

   15. lower-sqrt.f64 N/A

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

   16. inv-pow N/A

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

   17. lower-pow.f64 76.4

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

4. Applied rewrites 76.4%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift-sqrt.f64 N/A

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

   4. pow1/2 N/A

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

   5. pow-to-exp N/A

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

   6. lift-pow.f64 N/A

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

   7. pow-to-exp N/A

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

   8. rec-exp N/A

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

   9. prod-exp N/A

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

   10. lower-exp.f64 N/A

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

   11. rem-log-exp N/A

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

   12. pow-to-exp N/A

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

   13. log-pow N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-log.f64 N/A

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

   16. rem-log-exp N/A

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

   17. rec-exp N/A

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

   18. pow-to-exp N/A

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

   19. unpow-1 N/A

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

   20. remove-double-div N/A

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

   21. lower-log.f64 42.1

       \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{180} \cdot e^{\mathsf{fma}\left(0.5, \log \mathsf{PI}\left(\right), \color{blue}{\log angle}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

6. Applied rewrites 42.1%

   \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{180} \cdot \color{blue}{e^{\mathsf{fma}\left(0.5, \log \mathsf{PI}\left(\right), \log angle\right)}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

7. Final simplification 42.1%

   \[\leadsto {\left(\cos \left(e^{\mathsf{fma}\left(0.5, \log \mathsf{PI}\left(\right), \log angle\right)} \cdot \frac{\sqrt{\mathsf{PI}\left(\right)}}{180}\right) \cdot a\right)}^{2} + {\left(\sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2} \]
8. Add Preprocessing

Alternative 5: 79.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ {\left(\cos \left(\frac{1}{\frac{\frac{\frac{180}{t\_0}}{angle\_m}}{t\_0}}\right) \cdot a\right)}^{2} + {\left(\sin \left(\frac{angle\_m}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (sqrt (PI))))
   (+
    (pow (* (cos (/ 1.0 (/ (/ (/ 180.0 t_0) angle_m) t_0))) a) 2.0)
    (pow (* (sin (* (/ angle_m 180.0) (PI))) b) 2.0))))

Derivation

1. Initial program 76.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. lift-PI.f64 N/A

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

   6. add-sqr-sqrt N/A

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

   7. div-inv N/A

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

   8. times-frac N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lift-PI.f64 N/A

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

   15. lower-sqrt.f64 N/A

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

   16. inv-pow N/A

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

   17. lower-pow.f64 76.4

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

4. Applied rewrites 76.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-2neg N/A

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

   6. clear-num N/A

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

   7. frac-times N/A

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

   8. *-rgt-identity N/A

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

   9. distribute-lft-neg-out N/A

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

   10. frac-2neg N/A

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

   11. clear-num N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 N/A

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

6. Applied rewrites 76.4%

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

7. Final simplification 76.4%

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

Alternative 6: 79.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ {\left(\cos \left(\frac{-1}{\frac{-1}{angle\_m \cdot t\_0}} \cdot \frac{t\_0}{180}\right) \cdot a\right)}^{2} + {\left(\sin \left(\frac{angle\_m}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (sqrt (PI))))
   (+
    (pow (* (cos (* (/ -1.0 (/ -1.0 (* angle_m t_0))) (/ t_0 180.0))) a) 2.0)
    (pow (* (sin (* (/ angle_m 180.0) (PI))) b) 2.0))))

Derivation

1. Initial program 76.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. lift-PI.f64 N/A

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

   6. add-sqr-sqrt N/A

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

   7. div-inv N/A

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

   8. times-frac N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lift-PI.f64 N/A

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

   15. lower-sqrt.f64 N/A

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

   16. inv-pow N/A

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

   17. lower-pow.f64 76.4

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

4. Applied rewrites 76.4%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. frac-2neg N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f64 N/A

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

   6. clear-num N/A

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

   7. lift-/.f64 N/A

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

   8. distribute-neg-frac N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f64 76.4

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

   11. lift-/.f64 N/A

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

   12. clear-num N/A

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

   13. associate-/r/ N/A

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

   14. lift-pow.f64 N/A

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

   15. unpow-1 N/A

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

   16. remove-double-div N/A

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

   17. lower-*.f64 76.4

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

6. Applied rewrites 76.4%

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

7. Final simplification 76.4%

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

Alternative 7: 79.3% accurate, 0.9× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (sqrt (PI))))
   (+
    (pow (* (cos (* (* angle_m t_0) (/ t_0 180.0))) a) 2.0)
    (pow (* (sin (* (/ angle_m 180.0) (PI))) b) 2.0))))

