ab-angle->ABCF C

Percentage Accurate: 80.6% → 80.6%

Time: 10.9s

Alternatives: 14

Speedup: 1.3×

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

Specification

Math

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

FPCore

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

Initial Program: 80.6% 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: 80.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left({\cos \left(\frac{angle}{180} \cdot \sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}\right)}^{2} \cdot a, a, {\left(\sin \left(\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot b\right)}^{2}\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (fma
  (* (pow (cos (* (/ angle 180.0) (cbrt (pow (PI) 3.0)))) 2.0) a)
  a
  (pow (* (sin (* (* 0.005555555555555556 (PI)) angle)) b) 2.0)))

Derivation

1. Initial program 80.6%

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

3. Taylor expanded in angle around 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 80.6

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

5. Applied rewrites 80.6%

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

7. Applied rewrites 80.7%

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

   1. lift-PI.f64 N/A

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

   2. add-cbrt-cube N/A

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

   3. lower-cbrt.f64 N/A

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

   4. rem-cube-cbrt N/A

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

   5. add-cbrt-cube N/A

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

   6. lift-PI.f64 N/A

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

   7. lower-pow.f64 80.7

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

9. Applied rewrites 80.7%

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

Alternative 2: 80.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (sin (* (* 0.005555555555555556 (PI)) angle)) b)))
   (fma t_0 t_0 (pow (* (cos (* (/ angle 180.0) (PI))) a) 2.0))))

Derivation

1. Initial program 80.6%

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

3. Taylor expanded in angle around 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 80.6

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

5. Applied rewrites 80.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-fma.f64 80.7

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

7. Applied rewrites 80.7%

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

Alternative 3: 80.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (fma
  (* (pow (cos (fma (PI) (/ angle 180.0) (PI))) 2.0) a)
  a
  (pow (* (sin (* (* 0.005555555555555556 (PI)) angle)) b) 2.0)))

Derivation

1. Initial program 80.6%

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

3. Taylor expanded in angle around 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 80.6

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

5. Applied rewrites 80.6%

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

7. Applied rewrites 80.7%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. sqr-neg-rev N/A

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

   4. pow2 N/A

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

   5. lower-pow.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. cos-+PI-rev N/A

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

   11. lower-cos.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lower-fma.f64 80.7

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

9. Applied rewrites 80.7%

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

Alternative 4: 80.6% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (fma
  (* (+ 0.5 (* 0.5 (cos (* 2.0 (* (PI) (/ angle 180.0)))))) a)
  a
  (pow (* (sin (* (* 0.005555555555555556 (PI)) angle)) b) 2.0)))

Derivation

1. Initial program 80.6%

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

3. Taylor expanded in angle around 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 80.6

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

5. Applied rewrites 80.6%

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

7. Applied rewrites 80.7%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. sqr-cos-a N/A

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

   12. metadata-eval N/A

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

   13. lower-+.f64 N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. cos-2 N/A

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

   17. cos-sum N/A

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

   18. lower-*.f64 N/A

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

9. Applied rewrites 80.7%

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

Alternative 5: 80.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot b\\ \mathsf{fma}\left(t\_0, t\_0, {\left(1 \cdot a\right)}^{2}\right) \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (sin (* (* 0.005555555555555556 (PI)) angle)) b)))
   (fma t_0 t_0 (pow (* 1.0 a) 2.0))))

Derivation

1. Initial program 80.6%

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

3. Taylor expanded in angle around 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 80.6

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

5. Applied rewrites 80.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-fma.f64 80.7

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

7. Applied rewrites 80.7%

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

8. Taylor expanded in angle around 0

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

10. Applied rewrites 80.3%

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

Alternative 6: 50.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 6.9 \cdot 10^{-180}:\\ \;\;\;\;{\sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{2} \cdot \left(b \cdot b\right)\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(angle \cdot angle\right) \cdot b\right) \cdot b, -3.08641975308642 \cdot 10^{-5} \cdot \left(\frac{\left(a \cdot a\right) \cdot \mathsf{PI}\left(\right)}{b} \cdot \frac{\mathsf{PI}\left(\right)}{b} - \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;{\cos \left(-0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}^{2} \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 6.9e-180)
   (* (pow (sin (* (* (PI) 0.005555555555555556) angle)) 2.0) (* b b))
   (if (<= a 1.2e+38)
     (fma
      (* (* (* angle angle) b) b)
      (*
       -3.08641975308642e-5
       (- (* (/ (* (* a a) (PI)) b) (/ (PI) b)) (* (PI) (PI))))
      (* a a))
     (* (pow (cos (* -0.005555555555555556 (* (PI) angle))) 2.0) (* a a)))))

Derivation

1. Split input into 3 regimes

2. if a < 6.9000000000000003e-180

   1. Initial program 81.4%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 38.8

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

   5. Applied rewrites 38.8%

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

   if 6.9000000000000003e-180 < a < 1.20000000000000009e38

   1. Initial program 73.1%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 47.7%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 50.6%

