ab-angle->ABCF C

Percentage Accurate: 79.5% → 79.4%

Time: 12.4s

Alternatives: 18

Speedup: 1.0×

• Could not converge on a ground truth

  1. a = 4.903204134558972e-23
  2. b = 3.9633641187031625e+34
  3. angle = 8.637732900612023e+21

• 36.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 79.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\mathsf{PI}\left(\right)}^{0.25}\\ {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(t\_0 \cdot \left(t\_0 \cdot \left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right)\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (pow (PI) 0.25)))
   (+
    (pow (* a (cos (* (PI) (/ angle 180.0)))) 2.0)
    (pow
     (* b (sin (* t_0 (* t_0 (* (* (sqrt (PI)) angle) 0.005555555555555556)))))
     2.0))))

Derivation

1. Initial program 73.8%

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. add-sqr-sqrt N/A

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

   4. associate-*l* N/A

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

   5. rem-square-sqrt N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. pow1/2 N/A

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

   10. sqrt-pow1 N/A

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

   11. lower-pow.f64 N/A

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

   12. metadata-eval N/A

       \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left({\mathsf{PI}\left(\right)}^{\color{blue}{\frac{1}{4}}} \cdot \left(\sqrt{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)\right)}^{2} \]

   13. lower-*.f64 N/A

       \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left({\mathsf{PI}\left(\right)}^{\frac{1}{4}} \cdot \color{blue}{\left(\sqrt{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)}\right)\right)}^{2} \]

   14. lift-PI.f64 N/A

       \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left({\mathsf{PI}\left(\right)}^{\frac{1}{4}} \cdot \left(\sqrt{\sqrt{\color{blue}{\mathsf{PI}\left(\right)}}} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)\right)}^{2} \]

   15. pow1/2 N/A

       \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left({\mathsf{PI}\left(\right)}^{\frac{1}{4}} \cdot \left(\sqrt{\color{blue}{{\mathsf{PI}\left(\right)}^{\frac{1}{2}}}} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)\right)}^{2} \]

   16. sqrt-pow1 N/A

       \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left({\mathsf{PI}\left(\right)}^{\frac{1}{4}} \cdot \left(\color{blue}{{\mathsf{PI}\left(\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)\right)}^{2} \]

   17. lower-pow.f64 N/A

       \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left({\mathsf{PI}\left(\right)}^{\frac{1}{4}} \cdot \left(\color{blue}{{\mathsf{PI}\left(\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)\right)}^{2} \]

   18. metadata-eval N/A

       \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left({\mathsf{PI}\left(\right)}^{\frac{1}{4}} \cdot \left({\mathsf{PI}\left(\right)}^{\color{blue}{\frac{1}{4}}} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)\right)}^{2} \]

   19. lift-/.f64 N/A

       \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left({\mathsf{PI}\left(\right)}^{\frac{1}{4}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{4}} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\frac{angle}{180}}\right)\right)\right)\right)}^{2} \]

   20. div-inv N/A

       \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left({\mathsf{PI}\left(\right)}^{\frac{1}{4}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{4}} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right)\right)\right)}^{2} \]

   21. associate-*r* N/A

       \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left({\mathsf{PI}\left(\right)}^{\frac{1}{4}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{4}} \cdot \color{blue}{\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot angle\right) \cdot \frac{1}{180}\right)}\right)\right)\right)}^{2} \]

4. Applied rewrites 73.9%

   \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left({\mathsf{PI}\left(\right)}^{0.25} \cdot \left({\mathsf{PI}\left(\right)}^{0.25} \cdot \left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right)}\right)}^{2} \]
5. Add Preprocessing

