ab-angle->ABCF A

Percentage Accurate: 79.3% → 79.2%

Time: 14.7s

Alternatives: 17

Speedup: 1.3×

• Could not converge on a ground truth

  1. a = 2.758340683337559e-160
  2. b = 1.2776779258745173e+244
  3. angle = 2.3052097436744104e+182

• 36.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

Initial Program: 79.3% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 79.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\mathsf{PI}\left(\right)}\\ \mathsf{fma}\left(b \cdot {\cos \left(\left(0.005555555555555556 \cdot {t\_0}^{2}\right) \cdot \left(\left(-angle\right) \cdot t\_0\right)\right)}^{2}, b, {\left(\sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)}^{2}\right) \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (cbrt (PI))))
   (fma
    (*
     b
     (pow
      (cos (* (* 0.005555555555555556 (pow t_0 2.0)) (* (- angle) t_0)))
      2.0))
    b
    (pow (* (sin (* (* angle 0.005555555555555556) (PI))) a) 2.0))))

Derivation

1. Initial program 76.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

4. Applied rewrites 76.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. lift-/.f64 N/A

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

   8. un-div-inv N/A

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

   9. distribute-neg-frac2 N/A

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

   10. lift-PI.f64 N/A

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

   11. add-cube-cbrt N/A

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

   12. lift-/.f64 N/A

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

   13. div-inv N/A

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

   14. unpow-1 N/A

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

   15. lift-pow.f64 N/A

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

   16. times-frac N/A

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

   17. distribute-rgt-neg-in N/A

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

   18. lower-*.f64 N/A

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

6. Applied rewrites 76.5%

   \[\leadsto \mathsf{fma}\left({\cos \color{blue}{\left(\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot 0.005555555555555556\right) \cdot \left(-\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)}}^{2} \cdot b, b, {\left(\sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot a\right)}^{2}\right) \]

7. Final simplification 76.5%

   \[\leadsto \mathsf{fma}\left(b \cdot {\cos \left(\left(0.005555555555555556 \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}\right) \cdot \left(\left(-angle\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2}, b, {\left(\sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)}^{2}\right) \]
8. Add Preprocessing

Alternative 2: 79.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\mathsf{PI}\left(\right)}\\ \mathsf{fma}\left({\cos \left(\left(\left(0.005555555555555556 \cdot {t\_0}^{2}\right) \cdot t\_0\right) \cdot \left(-angle\right)\right)}^{2} \cdot b, b, {\left(\sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)}^{2}\right) \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (cbrt (PI))))
   (fma
    (*
     (pow
      (cos (* (* (* 0.005555555555555556 (pow t_0 2.0)) t_0) (- angle)))
      2.0)
     b)
    b
    (pow (* (sin (* (* angle 0.005555555555555556) (PI))) a) 2.0))))

Derivation

1. Initial program 76.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

4. Applied rewrites 76.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. lift-/.f64 N/A

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

   8. un-div-inv N/A

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

   9. distribute-neg-frac2 N/A

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

   10. lift-PI.f64 N/A

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

   11. add-cube-cbrt N/A

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

   12. lift-/.f64 N/A

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

   13. div-inv N/A

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

   14. unpow-1 N/A

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

   15. lift-pow.f64 N/A

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

   16. times-frac N/A

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

   17. distribute-rgt-neg-in N/A

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

   18. lower-*.f64 N/A

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

6. Applied rewrites 76.5%

   \[\leadsto \mathsf{fma}\left({\cos \color{blue}{\left(\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot 0.005555555555555556\right) \cdot \left(-\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)}}^{2} \cdot b, b, {\left(\sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot a\right)}^{2}\right) \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift-*.f64 N/A

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

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

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-neg.f64 76.5

       \[\leadsto \mathsf{fma}\left({\cos \left(\left(\left(0.005555555555555556 \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}\right) \cdot \color{blue}{\left(-\sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right) \cdot angle\right)}^{2} \cdot b, b, {\left(\sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot a\right)}^{2}\right) \]

8. Applied rewrites 76.5%

   \[\leadsto \mathsf{fma}\left({\cos \color{blue}{\left(\left(\left(0.005555555555555556 \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}\right) \cdot \left(-\sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot angle\right)}}^{2} \cdot b, b, {\left(\sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot a\right)}^{2}\right) \]

9. Final simplification 76.5%

   \[\leadsto \mathsf{fma}\left({\cos \left(\left(\left(0.005555555555555556 \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(-angle\right)\right)}^{2} \cdot b, b, {\left(\sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)}^{2}\right) \]
10. Add Preprocessing

