ab-angle->ABCF A

Percentage Accurate: 80.2% → 80.2%

Time: 12.1s

Alternatives: 8

Speedup: 1.9×

• Could not converge on a ground truth

  1. a = 1.4879705186589467e+280
  2. b = 7.85147976610769e-154
  3. angle = 5.687657170200855e+29

• 36.3% 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: 80.2% 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: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(1 \cdot b\right)}^{2} + {\left(\sin \left(\frac{1}{{\mathsf{PI}\left(\right)}^{-0.5}} \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \cdot a\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* 1.0 b) 2.0)
  (pow
   (*
    (sin
     (*
      (/ 1.0 (pow (PI) -0.5))
      (* angle (* 0.005555555555555556 (sqrt (PI))))))
    a)
   2.0)))

Derivation

1. Initial program 78.5%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 79.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. rem-square-sqrt N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r/ N/A

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

   11. clear-num N/A

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

   12. associate-*r/ N/A

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

   13. div-inv N/A

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

   14. times-frac N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. associate-*r/ N/A

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

   18. lift-/.f64 N/A

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

7. Applied rewrites 79.1%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot 0.005555555555555556\right) \cdot angle\right) \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{-0.5}}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

8. Final simplification 79.1%

   \[\leadsto {\left(1 \cdot b\right)}^{2} + {\left(\sin \left(\frac{1}{{\mathsf{PI}\left(\right)}^{-0.5}} \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \cdot a\right)}^{2} \]
9. Add Preprocessing

Alternative 2: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(\sin \left(\frac{angle}{{\mathsf{PI}\left(\right)}^{-0.5}} \cdot \left(0.005555555555555556 \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \cdot a\right)}^{2} + {\left(1 \cdot b\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow
   (*
    (sin (* (/ angle (pow (PI) -0.5)) (* 0.005555555555555556 (sqrt (PI)))))
    a)
   2.0)
  (pow (* 1.0 b) 2.0)))

Derivation

1. Initial program 78.5%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 79.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. rem-square-sqrt N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r/ N/A

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

   11. clear-num N/A

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

   12. un-div-inv N/A

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

   13. lift-*.f64 N/A

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

   14. div-inv N/A

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

   15. times-frac N/A

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

   16. lift-/.f64 N/A

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

   17. lower-*.f64 N/A

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

7. Applied rewrites 79.1%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot 0.005555555555555556\right) \cdot \frac{angle}{{\mathsf{PI}\left(\right)}^{-0.5}}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

8. Final simplification 79.1%

   \[\leadsto {\left(\sin \left(\frac{angle}{{\mathsf{PI}\left(\right)}^{-0.5}} \cdot \left(0.005555555555555556 \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \cdot a\right)}^{2} + {\left(1 \cdot b\right)}^{2} \]
9. Add Preprocessing

Alternative 3: 80.1% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 78.5%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 79.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 79.1

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

7. Applied rewrites 79.1%

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

8. Final simplification 79.1%

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

Alternative 4: 80.2% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 78.5%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 79.0%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
6. 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 79.0%

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

8. Final simplification 79.0%

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

Alternative 5: 68.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.75 \cdot 10^{-77}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot a\right)}^{2} + {\left(1 \cdot b\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.75e-77)
   (* b b)
   (+
    (pow (* (* (* 0.005555555555555556 (PI)) angle) a) 2.0)
    (pow (* 1.0 b) 2.0))))

Derivation

1. Split input into 2 regimes

2. if a < 1.75000000000000006e-77

   1. Initial program 77.1%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
   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 61.2

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

   5. Applied rewrites 61.2%

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

   if 1.75000000000000006e-77 < a

   1. Initial program 81.5%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 80.9%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 78.2

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

   8. Applied rewrites 78.2%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.75 \cdot 10^{-77}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot a\right)}^{2} + {\left(1 \cdot b\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.9% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;a \leq 2.3 \cdot 10^{-77}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;a \leq 2.4 \cdot 10^{+162}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot t\_0, angle \cdot angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(angle \cdot angle\right) \cdot a\right) \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (PI) (PI))))
   (if (<= a 2.3e-77)
     (* b b)
     (if (<= a 2.4e+162)
       (fma (* (* (* a a) 3.08641975308642e-5) t_0) (* angle angle) (* b b))
       (* (* (* (* (* angle angle) a) a) 3.08641975308642e-5) t_0)))))

Derivation

1. Split input into 3 regimes

2. if a < 2.29999999999999999e-77

   1. Initial program 77.1%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
   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 61.2

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

   5. Applied rewrites 61.2%

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

   if 2.29999999999999999e-77 < a < 2.40000000000000009e162

   1. Initial program 64.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 64.4

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 64.4

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 64.3

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

   4. Applied rewrites 64.3%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {b}^{2}} \]
   6. 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

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

   7. Applied rewrites 36.8%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 55.7%

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

   if 2.40000000000000009e162 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.6

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.6

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {b}^{2}} \]
   6. 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

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

   7. Applied rewrites 40.4%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 69.4%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.3 \cdot 10^{-77}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;a \leq 2.4 \cdot 10^{+162}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), angle \cdot angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(angle \cdot angle\right) \cdot a\right) \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.2% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.1 \cdot 10^{+63}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(angle \cdot angle\right) \cdot a\right) \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 2.1e+63)
   (* b b)
   (* (* (* (* (* angle angle) a) a) 3.08641975308642e-5) (* (PI) (PI)))))

Derivation

1. Split input into 2 regimes

2. if a < 2.1000000000000002e63

   1. Initial program 75.1%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
   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 59.0

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

   5. Applied rewrites 59.0%

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

   if 2.1000000000000002e63 < a

   1. Initial program 91.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 91.8

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 91.8

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 91.7

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

   4. Applied rewrites 91.7%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {b}^{2}} \]
   6. 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

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

   7. Applied rewrites 41.8%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 64.9%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.1 \cdot 10^{+63}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(angle \cdot angle\right) \cdot a\right) \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.6% 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 78.5%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 53.4

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

5. Applied rewrites 53.4%

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

Reproduce

herbie shell --seed 2024267 
(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)))