Alternative 2: 75.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \frac{angle}{180}\\ t_1 := {\left(a \cdot \cos t\_0\right)}^{2} + {\left(b \cdot \sin t\_0\right)}^{2}\\ t_2 := \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\\ t_3 := \sqrt{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-286}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(a - b\right) \cdot \left(b + a\right)\right) \cdot \left(\left(t\_2 \cdot -3.08641975308642 \cdot 10^{-5}\right) \cdot angle\right), angle, a \cdot a\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+202}:\\ \;\;\;\;\mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \left(\left(a \cdot a\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), a \cdot a\right) + {\left(b \cdot \sin \left(\left(\left(t\_3 \cdot angle\right) \cdot 0.005555555555555556\right) \cdot t\_3\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot b\right) \cdot t\_2, b, {\left(\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (PI) (/ angle 180.0)))
        (t_1 (+ (pow (* a (cos t_0)) 2.0) (pow (* b (sin t_0)) 2.0)))
        (t_2 (* (PI) (PI)))
        (t_3 (sqrt (PI))))
   (if (<= t_1 5e-286)
     (fma
      (* (* (- a b) (+ b a)) (* (* t_2 -3.08641975308642e-5) angle))
      angle
      (* a a))
     (if (<= t_1 1e+202)
       (+
        (fma
         (* -3.08641975308642e-5 (* angle angle))
         (* (* (* a a) (PI)) (PI))
         (* a a))
        (pow (* b (sin (* (* (* t_3 angle) 0.005555555555555556) t_3))) 2.0))
       (fma
        (* (* (* 3.08641975308642e-5 (* angle angle)) b) t_2)
        b
        (pow (* (cos (* (* 0.005555555555555556 angle) (PI))) a) 2.0))))))

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))) < 5.00000000000000037e-286

   1. Initial program 96.7%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   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 80.7%

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

   7. Applied rewrites 96.2%

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

   if 5.00000000000000037e-286 < (+.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))) < 9.999999999999999e201

   1. Initial program 55.0%

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

      1. *-rgt-identity N/A

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

      2. metadata-eval N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. times-frac N/A

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

      8. lift-PI.f64 N/A

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

      9. *-rgt-identity-rev N/A

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

      10. add-sqr-sqrt N/A

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

      11. times-frac N/A

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

      12. un-div-inv N/A

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

      13. clear-num N/A

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

      14. lift-/.f64 N/A

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

      15. /-rgt-identity N/A

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

      16. lower-*.f64 N/A

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

   4. Applied rewrites 55.3%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-PI.f64 N/A

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

      18. lower-PI.f64 N/A

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

      19. unpow2 N/A

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

   7. Applied rewrites 48.3%

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

   if 9.999999999999999e201 < (+.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 85.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 85.4%

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

   5. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. lower-PI.f64 84.0

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

   7. Applied rewrites 84.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot b\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, b, {\left(\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)}^{2}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 75.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \frac{angle}{180}\\ t_1 := {\left(b \cdot \sin t\_0\right)}^{2}\\ t_2 := {\left(a \cdot \cos t\_0\right)}^{2} + t\_1\\ t_3 := \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;t\_2 \leq 10^{-192}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(a - b\right) \cdot \left(b + a\right)\right) \cdot \left(\left(t\_3 \cdot -3.08641975308642 \cdot 10^{-5}\right) \cdot angle\right), angle, a \cdot a\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+202}:\\ \;\;\;\;\mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, \left(t\_3 \cdot angle\right) \cdot angle, 1\right) \cdot \left(a \cdot a\right) + t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot b\right) \cdot t\_3, b, {\left(\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (PI) (/ angle 180.0)))
        (t_1 (pow (* b (sin t_0)) 2.0))
        (t_2 (+ (pow (* a (cos t_0)) 2.0) t_1))
        (t_3 (* (PI) (PI))))
   (if (<= t_2 1e-192)
     (fma
      (* (* (- a b) (+ b a)) (* (* t_3 -3.08641975308642e-5) angle))
      angle
      (* a a))
     (if (<= t_2 1e+202)
       (+
        (* (fma -3.08641975308642e-5 (* (* t_3 angle) angle) 1.0) (* a a))
        t_1)
       (fma
        (* (* (* 3.08641975308642e-5 (* angle angle)) b) t_3)
        b
        (pow (* (cos (* (* 0.005555555555555556 angle) (PI))) a) 2.0))))))

