ab-angle->ABCF C

Percentage Accurate: 80.1% → 80.1%

Time: 12.1s

Alternatives: 17

Speedup: 1.0×

• Could not converge on a ground truth

  1. a = 4.7579930034793965e-15
  2. b = 6.43530853530374e-61
  3. angle = 6.862137534534348e+218

• 36.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

Initial Program: 80.1% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 80.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \mathsf{PI}\left(\right)\\ t_1 := \frac{angle}{180} \cdot \sinh t\_0\\ t_2 := \cosh t\_0\\ {\left(a \cdot \left(\cos \left(\frac{t\_2 \cdot angle}{-180}\right) \cdot \cos t\_1 - \sin \left(\frac{angle}{180} \cdot t\_2\right) \cdot \sin t\_1\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (log (PI))) (t_1 (* (/ angle 180.0) (sinh t_0))) (t_2 (cosh t_0)))
   (+
    (pow
     (*
      a
      (-
       (* (cos (/ (* t_2 angle) -180.0)) (cos t_1))
       (* (sin (* (/ angle 180.0) t_2)) (sin t_1))))
     2.0)
    (pow (* b (sin (* (PI) (/ angle 180.0)))) 2.0))))

Derivation

1. Initial program 77.0%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

   3. cos-neg-rev N/A

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

   4. remove-double-neg N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-PI.f64 N/A

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

   8. add-exp-log N/A

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

   9. sinh-+-cosh-rev N/A

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

   10. distribute-lft-in N/A

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

   11. cos-sum N/A

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

   12. lower--.f64 N/A

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

4. Applied rewrites 77.2%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

   3. lower-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*l/ N/A

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

   7. distribute-neg-frac2 N/A

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

   8. lower-/.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval 77.2

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

6. Applied rewrites 77.2%

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

Alternative 2: 80.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \mathsf{PI}\left(\right)\\ t_1 := \frac{angle}{180} \cdot \sinh t\_0\\ t_2 := \frac{angle}{180} \cdot \cosh t\_0\\ {\left(a \cdot \left(\cos t\_2 \cdot \cos t\_1 - \sin t\_2 \cdot \sin t\_1\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (log (PI)))
        (t_1 (* (/ angle 180.0) (sinh t_0)))
        (t_2 (* (/ angle 180.0) (cosh t_0))))
   (+
    (pow (* a (- (* (cos t_2) (cos t_1)) (* (sin t_2) (sin t_1)))) 2.0)
    (pow (* b (sin (* (PI) (/ angle 180.0)))) 2.0))))

Derivation

1. Initial program 77.0%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

   3. cos-neg-rev N/A

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

   4. remove-double-neg N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-PI.f64 N/A

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

   8. add-exp-log N/A

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

   9. sinh-+-cosh-rev N/A

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

   10. distribute-lft-in N/A

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

   11. cos-sum N/A

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

   12. lower--.f64 N/A

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

4. Applied rewrites 77.2%

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

Alternative 3: 80.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \mathsf{PI}\left(\right)\\ t_1 := \cosh t\_0\\ {\left(a \cdot \left(\cos \left(\frac{t\_1 \cdot angle}{-180}\right) \cdot \cos \left(\frac{angle}{180} \cdot \sinh t\_0\right) - \sin \left(\frac{angle}{180} \cdot t\_1\right) \cdot \sin \left(\left(\left(\mathsf{PI}\left(\right) - {\mathsf{PI}\left(\right)}^{-1}\right) \cdot 0.002777777777777778\right) \cdot angle\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (log (PI))) (t_1 (cosh t_0)))
   (+
    (pow
     (*
      a
      (-
       (* (cos (/ (* t_1 angle) -180.0)) (cos (* (/ angle 180.0) (sinh t_0))))
       (*
        (sin (* (/ angle 180.0) t_1))
        (sin (* (* (- (PI) (pow (PI) -1.0)) 0.002777777777777778) angle)))))
     2.0)
    (pow (* b (sin (* (PI) (/ angle 180.0)))) 2.0))))

Derivation

1. Initial program 77.0%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

   3. cos-neg-rev N/A

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

   4. remove-double-neg N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-PI.f64 N/A