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

   if 1.20000000000000009e38 < a

   1. Initial program 84.6%

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

   3. Taylor expanded in angle around 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 84.6

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

   5. Applied rewrites 84.6%

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

   7. Applied rewrites 84.6%

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

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. cos-neg-rev N/A

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

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

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

      6. metadata-eval N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 79.8

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

   10. Applied rewrites 79.8%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 6.9 \cdot 10^{-180}:\\ \;\;\;\;{\sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{2} \cdot \left(b \cdot b\right)\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(angle \cdot angle\right) \cdot b\right) \cdot b, -3.08641975308642 \cdot 10^{-5} \cdot \left(\frac{\left(a \cdot a\right) \cdot \mathsf{PI}\left(\right)}{b} \cdot \frac{\mathsf{PI}\left(\right)}{b} - \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;{\cos \left(-0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}^{2} \cdot \left(a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 6.9 \cdot 10^{-180}:\\ \;\;\;\;{\sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot 0.005555555555555556\right)}^{2} \cdot \left(b \cdot b\right)\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(angle \cdot angle\right) \cdot b\right) \cdot b, -3.08641975308642 \cdot 10^{-5} \cdot \left(\frac{\left(a \cdot a\right) \cdot \mathsf{PI}\left(\right)}{b} \cdot \frac{\mathsf{PI}\left(\right)}{b} - \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;{\cos \left(-0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}^{2} \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 6.9e-180)
   (* (pow (sin (* (* angle (PI)) 0.005555555555555556)) 2.0) (* b b))
   (if (<= a 1.2e+38)
     (fma
      (* (* (* angle angle) b) b)
      (*
       -3.08641975308642e-5
       (- (* (/ (* (* a a) (PI)) b) (/ (PI) b)) (* (PI) (PI))))
      (* a a))
     (* (pow (cos (* -0.005555555555555556 (* (PI) angle))) 2.0) (* a a)))))

Derivation

1. Split input into 3 regimes

2. if a < 6.9000000000000003e-180

   1. Initial program 81.4%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 55.2%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 36.3%

      \[\leadsto \left(\frac{a}{b} \cdot \frac{a}{b}\right) \cdot \left(\color{blue}{b} \cdot b\right) \]

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 38.6%

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

   if 6.9000000000000003e-180 < a < 1.20000000000000009e38

   1. Initial program 73.1%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 47.7%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 50.6%

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

   if 1.20000000000000009e38 < a

   1. Initial program 84.6%

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

   3. Taylor expanded in angle around 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 84.6

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

   5. Applied rewrites 84.6%

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

   7. Applied rewrites 84.6%

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

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. cos-neg-rev N/A

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

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

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

      6. metadata-eval N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 79.8

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

   10. Applied rewrites 79.8%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 6.9 \cdot 10^{-180}:\\ \;\;\;\;{\sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot 0.005555555555555556\right)}^{2} \cdot \left(b \cdot b\right)\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(angle \cdot angle\right) \cdot b\right) \cdot b, -3.08641975308642 \cdot 10^{-5} \cdot \left(\frac{\left(a \cdot a\right) \cdot \mathsf{PI}\left(\right)}{b} \cdot \frac{\mathsf{PI}\left(\right)}{b} - \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;{\cos \left(-0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}^{2} \cdot \left(a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.2 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(angle \cdot angle\right) \cdot b\right) \cdot b, -3.08641975308642 \cdot 10^{-5} \cdot \left(\frac{\left(a \cdot a\right) \cdot \mathsf{PI}\left(\right)}{b} \cdot \frac{\mathsf{PI}\left(\right)}{b} - \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;{\cos \left(-0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}^{2} \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.2e+38)
   (fma
    (* (* (* angle angle) b) b)
    (*
     -3.08641975308642e-5
     (- (* (/ (* (* a a) (PI)) b) (/ (PI) b)) (* (PI) (PI))))
    (* a a))
   (* (pow (cos (* -0.005555555555555556 (* (PI) angle))) 2.0) (* a a))))

Derivation

1. Split input into 2 regimes

2. if a < 1.20000000000000009e38

   1. Initial program 79.5%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 53.5%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 46.1%

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

   if 1.20000000000000009e38 < a

   1. Initial program 84.6%

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

   3. Taylor expanded in angle around 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 84.6

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

   5. Applied rewrites 84.6%

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

   7. Applied rewrites 84.6%

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

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. cos-neg-rev N/A

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

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

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

      6. metadata-eval N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 79.8

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

   10. Applied rewrites 79.8%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.2 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(angle \cdot angle\right) \cdot b\right) \cdot b, -3.08641975308642 \cdot 10^{-5} \cdot \left(\frac{\left(a \cdot a\right) \cdot \mathsf{PI}\left(\right)}{b} \cdot \frac{\mathsf{PI}\left(\right)}{b} - \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;{\cos \left(-0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}^{2} \cdot \left(a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.6% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (fma
  (* 1.0 a)
  a
  (pow (* (sin (* (* 0.005555555555555556 (PI)) angle)) b) 2.0)))

