ab-angle->ABCF A

Percentage Accurate: 79.7% → 79.7%

Time: 5.3s

Alternatives: 12

Speedup: 1.3×

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

Specification

Math

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

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: 79.7% 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: 79.7% accurate, 1.2× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (fma
  (+ 0.5 (* 0.5 (sin (+ (* (* (PI) (/ angle_m 180.0)) (- 2.0)) (/ (PI) 2.0)))))
  (* b b)
  (pow (* (sin (* (* (PI) angle_m) 0.005555555555555556)) a) 2.0)))

Derivation

1. Initial program 82.8%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-/.f64 N/A

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

4. Applied rewrites 82.8%

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

   1. lift-pow.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. unpow2 N/A

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

   7. sqr-cos-a N/A

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

   8. lower-+.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-PI.f64 82.8

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

6. Applied rewrites 82.8%

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

7. Taylor expanded in a around 0

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

   1. pow-prod-down N/A

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

   2. lower-pow.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 82.8

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

9. Applied rewrites 82.8%

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

    1. lift-cos.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-PI.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. lift-/.f64 N/A

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

    6. cos-neg-rev N/A

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

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

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

    8. lower-sin.f64 N/A

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

    9. lower-+.f64 N/A

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

    10. lower-neg.f64 N/A

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

    11. *-commutative N/A

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

    12. lower-*.f64 N/A

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

    13. lift-/.f64 N/A

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

    14. lift-*.f64 N/A

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

    15. lift-PI.f64 N/A

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

    16. lower-/.f64 N/A

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

    17. lift-PI.f64 82.8

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

11. Applied rewrites 82.8%

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

12. Final simplification 82.8%

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

Alternative 2: 79.7% accurate, 1.3× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (fma
  (+ 0.5 (* 0.5 (cos (* 2.0 (* (PI) (/ angle_m 180.0))))))
  (* b b)
  (pow (* (sin (* (* (PI) angle_m) 0.005555555555555556)) a) 2.0)))

Derivation

1. Initial program 82.8%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-/.f64 N/A

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

4. Applied rewrites 82.8%

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

   1. lift-pow.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. unpow2 N/A

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

   7. sqr-cos-a N/A

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

   8. lower-+.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-PI.f64 82.8

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

6. Applied rewrites 82.8%

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

7. Taylor expanded in a around 0

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

   1. pow-prod-down N/A

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

   2. lower-pow.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 82.8

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

9. Applied rewrites 82.8%

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

Alternative 3: 79.7% accurate, 1.3× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (fma
  (+ 0.5 (* 0.5 (cos (* 2.0 (* (PI) (* 0.005555555555555556 angle_m))))))
  (* b b)
  (pow (* (sin (* (* (PI) angle_m) 0.005555555555555556)) a) 2.0)))

Derivation

1. Initial program 82.8%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-/.f64 N/A

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

4. Applied rewrites 82.8%

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

   1. lift-pow.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. unpow2 N/A

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

   7. sqr-cos-a N/A

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

   8. lower-+.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-PI.f64 82.8

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

6. Applied rewrites 82.8%

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

7. Taylor expanded in a around 0

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

   1. pow-prod-down N/A

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

   2. lower-pow.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 82.8

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

9. Applied rewrites 82.8%

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

10. Taylor expanded in angle around 0

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

    1. lower-*.f64 82.8

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

12. Applied rewrites 82.8%

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

Alternative 4: 79.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right) \cdot a\\ \mathsf{fma}\left(t\_0, t\_0, b \cdot b\right) \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (sin (* (PI) (* 0.005555555555555556 angle_m))) a)))
   (fma t_0 t_0 (* b b))))

Derivation

1. Initial program 82.8%

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

3. Taylor expanded in angle around 0

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

   1. lower-*.f64 82.8

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

5. Applied rewrites 82.8%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lower-fma.f64 N/A

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

7. Applied rewrites 82.8%

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

8. Taylor expanded in angle around 0

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

   1. pow2 N/A

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

   2. lift-*.f64 82.7

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

10. Applied rewrites 82.7%

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

11. Final simplification 82.7%

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

Alternative 5: 79.6% accurate, 2.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+ (pow (* a (sin (* (* 0.005555555555555556 angle_m) (PI)))) 2.0) (* b b)))

