ab-angle->ABCF C

Percentage Accurate: 79.4% → 79.4%

Time: 17.8s

Alternatives: 15

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

Initial Program: 79.4% 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: 79.4% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 79.5%

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

   1. lift-*.f64 N/A

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

   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. clear-num N/A

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

   4. associate-*r/ N/A

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

   5. div-inv N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 79.6

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

4. Applied rewrites 79.6%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. inv-pow N/A

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

   4. pow-to-exp N/A

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

   5. lower-exp.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-log.f64 N/A

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

   8. lower-/.f64 44.7

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

6. Applied rewrites 44.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-log.f64 N/A

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

   4. log-pow N/A

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

   5. inv-pow N/A

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. log-div N/A

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

   9. lower--.f64 N/A

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

   10. lower-log.f64 N/A

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

   11. lower-log.f64 44.7

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

8. Applied rewrites 44.7%

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

Alternative 2: 79.4% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 79.5%

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

   1. lift-*.f64 N/A

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

   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. lower-/.f64 N/A

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

   5. lower-*.f64 79.6

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

4. Applied rewrites 79.6%

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

Alternative 3: 79.4% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 79.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. div-inv N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. metadata-eval 79.6

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

4. Applied rewrites 79.6%

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

5. Final simplification 79.6%

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

Alternative 4: 79.4% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 79.5%

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

   1. lift-*.f64 N/A

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

   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. lower-/.f64 N/A

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

   5. lower-*.f64 79.6

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

4. Applied rewrites 79.6%

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

5. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 79.2

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

7. Applied rewrites 79.2%

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

8. Final simplification 79.2%

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

Alternative 5: 79.4% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 79.5%

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

3. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{{a}^{2}} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
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 79.1

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

5. Applied rewrites 79.1%

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

Alternative 6: 75.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := 0.5 \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(angle\_m \cdot angle\_m\right) \cdot 3.08641975308642 \cdot 10^{-5}\right), b, a \cdot \left(a \cdot \left(0.5 + t\_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(a, a \cdot 1, b \cdot \left(b \cdot \left(0.5 - t\_0\right)\right)\right)}}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos (* 2.0 (* (PI) (* angle_m 0.005555555555555556)))))))
   (if (<= (/ angle_m 180.0) 2e+33)
     (fma
      (* (* b (* (PI) (PI))) (* (* angle_m angle_m) 3.08641975308642e-5))
      b
      (* a (* a (+ 0.5 t_0))))
     (/ 1.0 (/ 1.0 (fma a (* a 1.0) (* b (* b (- 0.5 t_0)))))))))

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 1.9999999999999999e33

   1. Initial program 87.6%

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

   4. Applied rewrites 62.0%

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

   5. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 N/A

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

      8. lower-PI.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 80.3

          \[\leadsto \mathsf{fma}\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot 3.08641975308642 \cdot 10^{-5}\right), b, a \cdot \left(a \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\right) \]

   7. Applied rewrites 80.3%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)}, b, a \cdot \left(a \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\right) \]

   if 1.9999999999999999e33 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 57.6%

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

   4. Applied rewrites 57.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(a, a \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right), b \cdot \left(b \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\right)}}} \]

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 57.8%

      \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(a, a \cdot \color{blue}{1}, b \cdot \left(b \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 65.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-84}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(angle\_m \cdot \left(\mathsf{PI}\left(\right) \cdot 0.011111111111111112\right)\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(angle\_m \cdot angle\_m\right) \cdot 3.08641975308642 \cdot 10^{-5}\right), b, a \cdot \left(a \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 4.2e-84)
   (* (* a a) (fma 0.5 (cos (* angle_m (* (PI) 0.011111111111111112))) 0.5))
   (fma
    (* (* b (* (PI) (PI))) (* (* angle_m angle_m) 3.08641975308642e-5))
    b
    (*
     a
     (*
      a
      (+
       0.5
       (* 0.5 (cos (* 2.0 (* (PI) (* angle_m 0.005555555555555556)))))))))))

