ab-angle->ABCF C

Percentage Accurate: 79.9% → 79.9%

Time: 11.9s

Alternatives: 14

Speedup: 0.6×

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

Specification

Math

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

FPCore

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

Initial Program: 79.9% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 79.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(angle\_m, -0.005555555555555556, 0.5\right) \cdot \mathsf{PI}\left(\right)\\ t_1 := \mathsf{fma}\left(angle\_m, -0.005555555555555556, 2\right) \cdot \mathsf{PI}\left(\right)\\ t_2 := \sin t\_1 \cdot \cos t\_0\\ t_3 := \cos t\_1 \cdot \sin t\_0\\ \mathsf{fma}\left(\frac{\frac{{t\_3}^{3} + {t\_2}^{3}}{\mathsf{fma}\left(t\_3, t\_3, t\_2 \cdot t\_2 - t\_3 \cdot t\_2\right)} + \sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right)}{2}, a \cdot a, {\left(\sin \left(\frac{angle\_m}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2}\right) \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (fma angle_m -0.005555555555555556 0.5) (PI)))
        (t_1 (* (fma angle_m -0.005555555555555556 2.0) (PI)))
        (t_2 (* (sin t_1) (cos t_0)))
        (t_3 (* (cos t_1) (sin t_0))))
   (fma
    (/
     (+
      (/
       (+ (pow t_3 3.0) (pow t_2 3.0))
       (fma t_3 t_3 (- (* t_2 t_2) (* t_3 t_2))))
      (sin (* 0.5 (PI))))
     2.0)
    (* a a)
    (pow (* (sin (* (/ angle_m 180.0) (PI))) b) 2.0))))

Derivation

1. Initial program 81.4%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. unpow-prod-down N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 81.4

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

4. Applied rewrites 81.4%

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

6. Applied rewrites 71.1%

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

7. Taylor expanded in angle around inf

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

   1. +-commutative N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

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

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

   5. +-commutative N/A

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

   6. lower-+.f64 N/A

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

9. Applied rewrites 81.5%

   \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), \mathsf{fma}\left(-0.005555555555555556, angle, 0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.005555555555555556, angle, 1\right), \mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right) + \sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right)}}{2}, a \cdot a, {\left(\sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2}\right) \]

11. Applied rewrites 81.5%

    \[\leadsto \mathsf{fma}\left(\frac{\frac{{\left(\cos \left(\mathsf{fma}\left(angle, -0.005555555555555556, 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(\mathsf{fma}\left(angle, -0.005555555555555556, 0.5\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{3} + {\left(\sin \left(\mathsf{fma}\left(angle, -0.005555555555555556, 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \cos \left(\mathsf{fma}\left(angle, -0.005555555555555556, 0.5\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{3}}{\mathsf{fma}\left(\cos \left(\mathsf{fma}\left(angle, -0.005555555555555556, 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(\mathsf{fma}\left(angle, -0.005555555555555556, 0.5\right) \cdot \mathsf{PI}\left(\right)\right), \cos \left(\mathsf{fma}\left(angle, -0.005555555555555556, 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(\mathsf{fma}\left(angle, -0.005555555555555556, 0.5\right) \cdot \mathsf{PI}\left(\right)\right), \left(\sin \left(\mathsf{fma}\left(angle, -0.005555555555555556, 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \cos \left(\mathsf{fma}\left(angle, -0.005555555555555556, 0.5\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(angle, -0.005555555555555556, 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \cos \left(\mathsf{fma}\left(angle, -0.005555555555555556, 0.5\right) \cdot \mathsf{PI}\left(\right)\right)\right) - \left(\cos \left(\mathsf{fma}\left(angle, -0.005555555555555556, 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(\mathsf{fma}\left(angle, -0.005555555555555556, 0.5\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(angle, -0.005555555555555556, 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \cos \left(\mathsf{fma}\left(angle, -0.005555555555555556, 0.5\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)} + \sin \color{blue}{\left(0.5 \cdot \mathsf{PI}\left(\right)\right)}}{2}, a \cdot a, {\left(\sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2}\right) \]
12. Add Preprocessing

Alternative 2: 79.9% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 81.4%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. unpow-prod-down N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 81.4

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

4. Applied rewrites 81.4%

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

6. Applied rewrites 71.1%

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

7. Taylor expanded in angle around inf

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

   1. +-commutative N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

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

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

   5. +-commutative N/A

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

   6. lower-+.f64 N/A

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

9. Applied rewrites 81.5%

   \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), \mathsf{fma}\left(-0.005555555555555556, angle, 0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.005555555555555556, angle, 1\right), \mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right) + \sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right)}}{2}, a \cdot a, {\left(\sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2}\right) \]
10. Step-by-step derivation

11. Applied rewrites 81.5%

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

Alternative 3: 79.8% accurate, 0.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 81.4%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. unpow-prod-down N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 81.4

