ab-angle->ABCF A

Percentage Accurate: 79.5% → 79.5%

Time: 17.2s

Alternatives: 12

Speedup: 2.5×

• 37.2% 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: 79.5% 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: 79.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := angle\_m \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot 0.005555555555555556\right)\right)\\ \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot \mathsf{fma}\left(0.5, \cos \left(\mathsf{PI}\left(\right) \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right), 0.5\right), b, t\_0 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \mathsf{fma}\left(\cos \left(angle\_m \cdot \left(\mathsf{PI}\left(\right) \cdot 0.011111111111111112\right)\right), -0.5, 0.5\right), a, \left(b \cdot b\right) \cdot \left(1 \cdot 1\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* angle_m (* (PI) (* a 0.005555555555555556)))))
   (if (<= (/ angle_m 180.0) 5e-5)
     (fma
      (* b (fma 0.5 (cos (* (PI) (* angle_m 0.011111111111111112))) 0.5))
      b
      (* t_0 t_0))
     (fma
      (* a (fma (cos (* angle_m (* (PI) 0.011111111111111112))) -0.5 0.5))
      a
      (* (* b b) (* 1.0 1.0))))))

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 5.00000000000000024e-5

   1. Initial program 87.3%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. lower-*.f64 82.3

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

   5. Applied rewrites 82.3%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. rem-square-sqrt N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. associate-*r* N/A

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

      11. sqrt-prod N/A

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

      12. pow1/2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. pow1/2 N/A

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

      17. lower-sqrt.f64 82.3

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

   7. Applied rewrites 82.3%

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

   9. Applied rewrites 82.3%

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

   if 5.00000000000000024e-5 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 58.2%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 58.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-*l/ N/A

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

      5. div-inv N/A

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

      6. times-frac N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 58.7

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

   7. Applied rewrites 58.7%

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

   9. Applied rewrites 58.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot \mathsf{fma}\left(\cos \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot 0.011111111111111112\right)\right), -0.5, 0.5\right), a, \left(b \cdot b\right) \cdot \left(1 \cdot 1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot \mathsf{fma}\left(0.5, \cos \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot 0.011111111111111112\right)\right), 0.5\right), b, \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot 0.005555555555555556\right)\right)\right) \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \mathsf{fma}\left(\cos \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot 0.011111111111111112\right)\right), -0.5, 0.5\right), a, \left(b \cdot b\right) \cdot \left(1 \cdot 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \sqrt[3]{\mathsf{PI}\left(\right)}\\ {\left(a \cdot \sin \left(\frac{angle\_m}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(t\_0 \cdot \left(t\_0 \cdot \left(t\_0 \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)\right)\right)}^{2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (cbrt (PI))))
   (+
    (pow (* a (sin (* (/ angle_m 180.0) (PI)))) 2.0)
    (pow
     (* b (cos (* t_0 (* t_0 (* t_0 (* angle_m 0.005555555555555556))))))
     2.0))))

Derivation

1. Initial program 78.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f64 N/A

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

   4. add-cube-cbrt N/A

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

   5. associate-*l* N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lower-cbrt.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lower-cbrt.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. div-inv N/A

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

   17. lower-*.f64 N/A

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

   18. metadata-eval N/A

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

   19. lift-PI.f64 N/A

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

   20. lower-cbrt.f64 78.7

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

4. Applied rewrites 78.7%

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

5. Final simplification 78.7%

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

Alternative 3: 79.5% accurate, 1.9× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* a (sin (* 0.005555555555555556 (/ (PI) (/ 1.0 angle_m))))) 2.0)
  (* b b)))

Derivation

1. Initial program 78.5%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 78.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. associate-*l/ N/A

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

   5. div-inv N/A

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

   6. times-frac N/A

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

   7. metadata-eval N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 78.7

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

7. Applied rewrites 78.7%

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

8. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 78.7

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

10. Applied rewrites 78.7%

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

Alternative 4: 79.5% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 78.5%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 78.6%

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

6. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 78.6

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

8. Applied rewrites 78.6%

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

Alternative 5: 72.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{-148}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 3 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(angle\_m \cdot \left(angle\_m \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \mathsf{fma}\left(\cos \left(angle\_m \cdot \left(\mathsf{PI}\left(\right) \cdot 0.011111111111111112\right)\right), -0.5, 0.5\right), a, \left(b \cdot b\right) \cdot \left(1 \cdot 1\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= (/ angle_m 180.0) 4e-148)
   (* b b)
   (if (<= (/ angle_m 180.0) 3e-25)
     (fma
      (* angle_m (* angle_m (* (PI) (PI))))
      (* 3.08641975308642e-5 (* a a))
      (* b b))
     (fma
      (* a (fma (cos (* angle_m (* (PI) 0.011111111111111112))) -0.5 0.5))
      a
      (* (* b b) (* 1.0 1.0))))))

