ab-angle->ABCF A

Percentage Accurate: 79.1% → 79.1%

Time: 27.1s

Alternatives: 14

Speedup: 1.2×

• 36.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

Initial Program: 79.1% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 79.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(0.005555555555555556, angle, 0.5\right)\right)\right)}^{2} \end{array} \]

FPCore

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

Derivation

1. Initial program 82.2%

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

   1. lift-cos.f64 N/A

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

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

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

   3. lower-sin.f64 N/A

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

   4. remove-double-neg N/A

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

   5. remove-double-neg N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lower-/.f64 82.3

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

4. Applied rewrites 82.3%

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

   1. lift-PI.f64 N/A

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

   2. add-cbrt-cube N/A

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

   3. lower-cbrt.f64 N/A

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

   4. rem-cube-cbrt N/A

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

   5. add-cbrt-cube N/A

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

   6. lift-PI.f64 N/A

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

   7. lower-pow.f64 82.3

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

6. Applied rewrites 82.3%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \color{blue}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{angle}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}^{2} \]
7. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. unpow3 N/A

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

   3. lift-*.f64 N/A

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

   4. lower-*.f64 82.3

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

8. Applied rewrites 82.3%

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

9. Taylor expanded in angle around inf

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

    1. lower-sin.f64 N/A

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

    2. associate-*r* N/A

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

    3. distribute-rgt-out N/A

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

    4. lower-*.f64 N/A

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

    5. lower-PI.f64 N/A

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

    6. lower-fma.f64 82.3

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

11. Applied rewrites 82.3%

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

12. Final simplification 82.3%

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

Alternative 2: 79.1% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 82.2%

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

   1. lift-cos.f64 N/A

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

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

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

   3. lower-sin.f64 N/A

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

   4. remove-double-neg N/A

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

   5. remove-double-neg N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lower-/.f64 82.3

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

4. Applied rewrites 82.3%

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

5. Taylor expanded in angle around inf

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

   1. lower-sin.f64 N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt-out N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. lower-fma.f64 82.3

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

7. Applied rewrites 82.3%

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

8. Final simplification 82.3%

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

Alternative 3: 79.1% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 82.2%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

4. Applied rewrites 82.1%

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

6. Applied rewrites 82.3%

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

Alternative 4: 79.1% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* angle (PI))))
   (fma
    (- 0.5 (* 0.5 (cos (* 2.0 (fma t_0 -0.005555555555555556 (/ (PI) 2.0))))))
    (* b b)
    (pow (* (sin (* 0.005555555555555556 t_0)) a) 2.0))))

Derivation

1. Initial program 82.2%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 56.6

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

5. Applied rewrites 56.6%

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

6. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. cos-neg-rev N/A

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

   6. lower-cos.f64 N/A

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

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

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

   8. metadata-eval N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-PI.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

8. Applied rewrites 71.3%

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

10. Applied rewrites 71.4%

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

12. Applied rewrites 82.2%

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

Alternative 5: 79.0% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+ (pow (* a (sin (* (/ angle 180.0) (PI)))) 2.0) (pow (* b 1.0) 2.0)))

Derivation

1. Initial program 82.2%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 81.0%

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

Alternative 6: 64.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 7 \cdot 10^{-146}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, b \cdot b, {\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2} \cdot \left(a \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 7e-146)
   (* b b)
   (fma
    1.0
    (* b b)
    (* (pow (sin (* (* (PI) angle) 0.005555555555555556)) 2.0) (* a a)))))

Derivation

1. Split input into 2 regimes

2. if angle < 7.0000000000000003e-146

   1. Initial program 83.0%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 56.1

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

   5. Applied rewrites 56.1%

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

   if 7.0000000000000003e-146 < angle

   1. Initial program 80.5%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.4

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

   5. Applied rewrites 57.4%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. cos-neg-rev N/A

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

      6. lower-cos.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-PI.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   8. Applied rewrites 79.4%

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

   9. Taylor expanded in angle around 0

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

   11. Applied rewrites 77.7%

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

Alternative 7: 79.0% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (fma (* 1.0 b) b (pow (* (sin (* (PI) (/ angle 180.0))) a) 2.0)))

Derivation

1. Initial program 82.2%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

4. Applied rewrites 82.1%

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

5. Taylor expanded in angle around 0

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

7. Applied rewrites 81.0%

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

Alternative 8: 57.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.16 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -3.08641975308642 \cdot 10^{-5}\right) \cdot angle\right), angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;{\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(0.005555555555555556, angle, 0.5\right)\right)}^{2} \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.16e+91)
   (fma
    (* (* (- b a) (+ b a)) (* (* (* (PI) (PI)) -3.08641975308642e-5) angle))
    angle
    (* b b))
   (* (pow (sin (* (PI) (fma 0.005555555555555556 angle 0.5))) 2.0) (* b b))))

