ab-angle->ABCF A

Percentage Accurate: 80.3% → 80.1%

Time: 16.1s

Alternatives: 14

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

Initial Program: 80.3% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 80.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := angle\_m \cdot \mathsf{PI}\left(\right)\\ t_1 := t\_0 \cdot -0.005555555555555556\\ {\left(a \cdot \sin \left(\frac{angle\_m}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \frac{\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(0.005555555555555556, t\_0, \mathsf{PI}\left(\right) \cdot -0.5\right)\right) + \cos \left(0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\_m\right)\right), 2, 2 \cdot \left(\sin \left(0.5 \cdot \mathsf{PI}\left(\right) - t\_1\right) + \sin \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), t\_1\right)\right)\right)\right)}{4}\right)}^{2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* angle_m (PI))) (t_1 (* t_0 -0.005555555555555556)))
   (+
    (pow (* a (sin (* (/ angle_m 180.0) (PI)))) 2.0)
    (pow
     (*
      b
      (/
       (fma
        (+
         (sin (fma 0.005555555555555556 t_0 (* (PI) -0.5)))
         (cos (* 0.005555555555555556 (* (PI) angle_m))))
        2.0
        (* 2.0 (+ (sin (- (* 0.5 (PI)) t_1)) (sin (fma 0.5 (PI) t_1)))))
       4.0))
     2.0))))

Derivation

1. Initial program 82.8%

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

   1. lift-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. cos-neg-rev N/A

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

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

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

   5. remove-double-neg N/A

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

   6. add-cube-cbrt N/A

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

   7. associate-/l* N/A

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

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

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

   9. sin-diff N/A

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

   10. lower--.f64 N/A

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

4. Applied rewrites 82.8%

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

6. Applied rewrites 82.8%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{\frac{\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right) \cdot -0.5\right)\right) + \cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right), 2, 2 \cdot \left(\sin \left(0.5 \cdot \mathsf{PI}\left(\right) - \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot -0.005555555555555556\right) + \sin \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot -0.005555555555555556\right)\right)\right)\right)}{4}}\right)}^{2} \]

7. Taylor expanded in angle around inf

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

   1. cos-neg-rev N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \frac{\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right) \cdot \frac{-1}{2}\right)\right) + \color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}, 2, 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right)\right)\right)\right)}{4}\right)}^{2} \]

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

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \frac{\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right) \cdot \frac{-1}{2}\right)\right) + \color{blue}{\sin \left(\left(\mathsf{neg}\left(\frac{-1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}, 2, 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right)\right)\right)\right)}{4}\right)}^{2} \]

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

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \frac{\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right) \cdot \frac{-1}{2}\right)\right) + \sin \left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{180}\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right), 2, 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right)\right)\right)\right)}{4}\right)}^{2} \]

   4. metadata-eval N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \frac{\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right) \cdot \frac{-1}{2}\right)\right) + \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right), 2, 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right)\right)\right)\right)}{4}\right)}^{2} \]

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

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

   6. lower-cos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-PI.f64 82.8

       \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \frac{\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right) \cdot -0.5\right)\right) + \cos \left(0.005555555555555556 \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot angle\right)\right), 2, 2 \cdot \left(\sin \left(0.5 \cdot \mathsf{PI}\left(\right) - \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot -0.005555555555555556\right) + \sin \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot -0.005555555555555556\right)\right)\right)\right)}{4}\right)}^{2} \]

9. Applied rewrites 82.8%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \frac{\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right) \cdot -0.5\right)\right) + \color{blue}{\cos \left(0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}, 2, 2 \cdot \left(\sin \left(0.5 \cdot \mathsf{PI}\left(\right) - \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot -0.005555555555555556\right) + \sin \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot -0.005555555555555556\right)\right)\right)\right)}{4}\right)}^{2} \]
10. Add Preprocessing

