ab-angle->ABCF C

Percentage Accurate: 79.7% → 79.7%

Time: 10.4s

Alternatives: 11

Speedup: 1.3×

• Could not converge on a ground truth

  1. a = 4.395189977725504e+192
  2. b = 8.114047235876369e+297
  3. angle = 4.097673385385107e+32

• 36.2% 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: 79.7% 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: 79.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(a \cdot \mathsf{fma}\left({\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot -0.005555555555555556\right)}^{4} - {\sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{4}, 0.5, 0.5\right), a, {\left(b \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (fma
  (*
   a
   (fma
    (-
     (pow (cos (* (* (PI) angle) -0.005555555555555556)) 4.0)
     (pow (sin (* (* (PI) 0.005555555555555556) angle)) 4.0))
    0.5
    0.5))
  a
  (pow (* b (sin (* (* 0.005555555555555556 angle) (PI)))) 2.0)))

Derivation

1. Initial program 78.0%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

4. Applied rewrites 78.1%

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

   1. lift-pow.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-2neg N/A

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

   5. metadata-eval N/A

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

   6. distribute-frac-neg N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/l* N/A

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

   9. div-inv N/A

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

   10. metadata-eval N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. cos-neg N/A

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

   15. lift-cos.f64 N/A

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

   16. unpow2 N/A

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

   17. lift-cos.f64 N/A

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

   18. lift-cos.f64 N/A

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

   19. sqr-cos-a N/A

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

6. Applied rewrites 78.1%

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

8. Applied rewrites 78.1%

   \[\leadsto \mathsf{fma}\left(\left(0.5 + 0.5 \cdot \color{blue}{\frac{{\cos \left(-0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4} - {\sin \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right)\right)}^{4}}{1}}\right) \cdot a, a, {\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2}\right) \]

9. Taylor expanded in angle around inf

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

    1. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} \cdot \left({\cos \left(\frac{-1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4} - {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}\right) + \frac{1}{2}\right)} \cdot a, a, {\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2}\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left({\cos \left(\frac{-1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4} - {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}\right) \cdot \frac{1}{2}} + \frac{1}{2}\right) \cdot a, a, {\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2}\right) \]

    3. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({\cos \left(\frac{-1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4} - {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}, \frac{1}{2}, \frac{1}{2}\right)} \cdot a, a, {\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2}\right) \]

11. Applied rewrites 78.1%

    \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({\cos \left(-0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}^{4} - {\sin \left(\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{4}, 0.5, 0.5\right)} \cdot a, a, {\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2}\right) \]

12. Final simplification 78.1%

    \[\leadsto \mathsf{fma}\left(a \cdot \mathsf{fma}\left({\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot -0.005555555555555556\right)}^{4} - {\sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{4}, 0.5, 0.5\right), a, {\left(b \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) \]
13. Add Preprocessing

Alternative 2: 79.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(\cos \left(\left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot 0.5 + 0.5\right) \cdot a, a, {\left(b \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (fma
  (* (+ (* (cos (* (* 0.011111111111111112 (PI)) angle)) 0.5) 0.5) a)
  a
  (pow (* b (sin (* (* 0.005555555555555556 angle) (PI)))) 2.0)))

Derivation

1. Initial program 78.0%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

4. Applied rewrites 78.1%

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

   1. lift-pow.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-2neg N/A

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

   5. metadata-eval N/A

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

   6. distribute-frac-neg N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/l* N/A

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

   9. div-inv N/A

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

   10. metadata-eval N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. cos-neg N/A

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

   15. lift-cos.f64 N/A

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

   16. unpow2 N/A

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

   17. lift-cos.f64 N/A

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

   18. lift-cos.f64 N/A

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

   19. sqr-cos-a N/A

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

6. Applied rewrites 78.1%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right)\right)} \cdot a, a, {\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2}\right) \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 78.1

      \[\leadsto \mathsf{fma}\left(\left(0.5 + 0.5 \cdot \cos \left(\left(\color{blue}{0.011111111111111112} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right) \cdot a, a, {\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2}\right) \]

8. Applied rewrites 78.1%

   \[\leadsto \mathsf{fma}\left(\left(0.5 + 0.5 \cdot \cos \color{blue}{\left(\left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}\right) \cdot a, a, {\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2}\right) \]

9. Final simplification 78.1%

   \[\leadsto \mathsf{fma}\left(\left(\cos \left(\left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot 0.5 + 0.5\right) \cdot a, a, {\left(b \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) \]
10. Add Preprocessing

