ab-angle->ABCF A

Percentage Accurate: 79.4% → 79.4%

Time: 19.0s

Alternatives: 16

Speedup: 1.0×

• 36.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

Initial Program: 79.4% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 79.4% accurate, 0.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 78.8%

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

3. Taylor expanded in angle around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-cos.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-PI.f64 78.9

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

5. Applied rewrites 78.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 78.9

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

   4. rem-exp-log N/A

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

   5. lift-log.f64 N/A

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

   6. *-rgt-identity N/A

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

   7. lift-*.f64 N/A

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

   8. sinh-+-cosh N/A

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

   9. lift-cosh.f64 N/A

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

   10. lift-sinh.f64 N/A

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

   11. flip-+ N/A

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

7. Applied rewrites 78.9%

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

8. Final simplification 78.9%

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

Alternative 2: 79.4% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 78.8%

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

3. Taylor expanded in angle around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-cos.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-PI.f64 78.9

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

5. Applied rewrites 78.9%

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

Alternative 3: 79.4% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 78.8%

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

3. Taylor expanded in angle around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-cos.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-PI.f64 78.9

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

5. Applied rewrites 78.9%

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

6. Taylor expanded in angle around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. rem-exp-log N/A

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

   4. rem-exp-log N/A

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

   5. exp-sum N/A

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

   6. +-commutative N/A

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

   7. lower-sin.f64 N/A

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

   8. +-commutative N/A

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

   9. exp-sum N/A

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

   10. rem-exp-log N/A

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

   11. rem-exp-log N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-PI.f64 78.9

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

8. Applied rewrites 78.9%

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

9. Final simplification 78.9%

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

Alternative 4: 79.3% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 78.8%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 78.6%

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

Alternative 5: 79.3% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 78.8%

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

3. Taylor expanded in angle around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-cos.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-PI.f64 78.9

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

5. Applied rewrites 78.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 78.9

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

   4. rem-exp-log N/A

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

   5. lift-log.f64 N/A

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

   6. *-rgt-identity N/A

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

   7. lift-*.f64 N/A

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

   8. sinh-+-cosh N/A

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

   9. lift-cosh.f64 N/A

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

   10. lift-sinh.f64 N/A

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

   11. flip-+ N/A

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

7. Applied rewrites 78.9%

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

8. Taylor expanded in angle around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-PI.f64 78.9

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

10. Applied rewrites 78.9%

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

11. Taylor expanded in angle around 0

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

13. Applied rewrites 78.6%

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

Alternative 6: 53.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;b \leq 9.2 \cdot 10^{-153}:\\ \;\;\;\;\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot a\right) \cdot angle\right) \cdot \left(angle \cdot a\right)\right) \cdot t\_0\\ \mathbf{elif}\;b \leq 1.26 \cdot 10^{+149}:\\ \;\;\;\;\left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\left(t\_0 \cdot a\right) \cdot a\right) + {\left(b \cdot \sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{angle}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{b}^{4}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (PI) (PI))))
   (if (<= b 9.2e-153)
     (* (* (* (* 3.08641975308642e-5 a) angle) (* angle a)) t_0)
     (if (<= b 1.26e+149)
       (+
        (* (* (* angle angle) 3.08641975308642e-5) (* (* t_0 a) a))
        (pow (* b (sin (fma (PI) (/ angle 180.0) (/ (PI) 2.0)))) 2.0))
       (sqrt (pow b 4.0))))))

Derivation

1. Split input into 3 regimes

2. if b < 9.19999999999999988e-153

   1. Initial program 79.2%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 44.8%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 46.9%

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

   10. Applied rewrites 48.9%

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

   if 9.19999999999999988e-153 < b < 1.26000000000000005e149

   1. Initial program 70.5%

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

      1. lift-cos.f64 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lower-/.f64 70.4

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

   4. Applied rewrites 70.4%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-PI.f64 N/A

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

      20. lower-PI.f64 58.1

          \[\leadsto \left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot a\right) \cdot a\right) + {\left(b \cdot \sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{angle}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}^{2} \]

