ab-angle->ABCF C

Percentage Accurate: 79.5% → 79.5%

Time: 10.4s

Alternatives: 14

Speedup: 1.2×

• 35.6% 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: 79.5% 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: 79.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 81.3%

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

3. Taylor expanded in angle around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-PI.f64 81.5

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

5. Applied rewrites 81.5%

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

   1. lift-cos.f64 N/A

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

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

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

   3. lower-sin.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lower-/.f64 81.5

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

7. Applied rewrites 81.5%

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

Alternative 2: 79.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 81.3%

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

3. Taylor expanded in angle around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-PI.f64 81.5

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

5. Applied rewrites 81.5%

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

Alternative 3: 79.5% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 81.3%

   \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
2. Add Preprocessing
3. 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. unpow2 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

4. Applied rewrites 81.3%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-cos.f64 N/A

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

   4. cos-neg-rev N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. cos-neg-rev N/A

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

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

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

   10. lift-cos.f64 N/A

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

   11. cos-neg-rev N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. cos-neg-rev N/A

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

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

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

   17. sqr-sin-a N/A

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

   18. lower--.f64 N/A

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

6. Applied rewrites 81.3%

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

Alternative 4: 79.5% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 81.3%

   \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
2. Add Preprocessing
3. 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. unpow2 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

4. Applied rewrites 81.3%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-cos.f64 N/A

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

   4. cos-neg-rev N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. cos-neg-rev N/A

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

   9. lift-cos.f64 N/A

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

   10. cos-neg-rev N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. cos-neg-rev N/A

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

   15. 1-sub-sin-rev N/A

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

   16. metadata-eval N/A

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

6. Applied rewrites 81.3%

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

   1. rem-cube-cbrt N/A

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

   2. lift-cbrt.f64 N/A

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

   3. cube-unmult N/A

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

   4. unpow2 N/A

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

   5. lift-pow.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 81.2

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

8. Applied rewrites 81.2%

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

10. Applied rewrites 81.3%

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

Alternative 5: 79.5% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 81.3%

   \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
2. Add Preprocessing
3. 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. unpow2 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

4. Applied rewrites 81.3%

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

6. Applied rewrites 81.3%

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

Alternative 6: 79.4% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 81.3%

   \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
2. Add Preprocessing
3. 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. unpow2 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

4. Applied rewrites 81.3%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-cos.f64 N/A

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

   4. cos-neg-rev N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. cos-neg-rev N/A

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

   9. lift-cos.f64 N/A

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

   10. cos-neg-rev N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. cos-neg-rev N/A

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

   15. 1-sub-sin-rev N/A

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

   16. metadata-eval N/A

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

6. Applied rewrites 81.3%

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

7. Taylor expanded in angle around inf

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

   1. unpow2 N/A

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

   2. 1-sub-sin-rev N/A

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

   3. unpow2 N/A

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

   4. lower-pow.f64 N/A

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

   5. cos-neg-rev N/A

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

   6. lower-cos.f64 N/A

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

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

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

   8. lower-*.f64 N/A

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

   9. metadata-eval N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-PI.f64 81.3

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

9. Applied rewrites 81.3%

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

11. Applied rewrites 81.3%

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

Alternative 7: 79.4% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 81.3%

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

3. Taylor expanded in angle around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-PI.f64 81.5

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

5. Applied rewrites 81.5%

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

6. Taylor expanded in angle around 0

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

8. Applied rewrites 81.0%

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

Alternative 8: 79.4% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 81.3%

   \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
2. Add Preprocessing
3. 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. unpow2 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

4. Applied rewrites 81.3%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-cos.f64 N/A

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

   4. cos-neg-rev N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. cos-neg-rev N/A

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

   9. lift-cos.f64 N/A

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

   10. cos-neg-rev N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. cos-neg-rev N/A

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

   15. 1-sub-sin-rev N/A

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

   16. metadata-eval N/A

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

6. Applied rewrites 81.3%

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

7. Taylor expanded in angle around 0

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

9. Applied rewrites 80.9%

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

Alternative 9: 63.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;b \leq 1.25 \cdot 10^{-37}:\\ \;\;\;\;{\cos \left(-0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\_m\right)\right)}^{2} \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;b \leq 4.1 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(t\_0 \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right) \cdot b, angle\_m \cdot angle\_m, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(b \cdot angle\_m\right) \cdot \left(b \cdot angle\_m\right)\right)\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (PI) (PI))))
   (if (<= b 1.25e-37)
     (* (pow (cos (* -0.005555555555555556 (* (PI) angle_m))) 2.0) (* a a))
     (if (<= b 4.1e+144)
       (fma
        (* (* (* t_0 3.08641975308642e-5) b) b)
        (* angle_m angle_m)
        (* a a))
       (* (* 3.08641975308642e-5 (* (* b angle_m) (* b angle_m))) t_0)))))

Derivation

1. Split input into 3 regimes

2. if b < 1.2499999999999999e-37

   1. Initial program 77.6%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   2. Add Preprocessing
   3. 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. unpow2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 77.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-cos.f64 N/A

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

      4. cos-neg-rev N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. cos-neg-rev N/A

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

      9. lift-cos.f64 N/A

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

      10. cos-neg-rev N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. cos-neg-rev N/A

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

      15. 1-sub-sin-rev N/A

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

      16. metadata-eval N/A

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

   6. Applied rewrites 77.6%

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

   7. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. 1-sub-sin-rev N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. cos-neg-rev N/A

