ab-angle->ABCF C

Percentage Accurate: 79.5% → 79.5%

Time: 9.6s

Alternatives: 15

Speedup: 1.4×

• Could not converge on a ground truth

  1. a = 2.0015043020978475e-176
  2. b = 1.994216753237052e-83
  3. angle = 8.14717723896072e+258

• 36.5% 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: 79.5% 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: 79.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\mathsf{PI}\left(\right)}\\ {\left(a \cdot \cos \left({t\_0}^{2} \cdot \left(t\_0 \cdot \frac{angle}{180}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (cbrt (PI))))
   (+
    (pow (* a (cos (* (pow t_0 2.0) (* t_0 (/ angle 180.0))))) 2.0)
    (pow (* b (sin (* (* (PI) 0.005555555555555556) angle))) 2.0))))

Derivation

1. Initial program 81.2%

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

3. Taylor expanded in angle around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-PI.f64 81.3

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

5. Applied rewrites 81.3%

   \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{\sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}\right)}^{2} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. add-cube-cbrt N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. lower-pow.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lower-cbrt.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lower-cbrt.f64 81.3

       \[\leadsto {\left(a \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\color{blue}{\sqrt[3]{\mathsf{PI}\left(\right)}} \cdot \frac{angle}{180}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)\right)}^{2} \]

7. Applied rewrites 81.3%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)\right)}^{2} \]
8. Add Preprocessing

Alternative 2: 71.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \frac{angle}{180}\\ t_1 := \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\\ t_2 := {\left(a \cdot \cos t\_0\right)}^{2} + {\left(b \cdot \sin t\_0\right)}^{2}\\ t_3 := \mathsf{PI}\left(\right) \cdot a\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{-309}:\\ \;\;\;\;{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2} \cdot \left(b \cdot b\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -3.08641975308642 \cdot 10^{-5}, t\_3 \cdot t\_3, a \cdot a\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right), t\_1, 0.005555555555555556\right)\right)\right) \cdot angle\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(t\_1 \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right) \cdot b, angle \cdot angle, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (PI) (/ angle 180.0)))
        (t_1 (* (PI) (PI)))
        (t_2 (+ (pow (* a (cos t_0)) 2.0) (pow (* b (sin t_0)) 2.0)))
        (t_3 (* (PI) a)))
   (if (<= t_2 2e-309)
     (* (pow (sin (* (* (PI) angle) 0.005555555555555556)) 2.0) (* b b))
     (if (<= t_2 1e+302)
       (+
        (fma (* (* angle angle) -3.08641975308642e-5) (* t_3 t_3) (* a a))
        (pow
         (*
          (*
           b
           (*
            (PI)
            (fma
             (* -2.8577960676726107e-8 (* angle angle))
             t_1
             0.005555555555555556)))
          angle)
         2.0))
       (fma
        (* (* (* t_1 3.08641975308642e-5) b) b)
        (* angle angle)
        (* a a))))))

Derivation

1. Split input into 3 regimes

2. if (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64))) < 1.9999999999999988e-309

   1. Initial program 95.0%

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

   3. Taylor expanded in angle around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 95.0

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

   5. Applied rewrites 95.0%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 95.3

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

   8. Applied rewrites 95.3%

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

   if 1.9999999999999988e-309 < (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64))) < 1.0000000000000001e302

   1. Initial program 66.7%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 62.1%

      \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\color{blue}{\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 0.005555555555555556\right)\right)\right) \cdot angle\right)}}^{2} \]

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}\right)} + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)}\right) + {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{32400} \cdot \color{blue}{\left(\left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} + {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      3. associate-*l* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}} + {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left({angle}^{2} \cdot \frac{-1}{32400}\right) \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)} + {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({angle}^{2} \cdot \frac{-1}{32400}, {a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}, {a}^{2}\right)} + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{angle}^{2} \cdot \frac{-1}{32400}}, {a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}, {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(angle \cdot angle\right)} \cdot \frac{-1}{32400}, {a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}, {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(angle \cdot angle\right)} \cdot \frac{-1}{32400}, {a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}, {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \color{blue}{\left(a \cdot a\right)} \cdot {\mathsf{PI}\left(\right)}^{2}, {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \left(a \cdot a\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      12. unswap-sqr N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \color{blue}{\left(a \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)}, {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \color{blue}{\left(a \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)}, {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot a\right)} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right), {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot a\right)} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right), {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      16. lower-PI.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot a\right) \cdot \left(a \cdot \mathsf{PI}\left(\right)\right), {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot a\right)}, {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot a\right)}, {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      19. lower-PI.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot a\right), {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      20. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\right), \color{blue}{a \cdot a}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      21. lower-*.f64 59.8

