ab-angle->ABCF B

Percentage Accurate: 53.4% → 66.8%

Time: 14.4s

Alternatives: 18

Speedup: 6.5×

• 32.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

Initial Program: 53.4% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 66.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(\left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \left(b\_m - a\right)\right) \cdot \left(b\_m + a\right)\\ \mathbf{if}\;b\_m \leq 6 \cdot 10^{+197}:\\ \;\;\;\;\cos \left(\frac{1}{\frac{\frac{180}{\mathsf{PI}\left(\right)}}{angle}}\right) \cdot t\_0\\ \mathbf{elif}\;b\_m \leq 6.7 \cdot 10^{+243}:\\ \;\;\;\;1 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(\left(\mathsf{fma}\left(0.011111111111111112 \cdot \mathsf{PI}\left(\right), b\_m - a, \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b\_m - a\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot -5.7155921353452215 \cdot 10^{-8}\right)\right) \cdot angle\right) \cdot \left(b\_m + a\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0
         (*
          (* (* 2.0 (sin (* (PI) (* 0.005555555555555556 angle)))) (- b_m a))
          (+ b_m a))))
   (if (<= b_m 6e+197)
     (* (cos (/ 1.0 (/ (/ 180.0 (PI)) angle))) t_0)
     (if (<= b_m 6.7e+243)
       (* 1.0 t_0)
       (*
        (cos (* (PI) (/ angle 180.0)))
        (*
         (*
          (fma
           (* 0.011111111111111112 (PI))
           (- b_m a)
           (*
            (* (pow (PI) 3.0) (- b_m a))
            (* (* angle angle) -5.7155921353452215e-8)))
          angle)
         (+ b_m a)))))))

Derivation

1. Split input into 3 regimes

2. if b < 6.0000000000000004e197

   1. Initial program 56.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 67.7

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

   4. Applied rewrites 69.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 70.3

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right)\right) \cdot \cos \left(\frac{1}{\frac{\color{blue}{\frac{180}{angle}}}{\mathsf{PI}\left(\right)}}\right) \]

   6. Applied rewrites 70.3%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}\right)} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 71.5

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right)\right) \cdot \cos \left(\frac{1}{\frac{\color{blue}{\frac{180}{\mathsf{PI}\left(\right)}}}{angle}}\right) \]

   8. Applied rewrites 71.5%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right)\right) \cdot \cos \left(\frac{1}{\color{blue}{\frac{\frac{180}{\mathsf{PI}\left(\right)}}{angle}}}\right) \]

   if 6.0000000000000004e197 < b < 6.6999999999999997e243

   1. Initial program 3.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 56.8

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

   4. Applied rewrites 42.9%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 100.0%

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

   if 6.6999999999999997e243 < b

   1. Initial program 34.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 77.7

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

   4. Applied rewrites 77.6%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \left(\left(a + b\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{17496000} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right) + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right)}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(a + b\right) \cdot \color{blue}{\left(\left(\frac{-1}{17496000} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right) + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right) \cdot angle\right)}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(a + b\right) \cdot \color{blue}{\left(\left(\frac{-1}{17496000} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right) + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right) \cdot angle\right)}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

