ab-angle->ABCF A

Percentage Accurate: 79.4% → 79.2%

Time: 9.0s

Alternatives: 8

Speedup: N/A×

• 36.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

Initial Program: 79.4% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 79.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;angle\_m \leq 1.95 \cdot 10^{-13}:\\ \;\;\;\;\left({\left(a \cdot angle\_m\right)}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} + {b}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right)}^{2}, a \cdot a, b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= angle_m 1.95e-13)
   (+
    (* (* (pow (* a angle_m) 2.0) (* (PI) (PI))) 3.08641975308642e-5)
    (pow b 2.0))
   (fma
    (pow (sin (* (PI) (* 0.005555555555555556 angle_m))) 2.0)
    (* a a)
    (* b b))))

Derivation

1. Split input into 2 regimes

2. if angle < 1.95000000000000002e-13

   1. Initial program 87.7%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 85.3

          \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   5. Applied rewrites 85.3%

      \[\leadsto \color{blue}{{\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. *-commutative N/A

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

      7. pow-prod-down N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-PI.f64 N/A

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

      17. lift-PI.f64 85.4

          \[\leadsto \left({\left(a \cdot angle\right)}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   7. Applied rewrites 85.4%

      \[\leadsto \left({\left(a \cdot angle\right)}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   8. Taylor expanded in angle around 0

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

      1. lift-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 85.3

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

      4. unpow1 85.3

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

      5. metadata-eval 85.3

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

      6. pow-flip 85.3

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

   10. Applied rewrites 85.3%

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

   if 1.95000000000000002e-13 < angle

   1. Initial program 62.5%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 62.9

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

   5. Applied rewrites 62.9%

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

      1. lift-+.f64 N/A

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

   7. Applied rewrites 62.9%

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

   8. Taylor expanded in angle around 0

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

      1. lower-*.f64 63.3

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

   10. Applied rewrites 63.3%

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

Alternative 2: 78.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;angle\_m \leq 7.4 \cdot 10^{-10}:\\ \;\;\;\;\left({\left(a \cdot angle\_m\right)}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} + {b}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\frac{angle\_m}{180} \cdot \mathsf{PI}\left(\right)\right)\right), a \cdot a, b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= angle_m 7.4e-10)
   (+
    (* (* (pow (* a angle_m) 2.0) (* (PI) (PI))) 3.08641975308642e-5)
    (pow b 2.0))
   (fma
    (- 0.5 (* 0.5 (cos (* 2.0 (* (/ angle_m 180.0) (PI))))))
    (* a a)
    (* b b))))

Derivation

1. Split input into 2 regimes

2. if angle < 7.4000000000000003e-10

   1. Initial program 87.8%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 85.4

          \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   5. Applied rewrites 85.4%

      \[\leadsto \color{blue}{{\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. *-commutative N/A

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

      7. pow-prod-down N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-PI.f64 N/A

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

      17. lift-PI.f64 85.5

          \[\leadsto \left({\left(a \cdot angle\right)}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   7. Applied rewrites 85.5%

      \[\leadsto \left({\left(a \cdot angle\right)}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   8. Taylor expanded in angle around 0

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

      1. lift-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 85.4

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

      4. unpow1 85.4

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

      5. metadata-eval 85.4

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

      6. pow-flip 85.4

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

   10. Applied rewrites 85.4%

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

   if 7.4000000000000003e-10 < angle

   1. Initial program 61.4%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.8

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

   5. Applied rewrites 61.8%

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

      1. lift-+.f64 N/A

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

   7. Applied rewrites 61.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. sqr-sin-a N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. lower-*.f64 61.8

          \[\leadsto \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}, a \cdot a, b \cdot b\right) \]

   9. Applied rewrites 61.8%

      \[\leadsto \mathsf{fma}\left(\color{blue}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}, a \cdot a, b \cdot b\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 79.2% accurate, 1.9× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+ (pow (* a (sin (/ (* (PI) angle_m) 180.0))) 2.0) (* b b)))

Derivation

1. Initial program 81.0%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 81.1

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

5. Applied rewrites 81.1%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. associate-*r/ N/A

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

   6. lower-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-PI.f64 81.2

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

7. Applied rewrites 81.2%

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

Alternative 4: 79.3% accurate, 1.9× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+ (pow (* a (sin (* (/ angle_m 180.0) (PI)))) 2.0) (* b b)))