Derivation

1. Initial program 82.8%

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

3. Taylor expanded in angle around 0

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

   1. lower-*.f64 82.8

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

5. Applied rewrites 82.8%

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

6. Taylor expanded in angle around 0

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

   1. *-commutative N/A

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

   2. lift-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. *-commutative N/A

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

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

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

   7. pow2 N/A

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

   8. lift-*.f64 82.7

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

8. Applied rewrites 82.7%

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

Alternative 6: 57.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot angle\_m\\ \mathbf{if}\;b \leq 5.8 \cdot 10^{-156}:\\ \;\;\;\;{\left(\sin \left(t\_0 \cdot 0.005555555555555556\right) \cdot a\right)}^{2}\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{+143}:\\ \;\;\;\;\left(\frac{{\left(t\_0 \cdot a\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}}{b \cdot b} + 1\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot t\_0\right), 0.5, 0.5\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (PI) angle_m)))
   (if (<= b 5.8e-156)
     (pow (* (sin (* t_0 0.005555555555555556)) a) 2.0)
     (if (<= b 2.3e+143)
       (*
        (+ (/ (* (pow (* t_0 a) 2.0) 3.08641975308642e-5) (* b b)) 1.0)
        (* b b))
       (* (fma (cos (* 0.011111111111111112 t_0)) 0.5 0.5) (* b b))))))

Derivation

1. Split input into 3 regimes

2. if b < 5.80000000000000041e-156

   1. Initial program 80.8%

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

   3. Taylor expanded in a around inf

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

      1. pow-prod-down N/A

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

      2. lower-pow.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 48.2

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

   5. Applied rewrites 48.2%

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

   if 5.80000000000000041e-156 < b < 2.3e143

   1. Initial program 78.6%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 58.7%

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

   6. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 59.8%

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

   9. Taylor expanded in a around inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. *-commutative N/A

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

       4. *-commutative N/A

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

       5. unpow-prod-down N/A

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

       6. lift-*.f64 N/A

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

       7. lift-PI.f64 N/A

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

       8. unpow-prod-down N/A

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

       9. *-commutative N/A

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

       10. lift-*.f64 N/A

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

       11. lift-pow.f64 N/A

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

       12. lower-*.f64 N/A

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

       13. pow2 N/A

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

       14. lift-*.f64 73.2

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

   11. Applied rewrites 73.2%

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

   if 2.3e143 < b

   1. Initial program 97.9%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
   2. Add Preprocessing
   3. 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. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

   4. Applied rewrites 97.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-cos-a N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-PI.f64 97.9

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

   6. Applied rewrites 97.9%

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

   7. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

   9. Applied rewrites 97.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right), 0.5, 0.5\right) \cdot \left(b \cdot b\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.8 \cdot 10^{-156}:\\ \;\;\;\;{\left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot a\right)}^{2}\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{+143}:\\ \;\;\;\;\left(\frac{{\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}}{b \cdot b} + 1\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right), 0.5, 0.5\right) \cdot \left(b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot angle\_m\\ t_1 := t\_0 \cdot a\\ \mathbf{if}\;b \leq 8.5 \cdot 10^{-161}:\\ \;\;\;\;\left(t\_0 \cdot \left(a \cdot t\_1\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{+143}:\\ \;\;\;\;\left(\frac{{t\_1}^{2} \cdot 3.08641975308642 \cdot 10^{-5}}{b \cdot b} + 1\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot t\_0\right), 0.5, 0.5\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (PI) angle_m)) (t_1 (* t_0 a)))
   (if (<= b 8.5e-161)
     (* (* t_0 (* a t_1)) 3.08641975308642e-5)
     (if (<= b 2.3e+143)
       (* (+ (/ (* (pow t_1 2.0) 3.08641975308642e-5) (* b b)) 1.0) (* b b))
       (* (fma (cos (* 0.011111111111111112 t_0)) 0.5 0.5) (* b b))))))

Derivation

1. Split input into 3 regimes

2. if b < 8.50000000000000054e-161

   1. Initial program 81.1%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 41.7%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 44.6

          \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5} \]