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

   8. add-exp-log N/A

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

   9. sinh-+-cosh-rev N/A

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

   10. distribute-lft-in N/A

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

   11. cos-sum N/A

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

   12. lower--.f64 N/A

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

4. Applied rewrites 77.2%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

   3. lower-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*l/ N/A

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

   7. distribute-neg-frac2 N/A

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

   8. lower-/.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval 77.2

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

6. Applied rewrites 77.2%

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

7. Taylor expanded in angle around inf

   \[\leadsto {\left(a \cdot \left(\cos \left(\frac{\cosh \log \mathsf{PI}\left(\right) \cdot angle}{-180}\right) \cdot \cos \left(\frac{angle}{180} \cdot \sinh \log \mathsf{PI}\left(\right)\right) - \sin \left(\frac{angle}{180} \cdot \cosh \log \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{360} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) - \frac{1}{\mathsf{PI}\left(\right)}\right)\right)\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto {\left(a \cdot \left(\cos \left(\frac{\cosh \log \mathsf{PI}\left(\right) \cdot angle}{-180}\right) \cdot \cos \left(\frac{angle}{180} \cdot \sinh \log \mathsf{PI}\left(\right)\right) - \sin \left(\frac{angle}{180} \cdot \cosh \log \mathsf{PI}\left(\right)\right) \cdot \sin \color{blue}{\left(\left(angle \cdot \left(\mathsf{PI}\left(\right) - \frac{1}{\mathsf{PI}\left(\right)}\right)\right) \cdot \frac{1}{360}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   2. associate-*r* N/A

      \[\leadsto {\left(a \cdot \left(\cos \left(\frac{\cosh \log \mathsf{PI}\left(\right) \cdot angle}{-180}\right) \cdot \cos \left(\frac{angle}{180} \cdot \sinh \log \mathsf{PI}\left(\right)\right) - \sin \left(\frac{angle}{180} \cdot \cosh \log \mathsf{PI}\left(\right)\right) \cdot \sin \color{blue}{\left(angle \cdot \left(\left(\mathsf{PI}\left(\right) - \frac{1}{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{360}\right)\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   3. *-commutative N/A

      \[\leadsto {\left(a \cdot \left(\cos \left(\frac{\cosh \log \mathsf{PI}\left(\right) \cdot angle}{-180}\right) \cdot \cos \left(\frac{angle}{180} \cdot \sinh \log \mathsf{PI}\left(\right)\right) - \sin \left(\frac{angle}{180} \cdot \cosh \log \mathsf{PI}\left(\right)\right) \cdot \sin \left(angle \cdot \color{blue}{\left(\frac{1}{360} \cdot \left(\mathsf{PI}\left(\right) - \frac{1}{\mathsf{PI}\left(\right)}\right)\right)}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   4. lower-sin.f64 N/A

      \[\leadsto {\left(a \cdot \left(\cos \left(\frac{\cosh \log \mathsf{PI}\left(\right) \cdot angle}{-180}\right) \cdot \cos \left(\frac{angle}{180} \cdot \sinh \log \mathsf{PI}\left(\right)\right) - \sin \left(\frac{angle}{180} \cdot \cosh \log \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\sin \left(angle \cdot \left(\frac{1}{360} \cdot \left(\mathsf{PI}\left(\right) - \frac{1}{\mathsf{PI}\left(\right)}\right)\right)\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   5. *-commutative N/A

      \[\leadsto {\left(a \cdot \left(\cos \left(\frac{\cosh \log \mathsf{PI}\left(\right) \cdot angle}{-180}\right) \cdot \cos \left(\frac{angle}{180} \cdot \sinh \log \mathsf{PI}\left(\right)\right) - \sin \left(\frac{angle}{180} \cdot \cosh \log \mathsf{PI}\left(\right)\right) \cdot \sin \left(angle \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) - \frac{1}{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{360}\right)}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   6. associate-*r* N/A