   7. Applied rewrites 58.1%

      \[\leadsto \color{blue}{\left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \cdot a\right)} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{angle}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}^{2} \]

   if 1.26000000000000005e149 < b

   1. Initial program 93.0%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 90.4

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

   5. Applied rewrites 90.4%

      \[\leadsto \color{blue}{b \cdot b} \]
   6. Step-by-step derivation

   7. Applied rewrites 92.4%

      \[\leadsto \sqrt{{b}^{4}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 53.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;b \leq 9.2 \cdot 10^{-153}:\\ \;\;\;\;\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot a\right) \cdot angle\right) \cdot \left(angle \cdot a\right)\right) \cdot t\_0\\ \mathbf{elif}\;b \leq 1.26 \cdot 10^{+149}:\\ \;\;\;\;\left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\left(t\_0 \cdot a\right) \cdot a\right) + {\left(b \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{b}^{4}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (PI) (PI))))
   (if (<= b 9.2e-153)
     (* (* (* (* 3.08641975308642e-5 a) angle) (* angle a)) t_0)
     (if (<= b 1.26e+149)
       (+
        (* (* (* angle angle) 3.08641975308642e-5) (* (* t_0 a) a))
        (pow (* b (cos (* (* (PI) 0.005555555555555556) angle))) 2.0))
       (sqrt (pow b 4.0))))))

Derivation

1. Split input into 3 regimes

2. if b < 9.19999999999999988e-153

   1. Initial program 79.2%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 44.8%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 46.9%

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

   10. Applied rewrites 48.9%

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

   if 9.19999999999999988e-153 < b < 1.26000000000000005e149

   1. Initial program 70.5%

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

   3. Taylor expanded in angle around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 70.5

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

   5. Applied rewrites 70.5%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-PI.f64 N/A

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

      20. lower-PI.f64 58.0

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

   8. Applied rewrites 58.0%

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

   if 1.26000000000000005e149 < b

   1. Initial program 93.0%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 90.4

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

   5. Applied rewrites 90.4%

      \[\leadsto \color{blue}{b \cdot b} \]
   6. Step-by-step derivation

   7. Applied rewrites 92.4%

      \[\leadsto \sqrt{{b}^{4}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 53.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9.2 \cdot 10^{-153}:\\ \;\;\;\;\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot a\right) \cdot angle\right) \cdot \left(angle \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{elif}\;b \leq 1.26 \cdot 10^{+149}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), a \cdot a, {\left(\cos \left(\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot b\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{b}^{4}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 9.2e-153)
   (* (* (* (* 3.08641975308642e-5 a) angle) (* angle a)) (* (PI) (PI)))
   (if (<= b 1.26e+149)
     (fma
      (* (* (* 3.08641975308642e-5 (* angle angle)) (PI)) (PI))
      (* a a)
      (pow (* (cos (* (* 0.005555555555555556 (PI)) angle)) b) 2.0))
     (sqrt (pow b 4.0)))))

Derivation

1. Split input into 3 regimes

2. if b < 9.19999999999999988e-153

   1. Initial program 79.2%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 44.8%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 46.9%

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

   10. Applied rewrites 48.9%

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

   if 9.19999999999999988e-153 < b < 1.26000000000000005e149

   1. Initial program 70.5%

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

   3. Taylor expanded in angle around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 70.5

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

   5. Applied rewrites 70.5%

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

      1. lift-+.f64 N/A

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

   7. Applied rewrites 64.9%

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

   8. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. lower-PI.f64 58.0

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

   10. Applied rewrites 58.0%

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

   if 1.26000000000000005e149 < b

   1. Initial program 93.0%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 90.4

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

   5. Applied rewrites 90.4%

      \[\leadsto \color{blue}{b \cdot b} \]
   6. Step-by-step derivation

   7. Applied rewrites 92.4%

      \[\leadsto \sqrt{{b}^{4}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 53.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;b \leq 9.2 \cdot 10^{-153}:\\ \;\;\;\;\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot a\right) \cdot angle\right) \cdot \left(angle \cdot a\right)\right) \cdot t\_0\\ \mathbf{elif}\;b \leq 1.26 \cdot 10^{+149}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot t\_0\right) \cdot a\right) \cdot a, angle \cdot angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{b}^{4}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (PI) (PI))))
   (if (<= b 9.2e-153)
     (* (* (* (* 3.08641975308642e-5 a) angle) (* angle a)) t_0)
     (if (<= b 1.26e+149)
       (fma (* (* (* 3.08641975308642e-5 t_0) a) a) (* angle angle) (* b b))
       (sqrt (pow b 4.0))))))

Derivation

1. Split input into 3 regimes

2. if b < 9.19999999999999988e-153

   1. Initial program 79.2%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 44.8%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 46.9%

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

   10. Applied rewrites 48.9%

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

   if 9.19999999999999988e-153 < b < 1.26000000000000005e149

   1. Initial program 70.5%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 56.6%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 58.2%

