ab-angle->ABCF A

Percentage Accurate: 80.2% → 80.0%

Time: 10.9s

Alternatives: 10

Speedup: N/A×

• Could not converge on a ground truth

  1. a = 4.6065760438252776e-290
  2. b = 4.661262043681911e-148
  3. angle = 3.9765858974306176e+154

• 37.4% 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: 80.0% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 75.5%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
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 75.6%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-/.f64 N/A

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

   4. lower-/.f64 75.7

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

7. Applied rewrites 75.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. associate-*l/ N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. lower-*.f64 75.7

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

9. Applied rewrites 75.7%

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

    1. lift-+.f64 N/A

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

    2. +-commutative N/A

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

    3. lift-pow.f64 N/A

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

    4. unpow2 N/A

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

    5. lift-*.f64 N/A

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

    6. *-commutative N/A

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

    7. associate-*r* N/A

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

    8. lower-fma.f64 N/A

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

11. Applied rewrites 75.7%

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

12. Final simplification 75.7%

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

Alternative 2: 67.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 4.8 \cdot 10^{-33}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(b \cdot 1\right)}^{2} + {\left(\left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right) \cdot a\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 4.8e-33)
   (* b b)
   (+
    (pow (* b 1.0) 2.0)
    (pow (* (* (* (PI) 0.005555555555555556) angle) a) 2.0))))

Derivation

1. Split input into 2 regimes

2. if a < 4.8e-33

   1. Initial program 72.6%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 55.7

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

   5. Applied rewrites 55.7%

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

   if 4.8e-33 < a

   1. Initial program 82.9%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 82.2%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 78.9

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

   8. Applied rewrites 78.9%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.8 \cdot 10^{-33}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(b \cdot 1\right)}^{2} + {\left(\left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right) \cdot a\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 4.8 \cdot 10^{-33}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(a \cdot \mathsf{PI}\left(\right)\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{2} + {\left(b \cdot 1\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 4.8e-33)
   (* b b)
   (+
    (pow (* (* (* a (PI)) 0.005555555555555556) angle) 2.0)
    (pow (* b 1.0) 2.0))))

Derivation

1. Split input into 2 regimes

2. if a < 4.8e-33

   1. Initial program 72.6%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 55.7

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

   5. Applied rewrites 55.7%

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

   if 4.8e-33 < a

   1. Initial program 82.9%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 82.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-/.f64 N/A

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

      4. lower-/.f64 82.2

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

   7. Applied rewrites 82.2%

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

   8. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 79.0

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

   10. Applied rewrites 79.0%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.8 \cdot 10^{-33}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(a \cdot \mathsf{PI}\left(\right)\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{2} + {\left(b \cdot 1\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.1% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 75.5%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
2. Add Preprocessing
3. 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. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

4. Applied rewrites 69.5%

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

6. Applied rewrites 75.6%

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

7. Taylor expanded in angle around 0

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

9. Applied rewrites 75.7%

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

10. Final simplification 75.7%

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

Alternative 5: 65.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 4.8 \cdot 10^{-33}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(\left(\left(angle \cdot angle\right) \cdot a\right) \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) + {\left(b \cdot 1\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 4.8e-33)
   (* b b)
   (+
    (* (* (PI) (PI)) (* (* (* (* angle angle) a) a) 3.08641975308642e-5))
    (pow (* b 1.0) 2.0))))

Derivation

1. Split input into 2 regimes

2. if a < 4.8e-33

   1. Initial program 72.6%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 55.7

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

   5. Applied rewrites 55.7%

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

   if 4.8e-33 < a

   1. Initial program 82.9%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 82.2%

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

   6. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. lower-PI.f64 70.6

          \[\leadsto \left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot \left(\left(angle \cdot angle\right) \cdot a\right)\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) + {\left(b \cdot 1\right)}^{2} \]

   8. Applied rewrites 70.6%

      \[\leadsto \color{blue}{\left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot \left(\left(angle \cdot angle\right) \cdot a\right)\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.8 \cdot 10^{-33}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(\left(\left(angle \cdot angle\right) \cdot a\right) \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) + {\left(b \cdot 1\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.7% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 4.8 \cdot 10^{-33}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;a \leq 1.75 \cdot 10^{+143}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), angle \cdot angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(a \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 4.8e-33)
   (* b b)
   (if (<= a 1.75e+143)
     (fma
      (* (* (* a a) 3.08641975308642e-5) (* (PI) (PI)))
      (* angle angle)
      (* b b))
     (* (pow (* (* a (PI)) angle) 2.0) 3.08641975308642e-5))))

Derivation

1. Split input into 3 regimes

2. if a < 4.8e-33

   1. Initial program 72.6%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 55.7

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

   5. Applied rewrites 55.7%

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

   if 4.8e-33 < a < 1.75000000000000004e143

   1. Initial program 74.1%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 43.6%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 64.3%

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

   if 1.75000000000000004e143 < a

   1. Initial program 94.3%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 14.7%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 40.6%

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

   10. Applied rewrites 60.1%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.8 \cdot 10^{-33}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;a \leq 1.75 \cdot 10^{+143}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), angle \cdot angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(a \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.6% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{-163}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+166}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), angle \cdot angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) 4e-163)
   (* b b)
   (if (<= (/ angle 180.0) 1e+166)
     (fma
      (* (* (* a a) 3.08641975308642e-5) (* (PI) (PI)))
      (* angle angle)
      (* b b))
     (* b b))))

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 3.99999999999999969e-163 or 9.9999999999999994e165 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 77.3%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 53.8

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

   5. Applied rewrites 53.8%

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

   if 3.99999999999999969e-163 < (/.f64 angle #s(literal 180 binary64)) < 9.9999999999999994e165

   1. Initial program 69.3%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 26.7%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 63.5%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{-163}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+166}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), angle \cdot angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.0% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.028:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5} \cdot a, a, -3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right), angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 0.028)
   (fma
    (*
     (* (* (PI) (PI)) angle)
     (fma (* 3.08641975308642e-5 a) a (* -3.08641975308642e-5 (* b b))))
    angle
    (* b b))
   (* b b)))

Derivation

1. Split input into 2 regimes

2. if b < 0.0280000000000000006

   1. Initial program 72.3%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 36.1%

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

   7. Applied rewrites 42.4%

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

   if 0.0280000000000000006 < b

   1. Initial program 86.4%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 76.4

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

   5. Applied rewrites 76.4%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.028:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5} \cdot a, a, -3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right), angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.0% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.6 \cdot 10^{-105}:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(\left(\left(angle \cdot angle\right) \cdot a\right) \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 6.6e-105)
   (* (* (PI) (PI)) (* (* (* (* angle angle) a) a) 3.08641975308642e-5))
   (* b b)))

Derivation

1. Split input into 2 regimes

2. if b < 6.5999999999999997e-105

   1. Initial program 74.5%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 35.0%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 31.6%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 33.2%

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

   if 6.5999999999999997e-105 < b

   1. Initial program 77.5%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
   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 63.2

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

   5. Applied rewrites 63.2%

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

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.6 \cdot 10^{-105}:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(\left(\left(angle \cdot angle\right) \cdot a\right) \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.0% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
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 53.6

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

5. Applied rewrites 53.6%

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

Reproduce

herbie shell --seed 2024296 
(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)))