Alternative 3: 79.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\mathsf{PI}\left(\right)}\\ {\left(\cos \left(\left(0.005555555555555556 \cdot {t\_0}^{2}\right) \cdot \left(angle \cdot t\_0\right)\right) \cdot b\right)}^{2} + {\left(\sin \left(\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot a\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (cbrt (PI))))
   (+
    (pow
     (* (cos (* (* 0.005555555555555556 (pow t_0 2.0)) (* angle t_0))) b)
     2.0)
    (pow (* (sin (* (* 0.005555555555555556 (PI)) angle)) a) 2.0))))

Derivation

1. Initial program 76.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. clear-num N/A

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

   5. lower-/.f64 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 76.4

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

4. Applied rewrites 76.4%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lift-PI.f64 N/A

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

   5. add-cube-cbrt N/A

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

   6. lift-/.f64 N/A

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

   7. div-inv N/A

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

   8. unpow-1 N/A

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

   9. lift-pow.f64 N/A

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

   10. times-frac N/A

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

   11. lower-*.f64 N/A

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

   12. div-inv N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 N/A

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

   15. pow2 N/A

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

   16. lower-pow.f64 N/A

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

   17. lift-PI.f64 N/A

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

   18. lower-cbrt.f64 N/A

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

   19. div-inv N/A

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

   20. lift-pow.f64 N/A

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

   21. unpow-1 N/A

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

   22. remove-double-div N/A

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

6. Applied rewrites 76.5%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot 0.005555555555555556\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)}\right)}^{2} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 76.5

      \[\leadsto {\color{blue}{\left(\sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)}}^{2} + {\left(b \cdot \cos \left(\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot 0.005555555555555556\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)\right)}^{2} \]

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. associate-*l* N/A

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

   9. lift-PI.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. lower-*.f64 76.5

       \[\leadsto {\left(\sin \left(\color{blue}{\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right)} \cdot angle\right) \cdot a\right)}^{2} + {\left(b \cdot \cos \left(\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot 0.005555555555555556\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)\right)}^{2} \]

8. Applied rewrites 76.5%

   \[\leadsto \color{blue}{{\left(\sin \left(\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot a\right)}^{2}} + {\left(b \cdot \cos \left(\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot 0.005555555555555556\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)\right)}^{2} \]

9. Final simplification 76.5%

   \[\leadsto {\left(\cos \left(\left(0.005555555555555556 \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}\right) \cdot \left(angle \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot b\right)}^{2} + {\left(\sin \left(\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot a\right)}^{2} \]
10. Add Preprocessing

Alternative 4: 79.3% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 76.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

4. Applied rewrites 76.3%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. metadata-eval N/A

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

   4. distribute-neg-frac N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/l/ N/A

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

   7. lift-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. distribute-neg-frac N/A

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

   12. lower-/.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. distribute-neg-frac N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f64 76.5

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

6. Applied rewrites 76.5%

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

7. Final simplification 76.5%

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

Alternative 5: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ {\left(\cos \left(\left(t\_0 \cdot t\_0\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot b\right)}^{2} + {\left(\sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (sqrt (PI))))
   (+
    (pow (* (cos (* (* t_0 t_0) (* angle 0.005555555555555556))) b) 2.0)
    (pow (* (sin (* (* angle 0.005555555555555556) (PI))) a) 2.0))))

Derivation

1. Initial program 76.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

4. Applied rewrites 76.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. lift-/.f64 N/A

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

   8. un-div-inv N/A

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

   9. distribute-neg-frac2 N/A

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

   10. lift-PI.f64 N/A

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

   11. add-cube-cbrt N/A

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

   12. lift-/.f64 N/A

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

   13. div-inv N/A

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

   14. unpow-1 N/A

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

   15. lift-pow.f64 N/A

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

   16. times-frac N/A

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

   17. distribute-rgt-neg-in N/A

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

   18. lower-*.f64 N/A

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

6. Applied rewrites 76.5%

   \[\leadsto \mathsf{fma}\left({\cos \color{blue}{\left(\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot 0.005555555555555556\right) \cdot \left(-\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)}}^{2} \cdot b, b, {\left(\sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot a\right)}^{2}\right) \]

8. Applied rewrites 76.3%

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

   1. lift-PI.f64 N/A

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

   2. add-sqr-sqrt N/A

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

   3. lower-*.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lower-sqrt.f64 76.4

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

10. Applied rewrites 76.4%

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

11. Final simplification 76.4%

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

Alternative 6: 79.2% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 76.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 76.4