   8. Applied rewrites 44.6%

      \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)}^{2} \cdot \color{blue}{3.08641975308642 \cdot 10^{-5}} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

         \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} \]

      3. lower-*.f64 44.6

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

   10. Applied rewrites 44.6%

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

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} \]

       3. associate-*l* N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(a \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\right)\right) \cdot \frac{1}{32400} \]

       4. lower-*.f64 N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(a \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\right)\right) \cdot \frac{1}{32400} \]

       5. lower-*.f64 46.3

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

   12. Applied rewrites 46.3%

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

   if 8.50000000000000054e-161 < b < 2.3e143

   1. Initial program 78.0%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 58.2%

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

   6. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 59.2%

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

   9. Taylor expanded in a around inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. *-commutative N/A

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

       4. *-commutative N/A

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

       5. unpow-prod-down N/A

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

       6. lift-*.f64 N/A

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

       7. lift-PI.f64 N/A

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

       8. unpow-prod-down N/A

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

       9. *-commutative N/A

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

       10. lift-*.f64 N/A

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

       11. lift-pow.f64 N/A

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

       12. lower-*.f64 N/A

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

       13. pow2 N/A

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

       14. lift-*.f64 71.7

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

   11. Applied rewrites 71.7%

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

   if 2.3e143 < b

   1. Initial program 97.9%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
   2. Add Preprocessing
   3. 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. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

   4. Applied rewrites 97.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-cos-a N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-PI.f64 97.9

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

   6. Applied rewrites 97.9%

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

   7. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

   9. Applied rewrites 97.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right), 0.5, 0.5\right) \cdot \left(b \cdot b\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.5 \cdot 10^{-161}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(a \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{+143}:\\ \;\;\;\;\left(\frac{{\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}}{b \cdot b} + 1\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right), 0.5, 0.5\right) \cdot \left(b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot angle\_m\\ \mathbf{if}\;a \leq 2.1 \cdot 10^{-76}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot t\_0\right), 0.5, 0.5\right) \cdot \left(b \cdot b\right)\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot a\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}, angle\_m \cdot angle\_m, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot \left(a \cdot \left(t\_0 \cdot a\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (PI) angle_m)))
   (if (<= a 2.1e-76)
     (* (fma (cos (* 0.011111111111111112 t_0)) 0.5 0.5) (* b b))
     (if (<= a 2.2e+148)
       (fma
        (* (pow (* (PI) a) 2.0) 3.08641975308642e-5)
        (* angle_m angle_m)
        (* b b))
       (* (* t_0 (* a (* t_0 a))) 3.08641975308642e-5)))))

Derivation

1. Split input into 3 regimes

2. if a < 2.09999999999999992e-76

   1. Initial program 81.1%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
   2. Add Preprocessing
   3. 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. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

   4. Applied rewrites 81.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-cos-a N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-PI.f64 81.1

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

   6. Applied rewrites 81.1%

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

   7. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

   9. Applied rewrites 66.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right), 0.5, 0.5\right) \cdot \left(b \cdot b\right)} \]

   if 2.09999999999999992e-76 < a < 2.1999999999999999e148

   1. Initial program 74.4%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 37.2%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow-prod-down N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-pow.f64 67.0

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

   8. Applied rewrites 67.0%

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

   if 2.1999999999999999e148 < a

   1. Initial program 99.6%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 40.2%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 81.9

          \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5} \]

   8. Applied rewrites 81.9%

      \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)}^{2} \cdot \color{blue}{3.08641975308642 \cdot 10^{-5}} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

         \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} \]

      3. lower-*.f64 81.9

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

   10. Applied rewrites 81.9%

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

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} \]

       3. associate-*l* N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(a \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\right)\right) \cdot \frac{1}{32400} \]

       4. lower-*.f64 N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(a \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\right)\right) \cdot \frac{1}{32400} \]