   7. Applied rewrites 99.9%

      \[\leadsto \left(\left(a + b\right) \cdot \color{blue}{\left(\mathsf{fma}\left(0.011111111111111112 \cdot \mathsf{PI}\left(\right), b - a, \left(-5.7155921353452215 \cdot 10^{-8} \cdot \left(angle \cdot angle\right)\right) \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right) \cdot angle\right)}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6 \cdot 10^{+197}:\\ \;\;\;\;\cos \left(\frac{1}{\frac{\frac{180}{\mathsf{PI}\left(\right)}}{angle}}\right) \cdot \left(\left(\left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\right)\\ \mathbf{elif}\;b \leq 6.7 \cdot 10^{+243}:\\ \;\;\;\;1 \cdot \left(\left(\left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(\left(\mathsf{fma}\left(0.011111111111111112 \cdot \mathsf{PI}\left(\right), b - a, \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot -5.7155921353452215 \cdot 10^{-8}\right)\right) \cdot angle\right) \cdot \left(b + a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 66.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot angle\\ t_1 := {b\_m}^{2} - {a}^{2}\\ t_2 := \left(b\_m - a\right) \cdot \left(b\_m + a\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot a\right) \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+276}:\\ \;\;\;\;\sin \left(\left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\left(a \cdot a\right) \cdot -0.011111111111111112, t\_0, \left(\left(\left(\mathsf{PI}\left(\right) \cdot b\_m\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot b\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(t\_0 \cdot 0.011111111111111112\right) \cdot t\_2\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (* (PI) angle))
        (t_1 (- (pow b_m 2.0) (pow a 2.0)))
        (t_2 (* (- b_m a) (+ b_m a))))
   (if (<= t_1 (- INFINITY))
     (* (* (* -0.011111111111111112 a) angle) (* (PI) a))
     (if (<= t_1 5e+276)
       (* (sin (* (* 0.011111111111111112 (PI)) angle)) t_2)
       (if (<= t_1 INFINITY)
         (fma
          (* (* a a) -0.011111111111111112)
          t_0
          (* (* (* (* (PI) b_m) angle) 0.011111111111111112) b_m))
         (* (sin (* t_0 0.011111111111111112)) t_2))))))

Derivation

1. Split input into 4 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -inf.0

   1. Initial program 48.5%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      9. lower-PI.f64 N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      11. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      12. difference-of-squares N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      14. lower-+.f64 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      15. lower--.f64 41.9

          \[\leadsto \left(angle \cdot \left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

   5. Applied rewrites 41.9%

      \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 41.9%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 63.2%

       \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 63.2%

       \[\leadsto \left(a \cdot \mathsf{PI}\left(\right)\right) \cdot \left(angle \cdot \color{blue}{\left(-0.011111111111111112 \cdot a\right)}\right) \]

   if -inf.0 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < 5.00000000000000001e276

   1. Initial program 64.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 64.8

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

   4. Applied rewrites 64.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 64.1

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right)\right) \cdot \cos \left(\frac{1}{\frac{\color{blue}{\frac{180}{angle}}}{\mathsf{PI}\left(\right)}}\right) \]

   6. Applied rewrites 64.1%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}\right)} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. inv-pow N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 32.2

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right)\right) \cdot \cos \left(e^{\color{blue}{\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right) \]

   8. Applied rewrites 32.2%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right)\right) \cdot \cos \color{blue}{\left(e^{\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)} \]

   10. Applied rewrites 65.2%

       \[\leadsto \color{blue}{\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(\left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)} \]

   if 5.00000000000000001e276 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < +inf.0

   1. Initial program 44.9%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      9. lower-PI.f64 N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      11. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      12. difference-of-squares N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      14. lower-+.f64 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      15. lower--.f64 47.1

          \[\leadsto \left(angle \cdot \left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

   5. Applied rewrites 47.1%

      \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \color{blue}{b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)} \]

   8. Applied rewrites 75.4%

      \[\leadsto \mathsf{fma}\left(-0.011111111111111112 \cdot \left(a \cdot a\right), \color{blue}{\mathsf{PI}\left(\right) \cdot angle}, \left(\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot b\right) \]