Derivation

1. Split input into 2 regimes

2. if b < 4.19999999999999996e-84

   1. Initial program 77.3%

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

   4. Applied rewrites 72.1%

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

   5. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 59.3

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

   7. Applied rewrites 59.3%

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

   if 4.19999999999999996e-84 < b

   1. Initial program 85.6%

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

   4. Applied rewrites 70.6%

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

   5. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 N/A

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

      8. lower-PI.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 79.7

          \[\leadsto \mathsf{fma}\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot 3.08641975308642 \cdot 10^{-5}\right), b, a \cdot \left(a \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\right) \]

   7. Applied rewrites 79.7%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)}, b, a \cdot \left(a \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-84}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot 0.011111111111111112\right)\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right), b, a \cdot \left(a \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-84}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(angle\_m \cdot \left(\mathsf{PI}\left(\right) \cdot 0.011111111111111112\right)\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle\_m \cdot angle\_m\right) \cdot \left(b \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right), b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\_m\right) \cdot 0.011111111111111112\right), 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 4.2e-84)
   (* (* a a) (fma 0.5 (cos (* angle_m (* (PI) 0.011111111111111112))) 0.5))
   (fma
    (* (* angle_m angle_m) (* b (* (* (PI) (PI)) 3.08641975308642e-5)))
    b
    (*
     (* a a)
     (fma 0.5 (cos (* (* (PI) angle_m) 0.011111111111111112)) 0.5)))))

Derivation

1. Split input into 2 regimes

2. if b < 4.19999999999999996e-84

   1. Initial program 77.3%

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

   4. Applied rewrites 72.1%

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

   5. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 59.3

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

   7. Applied rewrites 59.3%

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

   if 4.19999999999999996e-84 < b

   1. Initial program 85.6%

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

      1. lift-*.f64 N/A

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

      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. lower-/.f64 N/A

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

      5. lower-*.f64 85.7

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

   4. Applied rewrites 85.7%

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

   6. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(b \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right), b, \mathsf{fma}\left(0.5, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right), 0.5\right) \cdot \left(a \cdot a\right)\right)} \]

   7. Taylor expanded in angle around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}, b, \mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left({angle}^{2} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400}}, b, \mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{angle}^{2} \cdot \left(\left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)}, b, \mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left({angle}^{2} \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}, b, \mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}, b, \mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(angle \cdot angle\right)} \cdot \left(\frac{1}{32400} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), b, \mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(angle \cdot angle\right)} \cdot \left(\frac{1}{32400} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), b, \mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(\left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)}, b, \mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(b \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)\right)}, b, \mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \left(b \cdot \color{blue}{\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right), b, \mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(b \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}, b, \mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \left(b \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right), b, \mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \left(b \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right), b, \mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \left(b \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{32400}\right)\right), b, \mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \left(b \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{32400}\right)\right), b, \mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right) \]

      15. lower-PI.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \left(b \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right)\right), b, \mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right) \]

      16. lower-PI.f64 79.6

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \left(b \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right), b, \mathsf{fma}\left(0.5, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right), 0.5\right) \cdot \left(a \cdot a\right)\right) \]

   9. Applied rewrites 79.6%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(angle \cdot angle\right) \cdot \left(b \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)}, b, \mathsf{fma}\left(0.5, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right), 0.5\right) \cdot \left(a \cdot a\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-84}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot 0.011111111111111112\right)\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \left(b \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right), b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right), 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-84}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(angle\_m \cdot \left(\mathsf{PI}\left(\right) \cdot 0.011111111111111112\right)\right), 0.5\right)\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left(angle\_m \cdot angle\_m, \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(b \cdot \left(b \cdot 3.08641975308642 \cdot 10^{-5}\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot angle\_m\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(a, a \cdot -3.08641975308642 \cdot 10^{-5}, 3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right)\right), angle\_m, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 4.2e-84)
   (* (* a a) (fma 0.5 (cos (* angle_m (* (PI) 0.011111111111111112))) 0.5))
   (if (<= b 1.2e+147)
     (fma
      (* angle_m angle_m)
      (* (* (PI) (PI)) (* b (* b 3.08641975308642e-5)))
      (* a a))
     (fma
      (*
       (* (PI) angle_m)
       (*
        (PI)
        (fma a (* a -3.08641975308642e-5) (* 3.08641975308642e-5 (* b b)))))
      angle_m
      (* a a)))))