Derivation

1. Initial program 76.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. lift-PI.f64 N/A

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

   6. add-sqr-sqrt N/A

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

   7. div-inv N/A

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

   8. times-frac N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lift-PI.f64 N/A

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

   15. lower-sqrt.f64 N/A

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

   16. inv-pow N/A

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

   17. lower-pow.f64 76.4

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

4. Applied rewrites 76.4%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow-1 N/A

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

   5. remove-double-div N/A

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

   6. lower-*.f64 76.4

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

6. Applied rewrites 76.4%

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

7. Final simplification 76.4%

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

Alternative 8: 79.3% accurate, 1.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (sqrt (PI))))
   (+
    (pow (* (cos (* (* 0.005555555555555556 (* angle_m t_0)) t_0)) a) 2.0)
    (pow (* (sin (* (/ angle_m 180.0) (PI))) b) 2.0))))

Derivation

1. Initial program 76.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. lift-PI.f64 N/A

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

   6. add-sqr-sqrt N/A

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

   7. div-inv N/A

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

   8. times-frac N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lift-PI.f64 N/A

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

   15. lower-sqrt.f64 N/A

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

   16. inv-pow N/A

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

   17. lower-pow.f64 76.4

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

4. Applied rewrites 76.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. div-inv N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow-1 N/A

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

   7. remove-double-div N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. lift-/.f64 N/A

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

   11. div-inv N/A

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

   12. metadata-eval N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 76.3

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

6. Applied rewrites 76.4%

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

7. Final simplification 76.4%

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

Alternative 9: 79.3% accurate, 1.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* (cos (/ 1.0 (/ (/ 180.0 (PI)) angle_m))) a) 2.0)
  (pow (* (sin (* (/ angle_m 180.0) (PI))) b) 2.0)))

Derivation

1. Initial program 76.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. lift-PI.f64 N/A

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

   6. add-sqr-sqrt N/A

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

   7. div-inv N/A

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

   8. times-frac N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lift-PI.f64 N/A

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

   15. lower-sqrt.f64 N/A

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

   16. inv-pow N/A

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

   17. lower-pow.f64 76.4

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

4. Applied rewrites 76.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-2neg N/A

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

   6. clear-num N/A

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

   7. frac-times N/A

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

   8. *-rgt-identity N/A

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

   9. distribute-lft-neg-out N/A

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

   10. frac-2neg N/A

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

   11. clear-num N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 N/A

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

6. Applied rewrites 76.4%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift-/.f64 N/A

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

   6. clear-num N/A

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

   7. lift-/.f64 N/A

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

   8. frac-times N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f64 N/A

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

8. Applied rewrites 76.3%

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

9. Final simplification 76.3%

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

Alternative 10: 79.3% accurate, 1.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* (cos (/ 1.0 (/ (/ 180.0 angle_m) (PI)))) a) 2.0)
  (pow (* (sin (* (/ angle_m 180.0) (PI))) b) 2.0)))

Derivation

1. Initial program 76.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. clear-num N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 76.3

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

4. Applied rewrites 76.3%

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

5. Final simplification 76.3%

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

Alternative 11: 79.3% accurate, 1.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (/ angle_m 180.0) (PI))))
   (+ (pow (* (cos t_0) a) 2.0) (pow (* (sin t_0) b) 2.0))))

Derivation

1. Initial program 76.3%

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

3. Final simplification 76.3%

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

Alternative 12: 79.3% accurate, 1.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* (cos (* (* -0.005555555555555556 angle_m) (PI))) a) 2.0)
  (pow (* (sin (* (/ angle_m 180.0) (PI))) b) 2.0)))

Derivation

1. Initial program 76.3%

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

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. frac-2neg N/A

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

   6. distribute-frac-neg N/A

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

   7. distribute-lft-neg-out N/A

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

   8. cos-neg N/A

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

   9. lower-cos.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. div-inv N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval 76.3

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

4. Applied rewrites 76.3%

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

5. Final simplification 76.3%

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

Alternative 13: 79.3% accurate, 1.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (* 0.005555555555555556 angle_m) (PI))))
   (fma a (* (pow (cos t_0) 2.0) a) (pow (* (sin t_0) b) 2.0))))

Derivation

1. Initial program 76.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. lift-PI.f64 N/A

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

   6. add-sqr-sqrt N/A

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

   7. div-inv N/A

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

   8. times-frac N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lift-PI.f64 N/A