Derivation

1. Initial program 80.6%

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

3. Taylor expanded in angle around 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 80.6

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

5. Applied rewrites 80.6%

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

7. Applied rewrites 80.7%

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

8. Taylor expanded in angle around 0

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

10. Applied rewrites 80.3%

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

Alternative 10: 52.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.2 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(angle \cdot angle\right) \cdot b\right) \cdot b, -3.08641975308642 \cdot 10^{-5} \cdot \left(\frac{\left(a \cdot a\right) \cdot \mathsf{PI}\left(\right)}{b} \cdot \frac{\mathsf{PI}\left(\right)}{b} - \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, 0.011111111111111112, 0.5 \cdot \mathsf{PI}\left(\right)\right)\right), 0.5, 0.5\right) \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.2e+38)
   (fma
    (* (* (* angle angle) b) b)
    (*
     -3.08641975308642e-5
     (- (* (/ (* (* a a) (PI)) b) (/ (PI) b)) (* (PI) (PI))))
    (* a a))
   (*
    (fma (sin (fma (* (PI) angle) 0.011111111111111112 (* 0.5 (PI)))) 0.5 0.5)
    (* a a))))

Derivation

1. Split input into 2 regimes

2. if a < 1.20000000000000009e38

   1. Initial program 79.5%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 53.5%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 46.1%

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

   if 1.20000000000000009e38 < a

   1. Initial program 84.6%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 55.7%

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

   7. Applied rewrites 50.9%

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

   9. Applied rewrites 55.6%

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

   10. Taylor expanded in a around inf

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

   12. Applied rewrites 79.6%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.2 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(angle \cdot angle\right) \cdot b\right) \cdot b, -3.08641975308642 \cdot 10^{-5} \cdot \left(\frac{\left(a \cdot a\right) \cdot \mathsf{PI}\left(\right)}{b} \cdot \frac{\mathsf{PI}\left(\right)}{b} - \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, 0.011111111111111112, 0.5 \cdot \mathsf{PI}\left(\right)\right)\right), 0.5, 0.5\right) \cdot \left(a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 52.2% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.2 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(angle \cdot angle\right) \cdot b\right) \cdot b, -3.08641975308642 \cdot 10^{-5} \cdot \left(\frac{\left(a \cdot a\right) \cdot \mathsf{PI}\left(\right)}{b} \cdot \frac{\mathsf{PI}\left(\right)}{b} - \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.2e+38)
   (fma
    (* (* (* angle angle) b) b)
    (*
     -3.08641975308642e-5
     (- (* (/ (* (* a a) (PI)) b) (/ (PI) b)) (* (PI) (PI))))
    (* a a))
   (* a a)))

Derivation

1. Split input into 2 regimes

2. if a < 1.20000000000000009e38

   1. Initial program 79.5%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 53.5%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 46.1%

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

   if 1.20000000000000009e38 < a

   1. Initial program 84.6%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 79.9

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

   5. Applied rewrites 79.9%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.2 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(angle \cdot angle\right) \cdot b\right) \cdot b, -3.08641975308642 \cdot 10^{-5} \cdot \left(\frac{\left(a \cdot a\right) \cdot \mathsf{PI}\left(\right)}{b} \cdot \frac{\mathsf{PI}\left(\right)}{b} - \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 54.1% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 4 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, b \cdot b, -3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right), angle \cdot angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 4e+31)
   (fma
    (*
     (* (PI) (PI))
     (fma 3.08641975308642e-5 (* b b) (* -3.08641975308642e-5 (* a a))))
    (* angle angle)
    (* a a))
   (* a a)))

Derivation

1. Split input into 2 regimes

2. if a < 3.9999999999999999e31

   1. Initial program 79.3%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 79.4

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

   5. Applied rewrites 79.4%

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

   6. Taylor expanded in angle around 0

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

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

   8. Applied rewrites 47.1%

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

   if 3.9999999999999999e31 < a

   1. Initial program 85.1%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 78.9

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

   5. Applied rewrites 78.9%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, b \cdot b, -3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right), angle \cdot angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 54.0% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 4 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(\left(-3.08641975308642 \cdot 10^{-5} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(a \cdot a - b \cdot b\right), angle \cdot angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 4e+31)
   (fma
    (* (* -3.08641975308642e-5 (* (PI) (PI))) (- (* a a) (* b b)))
    (* angle angle)
    (* a a))
   (* a a)))

Derivation

1. Split input into 2 regimes

2. if a < 3.9999999999999999e31

   1. Initial program 79.3%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 47.1%

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

   if 3.9999999999999999e31 < a

   1. Initial program 85.1%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 78.9

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

   5. Applied rewrites 78.9%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(\left(-3.08641975308642 \cdot 10^{-5} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(a \cdot a - b \cdot b\right), angle \cdot angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.1% accurate, 74.7× speedup

Math

\[\begin{array}{l} \\ a \cdot a \end{array} \]

FPCore

(FPCore (a b angle) :precision binary64 (* a a))

C

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

Fortran

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

Java

public static double code(double a, double b, double angle) {
	return a * a;
}

Python

def code(a, b, angle):
	return a * a

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_] := N[(a * a), $MachinePrecision]

Derivation

1. Initial program 80.6%

   \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
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 53.3

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

5. Applied rewrites 53.3%

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

6. Final simplification 53.3%

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

Reproduce

herbie shell --seed 2024346 
(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)))