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))) < 1.0000000000000001e-192

   1. Initial program 81.7%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   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 68.0%

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

   7. Applied rewrites 78.1%

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

   if 1.0000000000000001e-192 < (+.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))) < 9.999999999999999e201

   1. Initial program 55.2%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-PI.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 48.7

          \[\leadsto \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot angle, 1\right) \cdot \color{blue}{\left(a \cdot a\right)} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   5. Applied rewrites 48.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot angle, 1\right) \cdot \left(a \cdot a\right)} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   if 9.999999999999999e201 < (+.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 85.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 85.4%

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

   5. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. lower-PI.f64 84.0

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

   7. Applied rewrites 84.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot b\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, b, {\left(\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)}^{2}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 79.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\mathsf{PI}\left(\right)}^{0.25}\\ {\left(1 \cdot a\right)}^{2} + {\left(b \cdot \sin \left(t\_0 \cdot \left(t\_0 \cdot \left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right)\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (pow (PI) 0.25)))
   (+
    (pow (* 1.0 a) 2.0)
    (pow
     (* b (sin (* t_0 (* t_0 (* (* (sqrt (PI)) angle) 0.005555555555555556)))))
     2.0))))

Derivation

1. Initial program 73.8%

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. add-sqr-sqrt N/A

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

   4. associate-*l* N/A

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

   5. rem-square-sqrt N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. pow1/2 N/A

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

   10. sqrt-pow1 N/A

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

   11. lower-pow.f64 N/A

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

   12. metadata-eval N/A

       \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left({\mathsf{PI}\left(\right)}^{\color{blue}{\frac{1}{4}}} \cdot \left(\sqrt{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)\right)}^{2} \]

   13. lower-*.f64 N/A

       \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left({\mathsf{PI}\left(\right)}^{\frac{1}{4}} \cdot \color{blue}{\left(\sqrt{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)}\right)\right)}^{2} \]

   14. lift-PI.f64 N/A

       \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left({\mathsf{PI}\left(\right)}^{\frac{1}{4}} \cdot \left(\sqrt{\sqrt{\color{blue}{\mathsf{PI}\left(\right)}}} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)\right)}^{2} \]

   15. pow1/2 N/A

       \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left({\mathsf{PI}\left(\right)}^{\frac{1}{4}} \cdot \left(\sqrt{\color{blue}{{\mathsf{PI}\left(\right)}^{\frac{1}{2}}}} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)\right)}^{2} \]

   16. sqrt-pow1 N/A

       \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left({\mathsf{PI}\left(\right)}^{\frac{1}{4}} \cdot \left(\color{blue}{{\mathsf{PI}\left(\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)\right)}^{2} \]

   17. lower-pow.f64 N/A

       \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left({\mathsf{PI}\left(\right)}^{\frac{1}{4}} \cdot \left(\color{blue}{{\mathsf{PI}\left(\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)\right)}^{2} \]

   18. metadata-eval N/A

       \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left({\mathsf{PI}\left(\right)}^{\frac{1}{4}} \cdot \left({\mathsf{PI}\left(\right)}^{\color{blue}{\frac{1}{4}}} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)\right)}^{2} \]

   19. lift-/.f64 N/A

       \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left({\mathsf{PI}\left(\right)}^{\frac{1}{4}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{4}} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\frac{angle}{180}}\right)\right)\right)\right)}^{2} \]

   20. div-inv N/A

       \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left({\mathsf{PI}\left(\right)}^{\frac{1}{4}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{4}} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right)\right)\right)}^{2} \]

   21. associate-*r* N/A

       \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left({\mathsf{PI}\left(\right)}^{\frac{1}{4}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{4}} \cdot \color{blue}{\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot angle\right) \cdot \frac{1}{180}\right)}\right)\right)\right)}^{2} \]

4. Applied rewrites 73.9%

   \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left({\mathsf{PI}\left(\right)}^{0.25} \cdot \left({\mathsf{PI}\left(\right)}^{0.25} \cdot \left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right)}\right)}^{2} \]
5. Step-by-step derivation