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

   15. lower-sqrt.f64 N/A

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

   16. inv-pow N/A

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

   17. lower-pow.f64 76.4

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

4. Applied rewrites 76.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-2neg N/A

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

   6. clear-num N/A

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

   7. frac-times N/A

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

   8. *-rgt-identity N/A

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

   9. distribute-lft-neg-out N/A

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

   10. frac-2neg N/A

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

   11. clear-num N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 N/A

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

6. Applied rewrites 76.4%

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

8. Applied rewrites 76.2%

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

9. Final simplification 76.2%

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

Alternative 14: 79.3% accurate, 1.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* (cos (* (* angle_m (PI)) -0.005555555555555556)) a) 2.0)
  (pow (* (sin (* (* 0.005555555555555556 (PI)) angle_m)) b) 2.0)))

Derivation

1. Initial program 76.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. lift-PI.f64 N/A

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

   6. add-sqr-sqrt N/A

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

   7. div-inv N/A

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

   8. times-frac N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lift-PI.f64 N/A

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

   15. lower-sqrt.f64 N/A

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

   16. inv-pow N/A

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

   17. lower-pow.f64 76.4

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

4. Applied rewrites 76.4%

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

6. Applied rewrites 76.2%

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

7. Final simplification 76.2%

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

Alternative 15: 79.3% accurate, 1.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* (sin (* (* angle_m (PI)) 0.005555555555555556)) b) 2.0)
  (pow (* (cos (* (* -0.005555555555555556 angle_m) (PI))) a) 2.0)))

Derivation

1. Initial program 76.3%

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

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. frac-2neg N/A

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

   6. distribute-frac-neg N/A

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

   7. distribute-lft-neg-out N/A

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

   8. cos-neg N/A

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

   9. lower-cos.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. div-inv N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval 76.3

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

4. Applied rewrites 76.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. div-inv N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 76.1

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

6. Applied rewrites 76.1%

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

7. Final simplification 76.1%

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

Alternative 16: 79.2% accurate, 1.9× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+ (* a a) (pow (* (sin (* (/ angle_m 180.0) (PI))) b) 2.0)))

Derivation

1. Initial program 76.3%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 75.8

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

5. Applied rewrites 75.8%

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

6. Final simplification 75.8%

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

Alternative 17: 55.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 1.15 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\_m\right) \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5} \cdot b, b, -3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right), angle\_m, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;{\cos \left(\left(angle\_m \cdot \mathsf{PI}\left(\right)\right) \cdot -0.005555555555555556\right)}^{2} \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 1.15e+20)
   (fma
    (*
     (* (* (PI) (PI)) angle_m)
     (fma (* 3.08641975308642e-5 b) b (* -3.08641975308642e-5 (* a a))))
    angle_m
    (* a a))
   (* (pow (cos (* (* angle_m (PI)) -0.005555555555555556)) 2.0) (* a a))))

Derivation

1. Split input into 2 regimes

2. if a < 1.15e20

   1. Initial program 73.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-PI.f64 N/A

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

      6. add-sqr-sqrt N/A

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

      7. div-inv N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. inv-pow N/A

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

      17. lower-pow.f64 73.3

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

   4. Applied rewrites 73.3%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 46.5%

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

   9. Applied rewrites 50.5%

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

   if 1.15e20 < a

   1. Initial program 86.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 86.8%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 78.6

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

   7. Applied rewrites 78.6%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.15 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5} \cdot b, b, -3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right), angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;{\cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot -0.005555555555555556\right)}^{2} \cdot \left(a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 79.2% accurate, 2.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+ (* a a) (pow (* (sin (* (* angle_m (PI)) 0.005555555555555556)) b) 2.0)))

Derivation

1. Initial program 76.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. div-inv N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 76.2

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

4. Applied rewrites 76.2%

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

5. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 75.7

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

7. Applied rewrites 75.7%

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

8. Final simplification 75.7%

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

Alternative 19: 55.5% accurate, 8.3× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 3.7 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\_m\right) \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5} \cdot b, b, -3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right), angle\_m, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 3.7e+144)
   (fma
    (*
     (* (* (PI) (PI)) angle_m)
     (fma (* 3.08641975308642e-5 b) b (* -3.08641975308642e-5 (* a a))))
    angle_m
    (* a a))
   (* a a)))

