ab-angle->ABCF C

Percentage Accurate: 79.8% → 79.7%

Time: 9.9s

Alternatives: 11

Speedup: N/A×

• Could not converge on a ground truth

  1. a = 6.400571720825848e+113
  2. b = 5.193508970485947e-134
  3. angle = 1.1805387580211728e+201

• 36.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

Initial Program: 79.8% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 79.7% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 76.9%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 77.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. associate-*r/ N/A

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

   5. div-inv N/A

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

   6. *-commutative N/A

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

   7. times-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. inv-pow N/A

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

   12. lower-pow.f64 77.3

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

7. Applied rewrites 77.3%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 77.3

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 77.3

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

9. Applied rewrites 77.3%

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

    1. lift-/.f64 N/A

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

    2. frac-2neg N/A

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

    3. lower-/.f64 N/A

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

    4. lower-neg.f64 N/A

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

    5. lift-pow.f64 N/A

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

    6. unpow-1 N/A

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

    7. distribute-neg-frac N/A

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

    8. metadata-eval N/A

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

    9. lower-/.f64 77.3

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

11. Applied rewrites 77.3%

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

12. Final simplification 77.3%

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

Alternative 2: 79.7% accurate, 1.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 76.9%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 77.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. associate-*r/ N/A

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

   5. div-inv N/A

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

   6. *-commutative N/A

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

   7. times-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. inv-pow N/A

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

   12. lower-pow.f64 77.3

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

7. Applied rewrites 77.3%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 77.3

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 77.3

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

9. Applied rewrites 77.3%

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

    1. lift-+.f64 N/A

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

11. Applied rewrites 77.3%

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

Alternative 3: 79.8% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 76.9%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 77.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. associate-*r/ N/A

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

   5. div-inv N/A

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

   6. *-commutative N/A

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

   7. times-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. inv-pow N/A

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

   12. lower-pow.f64 77.3

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

7. Applied rewrites 77.3%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 77.3

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 77.3

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

9. Applied rewrites 77.3%

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

    1. lift-/.f64 N/A

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

    2. div-inv N/A

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

    3. lift-pow.f64 N/A

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

    4. unpow-1 N/A

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

    5. remove-double-div N/A

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

    6. lower-*.f64 77.3

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

11. Applied rewrites 77.3%

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

Alternative 4: 79.8% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 76.9%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 77.2%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

7. Applied rewrites 77.2%

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

Alternative 5: 66.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.5 \cdot 10^{-35}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 \cdot a\right) \cdot 1\right) \cdot a + {\left(b \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 0.005555555555555556\right)\right) \cdot angle\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 3.5e-35)
   (* a a)
   (+
    (* (* (* 1.0 a) 1.0) a)
    (pow
     (*
      b
      (*
       (*
        (PI)
        (fma
         (* -2.8577960676726107e-8 (* angle angle))
         (* (PI) (PI))
         0.005555555555555556))
       angle))
     2.0))))

Derivation

1. Split input into 2 regimes

2. if b < 3.49999999999999996e-35

   1. Initial program 76.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 63.8

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

   5. Applied rewrites 63.8%

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

   if 3.49999999999999996e-35 < b

   1. Initial program 76.7%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 76.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-*r/ N/A

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

      5. div-inv N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. inv-pow N/A

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

      12. lower-pow.f64 76.7

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

   7. Applied rewrites 76.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 76.7

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 76.7

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

   9. Applied rewrites 76.7%

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

   10. Taylor expanded in angle around 0

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

       1. *-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. +-commutative N/A

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

       6. lower-*.f64 N/A

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

   12. Applied rewrites 73.1%

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

Alternative 6: 64.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.5 \cdot 10^{-35}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;b \leq 5.1 \cdot 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right) \cdot b, angle \cdot angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 3.5e-35)
   (* a a)
   (if (<= b 5.1e+167)
     (fma
      (* (* (* (* (PI) (PI)) 3.08641975308642e-5) b) b)
      (* angle angle)
      (* a a))
     (* (pow (* (* b (PI)) angle) 2.0) 3.08641975308642e-5))))

Derivation

1. Split input into 3 regimes

2. if b < 3.49999999999999996e-35

   1. Initial program 76.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 63.8

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

   5. Applied rewrites 63.8%

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

   if 3.49999999999999996e-35 < b < 5.10000000000000004e167

   1. Initial program 62.7%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 32.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 53.1%

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

   if 5.10000000000000004e167 < b

   1. Initial program 99.7%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 56.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 68.5%

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

   10. Applied rewrites 88.1%

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

Alternative 7: 62.1% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.5 \cdot 10^{-35}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right) \cdot b, angle \cdot angle, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 3.5e-35)
   (* a a)
   (fma
    (* (* (* (* (PI) (PI)) 3.08641975308642e-5) b) b)
    (* angle angle)
    (* a a))))

Derivation

1. Split input into 2 regimes

2. if b < 3.49999999999999996e-35

   1. Initial program 76.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 63.8

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

   5. Applied rewrites 63.8%

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

   if 3.49999999999999996e-35 < b

   1. Initial program 76.7%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 41.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 58.9%

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

Alternative 8: 60.9% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{+134}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.5e+134)
   (* a a)
   (* (* (* (* angle angle) 3.08641975308642e-5) b) (* (* (PI) (PI)) b))))

Derivation

1. Split input into 2 regimes

2. if b < 1.49999999999999998e134

   1. Initial program 74.7%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.2

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

   5. Applied rewrites 61.2%

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

   if 1.49999999999999998e134 < b

   1. Initial program 92.0%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 56.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 62.5%

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

   10. Applied rewrites 65.6%

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

Alternative 9: 60.2% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{+134}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right) \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.5e+134)
   (* a a)
   (* (* (* angle angle) 3.08641975308642e-5) (* (* (* (PI) (PI)) b) b))))

Derivation

1. Split input into 2 regimes

2. if b < 1.49999999999999998e134

   1. Initial program 74.7%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.2

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

   5. Applied rewrites 61.2%

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

   if 1.49999999999999998e134 < b

   1. Initial program 92.0%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 56.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 62.5%

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

Alternative 10: 60.2% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{+134}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.5e+134)
   (* a a)
   (* (* (PI) (PI)) (* (* b b) (* (* angle angle) 3.08641975308642e-5)))))

Derivation

1. Split input into 2 regimes

2. if b < 1.49999999999999998e134

   1. Initial program 74.7%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.2

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

   5. Applied rewrites 61.2%

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

   if 1.49999999999999998e134 < b

   1. Initial program 92.0%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 56.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 62.5%

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

   10. Applied rewrites 62.5%

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

Alternative 11: 57.1% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.9%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 56.0

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

5. Applied rewrites 56.0%

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

Reproduce

herbie shell --seed 2024309 
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* (PI) (/ angle 180.0)))) 2.0) (pow (* b (sin (* (PI) (/ angle 180.0)))) 2.0)))