ab-angle->ABCF C

Percentage Accurate: 79.3% → 79.3%

Time: 9.3s

Alternatives: 10

Speedup: 1.3×

• Could not converge on a ground truth

  1. a = 6.282080303677882e+57
  2. b = 7.569033908541267e-134
  3. angle = 7.047869501080159e+304

• 35.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

Initial Program: 79.3% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 79.3% accurate, 0.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 79.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. div-inv N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. inv-pow N/A

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

   10. lower-pow.f64 79.9

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

4. Applied rewrites 79.9%

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

Alternative 2: 79.2% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 79.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. div-inv N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. inv-pow N/A

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

   10. lower-pow.f64 79.9

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

4. Applied rewrites 79.9%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. div-inv N/A

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

   4. metadata-eval N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. frac-2neg N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f64 N/A

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

   11. metadata-eval N/A

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

   12. lift-pow.f64 N/A

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

   13. unpow-1 N/A

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

   14. distribute-neg-frac N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f64 79.8

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

6. Applied rewrites 79.8%

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

   1. lift-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. unpow-prod-down N/A

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

   4. lift-cos.f64 N/A

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

   5. cos-neg N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. associate-*r/ N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. distribute-frac-neg N/A

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

   12. metadata-eval N/A

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

   13. frac-2neg N/A

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

   14. lift-/.f64 N/A

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

   15. lift-cos.f64 N/A

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

   16. unpow-prod-down N/A

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

8. Applied rewrites 79.8%

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

Alternative 3: 79.2% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 79.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. div-inv N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 79.8

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

4. Applied rewrites 79.8%

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

Alternative 4: 79.3% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 79.7%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lower-cos.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-PI.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

5. Applied rewrites 67.4%

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

7. Applied rewrites 79.8%

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

Alternative 5: 79.3% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 79.7%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lower-cos.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-PI.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

5. Applied rewrites 67.4%

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

7. Applied rewrites 79.8%

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

9. Applied rewrites 79.7%

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

Alternative 6: 79.2% 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 79.7%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 79.7%

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

Alternative 7: 50.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.6 \cdot 10^{-159}:\\ \;\;\;\;{\sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{2} \cdot \left(b \cdot b\right)\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, b \cdot b, -3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right), angle \cdot angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.6e-159)
   (* (pow (sin (* (* (PI) 0.005555555555555556) angle)) 2.0) (* b b))
   (if (<= a 1.8e-6)
     (fma
      (*
       (* (PI) (PI))
       (fma 3.08641975308642e-5 (* b b) (* -3.08641975308642e-5 (* a a))))
      (* angle angle)
      (* a a))
     (* a a))))

Derivation

1. Split input into 3 regimes

2. if a < 1.6e-159

   1. Initial program 78.0%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-PI.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 39.7

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

   5. Applied rewrites 39.7%

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

   if 1.6e-159 < a < 1.79999999999999992e-6

   1. Initial program 80.3%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   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 63.6%

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

   if 1.79999999999999992e-6 < a

   1. Initial program 84.2%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 75.1

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

   5. Applied rewrites 75.1%

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

Alternative 8: 79.2% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 79.7%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lower-cos.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-PI.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

5. Applied rewrites 67.4%

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

7. Applied rewrites 79.8%

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

8. Taylor expanded in angle around 0

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

10. Applied rewrites 79.7%

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

Alternative 9: 53.0% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.8 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, b \cdot b, -3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right), angle \cdot angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.8e-6)
   (fma
    (*
     (* (PI) (PI))
     (fma 3.08641975308642e-5 (* b b) (* -3.08641975308642e-5 (* a a))))
    (* angle angle)
    (* a a))
   (* a a)))

Derivation

1. Split input into 2 regimes

2. if a < 1.79999999999999992e-6

   1. Initial program 78.4%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   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 46.1%

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

   if 1.79999999999999992e-6 < a

   1. Initial program 84.2%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 75.1

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

   5. Applied rewrites 75.1%

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

Alternative 10: 56.7% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 53.5

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

5. Applied rewrites 53.5%

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

Reproduce

herbie shell --seed 2024318 
(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)))