   if +inf.0 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

   4. Applied rewrites 74.2%

      \[\leadsto \color{blue}{\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot 0.011111111111111112\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -\infty:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot a\right) \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\right)\\ \mathbf{elif}\;{b}^{2} - {a}^{2} \leq 5 \cdot 10^{+276}:\\ \;\;\;\;\sin \left(\left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\\ \mathbf{elif}\;{b}^{2} - {a}^{2} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\left(a \cdot a\right) \cdot -0.011111111111111112, \mathsf{PI}\left(\right) \cdot angle, \left(\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \frac{angle}{180}\\ t_1 := \sqrt{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;\left(\sin t\_0 \cdot \left(\left({b\_m}^{2} - {a}^{2}\right) \cdot 2\right)\right) \cdot \cos t\_0 \leq -100:\\ \;\;\;\;\left(\left(\left(\sin \left(\left(t\_1 \cdot t\_1\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot 2\right) \cdot \left(b\_m - a\right)\right) \cdot \left(b\_m + a\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\frac{1}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}\right) \cdot \left(\left(\left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \left(b\_m - a\right)\right) \cdot \left(b\_m + a\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (* (PI) (/ angle 180.0))) (t_1 (sqrt (PI))))
   (if (<=
        (* (* (sin t_0) (* (- (pow b_m 2.0) (pow a 2.0)) 2.0)) (cos t_0))
        -100.0)
     (*
      (*
       (*
        (* (sin (* (* t_1 t_1) (* 0.005555555555555556 angle))) 2.0)
        (- b_m a))
       (+ b_m a))
      1.0)
     (*
      (cos (/ 1.0 (/ (/ 180.0 angle) (PI))))
      (*
       (* (* 2.0 (sin (* (PI) (* 0.005555555555555556 angle)))) (- b_m a))
       (+ b_m a))))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) < -100

   1. Initial program 51.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 64.0

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

   4. Applied rewrites 68.7%

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

      1. lift-PI.f64 N/A

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

      2. add-sqr-sqrt N/A

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

      3. lower-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lower-sqrt.f64 64.1

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

   6. Applied rewrites 64.1%

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

   7. Taylor expanded in angle around 0

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

   9. Applied rewrites 62.1%

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

   if -100 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64)))))

   1. Initial program 54.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 70.0

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

   4. Applied rewrites 69.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 71.3

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right)\right) \cdot \cos \left(\frac{1}{\frac{\color{blue}{\frac{180}{angle}}}{\mathsf{PI}\left(\right)}}\right) \]

   6. Applied rewrites 71.3%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \leq -100:\\ \;\;\;\;\left(\left(\left(\sin \left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot 2\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\frac{1}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}\right) \cdot \left(\left(\left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 33.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := {\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}\\ t_1 := \left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\\ \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+123}:\\ \;\;\;\;\cos \left(e^{\log t\_1}\right) \cdot \left(\left(\left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \left(b\_m - a\right)\right) \cdot \left(b\_m + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(b\_m - a\right) \cdot \left(b\_m + a\right)\right) \cdot 2\right) \cdot \left(\cos \left(\left(\left(t\_0 \cdot t\_0\right) \cdot 0.005555555555555556\right) \cdot angle\right) \cdot \sin t\_1\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (pow (cbrt (sqrt (PI))) 3.0))
        (t_1 (* (* (PI) 0.005555555555555556) angle)))
   (if (<= (/ angle 180.0) 5e+123)
     (*
      (cos (exp (log t_1)))
      (*
       (* (* 2.0 (sin (* (PI) (* 0.005555555555555556 angle)))) (- b_m a))
       (+ b_m a)))
     (*
      (* (* (- b_m a) (+ b_m a)) 2.0)
      (* (cos (* (* (* t_0 t_0) 0.005555555555555556) angle)) (sin t_1))))))

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.99999999999999974e123

   1. Initial program 57.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 75.0

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

   4. Applied rewrites 75.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 76.1

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right)\right) \cdot \cos \left(\frac{1}{\frac{\color{blue}{\frac{180}{angle}}}{\mathsf{PI}\left(\right)}}\right) \]

   6. Applied rewrites 76.1%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}\right)} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. inv-pow N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 36.0

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right)\right) \cdot \cos \left(e^{\color{blue}{\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right) \]

   8. Applied rewrites 36.0%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right)\right) \cdot \cos \color{blue}{\left(e^{\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)} \]
   9. Step-by-step derivation

      1. rem-log-exp N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. exp-to-pow N/A

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

      5. unpow-1 N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. div-inv N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/r/ N/A

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

      11. metadata-eval N/A

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

      12. associate-*l* N/A

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

      13. lift-PI.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-*.f64 36.0

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

   10. Applied rewrites 36.0%

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

   if 4.99999999999999974e123 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 28.4%