6. Applied rewrites 73.9%

   \[\leadsto \color{blue}{{\left(\cos \left(-0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot a\right)}^{2}} + {\left(b \cdot \sin \left({\mathsf{PI}\left(\right)}^{0.25} \cdot \left({\mathsf{PI}\left(\right)}^{0.25} \cdot \left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right)\right)}^{2} \]

7. Taylor expanded in angle around 0

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

9. Applied rewrites 73.9%

   \[\leadsto {\left(\color{blue}{1} \cdot a\right)}^{2} + {\left(b \cdot \sin \left({\mathsf{PI}\left(\right)}^{0.25} \cdot \left({\mathsf{PI}\left(\right)}^{0.25} \cdot \left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right)\right)}^{2} \]
10. Add Preprocessing

Alternative 5: 79.4% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 73.8%

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

   1. *-rgt-identity N/A

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

   2. metadata-eval N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. clear-num N/A

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

   6. un-div-inv N/A

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

   7. times-frac N/A

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

   8. lift-PI.f64 N/A

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

   9. *-rgt-identity-rev N/A

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

   10. add-sqr-sqrt N/A

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

   11. times-frac N/A

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

   12. un-div-inv N/A

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

   13. clear-num N/A

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

   14. lift-/.f64 N/A

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

   15. /-rgt-identity N/A

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

   16. lower-*.f64 N/A

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

4. Applied rewrites 73.9%

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

Alternative 6: 79.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (sqrt (PI))))
   (+
    (pow (* (cos (* -0.005555555555555556 (* angle (PI)))) a) 2.0)
    (pow (* b (sin (* (* (* t_0 angle) 0.005555555555555556) t_0))) 2.0))))

Derivation

1. Initial program 73.8%

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

   1. *-rgt-identity N/A

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

   2. metadata-eval N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. clear-num N/A

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

   6. un-div-inv N/A

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

   7. times-frac N/A

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

   8. lift-PI.f64 N/A

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

   9. *-rgt-identity-rev N/A

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

   10. add-sqr-sqrt N/A

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

   11. times-frac N/A

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

   12. un-div-inv N/A

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

   13. clear-num N/A

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

   14. lift-/.f64 N/A

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

   15. /-rgt-identity N/A

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

   16. lower-*.f64 N/A

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

4. Applied rewrites 73.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-cos.f64 N/A

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

   4. cos-neg-rev N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*r/ N/A

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

   8. lift-*.f64 N/A

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

   9. div-inv N/A

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

   10. metadata-eval N/A

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

   11. *-commutative N/A

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

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

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

   13. metadata-eval N/A

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

   14. lift-*.f64 N/A

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

   15. lift-cos.f64 N/A

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

   16. unpow1 N/A

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

   17. metadata-eval N/A

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

   18. sqrt-pow1 N/A

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

   19. pow2 N/A

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

   20. rem-sqrt-square N/A

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

   21. lower-*.f64 N/A

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

6. Applied rewrites 73.9%

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

Alternative 7: 79.4% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 73.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 73.9

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

4. Applied rewrites 73.9%

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

Alternative 8: 79.4% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 73.8%

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. add-sqr-sqrt N/A

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

   4. associate-*l* N/A

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

   5. rem-square-sqrt N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. pow1/2 N/A

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

   10. sqrt-pow1 N/A

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

   11. lower-pow.f64 N/A

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

   12. metadata-eval N/A

       \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left({\mathsf{PI}\left(\right)}^{\color{blue}{\frac{1}{4}}} \cdot \left(\sqrt{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)\right)}^{2} \]

   13. lower-*.f64 N/A

       \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left({\mathsf{PI}\left(\right)}^{\frac{1}{4}} \cdot \color{blue}{\left(\sqrt{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)}\right)\right)}^{2} \]

   14. lift-PI.f64 N/A

       \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left({\mathsf{PI}\left(\right)}^{\frac{1}{4}} \cdot \left(\sqrt{\sqrt{\color{blue}{\mathsf{PI}\left(\right)}}} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)\right)}^{2} \]