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -3.08641975308642 \cdot 10^{-5}, \left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\right), \color{blue}{a \cdot a}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 0.005555555555555556\right)\right)\right) \cdot angle\right)}^{2} \]

   8. Applied rewrites 59.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -3.08641975308642 \cdot 10^{-5}, \left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\right), a \cdot a\right)} + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 0.005555555555555556\right)\right)\right) \cdot angle\right)}^{2} \]

   if 1.0000000000000001e302 < (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64)))

   1. Initial program 99.0%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 34.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 93.7%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \leq 2 \cdot 10^{-309}:\\ \;\;\;\;{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2} \cdot \left(b \cdot b\right)\\ \mathbf{elif}\;{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \leq 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -3.08641975308642 \cdot 10^{-5}, \left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\right), a \cdot a\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 0.005555555555555556\right)\right)\right) \cdot angle\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right) \cdot b, angle \cdot angle, a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \frac{angle}{180}\\ t_1 := \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\\ t_2 := {\left(a \cdot \cos t\_0\right)}^{2} + {\left(b \cdot \sin t\_0\right)}^{2}\\ t_3 := \mathsf{PI}\left(\right) \cdot a\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{-309}:\\ \;\;\;\;{\sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{2} \cdot \left(b \cdot b\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -3.08641975308642 \cdot 10^{-5}, t\_3 \cdot t\_3, a \cdot a\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right), t\_1, 0.005555555555555556\right)\right)\right) \cdot angle\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(t\_1 \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right) \cdot b, angle \cdot angle, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (PI) (/ angle 180.0)))
        (t_1 (* (PI) (PI)))
        (t_2 (+ (pow (* a (cos t_0)) 2.0) (pow (* b (sin t_0)) 2.0)))
        (t_3 (* (PI) a)))
   (if (<= t_2 2e-309)
     (* (pow (sin (* (* (PI) 0.005555555555555556) angle)) 2.0) (* b b))
     (if (<= t_2 1e+302)
       (+
        (fma (* (* angle angle) -3.08641975308642e-5) (* t_3 t_3) (* a a))
        (pow
         (*
          (*
           b
           (*
            (PI)
            (fma
             (* -2.8577960676726107e-8 (* angle angle))
             t_1
             0.005555555555555556)))
          angle)
         2.0))
       (fma
        (* (* (* t_1 3.08641975308642e-5) b) b)
        (* angle angle)
        (* a a))))))

Derivation

1. Split input into 3 regimes

2. if (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64))) < 1.9999999999999988e-309

   1. Initial program 95.0%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 95.2

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

   5. Applied rewrites 95.2%

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

   if 1.9999999999999988e-309 < (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64))) < 1.0000000000000001e302

   1. Initial program 66.7%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 62.1%

      \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\color{blue}{\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 0.005555555555555556\right)\right)\right) \cdot angle\right)}}^{2} \]

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}\right)} + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)}\right) + {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{32400} \cdot \color{blue}{\left(\left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} + {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      3. associate-*l* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}} + {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left({angle}^{2} \cdot \frac{-1}{32400}\right) \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)} + {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({angle}^{2} \cdot \frac{-1}{32400}, {a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}, {a}^{2}\right)} + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{angle}^{2} \cdot \frac{-1}{32400}}, {a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}, {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(angle \cdot angle\right)} \cdot \frac{-1}{32400}, {a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}, {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(angle \cdot angle\right)} \cdot \frac{-1}{32400}, {a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}, {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \color{blue}{\left(a \cdot a\right)} \cdot {\mathsf{PI}\left(\right)}^{2}, {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \left(a \cdot a\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      12. unswap-sqr N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \color{blue}{\left(a \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)}, {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \color{blue}{\left(a \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)}, {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot a\right)} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right), {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot a\right)} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right), {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      16. lower-PI.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot a\right) \cdot \left(a \cdot \mathsf{PI}\left(\right)\right), {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot a\right)}, {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot a\right)}, {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      19. lower-PI.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot a\right), {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      20. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\right), \color{blue}{a \cdot a}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      21. lower-*.f64 59.8