      \[\leadsto {\left(a \cdot \left(\cos \left(\frac{\cosh \log \mathsf{PI}\left(\right) \cdot angle}{-180}\right) \cdot \cos \left(\frac{angle}{180} \cdot \sinh \log \mathsf{PI}\left(\right)\right) - \sin \left(\frac{angle}{180} \cdot \cosh \log \mathsf{PI}\left(\right)\right) \cdot \sin \color{blue}{\left(\left(angle \cdot \left(\mathsf{PI}\left(\right) - \frac{1}{\mathsf{PI}\left(\right)}\right)\right) \cdot \frac{1}{360}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   7. *-commutative N/A

      \[\leadsto {\left(a \cdot \left(\cos \left(\frac{\cosh \log \mathsf{PI}\left(\right) \cdot angle}{-180}\right) \cdot \cos \left(\frac{angle}{180} \cdot \sinh \log \mathsf{PI}\left(\right)\right) - \sin \left(\frac{angle}{180} \cdot \cosh \log \mathsf{PI}\left(\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{360} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) - \frac{1}{\mathsf{PI}\left(\right)}\right)\right)\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   8. *-commutative N/A

      \[\leadsto {\left(a \cdot \left(\cos \left(\frac{\cosh \log \mathsf{PI}\left(\right) \cdot angle}{-180}\right) \cdot \cos \left(\frac{angle}{180} \cdot \sinh \log \mathsf{PI}\left(\right)\right) - \sin \left(\frac{angle}{180} \cdot \cosh \log \mathsf{PI}\left(\right)\right) \cdot \sin \left(\frac{1}{360} \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) - \frac{1}{\mathsf{PI}\left(\right)}\right) \cdot angle\right)}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   9. associate-*r* N/A

      \[\leadsto {\left(a \cdot \left(\cos \left(\frac{\cosh \log \mathsf{PI}\left(\right) \cdot angle}{-180}\right) \cdot \cos \left(\frac{angle}{180} \cdot \sinh \log \mathsf{PI}\left(\right)\right) - \sin \left(\frac{angle}{180} \cdot \cosh \log \mathsf{PI}\left(\right)\right) \cdot \sin \color{blue}{\left(\left(\frac{1}{360} \cdot \left(\mathsf{PI}\left(\right) - \frac{1}{\mathsf{PI}\left(\right)}\right)\right) \cdot angle\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   10. lower-*.f64 N/A

       \[\leadsto {\left(a \cdot \left(\cos \left(\frac{\cosh \log \mathsf{PI}\left(\right) \cdot angle}{-180}\right) \cdot \cos \left(\frac{angle}{180} \cdot \sinh \log \mathsf{PI}\left(\right)\right) - \sin \left(\frac{angle}{180} \cdot \cosh \log \mathsf{PI}\left(\right)\right) \cdot \sin \color{blue}{\left(\left(\frac{1}{360} \cdot \left(\mathsf{PI}\left(\right) - \frac{1}{\mathsf{PI}\left(\right)}\right)\right) \cdot angle\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   11. *-commutative N/A

       \[\leadsto {\left(a \cdot \left(\cos \left(\frac{\cosh \log \mathsf{PI}\left(\right) \cdot angle}{-180}\right) \cdot \cos \left(\frac{angle}{180} \cdot \sinh \log \mathsf{PI}\left(\right)\right) - \sin \left(\frac{angle}{180} \cdot \cosh \log \mathsf{PI}\left(\right)\right) \cdot \sin \left(\color{blue}{\left(\left(\mathsf{PI}\left(\right) - \frac{1}{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{360}\right)} \cdot angle\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   12. lower-*.f64 N/A

       \[\leadsto {\left(a \cdot \left(\cos \left(\frac{\cosh \log \mathsf{PI}\left(\right) \cdot angle}{-180}\right) \cdot \cos \left(\frac{angle}{180} \cdot \sinh \log \mathsf{PI}\left(\right)\right) - \sin \left(\frac{angle}{180} \cdot \cosh \log \mathsf{PI}\left(\right)\right) \cdot \sin \left(\color{blue}{\left(\left(\mathsf{PI}\left(\right) - \frac{1}{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{360}\right)} \cdot angle\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   13. lower--.f64 N/A