Derivation

1. Split input into 3 regimes

2. if b < 4.19999999999999996e-84

   1. Initial program 77.3%

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

   4. Applied rewrites 72.1%

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

   5. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 59.3

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

   7. Applied rewrites 59.3%

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

   if 4.19999999999999996e-84 < b < 1.20000000000000001e147

   1. Initial program 77.7%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

      6. unpow1/2 N/A

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

      7. lower-sqrt.f64 70.3

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

      8. lift-/.f64 N/A

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

      9. div-inv N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 70.8

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

   4. Applied rewrites 70.8%

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

   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. lower-fma.f64 N/A

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

   7. Applied rewrites 40.2%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 74.8%

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

   if 1.20000000000000001e147 < b

   1. Initial program 95.0%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

      6. unpow1/2 N/A

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

      7. lower-sqrt.f64 71.7

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

      8. lift-/.f64 N/A

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

      9. div-inv N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 71.7

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

   4. Applied rewrites 71.7%

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

   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. lower-fma.f64 N/A

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

   7. Applied rewrites 60.4%

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

   9. Applied rewrites 79.4%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-84}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot 0.011111111111111112\right)\right), 0.5\right)\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(b \cdot \left(b \cdot 3.08641975308642 \cdot 10^{-5}\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(a, a \cdot -3.08641975308642 \cdot 10^{-5}, 3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right)\right), angle, a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.0% accurate, 7.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-84}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left(angle\_m \cdot angle\_m, \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(b \cdot \left(b \cdot 3.08641975308642 \cdot 10^{-5}\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot angle\_m\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(a, a \cdot -3.08641975308642 \cdot 10^{-5}, 3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right)\right), angle\_m, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 4.2e-84)
   (* a a)
   (if (<= b 1.2e+147)
     (fma
      (* angle_m angle_m)
      (* (* (PI) (PI)) (* b (* b 3.08641975308642e-5)))
      (* a a))
     (fma
      (*
       (* (PI) angle_m)
       (*
        (PI)
        (fma a (* a -3.08641975308642e-5) (* 3.08641975308642e-5 (* b b)))))
      angle_m
      (* a a)))))

Derivation

1. Split input into 3 regimes

2. if b < 4.19999999999999996e-84

   1. Initial program 77.3%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 58.7

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

   5. Applied rewrites 58.7%

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

   if 4.19999999999999996e-84 < b < 1.20000000000000001e147

   1. Initial program 77.7%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

      6. unpow1/2 N/A

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

      7. lower-sqrt.f64 70.3

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

      8. lift-/.f64 N/A

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

      9. div-inv N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 70.8

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

   4. Applied rewrites 70.8%

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

   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. lower-fma.f64 N/A

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

   7. Applied rewrites 40.2%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 74.8%

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

   if 1.20000000000000001e147 < b

   1. Initial program 95.0%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

      6. unpow1/2 N/A

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

      7. lower-sqrt.f64 71.7

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

      8. lift-/.f64 N/A

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

      9. div-inv N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 71.7

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

   4. Applied rewrites 71.7%

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

   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. lower-fma.f64 N/A

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

   7. Applied rewrites 60.4%

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

   9. Applied rewrites 79.4%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-84}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(b \cdot \left(b \cdot 3.08641975308642 \cdot 10^{-5}\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(a, a \cdot -3.08641975308642 \cdot 10^{-5}, 3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right)\right), angle, a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.5% accurate, 9.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;b \leq 4.2 \cdot 10^{-84}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(angle\_m \cdot angle\_m, t\_0 \cdot \left(b \cdot \left(b \cdot 3.08641975308642 \cdot 10^{-5}\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\left(angle\_m \cdot b\right) \cdot \left(angle\_m \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (PI) (PI))))
   (if (<= b 4.2e-84)
     (* a a)
     (if (<= b 8.2e+155)
       (fma
        (* angle_m angle_m)
        (* t_0 (* b (* b 3.08641975308642e-5)))
        (* a a))
       (* (* t_0 3.08641975308642e-5) (* (* angle_m b) (* angle_m b)))))))