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -3.08641975308642 \cdot 10^{-5}, \left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\right), \color{blue}{a \cdot a}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 0.005555555555555556\right)\right)\right) \cdot angle\right)}^{2} \]

   8. Applied rewrites 59.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -3.08641975308642 \cdot 10^{-5}, \left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\right), a \cdot a\right)} + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 0.005555555555555556\right)\right)\right) \cdot angle\right)}^{2} \]

   if 1.0000000000000001e302 < (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64)))

   1. Initial program 99.0%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 34.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 93.7%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \leq 2 \cdot 10^{-309}:\\ \;\;\;\;{\sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{2} \cdot \left(b \cdot b\right)\\ \mathbf{elif}\;{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \leq 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -3.08641975308642 \cdot 10^{-5}, \left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\right), a \cdot a\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 0.005555555555555556\right)\right)\right) \cdot angle\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right) \cdot b, angle \cdot angle, a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \frac{angle}{180}\\ t_1 := \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\\ t_2 := {\left(a \cdot \cos t\_0\right)}^{2} + {\left(b \cdot \sin t\_0\right)}^{2}\\ t_3 := \mathsf{PI}\left(\right) \cdot a\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{-309}:\\ \;\;\;\;{\cos \left(-0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}^{2} \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -3.08641975308642 \cdot 10^{-5}, t\_3 \cdot t\_3, a \cdot a\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right), t\_1, 0.005555555555555556\right)\right)\right) \cdot angle\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(t\_1 \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right) \cdot b, angle \cdot angle, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (PI) (/ angle 180.0)))
        (t_1 (* (PI) (PI)))
        (t_2 (+ (pow (* a (cos t_0)) 2.0) (pow (* b (sin t_0)) 2.0)))
        (t_3 (* (PI) a)))
   (if (<= t_2 2e-309)
     (* (pow (cos (* -0.005555555555555556 (* (PI) angle))) 2.0) (* a a))
     (if (<= t_2 1e+302)
       (+
        (fma (* (* angle angle) -3.08641975308642e-5) (* t_3 t_3) (* a a))
        (pow
         (*
          (*
           b
           (*
            (PI)
            (fma
             (* -2.8577960676726107e-8 (* angle angle))
             t_1
             0.005555555555555556)))
          angle)
         2.0))
       (fma
        (* (* (* t_1 3.08641975308642e-5) b) b)
        (* angle angle)
        (* a a))))))

Derivation

1. Split input into 3 regimes

2. if (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64))) < 1.9999999999999988e-309

   1. Initial program 95.0%

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

   3. Taylor expanded in angle around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 95.0

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

   5. Applied rewrites 95.0%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 53.0%

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

   10. Applied rewrites 52.9%

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

   11. Taylor expanded in a around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. cos-neg-rev N/A

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

       5. sin-+PI/2-rev N/A

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

       6. distribute-lft-neg-in N/A

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

       7. metadata-eval N/A

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

       8. sin-+PI/2-rev N/A

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

       9. cos-neg-rev N/A

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

       10. sin-+PI/2-rev N/A

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

       11. distribute-lft-neg-in N/A

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

       12. metadata-eval N/A

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

       13. sin-+PI/2-rev N/A

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

   13. Applied rewrites 95.0%

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

   if 1.9999999999999988e-309 < (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64))) < 1.0000000000000001e302

   1. Initial program 66.7%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 62.1%

      \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\color{blue}{\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 0.005555555555555556\right)\right)\right) \cdot angle\right)}}^{2} \]

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}\right)} + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)}\right) + {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{32400} \cdot \color{blue}{\left(\left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} + {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      3. associate-*l* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}} + {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left({angle}^{2} \cdot \frac{-1}{32400}\right) \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)} + {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({angle}^{2} \cdot \frac{-1}{32400}, {a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}, {a}^{2}\right)} + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{angle}^{2} \cdot \frac{-1}{32400}}, {a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}, {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(angle \cdot angle\right)} \cdot \frac{-1}{32400}, {a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}, {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(angle \cdot angle\right)} \cdot \frac{-1}{32400}, {a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}, {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \color{blue}{\left(a \cdot a\right)} \cdot {\mathsf{PI}\left(\right)}^{2}, {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \left(a \cdot a\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      12. unswap-sqr N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \color{blue}{\left(a \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)}, {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \color{blue}{\left(a \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)}, {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot a\right)} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right), {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot a\right)} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right), {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      16. lower-PI.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot a\right) \cdot \left(a \cdot \mathsf{PI}\left(\right)\right), {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot a\right)}, {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot a\right)}, {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      19. lower-PI.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot a\right), {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      20. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\right), \color{blue}{a \cdot a}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      21. lower-*.f64 59.8