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

   3. Taylor expanded in angle around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 31.5%

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

   7. Applied rewrites 49.0%

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

4. Final simplification 38.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+123}:\\ \;\;\;\;\cos \left(e^{\log \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}\right) \cdot \left(\left(\left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot 2\right) \cdot \left(\cos \left(\left(\left({\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3} \cdot {\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}\right) \cdot 0.005555555555555556\right) \cdot angle\right) \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 33.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(\left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \left(b\_m - a\right)\right) \cdot \left(b\_m + a\right)\\ \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+123}:\\ \;\;\;\;\cos \left(e^{\log \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0
         (*
          (* (* 2.0 (sin (* (PI) (* 0.005555555555555556 angle)))) (- b_m a))
          (+ b_m a))))
   (if (<= (/ angle 180.0) 5e+123)
     (* (cos (exp (log (* (* (PI) 0.005555555555555556) angle)))) t_0)
     (* 1.0 t_0))))

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.99999999999999974e123

   1. Initial program 57.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 75.0

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

   4. Applied rewrites 75.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 76.1

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right)\right) \cdot \cos \left(\frac{1}{\frac{\color{blue}{\frac{180}{angle}}}{\mathsf{PI}\left(\right)}}\right) \]

   6. Applied rewrites 76.1%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}\right)} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. inv-pow N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 36.0

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right)\right) \cdot \cos \left(e^{\color{blue}{\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right) \]

   8. Applied rewrites 36.0%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right)\right) \cdot \cos \color{blue}{\left(e^{\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)} \]
   9. Step-by-step derivation

      1. rem-log-exp N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. exp-to-pow N/A

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

      5. unpow-1 N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. div-inv N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/r/ N/A

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

      11. metadata-eval N/A

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

      12. associate-*l* N/A

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

      13. lift-PI.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-*.f64 36.0

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

   10. Applied rewrites 36.0%

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

   if 4.99999999999999974e123 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 28.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 28.4

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

   4. Applied rewrites 33.9%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 45.9%

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

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+123}:\\ \;\;\;\;\cos \left(e^{\log \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}\right) \cdot \left(\left(\left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(\left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := {b\_m}^{2} - {a}^{2}\\ t_1 := \left(\left(-0.011111111111111112 \cdot a\right) \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-182}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(b\_m \cdot b\_m\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (- (pow b_m 2.0) (pow a 2.0)))
        (t_1 (* (* (* -0.011111111111111112 a) angle) (* (PI) a))))
   (if (<= t_0 -1e-182)
     t_1
     (if (<= t_0 INFINITY)
       (* (* b_m b_m) (* (* (PI) angle) 0.011111111111111112))
       t_1))))

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -1e-182 or +inf.0 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 47.9%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      9. lower-PI.f64 N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      11. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      12. difference-of-squares N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      14. lower-+.f64 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      15. lower--.f64 49.3

          \[\leadsto \left(angle \cdot \left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

   5. Applied rewrites 49.3%

      \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 49.2%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 59.6%

       \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 59.7%

       \[\leadsto \left(a \cdot \mathsf{PI}\left(\right)\right) \cdot \left(angle \cdot \color{blue}{\left(-0.011111111111111112 \cdot a\right)}\right) \]

   if -1e-182 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < +inf.0

   1. Initial program 59.0%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      9. lower-PI.f64 N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      11. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      12. difference-of-squares N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      14. lower-+.f64 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      15. lower--.f64 56.4

          \[\leadsto \left(angle \cdot \left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

   5. Applied rewrites 56.4%

      \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 21.3%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 17.6%

       \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]