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

4. Applied rewrites 81.4%

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

6. Applied rewrites 71.1%

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

7. Taylor expanded in angle around inf

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

   1. +-commutative N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

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

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

   5. +-commutative N/A

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

   6. lower-+.f64 N/A

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

9. Applied rewrites 81.5%

   \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), \mathsf{fma}\left(-0.005555555555555556, angle, 0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.005555555555555556, angle, 1\right), \mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right) + \sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right)}}{2}, a \cdot a, {\left(\sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2}\right) \]
10. Step-by-step derivation

    1. lift-fma.f64 N/A

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

11. Applied rewrites 81.5%

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

Alternative 4: 79.8% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 81.4%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 81.6%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

7. Applied rewrites 81.6%

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

Alternative 5: 79.8% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 81.4%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 81.6%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

7. Applied rewrites 81.6%

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

8. Taylor expanded in angle around inf

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

   1. lower-sin.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 81.6

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

10. Applied rewrites 81.6%

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

Alternative 6: 66.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{-120}:\\ \;\;\;\;{\cos \left(\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\_m\right)}^{2} \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(1 \cdot a\right) \cdot 1, a, {\left(\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot 0.005555555555555556\right) \cdot angle\_m\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 8e-120)
   (* (pow (cos (* (* 0.005555555555555556 (PI)) angle_m)) 2.0) (* a a))
   (fma
    (* (* 1.0 a) 1.0)
    a
    (pow (* (* (* b (PI)) 0.005555555555555556) angle_m) 2.0))))

Derivation

1. Split input into 2 regimes

2. if b < 7.99999999999999983e-120

   1. Initial program 80.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. sin-sum N/A

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

      4. lift-cos.f64 N/A

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

      5. sin-PI/2 N/A

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

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

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

      7. lower--.f64 N/A

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

   4. Applied rewrites 80.5%

      \[\leadsto {\left(a \cdot \color{blue}{\left(0 \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right) - \cos \left(\mathsf{fma}\left(\frac{angle}{-180}, \mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right) \cdot 1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 61.1%

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

   if 7.99999999999999983e-120 < b

   1. Initial program 83.1%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 84.1%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   7. Applied rewrites 84.1%

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

   8. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 78.5

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

   10. Applied rewrites 78.5%

       \[\leadsto \mathsf{fma}\left(\left(1 \cdot a\right) \cdot 1, a, {\color{blue}{\left(\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot 0.005555555555555556\right) \cdot angle\right)}}^{2}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 66.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{-120}:\\ \;\;\;\;{\cos \left(-0.005555555555555556 \cdot \left(angle\_m \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(1 \cdot a\right) \cdot 1, a, {\left(\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot 0.005555555555555556\right) \cdot angle\_m\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 8e-120)
   (* (pow (cos (* -0.005555555555555556 (* angle_m (PI)))) 2.0) (* a a))
   (fma
    (* (* 1.0 a) 1.0)
    a
    (pow (* (* (* b (PI)) 0.005555555555555556) angle_m) 2.0))))

Derivation

1. Split input into 2 regimes

2. if b < 7.99999999999999983e-120

   1. Initial program 80.5%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 80.3%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   7. Applied rewrites 80.3%

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

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

   10. Applied rewrites 61.1%

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

   if 7.99999999999999983e-120 < b

   1. Initial program 83.1%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 84.1%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   7. Applied rewrites 84.1%

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

   8. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 78.5

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

   10. Applied rewrites 78.5%

       \[\leadsto \mathsf{fma}\left(\left(1 \cdot a\right) \cdot 1, a, {\color{blue}{\left(\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot 0.005555555555555556\right) \cdot angle\right)}}^{2}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 66.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 7.2 \cdot 10^{-120}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(-2 \cdot \left(\mathsf{fma}\left(-0.005555555555555556, angle\_m, 1\right) \cdot \mathsf{PI}\left(\right)\right)\right), 0.5, 0.5\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(1 \cdot a\right) \cdot 1, a, {\left(\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot 0.005555555555555556\right) \cdot angle\_m\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 7.2e-120)
   (*
    (fma
     (cos (* -2.0 (* (fma -0.005555555555555556 angle_m 1.0) (PI))))
     0.5
     0.5)
    (* a a))
   (fma
    (* (* 1.0 a) 1.0)
    a
    (pow (* (* (* b (PI)) 0.005555555555555556) angle_m) 2.0))))

Derivation

1. Split input into 2 regimes

2. if b < 7.2000000000000005e-120

   1. Initial program 80.5%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. unpow-prod-down N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 80.5

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

   4. Applied rewrites 80.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. add-cube-cbrt N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-cbrt.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. lower-cbrt.f64 80.7

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

   6. Applied rewrites 80.7%

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

   8. Applied rewrites 80.5%

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

   9. Taylor expanded in a around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 61.1%

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

   if 7.2000000000000005e-120 < b

   1. Initial program 83.1%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 84.1%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   7. Applied rewrites 84.1%