Derivation

1. Split input into 2 regimes

2. if a < 3.6999999999999997e144

   1. Initial program 72.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-PI.f64 N/A

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

      6. add-sqr-sqrt N/A

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

      7. div-inv N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. inv-pow N/A

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

      17. lower-pow.f64 73.0

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

   4. Applied rewrites 73.0%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 46.2%

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

   9. Applied rewrites 49.8%

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

   if 3.6999999999999997e144 < a

   1. Initial program 100.0%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.7 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5} \cdot b, b, -3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right), angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 61.8% accurate, 10.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 7.5 \cdot 10^{-77}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(b \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), angle\_m \cdot angle\_m, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 7.5e-77)
   (* a a)
   (fma
    (* (* (* b b) 3.08641975308642e-5) (* (PI) (PI)))
    (* angle_m angle_m)
    (* a a))))

Derivation

1. Split input into 2 regimes

2. if b < 7.5000000000000006e-77

   1. Initial program 76.6%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 59.5

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

   5. Applied rewrites 59.5%

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

   if 7.5000000000000006e-77 < b

   1. Initial program 75.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-PI.f64 N/A

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

      6. add-sqr-sqrt N/A

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

      7. div-inv N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. inv-pow N/A

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

      17. lower-pow.f64 75.5

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

   4. Applied rewrites 75.5%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 37.1%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 57.3%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.5 \cdot 10^{-77}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(b \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), angle \cdot angle, a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 61.0% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{+138}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right) \cdot \left(\left(\left(angle\_m \cdot angle\_m\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 6.5e+138)
   (* a a)
   (* (* (* (PI) (PI)) b) (* (* (* angle_m angle_m) 3.08641975308642e-5) b))))

Derivation

1. Split input into 2 regimes

2. if b < 6.50000000000000054e138

   1. Initial program 74.6%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.1

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

   5. Applied rewrites 57.1%

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

   if 6.50000000000000054e138 < b

   1. Initial program 88.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-PI.f64 N/A

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

      6. add-sqr-sqrt N/A

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

      7. div-inv N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. inv-pow N/A

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

      17. lower-pow.f64 88.6

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

   4. Applied rewrites 88.6%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 50.3%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 63.2%

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

   12. Applied rewrites 68.1%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{+138}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right) \cdot \left(\left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 60.4% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{+138}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right) \cdot b\right) \cdot \left(\left(3.08641975308642 \cdot 10^{-5} \cdot angle\_m\right) \cdot angle\_m\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 6.5e+138)
   (* a a)
   (* (* (* (* (PI) (PI)) b) b) (* (* 3.08641975308642e-5 angle_m) angle_m))))

Derivation

1. Split input into 2 regimes

2. if b < 6.50000000000000054e138

   1. Initial program 74.6%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.1

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

   5. Applied rewrites 57.1%

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

   if 6.50000000000000054e138 < b

   1. Initial program 88.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-PI.f64 N/A

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

      6. add-sqr-sqrt N/A

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

      7. div-inv N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. inv-pow N/A

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

      17. lower-pow.f64 88.6

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

   4. Applied rewrites 88.6%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 50.3%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 63.2%

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

   12. Applied rewrites 63.2%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{+138}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right) \cdot b\right) \cdot \left(\left(3.08641975308642 \cdot 10^{-5} \cdot angle\right) \cdot angle\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 60.4% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{+138}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right) \cdot b\right) \cdot \left(\left(angle\_m \cdot angle\_m\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 6.5e+138)
   (* a a)
   (* (* (* (* (PI) (PI)) b) b) (* (* angle_m angle_m) 3.08641975308642e-5))))

Derivation

1. Split input into 2 regimes

2. if b < 6.50000000000000054e138

   1. Initial program 74.6%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.1

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

   5. Applied rewrites 57.1%

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

   if 6.50000000000000054e138 < b

   1. Initial program 88.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-PI.f64 N/A

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

      6. add-sqr-sqrt N/A

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

      7. div-inv N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. inv-pow N/A

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

      17. lower-pow.f64 88.6

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

   4. Applied rewrites 88.6%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 50.3%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 63.2%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{+138}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right) \cdot b\right) \cdot \left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 56.8% accurate, 74.7× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ a \cdot a \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m) :precision binary64 (* a a))

C

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

Fortran

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

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return a * a;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return a * a

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(a * a), $MachinePrecision]

Derivation

1. Initial program 76.3%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 53.6

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

5. Applied rewrites 53.6%

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

Reproduce

herbie shell --seed 2024264 
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* (PI) (/ angle 180.0)))) 2.0) (pow (* b (sin (* (PI) (/ angle 180.0)))) 2.0)))