Derivation

1. Split input into 2 regimes

2. if b < 1.1600000000000001e91

   1. Initial program 80.1%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 43.5%

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

   7. Applied rewrites 51.1%

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

   if 1.1600000000000001e91 < b

   1. Initial program 90.6%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. remove-double-neg N/A

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

      5. remove-double-neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 91.1

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

   4. Applied rewrites 91.1%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 83.7

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

   7. Applied rewrites 83.7%

      \[\leadsto \color{blue}{{\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(0.005555555555555556, angle, 0.5\right)\right)}^{2} \cdot \left(b \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.16 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -3.08641975308642 \cdot 10^{-5}\right) \cdot angle\right), angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;{\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(0.005555555555555556, angle, 0.5\right)\right)}^{2} \cdot \left(b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.16 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -3.08641975308642 \cdot 10^{-5}\right) \cdot angle\right), angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;{\cos \left(-0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}^{2} \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.16e+91)
   (fma
    (* (* (- b a) (+ b a)) (* (* (* (PI) (PI)) -3.08641975308642e-5) angle))
    angle
    (* b b))
   (* (pow (cos (* -0.005555555555555556 (* (PI) angle))) 2.0) (* b b))))

Derivation

1. Split input into 2 regimes

2. if b < 1.1600000000000001e91

   1. Initial program 80.1%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 43.5%

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

   7. Applied rewrites 51.1%

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

   if 1.1600000000000001e91 < b

   1. Initial program 90.6%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 81.7

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

   5. Applied rewrites 81.7%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. cos-neg-rev N/A

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

      5. lower-cos.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 83.2

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

   8. Applied rewrites 83.2%

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

Alternative 10: 52.9% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.8 \cdot 10^{-163}:\\ \;\;\;\;\left(\left(\left(\left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot angle\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right)\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot a\right) \cdot a, angle \cdot angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 8.8e-163)
   (* (* (* (* (* a a) 3.08641975308642e-5) angle) (* angle (PI))) (PI))
   (if (<= b 9.8e+122)
     (fma
      (* (* (* 3.08641975308642e-5 (* (PI) (PI))) a) a)
      (* angle angle)
      (* b b))
     (* b b))))

Derivation

1. Split input into 3 regimes

2. if b < 8.80000000000000044e-163

   1. Initial program 80.7%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 39.4%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 34.0%

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

   10. Applied rewrites 40.0%

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

   if 8.80000000000000044e-163 < b < 9.7999999999999995e122

   1. Initial program 79.4%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 57.0%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 58.2%

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

   if 9.7999999999999995e122 < b

   1. Initial program 90.6%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 86.8

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

   5. Applied rewrites 86.8%

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

Alternative 11: 57.3% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7 \cdot 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -3.08641975308642 \cdot 10^{-5}\right) \cdot angle\right), angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 7e+122)
   (fma
    (* (* (- b a) (+ b a)) (* (* (* (PI) (PI)) -3.08641975308642e-5) angle))
    angle
    (* b b))
   (* b b)))

Derivation

1. Split input into 2 regimes

2. if b < 7.00000000000000028e122

   1. Initial program 80.4%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 43.7%

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

   7. Applied rewrites 51.1%

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

   if 7.00000000000000028e122 < b

   1. Initial program 90.6%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 86.8

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

   5. Applied rewrites 86.8%

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

Alternative 12: 61.1% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.1 \cdot 10^{+160}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot angle\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.1e+160)
   (* b b)
   (* (* (* (* (* a a) 3.08641975308642e-5) angle) (* angle (PI))) (PI))))

Derivation

1. Split input into 2 regimes

2. if a < 1.09999999999999996e160

   1. Initial program 79.5%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 62.2

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

   5. Applied rewrites 62.2%

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

   if 1.09999999999999996e160 < a

   1. Initial program 99.7%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 60.8%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 69.9%

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

   10. Applied rewrites 76.7%

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

Alternative 13: 60.2% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.25 \cdot 10^{+160}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right) \cdot \left(\left(\left(angle \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.25e+160)
   (* b b)
   (* (* 3.08641975308642e-5 (* a a)) (* (* (* angle angle) (PI)) (PI)))))

Derivation

1. Split input into 2 regimes

2. if a < 1.25e160

   1. Initial program 79.5%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 62.2

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

   5. Applied rewrites 62.2%

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

   if 1.25e160 < a

   1. Initial program 99.7%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 60.8%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 69.9%

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

Alternative 14: 56.5% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

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 = b * b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.2%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 56.6

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

5. Applied rewrites 56.6%

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

Reproduce

herbie shell --seed 2025005 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) (PI)))) 2.0) (pow (* b (cos (* (/ angle 180.0) (PI)))) 2.0)))