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 3.99999999999999974e-148

   1. Initial program 84.5%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 67.3

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

   5. Applied rewrites 67.3%

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

   if 3.99999999999999974e-148 < (/.f64 angle #s(literal 180 binary64)) < 2.9999999999999998e-25

   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. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

   5. Applied rewrites 67.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 96.5%

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

   if 2.9999999999999998e-25 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 60.8%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 60.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-*l/ N/A

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

      5. div-inv N/A

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

      6. times-frac N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 60.7

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

   7. Applied rewrites 60.7%

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

   9. Applied rewrites 60.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot \mathsf{fma}\left(\cos \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot 0.011111111111111112\right)\right), -0.5, 0.5\right), a, \left(b \cdot b\right) \cdot \left(1 \cdot 1\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{-148}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;\frac{angle}{180} \leq 3 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \mathsf{fma}\left(\cos \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot 0.011111111111111112\right)\right), -0.5, 0.5\right), a, \left(b \cdot b\right) \cdot \left(1 \cdot 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{-148}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(angle\_m \cdot \left(angle\_m \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot 0.5, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(angle\_m \cdot \mathsf{PI}\left(\right)\right)\right), -0.5, 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= (/ angle_m 180.0) 4e-148)
   (* b b)
   (if (<= (/ angle_m 180.0) 5e-5)
     (fma
      (* angle_m (* angle_m (* (PI) (PI))))
      (* 3.08641975308642e-5 (* a a))
      (* b b))
     (fma
      (* b 0.5)
      b
      (*
       (* a a)
       (fma (cos (* 0.011111111111111112 (* angle_m (PI)))) -0.5 0.5))))))

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 3.99999999999999974e-148

   1. Initial program 84.5%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 67.3

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

   5. Applied rewrites 67.3%

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

   if 3.99999999999999974e-148 < (/.f64 angle #s(literal 180 binary64)) < 5.00000000000000024e-5

   1. Initial program 99.7%

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

   3. 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. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

   5. Applied rewrites 66.7%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 95.8%

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

   if 5.00000000000000024e-5 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 58.2%

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

   4. Applied rewrites 58.1%

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

   6. Applied rewrites 41.2%

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

   7. Taylor expanded in b around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot b, b, \color{blue}{{a}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot b, b, \left(a \cdot a\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \frac{-1}{2}, \frac{1}{2}\right)}\right) \]

      7. lower-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\color{blue}{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}, \frac{-1}{2}, \frac{1}{2}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{90}\right)}, \frac{-1}{2}, \frac{1}{2}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{90}\right)}, \frac{-1}{2}, \frac{1}{2}\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{90}\right), \frac{-1}{2}, \frac{1}{2}\right)\right) \]

      11. lower-PI.f64 58.7

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

   9. Applied rewrites 58.7%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{-148}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot 0.5, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), -0.5, 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle\_m \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{fma}\left(b, b \cdot -3.08641975308642 \cdot 10^{-5}, 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right), angle\_m, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(0.011111111111111112 \cdot \left(angle\_m \cdot \mathsf{PI}\left(\right)\right)\right), 0.5\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 2.1e+63)
   (fma
    (*
     (* angle_m (* (PI) (PI)))
     (fma b (* b -3.08641975308642e-5) (* 3.08641975308642e-5 (* a a))))
    angle_m
    (* b b))
   (* (* b b) (fma 0.5 (cos (* 0.011111111111111112 (* angle_m (PI)))) 0.5))))

Derivation

1. Split input into 2 regimes

2. if b < 2.1000000000000002e63

   1. Initial program 76.1%

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

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

   5. Applied rewrites 46.9%

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

   7. Applied rewrites 55.3%

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

   if 2.1000000000000002e63 < b

   1. Initial program 91.7%

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

   4. Applied rewrites 79.2%

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

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 91.9

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

   7. Applied rewrites 91.9%

      \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot 0.011111111111111112\right), 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{fma}\left(b, b \cdot -3.08641975308642 \cdot 10^{-5}, 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right), angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(0.011111111111111112 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.4% accurate, 4.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := angle\_m \cdot \left(angle\_m \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\\ \mathbf{if}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{-148}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 7.5 \cdot 10^{+161}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right), b \cdot b\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+286}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \mathsf{fma}\left(a \cdot 3.08641975308642 \cdot 10^{-5}, a, b \cdot \left(b \cdot -3.08641975308642 \cdot 10^{-5}\right)\right), b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* angle_m (* angle_m (* (PI) (PI))))))
   (if (<= (/ angle_m 180.0) 4e-148)
     (* b b)
     (if (<= (/ angle_m 180.0) 7.5e+161)
       (fma t_0 (* 3.08641975308642e-5 (* a a)) (* b b))
       (if (<= (/ angle_m 180.0) 2e+286)
         (* b b)
         (fma
          t_0
          (fma (* a 3.08641975308642e-5) a (* b (* b -3.08641975308642e-5)))
          (* b b)))))))

