ab-angle->ABCF C

Percentage Accurate: 80.3% → 80.2%

Time: 11.7s

Alternatives: 12

Speedup: N/A×

• Could not converge on a ground truth

  1. a = 1.808092054810868e-247
  2. b = 2.8026197639808644e+134
  3. angle = 1.901445751054359e+34

• 36.3% 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: 80.3% 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: 80.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(a \cdot 1, a, {\left(b \cdot \sin \left(\frac{angle}{{\mathsf{PI}\left(\right)}^{-0.5}} \cdot \left(0.005555555555555556 \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right)}^{2}\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (fma
  (* a 1.0)
  a
  (pow
   (*
    b
    (sin (* (/ angle (pow (PI) -0.5)) (* 0.005555555555555556 (sqrt (PI))))))
   2.0)))

Derivation

1. Initial program 78.8%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

4. Applied rewrites 78.8%

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

5. Taylor expanded in angle around 0

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

7. Applied rewrites 79.0%

   \[\leadsto \mathsf{fma}\left(\color{blue}{1} \cdot a, a, {\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2}\right) \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. metadata-eval N/A

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

   5. div-inv N/A

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

   6. associate-*l/ N/A

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

   7. associate-/l* N/A

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

   8. rem-square-sqrt N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. associate-/l* N/A

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

   12. lift-/.f64 N/A

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

   13. associate-*l* N/A

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

   14. *-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. clear-num N/A

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

   18. un-div-inv N/A

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

   19. lift-*.f64 N/A

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

   20. div-inv N/A

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

   21. times-frac N/A

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

   22. lift-/.f64 N/A

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

9. Applied rewrites 79.0%

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

10. Final simplification 79.0%

    \[\leadsto \mathsf{fma}\left(a \cdot 1, a, {\left(b \cdot \sin \left(\frac{angle}{{\mathsf{PI}\left(\right)}^{-0.5}} \cdot \left(0.005555555555555556 \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right)}^{2}\right) \]
11. Add Preprocessing

Alternative 2: 80.3% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 78.8%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

4. Applied rewrites 78.8%

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

5. Taylor expanded in angle around 0

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

7. Applied rewrites 79.0%

   \[\leadsto \mathsf{fma}\left(\color{blue}{1} \cdot a, a, {\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2}\right) \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. metadata-eval N/A

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

   5. div-inv N/A

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

   6. clear-num N/A

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

   7. lift-/.f64 N/A

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

   8. associate-*l/ N/A

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

   9. lift-/.f64 N/A

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

   10. clear-num N/A

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

   11. frac-2neg N/A

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

   12. metadata-eval N/A

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

   13. associate-/r/ N/A

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

   14. distribute-neg-frac2 N/A

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

   15. unpow-1 N/A

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

   16. times-frac N/A

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

   17. lower-*.f64 N/A

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

   18. inv-pow N/A

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

   19. lower-pow.f64 N/A

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

   20. unpow-1 N/A

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

   21. distribute-neg-frac N/A

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

   22. metadata-eval N/A

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

   23. lower-/.f64 N/A

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

   24. div-inv N/A

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

   25. metadata-eval N/A

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

   26. lower-*.f64 79.0

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

9. Applied rewrites 79.0%

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

10. Final simplification 79.0%

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

Alternative 3: 80.3% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 78.8%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

4. Applied rewrites 78.8%

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

5. Taylor expanded in angle around 0

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

7. Applied rewrites 79.0%

   \[\leadsto \mathsf{fma}\left(\color{blue}{1} \cdot a, a, {\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2}\right) \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. metadata-eval N/A

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

   5. div-inv N/A

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

   6. associate-*l/ N/A

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

   7. associate-/l* N/A

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

   8. lift-PI.f64 N/A

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

   9. add-cube-cbrt N/A

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

   10. unpow2 N/A

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

   11. associate-/l* N/A

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

   12. *-commutative N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

9. Applied rewrites 79.0%

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

10. Final simplification 79.0%

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

Alternative 4: 80.3% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 78.8%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

4. Applied rewrites 78.8%

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

5. Taylor expanded in angle around 0

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

7. Applied rewrites 79.0%

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

8. Final simplification 79.0%

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

Alternative 5: 68.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.6 \cdot 10^{-15}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 1, a, {\left(\left(\left(\mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 0.005555555555555556\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot b\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 4.6e-15)
   (* a a)
   (fma
    (* a 1.0)
    a
    (pow
     (*
      (*
       (*
        (fma
         (* (* angle angle) -2.8577960676726107e-8)
         (* (PI) (PI))
         0.005555555555555556)
        (PI))
       angle)
      b)
     2.0))))