       5. lower-*.f64 84.8

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

   12. Applied rewrites 84.8%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.1 \cdot 10^{-76}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right), 0.5, 0.5\right) \cdot \left(b \cdot b\right)\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot a\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}, angle \cdot angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(a \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot angle\_m\\ \mathbf{if}\;a \leq 3.6 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot t\_0\right), 0.5, 0.5\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot \left(a \cdot \left(t\_0 \cdot a\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (PI) angle_m)))
   (if (<= a 3.6e+140)
     (* (fma (cos (* 0.011111111111111112 t_0)) 0.5 0.5) (* b b))
     (* (* t_0 (* a (* t_0 a))) 3.08641975308642e-5))))

Derivation

1. Split input into 2 regimes

2. if a < 3.6e140

   1. Initial program 80.0%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
   2. Add Preprocessing
   3. 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. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

   4. Applied rewrites 80.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-cos-a N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-PI.f64 80.0

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

   6. Applied rewrites 80.0%

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

   7. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

   9. Applied rewrites 65.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right), 0.5, 0.5\right) \cdot \left(b \cdot b\right)} \]

   if 3.6e140 < a

   1. Initial program 97.6%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 40.9%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 80.5

          \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5} \]

   8. Applied rewrites 80.5%

      \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)}^{2} \cdot \color{blue}{3.08641975308642 \cdot 10^{-5}} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

         \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} \]

      3. lower-*.f64 80.5

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

   10. Applied rewrites 80.5%

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

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} \]

       3. associate-*l* N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(a \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\right)\right) \cdot \frac{1}{32400} \]

       4. lower-*.f64 N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(a \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\right)\right) \cdot \frac{1}{32400} \]

       5. lower-*.f64 83.2

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

   12. Applied rewrites 83.2%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.6 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right), 0.5, 0.5\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(a \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.5% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot angle\_m\\ \mathbf{if}\;a \leq 3.6 \cdot 10^{+140}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot \left(a \cdot \left(t\_0 \cdot a\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (PI) angle_m)))
   (if (<= a 3.6e+140)
     (* b b)
     (* (* t_0 (* a (* t_0 a))) 3.08641975308642e-5))))

Derivation

1. Split input into 2 regimes

2. if a < 3.6e140

   1. Initial program 80.0%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 65.1

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

   5. Applied rewrites 65.1%

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

   if 3.6e140 < a

   1. Initial program 97.6%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 40.9%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 80.5

          \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5} \]

   8. Applied rewrites 80.5%

      \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)}^{2} \cdot \color{blue}{3.08641975308642 \cdot 10^{-5}} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

         \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} \]

      3. lower-*.f64 80.5

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

   10. Applied rewrites 80.5%

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

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} \]

       3. associate-*l* N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(a \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\right)\right) \cdot \frac{1}{32400} \]

       4. lower-*.f64 N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(a \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\right)\right) \cdot \frac{1}{32400} \]

       5. lower-*.f64 83.2

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

   12. Applied rewrites 83.2%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.6 \cdot 10^{+140}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(a \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.0% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \left(\mathsf{PI}\left(\right) \cdot angle\_m\right) \cdot a\\ \mathbf{if}\;a \leq 3.6 \cdot 10^{+140}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(t\_0 \cdot 3.08641975308642 \cdot 10^{-5}\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (* (PI) angle_m) a)))
   (if (<= a 3.6e+140) (* b b) (* t_0 (* t_0 3.08641975308642e-5)))))

Derivation

1. Split input into 2 regimes

2. if a < 3.6e140

   1. Initial program 80.0%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 65.1

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

   5. Applied rewrites 65.1%

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

   if 3.6e140 < a

   1. Initial program 97.6%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 40.9%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 80.5

          \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5} \]

   8. Applied rewrites 80.5%

      \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)}^{2} \cdot \color{blue}{3.08641975308642 \cdot 10^{-5}} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

         \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} \]

      3. lower-*.f64 80.5

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

   10. Applied rewrites 80.5%

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

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} \]

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 80.6

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

   12. Applied rewrites 80.6%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.6 \cdot 10^{+140}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right) \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.6% accurate, 74.7× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ b \cdot b \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m) :precision binary64 (* b b))

C

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

Fortran

angle_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return b * b;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return b * b

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(b * b), $MachinePrecision]

Derivation

1. Initial program 82.8%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 61.2

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

5. Applied rewrites 61.2%

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

6. Final simplification 61.2%

   \[\leadsto b \cdot b \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025037 
(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)))