   11. Taylor expanded in b around inf

       \[\leadsto {b}^{2} \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{90} \cdot \frac{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)}{b}\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 55.9%

       \[\leadsto \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot \color{blue}{\left(b \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -1 \cdot 10^{-182}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot a\right) \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\right)\\ \mathbf{elif}\;{b}^{2} - {a}^{2} \leq \infty:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot a\right) \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := {b\_m}^{2} - {a}^{2}\\ t_1 := \mathsf{PI}\left(\right) \cdot angle\\ t_2 := \left(t\_1 \cdot a\right) \cdot \left(-0.011111111111111112 \cdot a\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-182}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(b\_m \cdot b\_m\right) \cdot \left(t\_1 \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (- (pow b_m 2.0) (pow a 2.0)))
        (t_1 (* (PI) angle))
        (t_2 (* (* t_1 a) (* -0.011111111111111112 a))))
   (if (<= t_0 -1e-182)
     t_2
     (if (<= t_0 INFINITY) (* (* b_m b_m) (* t_1 0.011111111111111112)) t_2))))

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -1e-182 or +inf.0 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 47.9%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      9. lower-PI.f64 N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      11. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      12. difference-of-squares N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      14. lower-+.f64 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      15. lower--.f64 49.3

          \[\leadsto \left(angle \cdot \left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

   5. Applied rewrites 49.3%

      \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 49.2%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 59.6%

       \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]

   if -1e-182 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < +inf.0

   1. Initial program 59.0%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      9. lower-PI.f64 N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      11. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      12. difference-of-squares N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      14. lower-+.f64 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      15. lower--.f64 56.4

          \[\leadsto \left(angle \cdot \left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

   5. Applied rewrites 56.4%

      \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 21.3%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 17.6%

       \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]

   11. Taylor expanded in b around inf

       \[\leadsto {b}^{2} \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{90} \cdot \frac{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)}{b}\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 55.9%

       \[\leadsto \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot \color{blue}{\left(b \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -1 \cdot 10^{-182}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right) \cdot \left(-0.011111111111111112 \cdot a\right)\\ \mathbf{elif}\;{b}^{2} - {a}^{2} \leq \infty:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right) \cdot \left(-0.011111111111111112 \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(\left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \left(b\_m - a\right)\right) \cdot \left(b\_m + a\right)\\ \mathbf{if}\;b\_m \leq 6 \cdot 10^{+197}:\\ \;\;\;\;\cos \left(\frac{1}{\frac{\frac{180}{\mathsf{PI}\left(\right)}}{angle}}\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0
         (*
          (* (* 2.0 (sin (* (PI) (* 0.005555555555555556 angle)))) (- b_m a))
          (+ b_m a))))
   (if (<= b_m 6e+197)
     (* (cos (/ 1.0 (/ (/ 180.0 (PI)) angle))) t_0)
     (* 1.0 t_0))))

Derivation

1. Split input into 2 regimes

2. if b < 6.0000000000000004e197

   1. Initial program 56.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 67.7

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

   4. Applied rewrites 69.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 70.3

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right)\right) \cdot \cos \left(\frac{1}{\frac{\color{blue}{\frac{180}{angle}}}{\mathsf{PI}\left(\right)}}\right) \]

   6. Applied rewrites 70.3%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}\right)} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 71.5

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right)\right) \cdot \cos \left(\frac{1}{\frac{\color{blue}{\frac{180}{\mathsf{PI}\left(\right)}}}{angle}}\right) \]

   8. Applied rewrites 71.5%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right)\right) \cdot \cos \left(\frac{1}{\color{blue}{\frac{\frac{180}{\mathsf{PI}\left(\right)}}{angle}}}\right) \]

   if 6.0000000000000004e197 < b

   1. Initial program 26.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 71.8

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

   4. Applied rewrites 67.9%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 87.9%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6 \cdot 10^{+197}:\\ \;\;\;\;\cos \left(\frac{1}{\frac{\frac{180}{\mathsf{PI}\left(\right)}}{angle}}\right) \cdot \left(\left(\left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(\left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\\ t_1 := \left(\left(2 \cdot \sin t\_0\right) \cdot \left(b\_m - a\right)\right) \cdot \left(b\_m + a\right)\\ \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+65}:\\ \;\;\;\;\cos t\_0 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot t\_1\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (* (PI) (* 0.005555555555555556 angle)))
        (t_1 (* (* (* 2.0 (sin t_0)) (- b_m a)) (+ b_m a))))
   (if (<= (/ angle 180.0) 2e+65) (* (cos t_0) t_1) (* 1.0 t_1))))