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

4. Applied rewrites 76.4%

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

5. Final simplification 76.4%

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

Alternative 7: 79.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{-180}{angle}} \cdot 2\right) \cdot 0.5 + 0.5\right) \cdot b, b, {\left(\sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)}^{2}\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (fma
  (* (+ (* (cos (* (/ (PI) (/ -180.0 angle)) 2.0)) 0.5) 0.5) b)
  b
  (pow (* (sin (* (* angle 0.005555555555555556) (PI))) a) 2.0)))

Derivation

1. Initial program 76.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

4. Applied rewrites 76.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. lift-/.f64 N/A

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

   8. un-div-inv N/A

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

   9. distribute-neg-frac2 N/A

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

   10. lift-PI.f64 N/A

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

   11. add-cube-cbrt N/A

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

   12. lift-/.f64 N/A

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

   13. div-inv N/A

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

   14. unpow-1 N/A

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

   15. lift-pow.f64 N/A

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

   16. times-frac N/A

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

   17. distribute-rgt-neg-in N/A

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

   18. lower-*.f64 N/A

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

6. Applied rewrites 76.5%

   \[\leadsto \mathsf{fma}\left({\cos \color{blue}{\left(\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot 0.005555555555555556\right) \cdot \left(-\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)}}^{2} \cdot b, b, {\left(\sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot a\right)}^{2}\right) \]
7. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. sqr-cos-a N/A

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

   6. lift-*.f64 N/A

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

   7. lift-neg.f64 N/A

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

   8. distribute-rgt-neg-out N/A

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

8. Applied rewrites 76.3%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \frac{\mathsf{PI}\left(\right)}{\frac{-180}{angle}}\right)\right)} \cdot b, b, {\left(\sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot a\right)}^{2}\right) \]

9. Final simplification 76.3%

   \[\leadsto \mathsf{fma}\left(\left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{-180}{angle}} \cdot 2\right) \cdot 0.5 + 0.5\right) \cdot b, b, {\left(\sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)}^{2}\right) \]
10. Add Preprocessing

Alternative 8: 79.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(angle \cdot 0.005555555555555556\right) \cdot \mathsf{PI}\left(\right)\\ \mathsf{fma}\left(\left(\cos \left(t\_0 \cdot 2\right) \cdot 0.5 + 0.5\right) \cdot b, b, {\left(\sin t\_0 \cdot a\right)}^{2}\right) \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* angle 0.005555555555555556) (PI))))
   (fma (* (+ (* (cos (* t_0 2.0)) 0.5) 0.5) b) b (pow (* (sin t_0) a) 2.0))))

Derivation

1. Initial program 76.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

4. Applied rewrites 76.3%

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

   1. lift-pow.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac2 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-*l/ N/A

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

   9. lift-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. cos-neg N/A

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

   12. lift-cos.f64 N/A

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

   13. unpow2 N/A

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

   14. lift-cos.f64 N/A

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

   15. lift-cos.f64 N/A

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

   16. sqr-cos-a N/A

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

   17. lower-+.f64 N/A

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

   18. lower-*.f64 N/A

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

6. Applied rewrites 76.3%

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

7. Final simplification 76.3%

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

Alternative 9: 79.2% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 76.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

4. Applied rewrites 76.3%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. sqr-cos-a N/A

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

   6. lower-+.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. cos-2 N/A

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

   9. cos-sum N/A

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

   10. lower-cos.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. div-inv N/A

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

   13. lift-/.f64 N/A

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

   14. div-inv N/A

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

   15. distribute-lft-out N/A

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

   16. lower-*.f64 N/A

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

   17. metadata-eval N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval 76.3

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

6. Applied rewrites 76.3%

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

7. Final simplification 76.3%

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

Alternative 10: 61.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.5 \cdot 10^{+168}:\\ \;\;\;\;{\cos \left(\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(a \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.5e+168)
   (* (pow (cos (* (* 0.005555555555555556 (PI)) angle)) 2.0) (* b b))
   (* (pow (* (* a (PI)) angle) 2.0) 3.08641975308642e-5)))

Derivation

1. Split input into 2 regimes

2. if a < 1.4999999999999999e168

   1. Initial program 73.6%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 61.5

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

   5. Applied rewrites 61.5%

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

   if 1.4999999999999999e168 < a

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. div-inv N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. div-inv N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. inv-pow N/A

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

      14. lower-pow.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 74.2%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 74.2%

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

   12. Applied rewrites 86.3%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.5 \cdot 10^{+168}:\\ \;\;\;\;{\cos \left(\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(a \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.5 \cdot 10^{+168}:\\ \;\;\;\;{\cos \left(-0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(a \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.5e+168)
   (* (pow (cos (* -0.005555555555555556 (* angle (PI)))) 2.0) (* b b))
   (* (pow (* (* a (PI)) angle) 2.0) 3.08641975308642e-5)))

