ab-angle->ABCF C

Percentage Accurate: 79.6% → 79.6%

Time: 10.2s

Alternatives: 11

Speedup: 2.0×

• Could not converge on a ground truth

  1. a = 1.2466603003669427e+293
  2. b = 1.5979075951715443e+159
  3. angle = 1.4111020494147915e+30

• 36.6% 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: 79.6% 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: 79.6% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 80.4%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 80.8

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

5. Applied rewrites 80.8%

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

6. Final simplification 80.8%

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

Alternative 2: 79.6% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 80.4%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 80.8

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

5. Applied rewrites 80.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 80.8

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-/.f64 N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 80.7

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

7. Applied rewrites 80.7%

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

Alternative 3: 79.6% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 80.4%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 80.8

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

5. Applied rewrites 80.8%

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

6. Taylor expanded in angle around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-PI.f64 80.7

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

8. Applied rewrites 80.7%

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

9. Final simplification 80.7%

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

Alternative 4: 67.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.35 \cdot 10^{-63}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(\mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 0.005555555555555556\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot b\right)}^{2} + a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 2.35e-63)
   (* a a)
   (+
    (pow
     (*
      (*
       (*
        (fma
         (* (* angle angle) -2.8577960676726107e-8)
         (* (PI) (PI))
         0.005555555555555556)
        (PI))
       angle)
      b)
     2.0)
    (* a a))))

Derivation

1. Split input into 2 regimes

2. if b < 2.35e-63

   1. Initial program 77.8%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 62.1

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

   5. Applied rewrites 62.1%

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

   if 2.35e-63 < b

   1. Initial program 85.6%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 85.7

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

   5. Applied rewrites 85.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 85.7

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 85.6

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

   7. Applied rewrites 85.6%

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

   8. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   10. Applied rewrites 83.0%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.35 \cdot 10^{-63}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(\mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 0.005555555555555556\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot b\right)}^{2} + a \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.35 \cdot 10^{-63}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot {\left(b \cdot angle\right)}^{2}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) + a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 2.35e-63)
   (* a a)
   (+
    (* (* 3.08641975308642e-5 (pow (* b angle) 2.0)) (* (PI) (PI)))
    (* a a))))

Derivation

1. Split input into 2 regimes

2. if b < 2.35e-63

   1. Initial program 77.8%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 62.1

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

   5. Applied rewrites 62.1%

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

   if 2.35e-63 < b

   1. Initial program 85.6%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 85.7

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

   5. Applied rewrites 85.7%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-PI.f64 N/A

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

      17. lower-PI.f64 72.3

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

   8. Applied rewrites 72.3%

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

   10. Applied rewrites 73.4%

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

   12. Applied rewrites 83.3%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.35 \cdot 10^{-63}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot {\left(b \cdot angle\right)}^{2}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) + a \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.35 \cdot 10^{-63}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5} + a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 2.35e-63)
   (* a a)
   (+ (* (pow (* (* (PI) b) angle) 2.0) 3.08641975308642e-5) (* a a))))

Derivation

1. Split input into 2 regimes

2. if b < 2.35e-63

   1. Initial program 77.8%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 62.1

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

   5. Applied rewrites 62.1%

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

   if 2.35e-63 < b

   1. Initial program 85.6%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 85.7

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

   5. Applied rewrites 85.7%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-PI.f64 N/A

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

      17. lower-PI.f64 72.3

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

   8. Applied rewrites 72.3%

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

   10. Applied rewrites 73.4%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 73.4

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

   12. Applied rewrites 83.3%

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

Alternative 7: 67.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.35 \cdot 10^{-63}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{2} + a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 2.35e-63)
   (* a a)
   (+ (pow (* (* (* (PI) b) 0.005555555555555556) angle) 2.0) (* a a))))

Derivation

1. Split input into 2 regimes

2. if b < 2.35e-63

   1. Initial program 77.8%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 62.1

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

   5. Applied rewrites 62.1%

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

   if 2.35e-63 < b

   1. Initial program 85.6%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 85.7

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

   5. Applied rewrites 85.7%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 83.2

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

   8. Applied rewrites 83.2%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.35 \cdot 10^{-63}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{2} + a \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.4% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.35 \cdot 10^{-63}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot b\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), b, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 2.35e-63)
   (* a a)
   (fma
    (* (* (* 3.08641975308642e-5 (* angle angle)) b) (* (PI) (PI)))
    b
    (* a a))))

Derivation

1. Split input into 2 regimes

2. if b < 2.35e-63

   1. Initial program 77.8%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 62.1

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

   5. Applied rewrites 62.1%

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

   if 2.35e-63 < b

   1. Initial program 85.6%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 85.7

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

   5. Applied rewrites 85.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 85.7

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 85.6

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

   7. Applied rewrites 85.6%

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

      1. lift-+.f64 N/A

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

   9. Applied rewrites 83.9%

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

   10. Taylor expanded in angle around 0

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

       1. associate-*r* N/A

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

       2. associate-*r* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-PI.f64 N/A

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

       12. lower-PI.f64 77.1

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

   12. Applied rewrites 77.1%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.35 \cdot 10^{-63}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot b\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), b, a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.4% accurate, 11.5× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(\left(b \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot angle\right) \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) + a \cdot a \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (* (* (* (* (* b b) 3.08641975308642e-5) angle) angle) (* (PI) (PI)))
  (* a a)))

Derivation

1. Initial program 80.4%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 80.8

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

5. Applied rewrites 80.8%

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

6. Taylor expanded in angle around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. unpow2 N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-PI.f64 N/A

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

   17. lower-PI.f64 67.8

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

8. Applied rewrites 67.8%

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

10. Applied rewrites 68.1%

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

12. Applied rewrites 74.8%

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

13. Final simplification 74.8%

    \[\leadsto \left(\left(\left(\left(b \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot angle\right) \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) + a \cdot a \]
14. Add Preprocessing

Alternative 10: 70.4% accurate, 11.5× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(b \cdot b\right) \cdot angle\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) + a \cdot a \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (* (* (* (* b b) angle) (* 3.08641975308642e-5 angle)) (* (PI) (PI)))
  (* a a)))

Derivation

1. Initial program 80.4%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 80.8

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

5. Applied rewrites 80.8%

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

6. Taylor expanded in angle around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. unpow2 N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-PI.f64 N/A

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

   17. lower-PI.f64 67.8

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

8. Applied rewrites 67.8%

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

10. Applied rewrites 68.1%

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

12. Applied rewrites 74.8%

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

13. Final simplification 74.8%

    \[\leadsto \left(\left(\left(b \cdot b\right) \cdot angle\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) + a \cdot a \]
14. Add Preprocessing

Alternative 11: 57.3% 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 80.4%

   \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
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 57.6

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

5. Applied rewrites 57.6%

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

Reproduce

herbie shell --seed 2024271 
(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)))