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2e65

   1. Initial program 58.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 77.1

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

   4. Applied rewrites 77.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 77.2

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 77.8

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

   6. Applied rewrites 77.8%

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

   if 2e65 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 33.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 33.5

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

   4. Applied rewrites 38.7%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 47.6%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+65}:\\ \;\;\;\;\cos \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\left(\left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(\left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\\ \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+65}:\\ \;\;\;\;\cos t\_0 \cdot \left(\left(\left(\sin t\_0 \cdot \left(b\_m - a\right)\right) \cdot \left(b\_m + a\right)\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(\left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \left(b\_m - a\right)\right) \cdot \left(b\_m + a\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (* (* (PI) 0.005555555555555556) angle)))
   (if (<= (/ angle 180.0) 2e+65)
     (* (cos t_0) (* (* (* (sin t_0) (- b_m a)) (+ b_m a)) 2.0))
     (*
      1.0
      (*
       (* (* 2.0 (sin (* (PI) (* 0.005555555555555556 angle)))) (- b_m a))
       (+ b_m a))))))

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2e65

   1. Initial program 58.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 77.1

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

   4. Applied rewrites 77.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 77.8

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right)\right) \cdot \cos \left(\frac{1}{\frac{\color{blue}{\frac{180}{angle}}}{\mathsf{PI}\left(\right)}}\right) \]

   6. Applied rewrites 77.8%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}\right)} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. inv-pow N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 35.1

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right)\right) \cdot \cos \left(e^{\color{blue}{\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)} \cdot -1}\right) \]

   8. Applied rewrites 35.1%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right)\right) \cdot \cos \color{blue}{\left(e^{\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)} \]

   10. Applied rewrites 77.3%

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

   if 2e65 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 33.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 33.5

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

   4. Applied rewrites 38.7%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 47.6%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+65}:\\ \;\;\;\;\cos \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right) \cdot \left(\left(\left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(\left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+65}:\\ \;\;\;\;\left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(b\_m - a\right)\right) \cdot \left(b\_m + a\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(\left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \left(b\_m - a\right)\right) \cdot \left(b\_m + a\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= (/ angle 180.0) 2e+65)
   (* (* (sin (* (* (PI) angle) 0.011111111111111112)) (- b_m a)) (+ b_m a))
   (*
    1.0
    (*
     (* (* 2.0 (sin (* (PI) (* 0.005555555555555556 angle)))) (- b_m a))
     (+ b_m a)))))

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2e65

   1. Initial program 58.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 77.4%

      \[\leadsto \color{blue}{\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot 0.011111111111111112\right)\right)} \]

   if 2e65 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 33.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 33.5

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

   4. Applied rewrites 38.7%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 47.6%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+65}:\\ \;\;\;\;\left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(\left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+127}:\\ \;\;\;\;\left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(b\_m - a\right)\right) \cdot \left(b\_m + a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)\right) \cdot \left(\left(\left(b\_m - a\right) \cdot \left(b\_m + a\right)\right) \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= (/ angle 180.0) 5e+127)
   (* (* (sin (* (* (PI) angle) 0.011111111111111112)) (- b_m a)) (+ b_m a))
   (*
    (* 1.0 (sin (* (* (PI) 0.005555555555555556) angle)))
    (* (* (- b_m a) (+ b_m a)) 2.0))))

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 5.0000000000000004e127

   1. Initial program 57.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 74.9%

      \[\leadsto \color{blue}{\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot 0.011111111111111112\right)\right)} \]

   if 5.0000000000000004e127 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 28.4%