Derivation

1. Split input into 2 regimes

2. if a < 1.4999999999999999e168

   1. Initial program 73.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 73.6%

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

   5. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 61.3

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

   7. Applied rewrites 61.3%

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

   if 1.4999999999999999e168 < a

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. div-inv N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. div-inv N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. inv-pow N/A

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

      14. lower-pow.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 74.2%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 74.2%

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

   12. Applied rewrites 86.3%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.5 \cdot 10^{+168}:\\ \;\;\;\;{\cos \left(-0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(a \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 79.2% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 76.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

4. Applied rewrites 76.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. lift-/.f64 N/A

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

   8. un-div-inv N/A

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

   9. distribute-neg-frac2 N/A

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

   10. lift-PI.f64 N/A

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

   11. add-cube-cbrt N/A

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

   12. lift-/.f64 N/A

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

   13. div-inv N/A

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

   14. unpow-1 N/A

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

   15. lift-pow.f64 N/A

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

   16. times-frac N/A

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

   17. distribute-rgt-neg-in N/A

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

   18. lower-*.f64 N/A

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

6. Applied rewrites 76.5%

   \[\leadsto \mathsf{fma}\left({\cos \color{blue}{\left(\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot 0.005555555555555556\right) \cdot \left(-\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)}}^{2} \cdot b, b, {\left(\sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot a\right)}^{2}\right) \]

8. Applied rewrites 76.3%

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

9. Taylor expanded in angle around 0

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

    1. unpow2 N/A

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

    2. lower-*.f64 76.0

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

11. Applied rewrites 76.0%

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

12. Final simplification 76.0%

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

Alternative 13: 61.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.5 \cdot 10^{+168}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(a \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.5e+168)
   (* b b)
   (* (pow (* (* a (PI)) angle) 2.0) 3.08641975308642e-5)))

Derivation

1. Split input into 2 regimes

2. if a < 1.4999999999999999e168

   1. Initial program 73.6%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.0

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

   5. Applied rewrites 61.0%

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

   if 1.4999999999999999e168 < a

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. div-inv N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. div-inv N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. inv-pow N/A

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

      14. lower-pow.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 74.2%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 74.2%

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

   12. Applied rewrites 86.3%

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

Alternative 14: 55.1% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9.2 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5} \cdot a, a, -3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right), angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 9.2e+39)
   (fma
    (*
     (* (* (PI) (PI)) angle)
     (fma (* 3.08641975308642e-5 a) a (* -3.08641975308642e-5 (* b b))))
    angle
    (* b b))
   (* b b)))

Derivation

1. Split input into 2 regimes

2. if b < 9.20000000000000047e39

   1. Initial program 74.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. div-inv N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. div-inv N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. inv-pow N/A

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

      14. lower-pow.f64 74.4

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

   4. Applied rewrites 74.4%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 47.8%

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

   9. Applied rewrites 47.9%

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

   if 9.20000000000000047e39 < b

   1. Initial program 82.7%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 75.9

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

   5. Applied rewrites 75.9%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.2 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5} \cdot a, a, -3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right), angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 59.8% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 4.4 \cdot 10^{+177}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \cdot \left(\left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 4.4e+177)
   (* b b)
   (* (* (* (PI) (PI)) a) (* (* (* angle angle) 3.08641975308642e-5) a))))

Derivation

1. Split input into 2 regimes

2. if a < 4.3999999999999996e177

   1. Initial program 73.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 60.7

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

   5. Applied rewrites 60.7%

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

   if 4.3999999999999996e177 < a

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. div-inv N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. div-inv N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. inv-pow N/A

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

      14. lower-pow.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 79.2%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 79.2%

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

   12. Applied rewrites 79.6%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.4 \cdot 10^{+177}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \cdot \left(\left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 59.3% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.08 \cdot 10^{+175}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.08e+175)
   (* b b)
   (* (* (* (* (PI) (PI)) a) a) (* (* angle angle) 3.08641975308642e-5))))

Derivation

1. Split input into 2 regimes

2. if a < 1.08e175

   1. Initial program 73.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 60.7

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

   5. Applied rewrites 60.7%

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

   if 1.08e175 < a

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. div-inv N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. div-inv N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. inv-pow N/A

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

      14. lower-pow.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 79.2%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 79.2%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.08 \cdot 10^{+175}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 56.3% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.3%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 57.2

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

5. Applied rewrites 57.2%

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

Reproduce

herbie shell --seed 2024264 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) (PI)))) 2.0) (pow (* b (cos (* (/ angle 180.0) (PI)))) 2.0)))