   15. pow1/2 N/A

       \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left({\mathsf{PI}\left(\right)}^{\frac{1}{4}} \cdot \left(\sqrt{\color{blue}{{\mathsf{PI}\left(\right)}^{\frac{1}{2}}}} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)\right)}^{2} \]

   16. sqrt-pow1 N/A

       \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left({\mathsf{PI}\left(\right)}^{\frac{1}{4}} \cdot \left(\color{blue}{{\mathsf{PI}\left(\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)\right)}^{2} \]

   17. lower-pow.f64 N/A

       \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left({\mathsf{PI}\left(\right)}^{\frac{1}{4}} \cdot \left(\color{blue}{{\mathsf{PI}\left(\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)\right)}^{2} \]

   18. metadata-eval N/A

       \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left({\mathsf{PI}\left(\right)}^{\frac{1}{4}} \cdot \left({\mathsf{PI}\left(\right)}^{\color{blue}{\frac{1}{4}}} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)\right)}^{2} \]

   19. lift-/.f64 N/A

       \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left({\mathsf{PI}\left(\right)}^{\frac{1}{4}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{4}} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\frac{angle}{180}}\right)\right)\right)\right)}^{2} \]

   20. div-inv N/A

       \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left({\mathsf{PI}\left(\right)}^{\frac{1}{4}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{4}} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right)\right)\right)}^{2} \]

   21. associate-*r* N/A

       \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left({\mathsf{PI}\left(\right)}^{\frac{1}{4}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{4}} \cdot \color{blue}{\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot angle\right) \cdot \frac{1}{180}\right)}\right)\right)\right)}^{2} \]

4. Applied rewrites 73.9%

   \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left({\mathsf{PI}\left(\right)}^{0.25} \cdot \left({\mathsf{PI}\left(\right)}^{0.25} \cdot \left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right)}\right)}^{2} \]
5. Step-by-step derivation

6. Applied rewrites 73.9%

   \[\leadsto \color{blue}{{\left(\cos \left(-0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot a\right)}^{2}} + {\left(b \cdot \sin \left({\mathsf{PI}\left(\right)}^{0.25} \cdot \left({\mathsf{PI}\left(\right)}^{0.25} \cdot \left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right)\right)}^{2} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. pow2 N/A

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

   5. lift-pow.f64 N/A

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

   6. pow-pow N/A

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

   7. metadata-eval N/A

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

   8. pow1/2 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r* N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*l* N/A

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

   14. lift-sqrt.f64 N/A

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

   15. lift-sqrt.f64 N/A

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

   16. rem-square-sqrt N/A

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

8. Applied rewrites 73.9%

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

Alternative 9: 79.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* 0.005555555555555556 angle) (PI))))
   (fma (pow (cos t_0) 2.0) (* a a) (pow (* (sin t_0) b) 2.0))))

Derivation

1. Initial program 73.8%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. unpow-prod-down N/A

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

   6. lower-fma.f64 N/A

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

4. Applied rewrites 73.9%

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

Alternative 10: 79.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 73.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 73.8

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 73.8

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 73.8

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

   10. lift-/.f64 N/A

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

   11. clear-num N/A

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

   12. associate-/r/ N/A

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

   13. lower-*.f64 N/A

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

   14. metadata-eval 73.9

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

4. Applied rewrites 73.9%

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

Alternative 11: 79.5% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 73.8%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 73.8%

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

Alternative 12: 53.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 9.4 \cdot 10^{-160}:\\ \;\;\;\;{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2} \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b, b, {\left(\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 9.4e-160)
   (* (pow (sin (* (* (PI) angle) 0.005555555555555556)) 2.0) (* b b))
   (fma
    (* (* (* (* 3.08641975308642e-5 (* angle angle)) (PI)) (PI)) b)
    b
    (pow (* (cos (* (* 0.005555555555555556 angle) (PI))) a) 2.0))))