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

   3. Taylor expanded in angle around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 31.5%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 43.0%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+127}:\\ \;\;\;\;\left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 5 \cdot 10^{+284}:\\ \;\;\;\;\mathsf{fma}\left(\left(a \cdot a\right) \cdot -0.011111111111111112, \mathsf{PI}\left(\right) \cdot angle, \left(\left(\left(\mathsf{PI}\left(\right) \cdot b\_m\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot b\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot a\right) \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= (pow a 2.0) 5e+284)
   (fma
    (* (* a a) -0.011111111111111112)
    (* (PI) angle)
    (* (* (* (* (PI) b_m) angle) 0.011111111111111112) b_m))
   (* (* (* -0.011111111111111112 a) angle) (* (PI) a))))

Derivation

1. Split input into 2 regimes

2. if (pow.f64 a #s(literal 2 binary64)) < 4.9999999999999999e284

   1. Initial program 58.9%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      9. lower-PI.f64 N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      11. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      12. difference-of-squares N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      14. lower-+.f64 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      15. lower--.f64 55.4

          \[\leadsto \left(angle \cdot \left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

   5. Applied rewrites 55.4%

      \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \color{blue}{b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)} \]

   8. Applied rewrites 60.1%

      \[\leadsto \mathsf{fma}\left(-0.011111111111111112 \cdot \left(a \cdot a\right), \color{blue}{\mathsf{PI}\left(\right) \cdot angle}, \left(\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot b\right) \]

   if 4.9999999999999999e284 < (pow.f64 a #s(literal 2 binary64))

   1. Initial program 41.9%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      9. lower-PI.f64 N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      11. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      12. difference-of-squares N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      14. lower-+.f64 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      15. lower--.f64 47.3

          \[\leadsto \left(angle \cdot \left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

   5. Applied rewrites 47.3%

      \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.3%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 63.6%

       \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 63.7%

       \[\leadsto \left(a \cdot \mathsf{PI}\left(\right)\right) \cdot \left(angle \cdot \color{blue}{\left(-0.011111111111111112 \cdot a\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 5 \cdot 10^{+284}:\\ \;\;\;\;\mathsf{fma}\left(\left(a \cdot a\right) \cdot -0.011111111111111112, \mathsf{PI}\left(\right) \cdot angle, \left(\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot a\right) \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 57.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 2 \cdot 10^{+220}:\\ \;\;\;\;\left(\left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \left(\left(b\_m - a\right) \cdot \left(b\_m + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot a\right) \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= (pow a 2.0) 2e+220)
   (* (* (* 0.011111111111111112 (PI)) angle) (* (- b_m a) (+ b_m a)))
   (* (* (* -0.011111111111111112 a) angle) (* (PI) a))))

Derivation

1. Split input into 2 regimes

2. if (pow.f64 a #s(literal 2 binary64)) < 2e220

   1. Initial program 60.0%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      9. lower-PI.f64 N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      11. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      12. difference-of-squares N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      14. lower-+.f64 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      15. lower--.f64 56.3

          \[\leadsto \left(angle \cdot \left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

   5. Applied rewrites 56.3%

      \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   if 2e220 < (pow.f64 a #s(literal 2 binary64))

   1. Initial program 42.3%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      9. lower-PI.f64 N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      11. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      12. difference-of-squares N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      14. lower-+.f64 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      15. lower--.f64 46.8

          \[\leadsto \left(angle \cdot \left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

   5. Applied rewrites 46.8%

      \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.6%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 62.0%

       \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 62.0%

       \[\leadsto \left(a \cdot \mathsf{PI}\left(\right)\right) \cdot \left(angle \cdot \color{blue}{\left(-0.011111111111111112 \cdot a\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 2 \cdot 10^{+220}:\\ \;\;\;\;\left(\left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot a\right) \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 39.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot angle\\ \mathbf{if}\;{a}^{2} \leq 5 \cdot 10^{+159}:\\ \;\;\;\;t\_0 \cdot \left(\left(a \cdot a\right) \cdot -0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot a\right) \cdot \left(-0.011111111111111112 \cdot a\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (* (PI) angle)))
   (if (<= (pow a 2.0) 5e+159)
     (* t_0 (* (* a a) -0.011111111111111112))
     (* (* t_0 a) (* -0.011111111111111112 a)))))

