ab-angle->ABCF A

Percentage Accurate: 79.5% → 79.5%

Time: 26.2s

Alternatives: 12

Speedup: 1.9×

• 35.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

Initial Program: 79.5% 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, 0.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ t_1 := {\left(\sqrt[3]{t\_0}\right)}^{3}\\ t_2 := \frac{t\_0}{2}\\ {\left(a \cdot \sin \left(\frac{angle\_m}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \left(\sin \left(\frac{angle\_m}{-180} \cdot \mathsf{PI}\left(\right)\right) \cdot \cos \left(\sqrt{t\_1 \cdot t\_1} \cdot t\_2\right) + \cos \left(\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\_m\right) \cdot \sin \left(t\_0 \cdot t\_2\right)\right)\right)}^{2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (sqrt (PI))) (t_1 (pow (cbrt t_0) 3.0)) (t_2 (/ t_0 2.0)))
   (+
    (pow (* a (sin (* (/ angle_m 180.0) (PI)))) 2.0)
    (pow
     (*
      b
      (+
       (* (sin (* (/ angle_m -180.0) (PI))) (cos (* (sqrt (* t_1 t_1)) t_2)))
       (* (cos (* (* 0.005555555555555556 (PI)) angle_m)) (sin (* t_0 t_2)))))
     2.0))))

Derivation

1. Initial program 81.6%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. add-sqr-sqrt N/A

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

   5. associate-/l* N/A

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

   6. fp-cancel-sign-sub-inv N/A

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

   7. sin-diff N/A

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

   8. cos-neg-rev N/A

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

   9. lower--.f64 N/A

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

4. Applied rewrites 81.6%

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

5. Taylor expanded in angle around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-cos.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-PI.f64 81.7

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

7. Applied rewrites 81.7%

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

   1. rem-cube-cbrt N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. rem-square-sqrt N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. cbrt-prod N/A

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

   8. unpow-prod-down N/A

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

   9. lower-*.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. lower-cbrt.f64 N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-cbrt.f64 81.7

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

9. Applied rewrites 81.7%

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

10. Final simplification 81.7%

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

Alternative 2: 79.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \sqrt[3]{\mathsf{PI}\left(\right)}\\ t_1 := \sqrt{\mathsf{PI}\left(\right)}\\ t_2 := \frac{t\_1}{2}\\ {\left(a \cdot \sin \left(\frac{angle\_m}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \left(\sin \left(\frac{angle\_m}{-180} \cdot \mathsf{PI}\left(\right)\right) \cdot \cos \left(\sqrt{{t\_0}^{2} \cdot t\_0} \cdot t\_2\right) + \cos \left(\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\_m\right) \cdot \sin \left(t\_1 \cdot t\_2\right)\right)\right)}^{2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (cbrt (PI))) (t_1 (sqrt (PI))) (t_2 (/ t_1 2.0)))
   (+
    (pow (* a (sin (* (/ angle_m 180.0) (PI)))) 2.0)
    (pow
     (*
      b
      (+
       (*
        (sin (* (/ angle_m -180.0) (PI)))
        (cos (* (sqrt (* (pow t_0 2.0) t_0)) t_2)))
       (* (cos (* (* 0.005555555555555556 (PI)) angle_m)) (sin (* t_1 t_2)))))
     2.0))))

Derivation

1. Initial program 81.6%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. add-sqr-sqrt N/A

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

   5. associate-/l* N/A

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

   6. fp-cancel-sign-sub-inv N/A

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

   7. sin-diff N/A

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

   8. cos-neg-rev N/A

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

   9. lower--.f64 N/A

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

4. Applied rewrites 81.6%

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

5. Taylor expanded in angle around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-cos.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-PI.f64 81.7

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

7. Applied rewrites 81.7%

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

   1. lift-PI.f64 N/A

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

   2. add-cube-cbrt N/A

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

   3. lower-*.f64 N/A

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

   4. pow2 N/A

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

   5. lower-pow.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lower-cbrt.f64 81.7

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

9. Applied rewrites 81.7%

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

10. Final simplification 81.7%

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

Alternative 3: 79.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ {\left(a \cdot \sin \left(\frac{angle\_m}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \left(\sin \left(\frac{angle\_m}{-180} \cdot \mathsf{PI}\left(\right)\right) \cdot \cos \left(t\_0 \cdot \frac{t\_0}{2}\right) - \cos \left(\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\_m\right) \cdot -1\right)\right)}^{2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (sqrt (PI))))
   (+
    (pow (* a (sin (* (/ angle_m 180.0) (PI)))) 2.0)
    (pow
     (*
      b
      (-
       (* (sin (* (/ angle_m -180.0) (PI))) (cos (* t_0 (/ t_0 2.0))))
       (* (cos (* (* 0.005555555555555556 (PI)) angle_m)) -1.0)))
     2.0))))

Derivation

1. Initial program 81.6%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. add-sqr-sqrt N/A

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

   5. associate-/l* N/A

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

   6. fp-cancel-sign-sub-inv N/A

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

   7. sin-diff N/A

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

   8. cos-neg-rev N/A

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

   9. lower--.f64 N/A

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

4. Applied rewrites 81.6%

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

5. Taylor expanded in angle around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-cos.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-PI.f64 81.7