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 3.99999999999999974e-148 or 7.4999999999999995e161 < (/.f64 angle #s(literal 180 binary64)) < 2.00000000000000007e286

   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

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

      1. unpow2 N/A

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

      2. lower-*.f64 64.0

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

   5. Applied rewrites 64.0%

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

   if 3.99999999999999974e-148 < (/.f64 angle #s(literal 180 binary64)) < 7.4999999999999995e161

   1. Initial program 79.8%

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

   3. Taylor expanded in angle around 0

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

   5. Applied rewrites 43.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 75.6%

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

   if 2.00000000000000007e286 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 65.7%

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

   3. Taylor expanded in angle around 0

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

   5. Applied rewrites 57.6%

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

   7. Applied rewrites 57.6%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{-148}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;\frac{angle}{180} \leq 7.5 \cdot 10^{+161}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right), b \cdot b\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+286}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{fma}\left(a \cdot 3.08641975308642 \cdot 10^{-5}, a, b \cdot \left(b \cdot -3.08641975308642 \cdot 10^{-5}\right)\right), b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.0% accurate, 5.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(angle\_m \cdot \left(angle\_m \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right), b \cdot b\right)\\ \mathbf{if}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{-148}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 7.5 \cdot 10^{+161}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+286}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0
         (fma
          (* angle_m (* angle_m (* (PI) (PI))))
          (* 3.08641975308642e-5 (* a a))
          (* b b))))
   (if (<= (/ angle_m 180.0) 4e-148)
     (* b b)
     (if (<= (/ angle_m 180.0) 7.5e+161)
       t_0
       (if (<= (/ angle_m 180.0) 2e+286) (* b b) t_0)))))

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 3.99999999999999974e-148 or 7.4999999999999995e161 < (/.f64 angle #s(literal 180 binary64)) < 2.00000000000000007e286

   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

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

      1. unpow2 N/A

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

      2. lower-*.f64 64.0

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

   5. Applied rewrites 64.0%

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

   if 3.99999999999999974e-148 < (/.f64 angle #s(literal 180 binary64)) < 7.4999999999999995e161 or 2.00000000000000007e286 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 78.5%

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

   3. 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. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

   5. Applied rewrites 45.2%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 73.9%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{-148}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;\frac{angle}{180} \leq 7.5 \cdot 10^{+161}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right), b \cdot b\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+286}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right), b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.9% accurate, 8.3× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle\_m \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{fma}\left(b, b \cdot -3.08641975308642 \cdot 10^{-5}, 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right), angle\_m, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 2.1e+63)
   (fma
    (*
     (* angle_m (* (PI) (PI)))
     (fma b (* b -3.08641975308642e-5) (* 3.08641975308642e-5 (* a a))))
    angle_m
    (* b b))
   (* b b)))

Derivation

1. Split input into 2 regimes

2. if b < 2.1000000000000002e63

   1. Initial program 76.1%

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

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

   5. Applied rewrites 46.9%

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

   7. Applied rewrites 55.3%

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

   if 2.1000000000000002e63 < b

   1. Initial program 91.7%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 92.0

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

   5. Applied rewrites 92.0%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{fma}\left(b, b \cdot -3.08641975308642 \cdot 10^{-5}, 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right), angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.1% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := angle\_m \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;a \leq 9 \cdot 10^{+147}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;3.08641975308642 \cdot 10^{-5} \cdot \left(\left(a \cdot a\right) \cdot \left(t\_0 \cdot t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* angle_m (PI))))
   (if (<= a 9e+147) (* b b) (* 3.08641975308642e-5 (* (* a a) (* t_0 t_0))))))

Derivation

1. Split input into 2 regimes

2. if a < 9.00000000000000016e147

   1. Initial program 76.8%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 63.5

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

   5. Applied rewrites 63.5%

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

   if 9.00000000000000016e147 < a

   1. Initial program 91.3%

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

   3. Taylor expanded in angle around 0

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

   5. Applied rewrites 50.8%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 70.9%

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

Alternative 12: 56.7% 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 = abs(angle)
real(8) function code(a, b, angle_m)
    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 78.5%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 60.3

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

5. Applied rewrites 60.3%

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

Reproduce

herbie shell --seed 2024221 
(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)))