Derivation

1. Split input into 2 regimes

2. if a < 9.3999999999999995e-160

   1. Initial program 74.1%

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

   3. Taylor expanded in 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. 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 35.1

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

   5. Applied rewrites 35.1%

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

   if 9.3999999999999995e-160 < a

   1. Initial program 73.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 72.0%

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

   5. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. lower-PI.f64 68.7

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

   7. Applied rewrites 68.7%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot b, b, {\left(\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)}^{2}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 53.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.85 \cdot 10^{-159}:\\ \;\;\;\;{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2} \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot b\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), b, {\left(\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.85e-159)
   (* (pow (sin (* (* (PI) angle) 0.005555555555555556)) 2.0) (* b b))
   (fma
    (* (* (* 3.08641975308642e-5 (* angle angle)) b) (* (PI) (PI)))
    b
    (pow (* (cos (* (* 0.005555555555555556 angle) (PI))) a) 2.0))))

Derivation

1. Split input into 2 regimes

2. if a < 1.8499999999999999e-159

   1. Initial program 74.1%

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

   3. Taylor expanded in 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. 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 35.1

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

   5. Applied rewrites 35.1%

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

   if 1.8499999999999999e-159 < a

   1. Initial program 73.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 72.0%

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

   5. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. lower-PI.f64 68.7

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

   7. Applied rewrites 68.7%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot b\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, b, {\left(\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)}^{2}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 57.0% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.1 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(a - b\right) \cdot \left(b + a\right)\right) \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -3.08641975308642 \cdot 10^{-5}\right) \cdot angle\right), angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.1e+69)
   (fma
    (* (* (- a b) (+ b a)) (* (* (* (PI) (PI)) -3.08641975308642e-5) angle))
    angle
    (* a a))
   (* a a)))

Derivation

1. Split input into 2 regimes

2. if a < 1.1000000000000001e69

   1. Initial program 70.7%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   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 38.5%

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

   7. Applied rewrites 44.4%

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

   if 1.1000000000000001e69 < a

   1. Initial program 88.8%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 89.3

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

   5. Applied rewrites 89.3%

      \[\leadsto \color{blue}{a \cdot a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 64.8% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.2 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(\left(-3.08641975308642 \cdot 10^{-5} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(-b\right) \cdot b\right), angle \cdot angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.2e+69)
   (fma
    (* (* -3.08641975308642e-5 (* (PI) (PI))) (* (- b) b))
    (* angle angle)
    (* a a))
   (* a a)))

Derivation

1. Split input into 2 regimes

2. if a < 1.2000000000000001e69

   1. Initial program 70.7%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   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 38.5%

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

   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 53.8%

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

   if 1.2000000000000001e69 < a

   1. Initial program 88.8%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 89.3

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

   5. Applied rewrites 89.3%

      \[\leadsto \color{blue}{a \cdot a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 64.8% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.2 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), angle \cdot angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.2e+69)
   (fma
    (* (* (* 3.08641975308642e-5 (* b b)) (PI)) (PI))
    (* angle angle)
    (* a a))
   (* a a)))

Derivation

1. Split input into 2 regimes

2. if a < 1.2000000000000001e69

   1. Initial program 70.7%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   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 38.5%

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

   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 53.8%

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

   if 1.2000000000000001e69 < a

   1. Initial program 88.8%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 89.3

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

   5. Applied rewrites 89.3%

      \[\leadsto \color{blue}{a \cdot a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 58.7% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.18 \cdot 10^{+229}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right) \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.18e+229)
   (* a a)
   (* (* (* angle angle) 3.08641975308642e-5) (* (* (* (PI) (PI)) b) b))))

Derivation

1. Split input into 2 regimes

2. if b < 1.18e229

   1. Initial program 72.1%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 57.9

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

   5. Applied rewrites 57.9%

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

   if 1.18e229 < b

   1. Initial program 99.8%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 62.5%

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

   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 81.3%

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

Alternative 18: 56.3% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.8%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 56.4

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

5. Applied rewrites 56.4%

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

Reproduce

herbie shell --seed 2024305 
(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)))