Alternative 2: 80.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := angle\_m \cdot \mathsf{PI}\left(\right)\\ {\left(a \cdot \sin \left(\frac{angle\_m}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \frac{\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(0.005555555555555556, t\_0, \mathsf{PI}\left(\right) \cdot -0.5\right)\right) + \cos \left(0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\_m\right)\right), 2, 2 \cdot \left(\sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556, angle\_m, 0.5 \cdot \mathsf{PI}\left(\right)\right)\right) + \sin \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), t\_0 \cdot -0.005555555555555556\right)\right)\right)\right)}{4}\right)}^{2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* angle_m (PI))))
   (+
    (pow (* a (sin (* (/ angle_m 180.0) (PI)))) 2.0)
    (pow
     (*
      b
      (/
       (fma
        (+
         (sin (fma 0.005555555555555556 t_0 (* (PI) -0.5)))
         (cos (* 0.005555555555555556 (* (PI) angle_m))))
        2.0
        (*
         2.0
         (+
          (sin (fma (* (PI) 0.005555555555555556) angle_m (* 0.5 (PI))))
          (sin (fma 0.5 (PI) (* t_0 -0.005555555555555556))))))
       4.0))
     2.0))))

Derivation

1. Initial program 82.8%

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

   1. lift-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. cos-neg-rev N/A

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

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

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

   5. remove-double-neg N/A

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

   6. add-cube-cbrt N/A

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

   7. associate-/l* N/A

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

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

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

   9. sin-diff N/A

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

   10. lower--.f64 N/A

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

4. Applied rewrites 82.8%

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

6. Applied rewrites 82.8%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{\frac{\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right) \cdot -0.5\right)\right) + \cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right), 2, 2 \cdot \left(\sin \left(0.5 \cdot \mathsf{PI}\left(\right) - \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot -0.005555555555555556\right) + \sin \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot -0.005555555555555556\right)\right)\right)\right)}{4}}\right)}^{2} \]

7. Taylor expanded in angle around inf

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

   1. cos-neg-rev N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \frac{\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right) \cdot \frac{-1}{2}\right)\right) + \color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}, 2, 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right)\right)\right)\right)}{4}\right)}^{2} \]

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

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \frac{\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right) \cdot \frac{-1}{2}\right)\right) + \color{blue}{\sin \left(\left(\mathsf{neg}\left(\frac{-1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}, 2, 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right)\right)\right)\right)}{4}\right)}^{2} \]

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

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \frac{\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right) \cdot \frac{-1}{2}\right)\right) + \sin \left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{180}\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right), 2, 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right)\right)\right)\right)}{4}\right)}^{2} \]

   4. metadata-eval N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \frac{\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right) \cdot \frac{-1}{2}\right)\right) + \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right), 2, 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right)\right)\right)\right)}{4}\right)}^{2} \]

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

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

   6. lower-cos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-PI.f64 82.8

       \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \frac{\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right) \cdot -0.5\right)\right) + \cos \left(0.005555555555555556 \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot angle\right)\right), 2, 2 \cdot \left(\sin \left(0.5 \cdot \mathsf{PI}\left(\right) - \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot -0.005555555555555556\right) + \sin \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot -0.005555555555555556\right)\right)\right)\right)}{4}\right)}^{2} \]

9. Applied rewrites 82.8%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \frac{\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right) \cdot -0.5\right)\right) + \color{blue}{\cos \left(0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}, 2, 2 \cdot \left(\sin \left(0.5 \cdot \mathsf{PI}\left(\right) - \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot -0.005555555555555556\right) + \sin \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot -0.005555555555555556\right)\right)\right)\right)}{4}\right)}^{2} \]
10. Step-by-step derivation

    1. lift--.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. *-commutative N/A

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

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

       \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \frac{\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right) \cdot \frac{-1}{2}\right)\right) + \cos \left(\frac{1}{180} \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right), 2, 2 \cdot \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + \left(\mathsf{neg}\left(\frac{-1}{180}\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} + \sin \left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right)\right)\right)\right)}{4}\right)}^{2} \]