Alternative 3: 79.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right), 0.5, 0.5\right) \cdot a, a, {\left(b \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (fma
  (* (fma (cos (* 0.011111111111111112 (* (PI) angle))) 0.5 0.5) a)
  a
  (pow (* b (sin (* (* 0.005555555555555556 angle) (PI)))) 2.0)))

Derivation

1. Initial program 78.0%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

4. Applied rewrites 78.1%

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

   1. lift-pow.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-2neg N/A

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

   5. metadata-eval N/A

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

   6. distribute-frac-neg N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/l* N/A

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

   9. div-inv N/A

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

   10. metadata-eval N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. cos-neg N/A

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

   15. lift-cos.f64 N/A

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

   16. unpow2 N/A

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

   17. lift-cos.f64 N/A

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

   18. lift-cos.f64 N/A

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

   19. sqr-cos-a N/A

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

6. Applied rewrites 78.1%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right)\right)} \cdot a, a, {\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2}\right) \]
7. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 78.1

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. lift-PI.f64 N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f64 N/A

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

   14. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\color{blue}{\frac{1}{90}} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{2}, \frac{1}{2}\right) \cdot a, a, {\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2}\right) \]

   15. lift-PI.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right), \frac{1}{2}, \frac{1}{2}\right) \cdot a, a, {\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2}\right) \]

   16. lower-*.f64 78.1

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right), 0.5, 0.5\right) \cdot a, a, {\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2}\right) \]

8. Applied rewrites 78.1%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), 0.5, 0.5\right)} \cdot a, a, {\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2}\right) \]

9. Final simplification 78.1%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right), 0.5, 0.5\right) \cdot a, a, {\left(b \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) \]
10. Add Preprocessing

Alternative 4: 64.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;b \leq 1.05 \cdot 10^{-16}:\\ \;\;\;\;\left(a \cdot a\right) \cdot {\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot -0.005555555555555556\right)}^{2}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(\left(t\_0 \cdot angle\right) \cdot angle, \left(b \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(b \cdot angle\right) \cdot \left(b \cdot angle\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (PI) (PI))))
   (if (<= b 1.05e-16)
     (* (* a a) (pow (cos (* (* (PI) angle) -0.005555555555555556)) 2.0))
     (if (<= b 1.1e+151)
       (fma (* (* t_0 angle) angle) (* (* b b) 3.08641975308642e-5) (* a a))
       (* (* (* (* b angle) (* b angle)) 3.08641975308642e-5) t_0)))))

Derivation

1. Split input into 3 regimes

2. if b < 1.0500000000000001e-16

   1. Initial program 76.0%

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

      1. rem-exp-log N/A

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

      2. unpow1 N/A

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

      3. log-pow N/A

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

      4. lift-pow.f64 N/A

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

      5. log-pow N/A

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

      6. *-commutative N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. exp-1-e N/A

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

      10. lower-E.f64 N/A

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

      11. *-commutative N/A

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

      12. log-pow N/A

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

      13. lift-pow.f64 N/A

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

      14. lower-log.f64 74.4

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

   4. Applied rewrites 74.4%

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

   5. Taylor expanded in b around 0

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

      1. log-E N/A

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

      2. unpow1 N/A

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

      3. *-commutative N/A

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

      4. rem-exp-log N/A

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

      5. *-rgt-identity N/A

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

      6. log-E N/A

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

      7. lower-*.f64 N/A

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

   7. Applied rewrites 59.9%

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

   if 1.0500000000000001e-16 < b < 1.10000000000000003e151

   1. Initial program 61.0%

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

      1. rem-exp-log N/A

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

      2. unpow1 N/A

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

      3. log-pow N/A

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

      4. lift-pow.f64 N/A

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

      5. log-pow N/A

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

      6. *-commutative N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. exp-1-e N/A

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

      10. lower-E.f64 N/A

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

      11. *-commutative N/A

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

      12. log-pow N/A

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

      13. lift-pow.f64 N/A

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

      14. lower-log.f64 59.7

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

   4. Applied rewrites 59.7%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 26.1%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 43.8%

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

   if 1.10000000000000003e151 < b

   1. Initial program 99.5%

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

      1. rem-exp-log N/A

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

      2. unpow1 N/A

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

      3. log-pow N/A

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

      4. lift-pow.f64 N/A

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

      5. log-pow N/A

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

      6. *-commutative N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. exp-1-e N/A