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -3.08641975308642 \cdot 10^{-5}, \left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\right), \color{blue}{a \cdot a}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 0.005555555555555556\right)\right)\right) \cdot angle\right)}^{2} \]

   8. Applied rewrites 59.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -3.08641975308642 \cdot 10^{-5}, \left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\right), a \cdot a\right)} + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 0.005555555555555556\right)\right)\right) \cdot angle\right)}^{2} \]

   if 1.0000000000000001e302 < (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64)))

   1. Initial program 99.0%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 34.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 93.7%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \leq 2 \cdot 10^{-309}:\\ \;\;\;\;{\cos \left(-0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}^{2} \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \leq 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -3.08641975308642 \cdot 10^{-5}, \left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\right), a \cdot a\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 0.005555555555555556\right)\right)\right) \cdot angle\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right) \cdot b, angle \cdot angle, a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\\ t_1 := \mathsf{PI}\left(\right) \cdot \frac{angle}{180}\\ t_2 := {\left(a \cdot \cos t\_1\right)}^{2} + {\left(b \cdot \sin t\_1\right)}^{2}\\ t_3 := \mathsf{PI}\left(\right) \cdot a\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{-309}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;t\_2 \leq 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -3.08641975308642 \cdot 10^{-5}, t\_3 \cdot t\_3, a \cdot a\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right), t\_0, 0.005555555555555556\right)\right)\right) \cdot angle\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(t\_0 \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right) \cdot b, angle \cdot angle, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (PI) (PI)))
        (t_1 (* (PI) (/ angle 180.0)))
        (t_2 (+ (pow (* a (cos t_1)) 2.0) (pow (* b (sin t_1)) 2.0)))
        (t_3 (* (PI) a)))
   (if (<= t_2 2e-309)
     (* a a)
     (if (<= t_2 1e+302)
       (+
        (fma (* (* angle angle) -3.08641975308642e-5) (* t_3 t_3) (* a a))
        (pow
         (*
          (*
           b
           (*
            (PI)
            (fma
             (* -2.8577960676726107e-8 (* angle angle))
             t_0
             0.005555555555555556)))
          angle)
         2.0))
       (fma
        (* (* (* t_0 3.08641975308642e-5) b) b)
        (* angle angle)
        (* a a))))))

Derivation

1. Split input into 3 regimes

2. if (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64))) < 1.9999999999999988e-309

   1. Initial program 95.0%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 94.9

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

   5. Applied rewrites 94.9%

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

   if 1.9999999999999988e-309 < (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64))) < 1.0000000000000001e302

   1. Initial program 66.7%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 62.1%

      \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\color{blue}{\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 0.005555555555555556\right)\right)\right) \cdot angle\right)}}^{2} \]

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}\right)} + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)}\right) + {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{32400} \cdot \color{blue}{\left(\left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} + {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      3. associate-*l* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}} + {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left({angle}^{2} \cdot \frac{-1}{32400}\right) \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)} + {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({angle}^{2} \cdot \frac{-1}{32400}, {a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}, {a}^{2}\right)} + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{angle}^{2} \cdot \frac{-1}{32400}}, {a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}, {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(angle \cdot angle\right)} \cdot \frac{-1}{32400}, {a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}, {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(angle \cdot angle\right)} \cdot \frac{-1}{32400}, {a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}, {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \color{blue}{\left(a \cdot a\right)} \cdot {\mathsf{PI}\left(\right)}^{2}, {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \left(a \cdot a\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      12. unswap-sqr N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \color{blue}{\left(a \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)}, {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \color{blue}{\left(a \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)}, {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot a\right)} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right), {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot a\right)} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right), {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      16. lower-PI.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot a\right) \cdot \left(a \cdot \mathsf{PI}\left(\right)\right), {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot a\right)}, {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot a\right)}, {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      19. lower-PI.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot a\right), {a}^{2}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      20. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot \frac{-1}{32400}, \left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\right), \color{blue}{a \cdot a}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right) \cdot angle\right)}^{2} \]

