ab-angle->ABCF A

Percentage Accurate: 80.0% → 80.0%

Time: 12.8s

Alternatives: 7

Speedup: N/A×

• 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: 80.0% 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

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

FPCore

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

Derivation

1. Initial program 82.4%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 82.7%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

7. Applied rewrites 82.0%

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

9. Applied rewrites 82.7%

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

Alternative 2: 79.9% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 82.4%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 82.7%

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

6. Taylor expanded in angle around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-PI.f64 82.7

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

8. Applied rewrites 82.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

10. Applied rewrites 82.7%

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

Alternative 3: 67.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.08 \cdot 10^{-94}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot \left(\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.08e-94)
   (* b b)
   (+
    (pow (* a (* (* 0.005555555555555556 (PI)) angle)) 2.0)
    (pow (* b 1.0) 2.0))))

Derivation

1. Split input into 2 regimes

2. if a < 1.08e-94

   1. Initial program 83.2%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.1

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

   5. Applied rewrites 61.1%

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

   if 1.08e-94 < a

   1. Initial program 81.0%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 80.4%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 77.3

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

   8. Applied rewrites 77.3%

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

Alternative 4: 65.1% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.08 \cdot 10^{-94}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), angle \cdot angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(a \cdot angle\right) \cdot \left(a \cdot angle\right)\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.08e-94)
   (* b b)
   (if (<= a 1.3e+146)
     (fma
      (* (* (* 3.08641975308642e-5 (* a a)) (PI)) (PI))
      (* angle angle)
      (* b b))
     (* (* 3.08641975308642e-5 (* (* a angle) (* a angle))) (* (PI) (PI))))))

Derivation

1. Split input into 3 regimes

2. if a < 1.08e-94

   1. Initial program 83.2%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.1

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

   5. Applied rewrites 61.1%

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

   if 1.08e-94 < a < 1.30000000000000007e146

   1. Initial program 74.2%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
   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 38.1%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 69.0%

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

   if 1.30000000000000007e146 < a

   1. Initial program 97.1%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
   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 61.5%

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

   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 76.5%

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

   10. Applied rewrites 73.0%

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

Alternative 5: 62.7% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.35 \cdot 10^{+59}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(a \cdot angle\right) \cdot \left(a \cdot angle\right)\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.35e+59)
   (* b b)
   (* (* 3.08641975308642e-5 (* (* a angle) (* a angle))) (* (PI) (PI)))))

Derivation

1. Split input into 2 regimes

2. if a < 1.3500000000000001e59

   1. Initial program 81.7%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.3

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

   5. Applied rewrites 61.3%

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

   if 1.3500000000000001e59 < a

   1. Initial program 85.8%

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

   3. 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 52.8%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 65.0%

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

   10. Applied rewrites 62.7%

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

Alternative 6: 61.3% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.4 \cdot 10^{+129}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot \left(a \cdot \left(angle \cdot angle\right)\right)\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 2.4e+129)
   (* b b)
   (* (* 3.08641975308642e-5 (* a (* a (* angle angle)))) (* (PI) (PI)))))

Derivation

1. Split input into 2 regimes

2. if a < 2.3999999999999999e129

   1. Initial program 81.1%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 60.4

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

   5. Applied rewrites 60.4%

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

   if 2.3999999999999999e129 < a

   1. Initial program 91.9%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 56.1%

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

   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 72.8%

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

   10. Applied rewrites 63.1%

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

Alternative 7: 57.4% 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 82.4%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
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 56.5

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

5. Applied rewrites 56.5%

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

Reproduce

herbie shell --seed 2024326 
(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)))