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

      10. lower-E.f64 N/A

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

      11. *-commutative N/A

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

      12. log-pow N/A

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

      13. lift-pow.f64 N/A

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

      14. lower-log.f64 98.6

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 71.0%

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

   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 69.6%

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

   12. Applied rewrites 82.9%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{-16}:\\ \;\;\;\;\left(a \cdot a\right) \cdot {\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot -0.005555555555555556\right)}^{2}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot angle, \left(b \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(b \cdot angle\right) \cdot \left(b \cdot angle\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.6% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 78.0%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

4. Applied rewrites 78.1%

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

5. Taylor expanded in angle around 0

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

7. Applied rewrites 77.4%

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

8. Final simplification 77.4%

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

Alternative 6: 64.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;b \leq 1.05 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right), 0.5, 0.5\right) \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(\left(t\_0 \cdot angle\right) \cdot angle, \left(b \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(b \cdot angle\right) \cdot \left(b \cdot angle\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (PI) (PI))))
   (if (<= b 1.05e-16)
     (* (fma (cos (* 0.011111111111111112 (* (PI) angle))) 0.5 0.5) (* a a))
     (if (<= b 1.1e+151)
       (fma (* (* t_0 angle) angle) (* (* b b) 3.08641975308642e-5) (* a a))
       (* (* (* (* b angle) (* b angle)) 3.08641975308642e-5) t_0)))))

Derivation

1. Split input into 3 regimes

2. if b < 1.0500000000000001e-16

   1. Initial program 76.0%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 76.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. distribute-frac-neg N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. div-inv N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. cos-neg N/A

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

      15. lift-cos.f64 N/A

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

      16. unpow2 N/A

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

      17. lift-cos.f64 N/A

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

      18. lift-cos.f64 N/A

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

      19. sqr-cos-a N/A

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

   6. Applied rewrites 76.2%

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

   7. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 59.9

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

   9. Applied rewrites 59.9%

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

   if 1.0500000000000001e-16 < b < 1.10000000000000003e151

   1. Initial program 61.0%

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

      1. rem-exp-log N/A

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

      2. unpow1 N/A

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

      3. log-pow N/A

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

      4. lift-pow.f64 N/A

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

      5. log-pow N/A

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

      6. *-commutative N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. exp-1-e N/A

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

      10. lower-E.f64 N/A

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

      11. *-commutative N/A

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

      12. log-pow N/A

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

      13. lift-pow.f64 N/A

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

      14. lower-log.f64 59.7

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

   4. Applied rewrites 59.7%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 26.1%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 43.8%

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

   if 1.10000000000000003e151 < b

   1. Initial program 99.5%

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

      1. rem-exp-log N/A

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

      2. unpow1 N/A

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

      3. log-pow N/A

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

      4. lift-pow.f64 N/A

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

      5. log-pow N/A

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

      6. *-commutative N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. exp-1-e N/A

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

      10. lower-E.f64 N/A

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

      11. *-commutative N/A

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

      12. log-pow N/A

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

      13. lift-pow.f64 N/A

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

      14. lower-log.f64 98.6

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 71.0%

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

   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 69.6%

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

   12. Applied rewrites 82.9%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right), 0.5, 0.5\right) \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot angle, \left(b \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(b \cdot angle\right) \cdot \left(b \cdot angle\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.2% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;b \leq 1.05 \cdot 10^{-16}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(\left(t\_0 \cdot angle\right) \cdot angle, \left(b \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(b \cdot angle\right) \cdot \left(b \cdot angle\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (PI) (PI))))
   (if (<= b 1.05e-16)
     (* a a)
     (if (<= b 1.1e+151)
       (fma (* (* t_0 angle) angle) (* (* b b) 3.08641975308642e-5) (* a a))
       (* (* (* (* b angle) (* b angle)) 3.08641975308642e-5) t_0)))))

Derivation

1. Split input into 3 regimes

2. if b < 1.0500000000000001e-16

   1. Initial program 76.0%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 59.3

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

   5. Applied rewrites 59.3%

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

   if 1.0500000000000001e-16 < b < 1.10000000000000003e151

   1. Initial program 61.0%

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

      1. rem-exp-log N/A

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

      2. unpow1 N/A

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

      3. log-pow N/A

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

      4. lift-pow.f64 N/A

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

      5. log-pow N/A

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

      6. *-commutative N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. exp-1-e N/A

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

      10. lower-E.f64 N/A

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

      11. *-commutative N/A

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

      12. log-pow N/A

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

      13. lift-pow.f64 N/A

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

      14. lower-log.f64 59.7

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

   4. Applied rewrites 59.7%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 26.1%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 43.8%