       \[\leadsto {\left(a \cdot \left(\cos \left(\frac{\cosh \log \mathsf{PI}\left(\right) \cdot angle}{-180}\right) \cdot \cos \left(\frac{angle}{180} \cdot \sinh \log \mathsf{PI}\left(\right)\right) - \sin \left(\frac{angle}{180} \cdot \cosh \log \mathsf{PI}\left(\right)\right) \cdot \sin \left(\left(\color{blue}{\left(\mathsf{PI}\left(\right) - \frac{1}{\mathsf{PI}\left(\right)}\right)} \cdot \frac{1}{360}\right) \cdot angle\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   14. lower-PI.f64 N/A

       \[\leadsto {\left(a \cdot \left(\cos \left(\frac{\cosh \log \mathsf{PI}\left(\right) \cdot angle}{-180}\right) \cdot \cos \left(\frac{angle}{180} \cdot \sinh \log \mathsf{PI}\left(\right)\right) - \sin \left(\frac{angle}{180} \cdot \cosh \log \mathsf{PI}\left(\right)\right) \cdot \sin \left(\left(\left(\color{blue}{\mathsf{PI}\left(\right)} - \frac{1}{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{360}\right) \cdot angle\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   15. lower-/.f64 N/A

       \[\leadsto {\left(a \cdot \left(\cos \left(\frac{\cosh \log \mathsf{PI}\left(\right) \cdot angle}{-180}\right) \cdot \cos \left(\frac{angle}{180} \cdot \sinh \log \mathsf{PI}\left(\right)\right) - \sin \left(\frac{angle}{180} \cdot \cosh \log \mathsf{PI}\left(\right)\right) \cdot \sin \left(\left(\left(\mathsf{PI}\left(\right) - \color{blue}{\frac{1}{\mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{360}\right) \cdot angle\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   16. lower-PI.f64 77.2

       \[\leadsto {\left(a \cdot \left(\cos \left(\frac{\cosh \log \mathsf{PI}\left(\right) \cdot angle}{-180}\right) \cdot \cos \left(\frac{angle}{180} \cdot \sinh \log \mathsf{PI}\left(\right)\right) - \sin \left(\frac{angle}{180} \cdot \cosh \log \mathsf{PI}\left(\right)\right) \cdot \sin \left(\left(\left(\mathsf{PI}\left(\right) - \frac{1}{\color{blue}{\mathsf{PI}\left(\right)}}\right) \cdot 0.002777777777777778\right) \cdot angle\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

9. Applied rewrites 77.2%

   \[\leadsto {\left(a \cdot \left(\cos \left(\frac{\cosh \log \mathsf{PI}\left(\right) \cdot angle}{-180}\right) \cdot \cos \left(\frac{angle}{180} \cdot \sinh \log \mathsf{PI}\left(\right)\right) - \sin \left(\frac{angle}{180} \cdot \cosh \log \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\sin \left(\left(\left(\mathsf{PI}\left(\right) - \frac{1}{\mathsf{PI}\left(\right)}\right) \cdot 0.002777777777777778\right) \cdot angle\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

10. Final simplification 77.2%

    \[\leadsto {\left(a \cdot \left(\cos \left(\frac{\cosh \log \mathsf{PI}\left(\right) \cdot angle}{-180}\right) \cdot \cos \left(\frac{angle}{180} \cdot \sinh \log \mathsf{PI}\left(\right)\right) - \sin \left(\frac{angle}{180} \cdot \cosh \log \mathsf{PI}\left(\right)\right) \cdot \sin \left(\left(\left(\mathsf{PI}\left(\right) - {\mathsf{PI}\left(\right)}^{-1}\right) \cdot 0.002777777777777778\right) \cdot angle\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
11. Add Preprocessing

Alternative 4: 80.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \cos \left(\left(\frac{\sqrt{\mathsf{PI}\left(\right)} \cdot angle}{180} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot {\mathsf{PI}\left(\right)}^{0.16666666666666666}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow
   (*
    a
    (cos
     (*
      (* (/ (* (sqrt (PI)) angle) 180.0) (cbrt (PI)))
      (pow (PI) 0.16666666666666666))))
   2.0)
  (pow (* b (sin (* (PI) (/ angle 180.0)))) 2.0)))