Derivation

1. Split input into 2 regimes

2. if b < 4.59999999999999981e-15

   1. Initial program 77.2%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 57.5

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

   5. Applied rewrites 57.5%

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

   if 4.59999999999999981e-15 < b

   1. Initial program 82.3%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 82.3%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 82.3%

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

   8. Taylor expanded in angle around 0

      \[\leadsto \mathsf{fma}\left(1 \cdot a, a, {\left(\color{blue}{\left(angle \cdot \left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot b\right)}^{2}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(1 \cdot a, a, {\left(\color{blue}{\left(\left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)} \cdot b\right)}^{2}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(1 \cdot a, a, {\left(\left(\color{blue}{\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + \frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)} \cdot angle\right) \cdot b\right)}^{2}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(1 \cdot a, a, {\left(\left(\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + \frac{-1}{34992000} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{3} \cdot {angle}^{2}\right)}\right) \cdot angle\right) \cdot b\right)}^{2}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(1 \cdot a, a, {\left(\left(\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + \color{blue}{\left(\frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {angle}^{2}}\right) \cdot angle\right) \cdot b\right)}^{2}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(1 \cdot a, a, {\left(\color{blue}{\left(\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + \left(\frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {angle}^{2}\right) \cdot angle\right)} \cdot b\right)}^{2}\right) \]

   10. Applied rewrites 80.1%

       \[\leadsto \mathsf{fma}\left(1 \cdot a, a, {\left(\color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 0.005555555555555556\right)\right) \cdot angle\right)} \cdot b\right)}^{2}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.6 \cdot 10^{-15}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 1, a, {\left(\left(\left(\mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 0.005555555555555556\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot b\right)}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.6 \cdot 10^{-15}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 1, a, {\left(\left(\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot b\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 4.6e-15)
   (* a a)
   (fma (* a 1.0) a (pow (* (* (* 0.005555555555555556 (PI)) angle) b) 2.0))))

Derivation

1. Split input into 2 regimes

2. if b < 4.59999999999999981e-15

   1. Initial program 77.2%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 57.5

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

   5. Applied rewrites 57.5%

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

   if 4.59999999999999981e-15 < b

   1. Initial program 82.3%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 82.3%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 82.3%

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

   8. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 80.3

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

   10. Applied rewrites 80.3%

       \[\leadsto \mathsf{fma}\left(1 \cdot a, a, {\left(\color{blue}{\left(\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)} \cdot b\right)}^{2}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.6 \cdot 10^{-15}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 1, a, {\left(\left(\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot b\right)}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.2% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 7.5 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5} \cdot b, b, \left(-3.08641975308642 \cdot 10^{-5} \cdot a\right) \cdot a\right), angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 7.5e+70)
   (fma
    (*
     (* (* (PI) (PI)) angle)
     (fma (* 3.08641975308642e-5 b) b (* (* -3.08641975308642e-5 a) a)))
    angle
    (* a a))
   (* a a)))

Derivation

1. Split input into 2 regimes

2. if a < 7.50000000000000031e70

   1. Initial program 76.8%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   2. Add Preprocessing
   3. 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. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-PI.f64 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*l* N/A

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

      8. div-inv N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lower-cbrt.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. lower-pow.f64 N/A

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

      17. lift-PI.f64 N/A

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

      18. lower-cbrt.f64 N/A

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

      19. inv-pow N/A

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

      20. lower-pow.f64 76.9

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

   4. Applied rewrites 76.9%

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

   5. Taylor expanded in angle around 0

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

      1. *-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

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

   7. Applied rewrites 44.9%

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

   9. Applied rewrites 49.6%

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

   if 7.50000000000000031e70 < a

   1. Initial program 89.9%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 87.6

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

   5. Applied rewrites 87.6%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 7.5 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5} \cdot b, b, \left(-3.08641975308642 \cdot 10^{-5} \cdot a\right) \cdot a\right), angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.6% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.6 \cdot 10^{-15}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(b \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), angle \cdot angle, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 4.6e-15)
   (* a a)
   (fma
    (* (* (* b b) 3.08641975308642e-5) (* (PI) (PI)))
    (* angle angle)
    (* a a))))