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

   if 1.26000000000000005e149 < b

   1. Initial program 93.0%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 90.4

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

   5. Applied rewrites 90.4%

      \[\leadsto \color{blue}{b \cdot b} \]
   6. Step-by-step derivation

   7. Applied rewrites 92.4%

      \[\leadsto \sqrt{{b}^{4}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 53.4% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;b \leq 9.2 \cdot 10^{-153}:\\ \;\;\;\;\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot a\right) \cdot angle\right) \cdot \left(angle \cdot a\right)\right) \cdot t\_0\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot t\_0\right) \cdot a\right) \cdot a, angle \cdot angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (PI) (PI))))
   (if (<= b 9.2e-153)
     (* (* (* (* 3.08641975308642e-5 a) angle) (* angle a)) t_0)
     (if (<= b 1.15e+154)
       (fma (* (* (* 3.08641975308642e-5 t_0) a) a) (* angle angle) (* b b))
       (* b b)))))

Derivation

1. Split input into 3 regimes

2. if b < 9.19999999999999988e-153

   1. Initial program 79.2%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 44.8%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 46.9%

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

   10. Applied rewrites 48.9%

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

   if 9.19999999999999988e-153 < b < 1.15e154

   1. Initial program 68.7%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 55.0%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 56.7%

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

   if 1.15e154 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{b \cdot b} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 62.3% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.2 \cdot 10^{+124}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot a\right) \cdot angle\right) \cdot \left(angle \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 3.2e+124)
   (* b b)
   (* (* (* (* 3.08641975308642e-5 a) angle) (* angle a)) (* (PI) (PI)))))

Derivation

1. Split input into 2 regimes

2. if a < 3.19999999999999993e124

   1. Initial program 75.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 58.2

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

   5. Applied rewrites 58.2%

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

   if 3.19999999999999993e124 < a

   1. Initial program 92.1%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 55.6%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 79.5%

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

   10. Applied rewrites 81.5%

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

Alternative 12: 62.3% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.2 \cdot 10^{+124}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot a\right) \cdot \left(angle \cdot a\right)\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 3.2e+124)
   (* b b)
   (* (* 3.08641975308642e-5 (* (* angle a) (* angle a))) (* (PI) (PI)))))

Derivation

1. Split input into 2 regimes

2. if a < 3.19999999999999993e124

   1. Initial program 75.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 58.2

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

   5. Applied rewrites 58.2%

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

   if 3.19999999999999993e124 < a

   1. Initial program 92.1%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 55.6%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 79.5%

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

   10. Applied rewrites 81.5%

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

Alternative 13: 61.9% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.2 \cdot 10^{+124}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot \left(\left(a \cdot angle\right) \cdot angle\right)\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 3.2e+124)
   (* b b)
   (* (* 3.08641975308642e-5 (* a (* (* a angle) angle))) (* (PI) (PI)))))

Derivation

1. Split input into 2 regimes

2. if a < 3.19999999999999993e124

   1. Initial program 75.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 58.2

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

   5. Applied rewrites 58.2%

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

   if 3.19999999999999993e124 < a

   1. Initial program 92.1%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 55.6%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 79.5%

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

Alternative 14: 61.9% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.2 \cdot 10^{+124}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot a\right) \cdot \left(\left(\left(angle \cdot a\right) \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 3.2e+124)
   (* b b)
   (* (* 3.08641975308642e-5 a) (* (* (* angle a) angle) (* (PI) (PI))))))

Derivation

1. Split input into 2 regimes

2. if a < 3.19999999999999993e124

   1. Initial program 75.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 58.2

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

   5. Applied rewrites 58.2%

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

   if 3.19999999999999993e124 < a

   1. Initial program 92.1%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 55.6%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 79.5%

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

   10. Applied rewrites 79.5%

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

Alternative 15: 60.0% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.2 \cdot 10^{+124}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right) \cdot \left(\left(\left(angle \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 3.2e+124)
   (* b b)
   (* (* 3.08641975308642e-5 (* a a)) (* (* (* angle angle) (PI)) (PI)))))

Derivation

1. Split input into 2 regimes

2. if a < 3.19999999999999993e124

   1. Initial program 75.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 58.2

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

   5. Applied rewrites 58.2%

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

   if 3.19999999999999993e124 < a

   1. Initial program 92.1%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 55.6%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 79.5%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 66.6%

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

Alternative 16: 56.2% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 52.3

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

5. Applied rewrites 52.3%

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

Reproduce

herbie shell --seed 2024340 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) (PI)))) 2.0) (pow (* b (cos (* (/ angle 180.0) (PI)))) 2.0)))