Derivation

1. Initial program 77.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f64 N/A

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

   4. add-sqr-sqrt N/A

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

   5. associate-*r* N/A

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

   6. add-cube-cbrt N/A

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

   7. sqrt-prod N/A

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

   8. pow2 N/A

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

   9. sqrt-pow1 N/A

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

   10. metadata-eval N/A

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

   11. unpow1 N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f64 N/A

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

4. Applied rewrites 77.2%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\left(\frac{\sqrt{\mathsf{PI}\left(\right)} \cdot angle}{180} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot {\mathsf{PI}\left(\right)}^{0.16666666666666666}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
5. Add Preprocessing

Alternative 5: 80.1% accurate, 0.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 77.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f64 N/A

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

   4. add-sqr-sqrt N/A

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

   5. sqr-neg-rev N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-neg.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lower-sqrt.f64 77.1

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

4. Applied rewrites 77.1%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 77.1

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

6. Applied rewrites 77.1%

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

7. Final simplification 77.1%

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

Alternative 6: 80.1% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 77.0%

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

3. Taylor expanded in angle around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-cos.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-PI.f64 77.1

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

5. Applied rewrites 77.1%

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

Alternative 7: 80.1% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 77.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 77.0

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

4. Applied rewrites 77.0%

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

5. Taylor expanded in angle around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-PI.f64 77.0

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

7. Applied rewrites 77.0%

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

8. Taylor expanded in angle around inf

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

   1. cos-neg-rev N/A

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

   2. lower-cos.f64 N/A

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

   3. distribute-lft-neg-in N/A

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

   4. metadata-eval N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-PI.f64 77.1

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

10. Applied rewrites 77.1%

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

11. Final simplification 77.1%

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

Alternative 8: 80.1% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 77.0%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 77.0%

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

Alternative 9: 63.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot angle\\ \mathbf{if}\;angle \leq 3.9:\\ \;\;\;\;\mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, t\_0, 1\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\\ \mathbf{elif}\;angle \leq 1.5 \cdot 10^{+172}:\\ \;\;\;\;{\left(a \cdot \cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{180}\right)\right)}^{2} + \left(3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{\log a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* (* (PI) (PI)) angle) angle)))
   (if (<= angle 3.9)
     (+
      (* (fma -3.08641975308642e-5 t_0 1.0) (* a a))
      (pow (* b (sin (* (PI) (/ angle 180.0)))) 2.0))
     (if (<= angle 1.5e+172)
       (+
        (pow (* a (cos (/ (* angle (PI)) 180.0))) 2.0)
        (* (* 3.08641975308642e-5 (* b b)) t_0))
       (exp (* (log a) 2.0))))))

Derivation

1. Split input into 3 regimes

2. if angle < 3.89999999999999991

   1. Initial program 84.4%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-PI.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 65.2

          \[\leadsto \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot angle, 1\right) \cdot \color{blue}{\left(a \cdot a\right)} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   5. Applied rewrites 65.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot angle, 1\right) \cdot \left(a \cdot a\right)} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   if 3.89999999999999991 < angle < 1.5e172

   1. Initial program 51.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 51.6

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

   4. Applied rewrites 51.6%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-PI.f64 N/A

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

      18. lower-PI.f64 45.8

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

   7. Applied rewrites 45.8%

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

   if 1.5e172 < angle

   1. Initial program 56.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 46.5

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

   5. Applied rewrites 46.5%

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

   7. Applied rewrites 12.8%

      \[\leadsto e^{\log a \cdot 2} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 62.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.5 \cdot 10^{+96}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 5.5e+96)
   (* a a)
   (* (pow (* (* b (PI)) angle) 2.0) 3.08641975308642e-5)))

Derivation

1. Split input into 2 regimes

2. if b < 5.5000000000000002e96

   1. Initial program 75.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 68.8

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

   5. Applied rewrites 68.8%

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

   if 5.5000000000000002e96 < b

   1. Initial program 82.2%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 44.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 53.8%