Derivation

1. Initial program 81.0%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 81.1

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

5. Applied rewrites 81.1%

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

Alternative 5: 78.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;angle\_m \leq 7.4 \cdot 10^{-10}:\\ \;\;\;\;{\left(\left(\left(angle\_m \cdot a\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.005555555555555556\right)}^{2} + b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\frac{angle\_m}{180} \cdot \mathsf{PI}\left(\right)\right)\right), a \cdot a, b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= angle_m 7.4e-10)
   (+ (pow (* (* (* angle_m a) (PI)) 0.005555555555555556) 2.0) (* b b))
   (fma
    (- 0.5 (* 0.5 (cos (* 2.0 (* (/ angle_m 180.0) (PI))))))
    (* a a)
    (* b b))))

Derivation

1. Split input into 2 regimes

2. if angle < 7.4000000000000003e-10

   1. Initial program 87.8%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 87.8

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

   5. Applied rewrites 87.8%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 85.4

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

   8. Applied rewrites 85.4%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 85.4

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

   10. Applied rewrites 85.4%

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

   if 7.4000000000000003e-10 < angle

   1. Initial program 61.4%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.8

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

   5. Applied rewrites 61.8%

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

      1. lift-+.f64 N/A

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

   7. Applied rewrites 61.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. sqr-sin-a N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. lower-*.f64 61.8

          \[\leadsto \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}, a \cdot a, b \cdot b\right) \]

   9. Applied rewrites 61.8%

      \[\leadsto \mathsf{fma}\left(\color{blue}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}, a \cdot a, b \cdot b\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 66.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 2.45 \cdot 10^{-123}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(angle\_m \cdot a\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.005555555555555556\right)}^{2} + b \cdot b\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 2.45e-123)
   (* b b)
   (+ (pow (* (* (* angle_m a) (PI)) 0.005555555555555556) 2.0) (* b b))))

Derivation

1. Split input into 2 regimes

2. if a < 2.4499999999999999e-123

   1. Initial program 80.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 64.3

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

   5. Applied rewrites 64.3%

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

   if 2.4499999999999999e-123 < a

   1. Initial program 81.3%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 81.4

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

   5. Applied rewrites 81.4%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 78.7

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

   8. Applied rewrites 78.7%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 78.7

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

   10. Applied rewrites 78.7%

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

Alternative 7: 66.6% accurate, 9.3× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 2.45 \cdot 10^{-123}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.005555555555555556 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\_m\right) \cdot a\right), 0.005555555555555556 \cdot \left(\left(a \cdot angle\_m\right) \cdot \mathsf{PI}\left(\right)\right), b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 2.45e-123)
   (* b b)
   (fma
    (* 0.005555555555555556 (* (* (PI) angle_m) a))
    (* 0.005555555555555556 (* (* a angle_m) (PI)))
    (* b b))))

Derivation

1. Split input into 2 regimes

2. if a < 2.4499999999999999e-123

   1. Initial program 80.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 64.3

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

   5. Applied rewrites 64.3%

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

   if 2.4499999999999999e-123 < a

   1. Initial program 81.3%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 81.4

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

   5. Applied rewrites 81.4%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 78.7

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

   8. Applied rewrites 78.7%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 78.7

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

   10. Applied rewrites 78.7%

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

       1. lift-*.f64 N/A

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

       2. lift-PI.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lift-PI.f64 78.7

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

   12. Applied rewrites 78.7%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.45 \cdot 10^{-123}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.005555555555555556 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right), 0.005555555555555556 \cdot \left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right), b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.9% accurate, 74.7× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ b \cdot b \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m) :precision binary64 (* b b))

C

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

Fortran

angle_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle_m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle_m
    code = b * b
end function

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return b * b;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return b * b

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(b * b), $MachinePrecision]

Derivation

1. Initial program 81.0%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 56.6

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

5. Applied rewrites 56.6%

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

Reproduce

herbie shell --seed 2025045 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) (PI)))) 2.0) (pow (* b (cos (* (/ angle 180.0) (PI)))) 2.0)))