      21. lower-*.f64 59.8

          \[\leadsto \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -3.08641975308642 \cdot 10^{-5}, \left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\right), \color{blue}{a \cdot a}\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 0.005555555555555556\right)\right)\right) \cdot angle\right)}^{2} \]

   8. Applied rewrites 59.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -3.08641975308642 \cdot 10^{-5}, \left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\right), a \cdot a\right)} + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 0.005555555555555556\right)\right)\right) \cdot angle\right)}^{2} \]

   if 1.0000000000000001e302 < (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64)))

   1. Initial program 99.0%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 34.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 93.7%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \leq 2 \cdot 10^{-309}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \leq 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -3.08641975308642 \cdot 10^{-5}, \left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\right), a \cdot a\right) + {\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 0.005555555555555556\right)\right)\right) \cdot angle\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right) \cdot b, angle \cdot angle, a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left({\sin \left(\mathsf{fma}\left(0.005555555555555556, angle, 0.5\right) \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot a, a, {\left(b \cdot \sin \left(\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right)}^{2}\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (fma
  (* (pow (sin (* (fma 0.005555555555555556 angle 0.5) (PI))) 2.0) a)
  a
  (pow (* b (sin (* (* 0.005555555555555556 (PI)) angle))) 2.0)))

Derivation

1. Initial program 81.2%

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

   1. lift-cos.f64 N/A

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

   2. sin-+PI/2-rev N/A

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

   3. lower-sin.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lower-/.f64 81.1

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

4. Applied rewrites 81.1%

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

5. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. associate-*r* N/A

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

   6. distribute-rgt-out N/A

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

   7. lower-*.f64 N/A

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

   8. lower-PI.f64 N/A

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

   9. lower-fma.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

7. Applied rewrites 72.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(0.005555555555555556, angle, 0.5\right)\right)}^{2}, a \cdot a, {\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2} \cdot \left(b \cdot b\right)\right)} \]
8. Step-by-step derivation

9. Applied rewrites 81.3%

   \[\leadsto \mathsf{fma}\left({\sin \left(\mathsf{fma}\left(0.005555555555555556, angle, 0.5\right) \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot a, \color{blue}{a}, {\left(b \cdot \sin \left(\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right)}^{2}\right) \]
10. Add Preprocessing

Alternative 7: 79.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 81.2%

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

3. Taylor expanded in angle around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-PI.f64 81.3

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

5. Applied rewrites 81.3%

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

6. Taylor expanded in angle around inf

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

   1. cos-neg-rev N/A

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

   2. lower-cos.f64 N/A

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

   3. distribute-lft-neg-in N/A

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

   4. lower-*.f64 N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-PI.f64 81.2

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

8. Applied rewrites 81.2%

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

Alternative 8: 79.5% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 81.2%

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

3. Taylor expanded in angle around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-PI.f64 81.3

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

5. Applied rewrites 81.3%

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

6. Taylor expanded in angle around 0

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

8. Applied rewrites 81.0%

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

Alternative 9: 79.5% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 81.2%

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

   1. unpow1 N/A

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

   2. pow-to-exp N/A

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

   3. lower-exp.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-log.f64 43.3

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 43.3

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 43.3

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

4. Applied rewrites 43.3%

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

5. Taylor expanded in angle around 0

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

7. Applied rewrites 43.3%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lift-pow.f64 N/A

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

   8. lift-exp.f64 N/A

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

9. Applied rewrites 80.9%

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

Alternative 10: 66.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.9 \cdot 10^{-67}:\\ \;\;\;\;{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle, 0.5\right) \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot 1\right)}^{2} + {\left(b \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.9e-67)
   (* (pow (sin (* (fma 0.005555555555555556 angle 0.5) (PI))) 2.0) (* a a))
   (+
    (pow (* a 1.0) 2.0)
    (pow (* b (* (* (PI) angle) 0.005555555555555556)) 2.0))))

Derivation

1. Split input into 2 regimes

2. if b < 1.89999999999999994e-67

   1. Initial program 80.7%

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

      1. lift-cos.f64 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lower-/.f64 80.8

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

   4. Applied rewrites 80.8%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

   7. Applied rewrites 75.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(0.005555555555555556, angle, 0.5\right)\right)}^{2}, a \cdot a, {\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2} \cdot \left(b \cdot b\right)\right)} \]