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

7. Applied rewrites 81.7%

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

   1. lift-sin.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-neg.f64 N/A

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

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

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

   5. sin-neg N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*r/ N/A

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

   8. lift-sqrt.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. rem-square-sqrt N/A

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

   11. lift-PI.f64 N/A

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

   12. sin-PI/2 N/A

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

   13. metadata-eval 81.7

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

9. Applied rewrites 81.7%

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

10. Final simplification 81.7%

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

Alternative 4: 79.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 81.6%

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

3. Taylor expanded in angle around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-cos.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-PI.f64 81.7

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

5. Applied rewrites 81.7%

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

6. Final simplification 81.7%

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

Alternative 5: 79.5% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 81.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

4. Applied rewrites 81.6%

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

6. Applied rewrites 81.6%

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

Alternative 6: 79.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \mathsf{fma}\left(1 \cdot b, b, {\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle\_m}{180}\right) \cdot a\right)}^{2}\right) \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (fma (* 1.0 b) b (pow (* (sin (* (PI) (/ angle_m 180.0))) a) 2.0)))

Derivation

1. Initial program 81.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

4. Applied rewrites 81.6%

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

5. Taylor expanded in angle around 0

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

7. Applied rewrites 81.1%

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

Alternative 7: 57.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.16 \cdot 10^{+89}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(-angle\_m\right), -angle\_m, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;{\cos \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\_m\right)}^{2} \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 1.16e+89)
   (fma
    (*
     (* (* (- b a) (+ b a)) (* (* (PI) (PI)) -3.08641975308642e-5))
     (- angle_m))
    (- angle_m)
    (* b b))
   (* (pow (cos (* (* (PI) 0.005555555555555556) angle_m)) 2.0) (* b b))))

Derivation

1. Split input into 2 regimes

2. if b < 1.16e89

   1. Initial program 77.8%

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

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 44.6%

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

   7. Applied rewrites 50.6%

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

   if 1.16e89 < b

   1. Initial program 96.7%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 93.2

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

   5. Applied rewrites 93.2%

      \[\leadsto \color{blue}{{\cos \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{2} \cdot \left(b \cdot b\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 57.7% accurate, 8.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.12 \cdot 10^{+89}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(-angle\_m\right), -angle\_m, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 1.12e+89)
   (fma
    (*
     (* (* (- b a) (+ b a)) (* (* (PI) (PI)) -3.08641975308642e-5))
     (- angle_m))
    (- angle_m)
    (* b b))
   (* b b)))

Derivation

1. Split input into 2 regimes

2. if b < 1.11999999999999995e89

   1. Initial program 77.8%

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

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 44.6%

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

   7. Applied rewrites 50.6%

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

   if 1.11999999999999995e89 < b

   1. Initial program 96.7%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
   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 91.8

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

   5. Applied rewrites 91.8%

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

Alternative 9: 63.0% accurate, 9.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;a \leq 140:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;a \leq 3.1 \cdot 10^{+137}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot t\_0\right) \cdot a\right) \cdot a, angle\_m \cdot angle\_m, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot \left(\left(angle\_m \cdot angle\_m\right) \cdot a\right)\right)\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (PI) (PI))))
   (if (<= a 140.0)
     (* b b)
     (if (<= a 3.1e+137)
       (fma
        (* (* (* 3.08641975308642e-5 t_0) a) a)
        (* angle_m angle_m)
        (* b b))
       (* (* 3.08641975308642e-5 (* a (* (* angle_m angle_m) a))) t_0)))))

Derivation

1. Split input into 3 regimes

2. if a < 140

   1. Initial program 81.2%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 60.6

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

   5. Applied rewrites 60.6%

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

   if 140 < a < 3.0999999999999999e137

   1. Initial program 71.6%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 32.1%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 54.9%

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

   if 3.0999999999999999e137 < a

   1. Initial program 91.2%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 39.2%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 66.2%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 66.8%

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

Alternative 10: 57.6% accurate, 9.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.12 \cdot 10^{+89}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -3.08641975308642 \cdot 10^{-5}\right) \cdot angle\_m\right), angle\_m, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 1.12e+89)
   (fma
    (* (* (- b a) (+ b a)) (* (* (* (PI) (PI)) -3.08641975308642e-5) angle_m))
    angle_m
    (* b b))
   (* b b)))

Derivation

1. Split input into 2 regimes

2. if b < 1.11999999999999995e89

   1. Initial program 77.8%

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

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 44.6%

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

   7. Applied rewrites 50.1%

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

   if 1.11999999999999995e89 < b

   1. Initial program 96.7%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
   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 91.8

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

   5. Applied rewrites 91.8%

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

Alternative 11: 61.5% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 1.05 \cdot 10^{+117}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot \left(\left(angle\_m \cdot angle\_m\right) \cdot a\right)\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 1.05e+117)
   (* b b)
   (* (* 3.08641975308642e-5 (* a (* (* angle_m angle_m) a))) (* (PI) (PI)))))

Derivation

1. Split input into 2 regimes

2. if a < 1.0500000000000001e117

   1. Initial program 80.1%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 58.7

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

   5. Applied rewrites 58.7%

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

   if 1.0500000000000001e117 < a

   1. Initial program 88.7%

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

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 39.6%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 57.5%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 58.0%

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

Alternative 12: 57.8% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.6%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
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 54.9

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

5. Applied rewrites 54.9%

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

Reproduce

herbie shell --seed 2024359 
(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)))