ab-angle->ABCF A

Percentage Accurate: 80.4% → 80.4%

Time: 10.6s

Alternatives: 9

Speedup: N/A×

• Could not converge on a ground truth

  1. a = 2.9034397470685337e-10
  2. b = 2.613304488515468e-308
  3. angle = 2.81935852094329e+36

• 36.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

Initial Program: 80.4% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 80.4% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 79.6%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 79.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. lower-fma.f64 N/A

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

7. Applied rewrites 80.0%

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

Alternative 2: 80.3% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 79.6%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 79.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. lower-fma.f64 N/A

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

7. Applied rewrites 80.0%

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

   1. lift-sin.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. remove-double-div N/A

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

   4. unpow-1 N/A

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

   5. lift-pow.f64 N/A

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

   6. div-inv N/A

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

   7. frac-2neg N/A

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

   8. distribute-frac-neg2 N/A

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

   9. sin-neg N/A

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

   10. lift-*.f64 N/A

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

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

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lift-pow.f64 N/A

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

   15. unpow-1 N/A

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

   16. div-inv N/A

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

   17. unpow-1 N/A

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

   18. lift-pow.f64 N/A

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

   19. times-frac N/A

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

   20. metadata-eval N/A

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

   21. metadata-eval N/A

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

   22. metadata-eval N/A

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

   23. div-inv N/A

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

   24. lift-pow.f64 N/A

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

   25. unpow-1 N/A

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

9. Applied rewrites 80.0%

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

    1. lift-pow.f64 N/A

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

    2. unpow2 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-neg.f64 N/A

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

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

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

    6. lift-*.f64 N/A

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

    7. lift-neg.f64 N/A

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

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

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

    9. sqr-neg N/A

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

    10. pow2 N/A

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

    11. lower-pow.f64 N/A

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

    12. *-commutative N/A

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

    13. lower-*.f64 80.0

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

    14. lift-*.f64 N/A

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

    15. *-commutative N/A

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

    16. lower-*.f64 80.0

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

11. Applied rewrites 80.0%

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

Alternative 3: 67.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 4500000:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(1 \cdot b\right) \cdot 1, b, {\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 0.005555555555555556\right)\right) \cdot angle\right) \cdot a\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 4500000.0)
   (* b b)
   (fma
    (* (* 1.0 b) 1.0)
    b
    (pow
     (*
      (*
       (*
        (PI)
        (fma
         (* -2.8577960676726107e-8 (* angle angle))
         (* (PI) (PI))
         0.005555555555555556))
       angle)
      a)
     2.0))))

Derivation

1. Split input into 2 regimes

2. if a < 4.5e6

   1. Initial program 79.1%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 63.5

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

   5. Applied rewrites 63.5%

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

   if 4.5e6 < a

   1. Initial program 80.9%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 80.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 81.0%

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

   8. Taylor expanded in angle around 0

      \[\leadsto \mathsf{fma}\left(\left(1 \cdot b\right) \cdot 1, b, {\left(\color{blue}{\left(angle \cdot \left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot a\right)}^{2}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(1 \cdot b\right) \cdot 1, b, {\left(\color{blue}{\left(\left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)} \cdot a\right)}^{2}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(1 \cdot b\right) \cdot 1, b, {\left(\left(\left(\color{blue}{\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot \frac{-1}{34992000}} + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot a\right)}^{2}\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(1 \cdot b\right) \cdot 1, b, {\left(\left(\left(\color{blue}{{angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \frac{-1}{34992000}\right)} + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot a\right)}^{2}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(1 \cdot b\right) \cdot 1, b, {\left(\left(\left({angle}^{2} \cdot \color{blue}{\left(\frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right)} + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot a\right)}^{2}\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(1 \cdot b\right) \cdot 1, b, {\left(\left(\color{blue}{\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)} \cdot angle\right) \cdot a\right)}^{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(1 \cdot b\right) \cdot 1, b, {\left(\color{blue}{\left(\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot angle\right)} \cdot a\right)}^{2}\right) \]

   10. Applied rewrites 76.2%

       \[\leadsto \mathsf{fma}\left(\left(1 \cdot b\right) \cdot 1, b, {\left(\color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 0.005555555555555556\right)\right) \cdot angle\right)} \cdot a\right)}^{2}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 65.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 4500000:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;a \leq 1.75 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, {\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot \left(angle \cdot angle\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(a \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 4500000.0)
   (* b b)
   (if (<= a 1.75e+139)
     (fma
      (* a a)
      (* (pow (* 0.005555555555555556 (PI)) 2.0) (* angle angle))
      (* b b))
     (* (pow (* (* a (PI)) angle) 2.0) 3.08641975308642e-5))))