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

   8. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 78.5

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

   10. Applied rewrites 78.5%

       \[\leadsto \mathsf{fma}\left(\left(1 \cdot a\right) \cdot 1, a, {\color{blue}{\left(\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot 0.005555555555555556\right) \cdot angle\right)}}^{2}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 57.8% accurate, 8.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 3.5 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(a - b\right) \cdot \left(b + a\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(-angle\_m\right), -angle\_m, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 3.5e+82)
   (fma
    (*
     (* (* (- a b) (+ b a)) (* (* (PI) (PI)) -3.08641975308642e-5))
     (- angle_m))
    (- angle_m)
    (* a a))
   (* a a)))

Derivation

1. Split input into 2 regimes

2. if a < 3.5e82

   1. Initial program 78.4%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 48.0%

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

   7. Applied rewrites 51.5%

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

   if 3.5e82 < a

   1. Initial program 93.6%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 91.8

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

   5. Applied rewrites 91.8%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.5 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(a - b\right) \cdot \left(b + a\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(-angle\right), -angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.4% accurate, 9.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 7.5 \cdot 10^{-92}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), angle\_m \cdot angle\_m, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(b \cdot angle\_m\right) \cdot angle\_m\right) \cdot b\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 7.5e-92)
   (* a a)
   (if (<= b 3.2e+152)
     (fma
      (* (* (* 3.08641975308642e-5 (* b b)) (PI)) (PI))
      (* angle_m angle_m)
      (* a a))
     (*
      (* 3.08641975308642e-5 (* (* (* b angle_m) angle_m) b))
      (* (PI) (PI))))))

Derivation

1. Split input into 3 regimes

2. if b < 7.5000000000000005e-92

   1. Initial program 80.5%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.8

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

   5. Applied rewrites 61.8%

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

   if 7.5000000000000005e-92 < b < 3.20000000000000005e152

   1. Initial program 74.3%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 33.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 61.2%

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

   if 3.20000000000000005e152 < b

   1. Initial program 97.3%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 49.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 73.6%

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

   10. Applied rewrites 76.7%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.5 \cdot 10^{-92}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), angle \cdot angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(b \cdot angle\right) \cdot angle\right) \cdot b\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.8% accurate, 9.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 3.5 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(a - b\right) \cdot \left(b + a\right)\right) \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -3.08641975308642 \cdot 10^{-5}\right) \cdot angle\_m\right), angle\_m, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 3.5e+82)
   (fma
    (* (* (- a b) (+ b a)) (* (* (* (PI) (PI)) -3.08641975308642e-5) angle_m))
    angle_m
    (* a a))
   (* a a)))

Derivation

1. Split input into 2 regimes

2. if a < 3.5e82

   1. Initial program 78.4%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 48.0%

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

   7. Applied rewrites 51.5%

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

   if 3.5e82 < a

   1. Initial program 93.6%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 91.8

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

   5. Applied rewrites 91.8%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.5 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(a - b\right) \cdot \left(b + a\right)\right) \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -3.08641975308642 \cdot 10^{-5}\right) \cdot angle\right), angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.3% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 5.4 \cdot 10^{+162}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(b \cdot angle\_m\right) \cdot angle\_m\right) \cdot b\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 5.4e+162)
   (* a a)
   (* (* 3.08641975308642e-5 (* (* (* b angle_m) angle_m) b)) (* (PI) (PI)))))

Derivation

1. Split input into 2 regimes

2. if b < 5.4000000000000003e162

   1. Initial program 79.1%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 60.5

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

   5. Applied rewrites 60.5%

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

   if 5.4000000000000003e162 < b

   1. Initial program 99.9%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 55.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 79.9%

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

   10. Applied rewrites 83.3%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.4 \cdot 10^{+162}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(b \cdot angle\right) \cdot angle\right) \cdot b\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.1% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 5.4 \cdot 10^{+162}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle\_m \cdot angle\_m\right) \cdot b\right) \cdot b\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 5.4e+162)
   (* a a)
   (* (* 3.08641975308642e-5 (* (* (* angle_m angle_m) b) b)) (* (PI) (PI)))))

Derivation

1. Split input into 2 regimes

2. if b < 5.4000000000000003e162

   1. Initial program 79.1%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 60.5

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

   5. Applied rewrites 60.5%

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

   if 5.4000000000000003e162 < b

   1. Initial program 99.9%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 55.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 79.9%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.4 \cdot 10^{+162}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot angle\right) \cdot b\right) \cdot b\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 56.9% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.4%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 57.4

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

5. Applied rewrites 57.4%

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

6. Final simplification 57.4%

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

Reproduce

herbie shell --seed 2024344 
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* (PI) (/ angle 180.0)))) 2.0) (pow (* b (sin (* (PI) (/ angle 180.0)))) 2.0)))