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-PI.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 62.6

          \[\leadsto {\sin \left(\mathsf{fma}\left(0.005555555555555556, angle, 0.5\right) \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot \color{blue}{\left(a \cdot a\right)} \]

   10. Applied rewrites 62.6%

       \[\leadsto \color{blue}{{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle, 0.5\right) \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot \left(a \cdot a\right)} \]

   if 1.89999999999999994e-67 < b

   1. Initial program 82.1%

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

   3. Taylor expanded in angle around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 82.1

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

   5. Applied rewrites 82.1%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 79.9%

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

   9. Taylor expanded in angle around 0

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

   11. Applied rewrites 79.9%

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

Alternative 11: 53.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.6 \cdot 10^{-155}:\\ \;\;\;\;angle \cdot \left(angle \cdot \left({\left(b \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\\ \mathbf{elif}\;a \leq 2.5 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(\left(-3.08641975308642 \cdot 10^{-5} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(-b\right) \cdot b\right), angle \cdot angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 3.6e-155)
   (* angle (* angle (* (pow (* b (PI)) 2.0) 3.08641975308642e-5)))
   (if (<= a 2.5e+142)
     (fma
      (* (* -3.08641975308642e-5 (* (PI) (PI))) (* (- b) b))
      (* angle angle)
      (* a a))
     (* a a))))

Derivation

1. Split input into 3 regimes

2. if a < 3.59999999999999989e-155

   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

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 45.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 37.6%

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

   10. Applied rewrites 48.2%

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

   if 3.59999999999999989e-155 < a < 2.5000000000000001e142

   1. Initial program 72.8%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 62.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 63.7%

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

   if 2.5000000000000001e142 < a

   1. Initial program 100.0%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.6 \cdot 10^{-155}:\\ \;\;\;\;angle \cdot \left(angle \cdot \left({\left(b \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\\ \mathbf{elif}\;a \leq 2.5 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(\left(-3.08641975308642 \cdot 10^{-5} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(-b\right) \cdot b\right), angle \cdot angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.2% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.9 \cdot 10^{-67}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-3.08641975308642 \cdot 10^{-5} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(-b\right) \cdot b\right), angle \cdot angle, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.9e-67)
   (* a a)
   (fma
    (* (* -3.08641975308642e-5 (* (PI) (PI))) (* (- b) b))
    (* angle angle)
    (* a a))))

Derivation

1. Split input into 2 regimes

2. if b < 1.89999999999999994e-67

   1. Initial program 80.7%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 62.1

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

   5. Applied rewrites 62.1%

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

   if 1.89999999999999994e-67 < b

   1. Initial program 82.1%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 45.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 65.8%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.9 \cdot 10^{-67}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-3.08641975308642 \cdot 10^{-5} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(-b\right) \cdot b\right), angle \cdot angle, a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.2% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.9 \cdot 10^{-67}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right) \cdot b, angle \cdot angle, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.9e-67)
   (* a a)
   (fma
    (* (* (* (* (PI) (PI)) 3.08641975308642e-5) b) b)
    (* angle angle)
    (* a a))))

Derivation

1. Split input into 2 regimes

2. if b < 1.89999999999999994e-67

   1. Initial program 80.7%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 62.1

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

   5. Applied rewrites 62.1%

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

   if 1.89999999999999994e-67 < b

   1. Initial program 82.1%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 45.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 65.8%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.9 \cdot 10^{-67}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right) \cdot b, angle \cdot angle, a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.2% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{+125}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot angle\right) \cdot b\right) \cdot b\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 8e+125)
   (* a a)
   (* (* 3.08641975308642e-5 (* (* (* angle angle) b) b)) (* (PI) (PI)))))

Derivation

1. Split input into 2 regimes

2. if b < 7.9999999999999994e125

   1. Initial program 80.1%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.3

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

   5. Applied rewrites 61.3%

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

   if 7.9999999999999994e125 < b

   1. Initial program 86.8%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 44.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 59.3%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 69.0%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{+125}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot angle\right) \cdot b\right) \cdot b\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 56.4% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, angle)
use fmin_fmax_functions
    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 81.2%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 56.0

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

5. Applied rewrites 56.0%

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

6. Final simplification 56.0%

   \[\leadsto a \cdot a \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025010 
(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)))