Derivation

1. Split input into 3 regimes

2. if a < 4.5e6

   1. Initial program 79.1%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 63.5

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

   5. Applied rewrites 63.5%

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

   if 4.5e6 < a < 1.74999999999999989e139

   1. Initial program 67.5%

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

   4. Applied rewrites 25.3%

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

   5. Taylor expanded in angle around 0

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

      1. +-commutative N/A

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

      2. exp-sum N/A

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

      3. rem-exp-log N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

   7. Applied rewrites 56.6%

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

   if 1.74999999999999989e139 < a

   1. Initial program 96.9%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 46.2%

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

   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 65.5%

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

   10. Applied rewrites 93.8%

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

Alternative 5: 67.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 4500000:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(1 \cdot b\right) \cdot 1, b, {\left(\left(\left(-0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \left(-a\right)\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 4500000.0)
   (* b b)
   (fma
    (* (* 1.0 b) 1.0)
    b
    (pow (* (* (* -0.005555555555555556 (PI)) angle) (- a)) 2.0))))

Derivation

1. Split input into 2 regimes

2. if a < 4.5e6

   1. Initial program 79.1%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 63.5

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

   5. Applied rewrites 63.5%

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

   if 4.5e6 < a

   1. Initial program 80.9%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 80.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 81.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. remove-double-div N/A

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

      4. unpow-1 N/A

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

      5. lift-pow.f64 N/A

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

      6. div-inv N/A

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

      7. frac-2neg N/A

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

      8. distribute-frac-neg2 N/A

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

      9. sin-neg N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow-1 N/A

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

      16. div-inv N/A

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

      17. unpow-1 N/A

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

      18. lift-pow.f64 N/A

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

      19. times-frac N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. div-inv N/A

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

      24. lift-pow.f64 N/A

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

      25. unpow-1 N/A

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

   9. Applied rewrites 80.8%

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

   10. Taylor expanded in angle around 0

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-PI.f64 76.3

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

   12. Applied rewrites 76.3%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4500000:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(1 \cdot b\right) \cdot 1, b, {\left(\left(\left(-0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \left(-a\right)\right)}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 4500000:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;a \leq 1.75 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), angle \cdot angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(a \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 4500000.0)
   (* b b)
   (if (<= a 1.75e+139)
     (fma
      (* (* (* 3.08641975308642e-5 (* a a)) (PI)) (PI))
      (* angle angle)
      (* b b))
     (* (pow (* (* a (PI)) angle) 2.0) 3.08641975308642e-5))))

Derivation

1. Split input into 3 regimes

2. if a < 4.5e6

   1. Initial program 79.1%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 63.5

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

   5. Applied rewrites 63.5%

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

   if 4.5e6 < a < 1.74999999999999989e139

   1. Initial program 67.5%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {b}^{2}} \]
   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 26.6%

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

   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 56.6%

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

   if 1.74999999999999989e139 < a

   1. Initial program 96.9%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 46.2%

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

   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 65.5%

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

   10. Applied rewrites 93.8%

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

Alternative 7: 63.9% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 4500000:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), angle \cdot angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot \left(\left(angle \cdot angle\right) \cdot a\right)\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 4500000.0)
   (* b b)
   (if (<= a 1.35e+154)
     (fma
      (* (* (* 3.08641975308642e-5 (* a a)) (PI)) (PI))
      (* angle angle)
      (* b b))
     (* (* 3.08641975308642e-5 (* a (* (* angle angle) a))) (* (PI) (PI))))))

Derivation

1. Split input into 3 regimes

2. if a < 4.5e6

   1. Initial program 79.1%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 63.5

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

   5. Applied rewrites 63.5%

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

   if 4.5e6 < a < 1.35000000000000003e154

   1. Initial program 66.1%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
   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 26.1%

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

   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 55.4%

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

   if 1.35000000000000003e154 < a

   1. Initial program 99.6%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {b}^{2}} \]
   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 47.4%

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

   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 67.4%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 77.3%

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

Alternative 8: 62.5% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 7 \cdot 10^{+135}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot \left(\left(angle \cdot angle\right) \cdot a\right)\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 7e+135)
   (* b b)
   (* (* 3.08641975308642e-5 (* a (* (* angle angle) a))) (* (PI) (PI)))))

Derivation

1. Split input into 2 regimes

2. if a < 7.0000000000000005e135

   1. Initial program 77.3%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 60.5

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

   5. Applied rewrites 60.5%

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

   if 7.0000000000000005e135 < a

   1. Initial program 95.4%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
   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 45.0%

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

   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 63.7%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 73.0%

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

Alternative 9: 58.0% 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

real(8) function code(a, b, angle)
    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 79.6%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
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 55.9

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

5. Applied rewrites 55.9%

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

Reproduce

herbie shell --seed 2024302 
(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)))