    5. metadata-eval N/A

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

    6. +-commutative N/A

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

    7. lift-*.f64 N/A

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

    8. *-commutative N/A

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

    9. associate-*r* N/A

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

    10. lower-fma.f64 N/A

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

    11. *-commutative N/A

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

    12. lower-*.f64 82.7

        \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \frac{\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right) \cdot -0.5\right)\right) + \cos \left(0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right), 2, 2 \cdot \left(\sin \left(\mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot 0.005555555555555556}, angle, 0.5 \cdot \mathsf{PI}\left(\right)\right)\right) + \sin \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot -0.005555555555555556\right)\right)\right)\right)}{4}\right)}^{2} \]

11. Applied rewrites 82.7%

    \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \frac{\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right) \cdot -0.5\right)\right) + \cos \left(0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right), 2, 2 \cdot \left(\sin \color{blue}{\left(\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556, angle, 0.5 \cdot \mathsf{PI}\left(\right)\right)\right)} + \sin \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot -0.005555555555555556\right)\right)\right)\right)}{4}\right)}^{2} \]
12. Add Preprocessing

Alternative 3: 79.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 82.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 82.8

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

4. Applied rewrites 82.8%

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

6. Applied rewrites 82.8%

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

Alternative 4: 80.3% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 82.8%

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

   1. lift-+.f64 N/A

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

   2. +-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. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. unpow-prod-down N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 82.8

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

4. Applied rewrites 82.8%

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

Alternative 5: 80.3% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 82.8%

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

Alternative 6: 80.3% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 82.8%

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. add-sqr-sqrt N/A

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

   4. sqr-neg-rev N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-neg.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. lower-sqrt.f64 82.7

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

4. Applied rewrites 82.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lift-neg.f64 N/A

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

   7. lift-neg.f64 N/A

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

   8. sqr-neg-rev N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. rem-square-sqrt N/A

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

   12. *-commutative N/A

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

   13. lift-/.f64 N/A

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

   14. lift-PI.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. lift-PI.f64 N/A

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

   17. rem-square-sqrt N/A

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

   18. lift-sqrt.f64 N/A

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

   19. lift-sqrt.f64 N/A

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

   20. associate-*r* N/A

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

6. Applied rewrites 82.7%

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

7. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 82.8

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

9. Applied rewrites 82.8%

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

Alternative 7: 80.3% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 82.8%

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

3. Taylor expanded in angle around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-PI.f64 82.8

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

5. Applied rewrites 82.8%

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

Alternative 8: 80.3% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 82.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 82.8

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

4. Applied rewrites 82.8%

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

5. Taylor expanded in angle around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-PI.f64 82.8

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

7. Applied rewrites 82.8%

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

Alternative 9: 79.5% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 82.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 82.8

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

4. Applied rewrites 82.8%

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

6. Applied rewrites 82.8%

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

7. Taylor expanded in angle around 0

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

9. Applied rewrites 82.6%

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

Alternative 10: 80.2% accurate, 1.3× speedup

Math

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

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 1.0) 2.0)))

Derivation

1. Initial program 82.8%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 82.6%

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

Alternative 11: 67.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 1.9 \cdot 10^{-57}:\\ \;\;\;\;{\cos \left(0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\_m\right)\right)}^{2} \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(a \cdot angle\_m\right) \cdot angle\_m\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), a, {\left(\cos \left(\frac{angle\_m \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot b\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 1.9e-57)
   (* (pow (cos (* 0.005555555555555556 (* (PI) angle_m))) 2.0) (* b b))
   (fma
    (* (* 3.08641975308642e-5 (* (* a angle_m) angle_m)) (* (PI) (PI)))
    a
    (pow (* (cos (/ (* angle_m (PI)) -180.0)) b) 2.0))))

Derivation

1. Split input into 2 regimes

2. if a < 1.8999999999999999e-57

   1. Initial program 81.3%

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

   3. Taylor expanded in angle around 0

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

      1. *-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 46.0%

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

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}} \]
   7. 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. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