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

   if 1.10000000000000003e151 < b

   1. Initial program 99.5%

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

      1. rem-exp-log N/A

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

      2. unpow1 N/A

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

      3. log-pow N/A

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

      4. lift-pow.f64 N/A

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

      5. log-pow N/A

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

      6. *-commutative N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. exp-1-e N/A

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

      10. lower-E.f64 N/A

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

      11. *-commutative N/A

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

      12. log-pow N/A

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

      13. lift-pow.f64 N/A

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

      14. lower-log.f64 98.6

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 71.0%

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

   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 69.6%

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

   12. Applied rewrites 82.9%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{-16}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot angle, \left(b \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(b \cdot angle\right) \cdot \left(b \cdot angle\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.7% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{+134}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(b \cdot angle\right) \cdot \left(b \cdot angle\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 2.1e+134)
   (* a a)
   (* (* (* (* b angle) (* b angle)) 3.08641975308642e-5) (* (PI) (PI)))))

Derivation

1. Split input into 2 regimes

2. if b < 2.1000000000000001e134

   1. Initial program 73.7%

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

   3. 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 55.4

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

   5. Applied rewrites 55.4%

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

   if 2.1000000000000001e134 < b

   1. Initial program 97.7%

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

      1. rem-exp-log N/A

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

      2. unpow1 N/A

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

      3. log-pow N/A

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

      4. lift-pow.f64 N/A

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

      5. log-pow N/A

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

      6. *-commutative N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. exp-1-e N/A

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

      10. lower-E.f64 N/A

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

      11. *-commutative N/A

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

      12. log-pow N/A

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

      13. lift-pow.f64 N/A

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

      14. lower-log.f64 96.8

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

   4. Applied rewrites 96.8%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 70.3%

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

   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 68.9%

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

   12. Applied rewrites 81.6%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{+134}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(b \cdot angle\right) \cdot \left(b \cdot angle\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.4% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{+134}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(b \cdot angle\right) \cdot b\right) \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 2.1e+134)
   (* a a)
   (* (* (* (* (* b angle) b) angle) 3.08641975308642e-5) (* (PI) (PI)))))

Derivation

1. Split input into 2 regimes

2. if b < 2.1000000000000001e134

   1. Initial program 73.7%

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

   3. 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 55.4

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

   5. Applied rewrites 55.4%

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

   if 2.1000000000000001e134 < b

   1. Initial program 97.7%

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

      1. rem-exp-log N/A

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

      2. unpow1 N/A

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

      3. log-pow N/A

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

      4. lift-pow.f64 N/A

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

      5. log-pow N/A

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

      6. *-commutative N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. exp-1-e N/A

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

      10. lower-E.f64 N/A

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

      11. *-commutative N/A

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

      12. log-pow N/A

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

      13. lift-pow.f64 N/A

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

      14. lower-log.f64 96.8

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

   4. Applied rewrites 96.8%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 70.3%

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

   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 68.9%

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

   12. Applied rewrites 81.4%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{+134}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(b \cdot angle\right) \cdot b\right) \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.6% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{+134}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(b \cdot b\right) \cdot angle\right) \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 2.1e+134)
   (* a a)
   (* (* (* (* (* b b) angle) angle) 3.08641975308642e-5) (* (PI) (PI)))))

Derivation

1. Split input into 2 regimes

2. if b < 2.1000000000000001e134

   1. Initial program 73.7%

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

   3. 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 55.4

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

   5. Applied rewrites 55.4%

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

   if 2.1000000000000001e134 < b

   1. Initial program 97.7%

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

      1. rem-exp-log N/A

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

      2. unpow1 N/A

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

      3. log-pow N/A

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

      4. lift-pow.f64 N/A

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

      5. log-pow N/A

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

      6. *-commutative N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. exp-1-e N/A

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

      10. lower-E.f64 N/A

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

      11. *-commutative N/A

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

      12. log-pow N/A

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

      13. lift-pow.f64 N/A

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

      14. lower-log.f64 96.8

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

   4. Applied rewrites 96.8%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 70.3%

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

   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 68.9%

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

   12. Applied rewrites 75.2%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{+134}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(b \cdot b\right) \cdot angle\right) \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.9% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.0%

   \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
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 51.8

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

5. Applied rewrites 51.8%

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

Reproduce

herbie shell --seed 2024288 
(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)))