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

      8. lower-cos.f64 N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 62.0

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

   9. Applied rewrites 62.0%

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

   if 1.2499999999999999e-37 < b < 4.10000000000000001e144

   1. Initial program 82.4%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 39.2%

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

   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 67.3%

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

   if 4.10000000000000001e144 < b

   1. Initial program 97.8%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 46.0%

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

   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 83.2%

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

   10. Applied rewrites 92.9%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.25 \cdot 10^{-37}:\\ \;\;\;\;{\cos \left(-0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}^{2} \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;b \leq 4.1 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right) \cdot b, angle \cdot angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(b \cdot angle\right) \cdot \left(b \cdot angle\right)\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;b \leq 1.25 \cdot 10^{-37}:\\ \;\;\;\;{\cos \left(\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\_m\right)}^{2} \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;b \leq 4.1 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(t\_0 \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right) \cdot b, angle\_m \cdot angle\_m, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(b \cdot angle\_m\right) \cdot \left(b \cdot angle\_m\right)\right)\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (PI) (PI))))
   (if (<= b 1.25e-37)
     (* (pow (cos (* (* 0.005555555555555556 (PI)) angle_m)) 2.0) (* a a))
     (if (<= b 4.1e+144)
       (fma
        (* (* (* t_0 3.08641975308642e-5) b) b)
        (* angle_m angle_m)
        (* a a))
       (* (* 3.08641975308642e-5 (* (* b angle_m) (* b angle_m))) t_0)))))

Derivation

1. Split input into 3 regimes

2. if b < 1.2499999999999999e-37

   1. Initial program 77.6%

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

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

      4. metadata-eval N/A

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

      5. unpow1 N/A

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

      6. metadata-eval N/A

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

      7. sqr-pow N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 35.0%

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

   5. Taylor expanded in a around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. rem-square-sqrt N/A

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

      7. mul-1-neg N/A

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

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

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

      9. distribute-rgt-neg-in N/A

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

      10. unpow2 N/A

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

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

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

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

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

      13. sqr-neg-rev N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 61.9%

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

   if 1.2499999999999999e-37 < b < 4.10000000000000001e144

   1. Initial program 82.4%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 39.2%

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

   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 67.3%

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

   if 4.10000000000000001e144 < b

   1. Initial program 97.8%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 46.0%

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

   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 83.2%

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

   10. Applied rewrites 92.9%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.25 \cdot 10^{-37}:\\ \;\;\;\;{\cos \left(\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;b \leq 4.1 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right) \cdot b, angle \cdot angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(b \cdot angle\right) \cdot \left(b \cdot angle\right)\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.2% accurate, 9.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;b \leq 1.25 \cdot 10^{-37}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;b \leq 4.1 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(t\_0 \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right) \cdot b, angle\_m \cdot angle\_m, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(b \cdot angle\_m\right) \cdot \left(b \cdot angle\_m\right)\right)\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (PI) (PI))))
   (if (<= b 1.25e-37)
     (* a a)
     (if (<= b 4.1e+144)
       (fma
        (* (* (* t_0 3.08641975308642e-5) b) b)
        (* angle_m angle_m)
        (* a a))
       (* (* 3.08641975308642e-5 (* (* b angle_m) (* b angle_m))) t_0)))))

Derivation

1. Split input into 3 regimes

2. if b < 1.2499999999999999e-37

   1. Initial program 77.6%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 61.8

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

   5. Applied rewrites 61.8%

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

   if 1.2499999999999999e-37 < b < 4.10000000000000001e144

   1. Initial program 82.4%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 39.2%

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

   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 67.3%

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

   if 4.10000000000000001e144 < b

   1. Initial program 97.8%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 46.0%

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

   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 83.2%

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

   10. Applied rewrites 92.9%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.25 \cdot 10^{-37}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;b \leq 4.1 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right) \cdot b, angle \cdot angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(b \cdot angle\right) \cdot \left(b \cdot angle\right)\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.0% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.75 \cdot 10^{+123}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(b \cdot angle\_m\right) \cdot \left(b \cdot angle\_m\right)\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 1.75e+123)
   (* a a)
   (* (* 3.08641975308642e-5 (* (* b angle_m) (* b angle_m))) (* (PI) (PI)))))

Derivation

1. Split input into 2 regimes

2. if b < 1.75e123

   1. Initial program 78.1%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 61.1

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

   5. Applied rewrites 61.1%

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

   if 1.75e123 < b

   1. Initial program 97.9%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 46.3%

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

   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 79.4%

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

   10. Applied rewrites 90.9%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.75 \cdot 10^{+123}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(b \cdot angle\right) \cdot \left(b \cdot angle\right)\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.5% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.75 \cdot 10^{+123}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle\_m \cdot \left(angle\_m \cdot \left(b \cdot b\right)\right)\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 1.75e+123)
   (* a a)
   (* (* 3.08641975308642e-5 (* angle_m (* angle_m (* b b)))) (* (PI) (PI)))))

Derivation

1. Split input into 2 regimes

2. if b < 1.75e123

   1. Initial program 78.1%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 61.1

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

   5. Applied rewrites 61.1%

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

   if 1.75e123 < b

   1. Initial program 97.9%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 46.3%

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

   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 79.4%

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

   10. Applied rewrites 72.8%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.75 \cdot 10^{+123}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 56.2% 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 =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle_m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle_m
    code = 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 81.3%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 56.5

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

5. Applied rewrites 56.5%

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

6. Final simplification 56.5%

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

Reproduce

herbie shell --seed 2024359 
(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)))