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

   10. Applied rewrites 69.0%

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

Alternative 11: 57.8% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 4.6 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(a - b\right) \cdot \left(b + a\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(-angle\right), -angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 4.6e+95)
   (fma
    (*
     (* (* (- a b) (+ b a)) (* (* (PI) (PI)) -3.08641975308642e-5))
     (- angle))
    (- angle)
    (* a a))
   (* a a)))

Derivation

1. Split input into 2 regimes

2. if a < 4.59999999999999994e95

   1. Initial program 75.6%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 43.6%

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

   7. Applied rewrites 48.9%

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

   if 4.59999999999999994e95 < a

   1. Initial program 85.4%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 85.0

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

   5. Applied rewrites 85.0%

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

Alternative 12: 57.8% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 4.6 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(a - b\right) \cdot \left(b + a\right)\right) \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -3.08641975308642 \cdot 10^{-5}\right) \cdot angle\right), angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 4.6e+95)
   (fma
    (* (* (- a b) (+ b a)) (* (* (* (PI) (PI)) -3.08641975308642e-5) angle))
    angle
    (* a a))
   (* a a)))

Derivation

1. Split input into 2 regimes

2. if a < 4.59999999999999994e95

   1. Initial program 75.6%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 43.6%

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

   7. Applied rewrites 48.9%

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

   if 4.59999999999999994e95 < a

   1. Initial program 85.4%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 85.0

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

   5. Applied rewrites 85.0%

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

Alternative 13: 62.5% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.7 \cdot 10^{-54}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-3.08641975308642 \cdot 10^{-5} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(-b\right) \cdot b\right), angle \cdot angle, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.7e-54)
   (* a a)
   (fma
    (* (* -3.08641975308642e-5 (* (PI) (PI))) (* (- b) b))
    (* angle angle)
    (* a a))))

Derivation

1. Split input into 2 regimes

2. if b < 1.69999999999999994e-54

   1. Initial program 75.2%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 68.0

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

   5. Applied rewrites 68.0%

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

   if 1.69999999999999994e-54 < b

   1. Initial program 81.8%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 46.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 65.6%

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

Alternative 14: 62.5% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.7 \cdot 10^{-54}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), angle \cdot angle, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.7e-54)
   (* a a)
   (fma
    (* (* (* 3.08641975308642e-5 (* b b)) (PI)) (PI))
    (* angle angle)
    (* a a))))

Derivation

1. Split input into 2 regimes

2. if b < 1.69999999999999994e-54

   1. Initial program 75.2%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 68.0

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

   5. Applied rewrites 68.0%

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

   if 1.69999999999999994e-54 < b

   1. Initial program 81.8%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 46.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 65.6%

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

Alternative 15: 61.1% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9 \cdot 10^{+96}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot b\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 9e+96)
   (* a a)
   (* (* (* (* 3.08641975308642e-5 (* angle angle)) b) (* (PI) (PI))) b)))

Derivation

1. Split input into 2 regimes

2. if b < 8.99999999999999914e96

   1. Initial program 75.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 68.8

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

   5. Applied rewrites 68.8%

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

   if 8.99999999999999914e96 < b

   1. Initial program 82.2%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 44.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 53.8%

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

   10. Applied rewrites 54.3%

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

Alternative 16: 60.4% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9 \cdot 10^{+96}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right) \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot angle\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 9e+96)
   (* a a)
   (* (* 3.08641975308642e-5 (* b b)) (* (* (* (PI) (PI)) angle) angle))))

Derivation

1. Split input into 2 regimes

2. if b < 8.99999999999999914e96

   1. Initial program 75.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 68.8

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

   5. Applied rewrites 68.8%

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

   if 8.99999999999999914e96 < b

   1. Initial program 82.2%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 44.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 53.8%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 55.9%

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

Alternative 17: 56.9% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 60.0

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

5. Applied rewrites 60.0%

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

Reproduce

herbie shell --seed 2024353 
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* (PI) (/ angle 180.0)))) 2.0) (pow (* b (sin (* (PI) (/ angle 180.0)))) 2.0)))