Derivation

1. Split input into 2 regimes

2. if b < 4.59999999999999981e-15

   1. Initial program 77.2%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 57.5

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

   5. Applied rewrites 57.5%

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

   if 4.59999999999999981e-15 < b

   1. Initial program 82.3%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   2. Add Preprocessing
   3. 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. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-PI.f64 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*l* N/A

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

      8. div-inv N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lower-cbrt.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. lower-pow.f64 N/A

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

      17. lift-PI.f64 N/A

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

      18. lower-cbrt.f64 N/A

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

      19. inv-pow N/A

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

      20. lower-pow.f64 82.2

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

   4. Applied rewrites 82.2%

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

   5. Taylor expanded in angle around 0

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

      1. *-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

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

   7. Applied rewrites 39.8%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 67.2%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.6 \cdot 10^{-15}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(b \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), angle \cdot angle, a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.7% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.6 \cdot 10^{-15}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b\right) \cdot b, angle \cdot angle, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 4.6e-15)
   (* a a)
   (fma
    (* (* (* 3.08641975308642e-5 (* (PI) (PI))) b) b)
    (* angle angle)
    (* a a))))

Derivation

1. Split input into 2 regimes

2. if b < 4.59999999999999981e-15

   1. Initial program 77.2%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 57.5

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

   5. Applied rewrites 57.5%

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

   if 4.59999999999999981e-15 < b

   1. Initial program 82.3%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   2. Add Preprocessing
   3. 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. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-PI.f64 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*l* N/A

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

      8. div-inv N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lower-cbrt.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. lower-pow.f64 N/A

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

      17. lift-PI.f64 N/A

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

      18. lower-cbrt.f64 N/A

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

      19. inv-pow N/A

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

      20. lower-pow.f64 82.2

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

   4. Applied rewrites 82.2%

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

   5. Taylor expanded in angle around 0

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

      1. *-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

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

   7. Applied rewrites 39.8%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 67.2%

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

Alternative 10: 61.9% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{+137}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right) \cdot b\right) \cdot \left(\left(3.08641975308642 \cdot 10^{-5} \cdot angle\right) \cdot angle\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 3e+137)
   (* a a)
   (* (* (* (* (PI) (PI)) b) b) (* (* 3.08641975308642e-5 angle) angle))))

Derivation

1. Split input into 2 regimes

2. if b < 3.0000000000000001e137

   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

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

      1. unpow2 N/A

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

      2. lower-*.f64 54.3

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

   5. Applied rewrites 54.3%

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

   if 3.0000000000000001e137 < b

   1. Initial program 94.0%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-PI.f64 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*l* N/A

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

      8. div-inv N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lower-cbrt.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. lower-pow.f64 N/A

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

      17. lift-PI.f64 N/A

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

      18. lower-cbrt.f64 N/A

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

      19. inv-pow N/A

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

      20. lower-pow.f64 94.0

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

   4. Applied rewrites 94.0%

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

   5. Taylor expanded in angle around 0

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

      1. *-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

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

   7. Applied rewrites 41.8%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 73.7%

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

   12. Applied rewrites 73.7%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{+137}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right) \cdot b\right) \cdot \left(\left(3.08641975308642 \cdot 10^{-5} \cdot angle\right) \cdot angle\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.9% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{+137}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right) \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 3e+137)
   (* a a)
   (* (* 3.08641975308642e-5 (* angle angle)) (* (* (* (PI) (PI)) b) b))))

Derivation

1. Split input into 2 regimes

2. if b < 3.0000000000000001e137

   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

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

      1. unpow2 N/A

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

      2. lower-*.f64 54.3

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

   5. Applied rewrites 54.3%

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

   if 3.0000000000000001e137 < b

   1. Initial program 94.0%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-PI.f64 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*l* N/A

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

      8. div-inv N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lower-cbrt.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. lower-pow.f64 N/A

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

      17. lift-PI.f64 N/A

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

      18. lower-cbrt.f64 N/A

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

      19. inv-pow N/A

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

      20. lower-pow.f64 94.0

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

   4. Applied rewrites 94.0%

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

   5. Taylor expanded in angle around 0

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

      1. *-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

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

   7. Applied rewrites 41.8%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 73.7%

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

Alternative 12: 58.1% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 51.9

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

5. Applied rewrites 51.9%

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

Reproduce

herbie shell --seed 2024268 
(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)))