Derivation

1. Split input into 3 regimes

2. if b < 4.19999999999999996e-84

   1. Initial program 77.3%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 58.7

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

   5. Applied rewrites 58.7%

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

   if 4.19999999999999996e-84 < b < 8.1999999999999996e155

   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. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

      6. unpow1/2 N/A

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

      7. lower-sqrt.f64 67.6

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

      8. lift-/.f64 N/A

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

      9. div-inv N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 68.0

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

   4. Applied rewrites 68.0%

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

   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. lower-fma.f64 N/A

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

   7. Applied rewrites 37.9%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 70.0%

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

   if 8.1999999999999996e155 < b

   1. Initial program 99.9%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

      6. unpow1/2 N/A

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

      7. lower-sqrt.f64 75.8

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

      8. lift-/.f64 N/A

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

      9. div-inv N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 75.8

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

   4. Applied rewrites 75.8%

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

   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. lower-fma.f64 N/A

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

   7. Applied rewrites 65.8%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 76.6%

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

   12. Applied rewrites 83.4%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-84}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(b \cdot \left(b \cdot 3.08641975308642 \cdot 10^{-5}\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\left(angle \cdot b\right) \cdot \left(angle \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.8% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.75 \cdot 10^{+156}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\left(angle\_m \cdot b\right) \cdot \left(angle\_m \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 1.75e+156)
   (* a a)
   (* (* (* (PI) (PI)) 3.08641975308642e-5) (* (* angle_m b) (* angle_m b)))))

Derivation

1. Split input into 2 regimes

2. if b < 1.7500000000000002e156

   1. Initial program 77.0%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 59.8

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

   5. Applied rewrites 59.8%

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

   if 1.7500000000000002e156 < b

   1. Initial program 99.9%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

      6. unpow1/2 N/A

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

      7. lower-sqrt.f64 74.9

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

      8. lift-/.f64 N/A

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

      9. div-inv N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 74.9

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

   4. Applied rewrites 74.9%

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

   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. lower-fma.f64 N/A

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

   7. Applied rewrites 68.1%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 79.2%

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

   12. Applied rewrites 86.3%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.75 \cdot 10^{+156}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\left(angle \cdot b\right) \cdot \left(angle \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.8% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.75 \cdot 10^{+156}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(angle\_m \cdot b\right) \cdot \left(angle\_m \cdot \left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 1.75e+156)
   (* a a)
   (* (* angle_m b) (* angle_m (* b (* (PI) (* (PI) 3.08641975308642e-5)))))))

Derivation

1. Split input into 2 regimes

2. if b < 1.7500000000000002e156

   1. Initial program 77.0%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 59.8

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

   5. Applied rewrites 59.8%

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

   if 1.7500000000000002e156 < b

   1. Initial program 99.9%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

      6. unpow1/2 N/A

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

      7. lower-sqrt.f64 74.9

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

      8. lift-/.f64 N/A

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

      9. div-inv N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 74.9

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

   4. Applied rewrites 74.9%

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

   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. lower-fma.f64 N/A

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

   7. Applied rewrites 68.1%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 79.2%

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

   12. Applied rewrites 86.2%

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

Alternative 14: 61.3% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.75 \cdot 10^{+156}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(b \cdot \left(b \cdot \left(angle\_m \cdot angle\_m\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 1.75e+156)
   (* a a)
   (* (PI) (* (* (PI) 3.08641975308642e-5) (* b (* b (* angle_m angle_m)))))))

Derivation

1. Split input into 2 regimes

2. if b < 1.7500000000000002e156

   1. Initial program 77.0%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 59.8

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

   5. Applied rewrites 59.8%

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

   if 1.7500000000000002e156 < b

   1. Initial program 99.9%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

      6. unpow1/2 N/A

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

      7. lower-sqrt.f64 74.9

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

      8. lift-/.f64 N/A

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

      9. div-inv N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 74.9

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

   4. Applied rewrites 74.9%

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

   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. lower-fma.f64 N/A

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

   7. Applied rewrites 68.1%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 79.2%

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

   12. Applied rewrites 79.3%

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

Alternative 15: 56.6% 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 79.5%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 55.9

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

5. Applied rewrites 55.9%

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

Reproduce

herbie shell --seed 2024221 
(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)))