      5. metadata-eval N/A

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

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

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

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

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

      8. cos-neg-rev N/A

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

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

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

      10. metadata-eval N/A

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

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

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

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

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

      13. cos-neg-rev N/A

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

   8. Applied rewrites 64.9%

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

   if 1.8999999999999999e-57 < a

   1. Initial program 86.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 86.1

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

   4. Applied rewrites 86.1%

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

   6. Applied rewrites 86.1%

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

   7. Taylor expanded in angle around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{32400} \cdot \left(a \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}, a, {\left(\cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot b\right)}^{2}\right) \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{32400} \cdot \color{blue}{\left(\left(a \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, a, {\left(\cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot b\right)}^{2}\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{32400} \cdot \left(a \cdot {angle}^{2}\right)\right) \cdot {\mathsf{PI}\left(\right)}^{2}}, a, {\left(\cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot b\right)}^{2}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{32400} \cdot \left(a \cdot {angle}^{2}\right)\right) \cdot {\mathsf{PI}\left(\right)}^{2}}, a, {\left(\cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot b\right)}^{2}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{32400} \cdot \left(a \cdot {angle}^{2}\right)\right)} \cdot {\mathsf{PI}\left(\right)}^{2}, a, {\left(\cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot b\right)}^{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(a \cdot \color{blue}{\left(angle \cdot angle\right)}\right)\right) \cdot {\mathsf{PI}\left(\right)}^{2}, a, {\left(\cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot b\right)}^{2}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \color{blue}{\left(\left(a \cdot angle\right) \cdot angle\right)}\right) \cdot {\mathsf{PI}\left(\right)}^{2}, a, {\left(\cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot b\right)}^{2}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \color{blue}{\left(\left(a \cdot angle\right) \cdot angle\right)}\right) \cdot {\mathsf{PI}\left(\right)}^{2}, a, {\left(\cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot b\right)}^{2}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(\color{blue}{\left(a \cdot angle\right)} \cdot angle\right)\right) \cdot {\mathsf{PI}\left(\right)}^{2}, a, {\left(\cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot b\right)}^{2}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(\left(a \cdot angle\right) \cdot angle\right)\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, a, {\left(\cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot b\right)}^{2}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(\left(a \cdot angle\right) \cdot angle\right)\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, a, {\left(\cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot b\right)}^{2}\right) \]

      11. lower-PI.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(\left(a \cdot angle\right) \cdot angle\right)\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right), a, {\left(\cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot b\right)}^{2}\right) \]

      12. lower-PI.f64 83.8

          \[\leadsto \mathsf{fma}\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(a \cdot angle\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right), a, {\left(\cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot b\right)}^{2}\right) \]

   9. Applied rewrites 83.8%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(a \cdot angle\right) \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, a, {\left(\cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot b\right)}^{2}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 57.6% accurate, 9.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 3.4 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -3.08641975308642 \cdot 10^{-5}\right) \cdot angle\_m\right), angle\_m, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 3.4e+26)
   (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 < 3.4000000000000003e26

   1. Initial program 84.4%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      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 49.2%

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

   7. Applied rewrites 55.8%

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

   if 3.4000000000000003e26 < b

   1. Initial program 78.2%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
   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 70.1

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

   5. Applied rewrites 70.1%

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

Alternative 13: 61.8% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 9.4 \cdot 10^{+120}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(a \cdot a\right) \cdot angle\_m\right) \cdot angle\_m\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 9.4e+120)
   (* b b)
   (* (* 3.08641975308642e-5 (* (* (* a a) angle_m) angle_m)) (* (PI) (PI)))))

Derivation

1. Split input into 2 regimes

2. if a < 9.39999999999999987e120

   1. Initial program 80.0%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 64.7

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

   5. Applied rewrites 64.7%

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

   if 9.39999999999999987e120 < a

   1. Initial program 94.9%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {b}^{2}} \]
   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 38.7%

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

   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 55.4%

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

   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 66.6%

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

Alternative 14: 57.4% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.8%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 60.7

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

5. Applied rewrites 60.7%

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

Reproduce

herbie shell --seed 2024354 
(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)))