Derivation

1. Split input into 2 regimes

2. if (pow.f64 a #s(literal 2 binary64)) < 5.00000000000000003e159

   1. Initial program 61.0%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      9. lower-PI.f64 N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      11. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      12. difference-of-squares N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      14. lower-+.f64 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      15. lower--.f64 56.6

          \[\leadsto \left(angle \cdot \left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

   5. Applied rewrites 56.6%

      \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 28.4%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \]

   if 5.00000000000000003e159 < (pow.f64 a #s(literal 2 binary64))

   1. Initial program 42.9%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      9. lower-PI.f64 N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      11. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      12. difference-of-squares N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      14. lower-+.f64 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      15. lower--.f64 47.5

          \[\leadsto \left(angle \cdot \left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

   5. Applied rewrites 47.5%

      \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.3%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 58.1%

       \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 5 \cdot 10^{+159}:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(\left(a \cdot a\right) \cdot -0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right) \cdot \left(-0.011111111111111112 \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 67.6% accurate, 3.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(b\_m - a\right)\right) \cdot \left(b\_m + a\right) \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (* (* (sin (* (* (PI) angle) 0.011111111111111112)) (- b_m a)) (+ b_m a)))

Derivation

1. Initial program 53.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lift--.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. lift-pow.f64 N/A

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

   11. unpow2 N/A

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

   12. difference-of-squares N/A

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

   13. associate-*l* N/A

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

   14. lower-*.f64 N/A

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

4. Applied rewrites 68.0%

   \[\leadsto \color{blue}{\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot 0.011111111111111112\right)\right)} \]

5. Final simplification 68.0%

   \[\leadsto \left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]
6. Add Preprocessing

Alternative 17: 57.2% accurate, 6.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 4 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(\left(a \cdot a\right) \cdot -0.011111111111111112, \mathsf{PI}\left(\right) \cdot angle, \left(\left(\left(\mathsf{PI}\left(\right) \cdot b\_m\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot b\_m\right)\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{+241}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot a\right) \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \left(-1.1431184270690443 \cdot 10^{-7} \cdot \left(angle \cdot angle\right)\right) \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right) \cdot \left(\left(\left(b\_m - a\right) \cdot \left(b\_m + a\right)\right) \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= a 4e+142)
   (fma
    (* (* a a) -0.011111111111111112)
    (* (PI) angle)
    (* (* (* (* (PI) b_m) angle) 0.011111111111111112) b_m))
   (if (<= a 1.9e+241)
     (* (* (* -0.011111111111111112 a) angle) (* (PI) a))
     (*
      (*
       (fma
        (* (PI) (PI))
        (* (* -1.1431184270690443e-7 (* angle angle)) (PI))
        (* (PI) 0.005555555555555556))
       angle)
      (* (* (- b_m a) (+ b_m a)) 2.0)))))

Derivation

1. Split input into 3 regimes

2. if a < 4.0000000000000002e142

   1. Initial program 56.0%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      9. lower-PI.f64 N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      11. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      12. difference-of-squares N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      14. lower-+.f64 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      15. lower--.f64 52.8

          \[\leadsto \left(angle \cdot \left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

   5. Applied rewrites 52.8%

      \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \color{blue}{b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)} \]

   8. Applied rewrites 54.7%

      \[\leadsto \mathsf{fma}\left(-0.011111111111111112 \cdot \left(a \cdot a\right), \color{blue}{\mathsf{PI}\left(\right) \cdot angle}, \left(\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot b\right) \]

   if 4.0000000000000002e142 < a < 1.89999999999999986e241

   1. Initial program 33.6%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      9. lower-PI.f64 N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      11. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      12. difference-of-squares N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      14. lower-+.f64 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      15. lower--.f64 51.2

          \[\leadsto \left(angle \cdot \left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

   5. Applied rewrites 51.2%

      \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 55.2%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 74.0%

       \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 74.0%

       \[\leadsto \left(a \cdot \mathsf{PI}\left(\right)\right) \cdot \left(angle \cdot \color{blue}{\left(-0.011111111111111112 \cdot a\right)}\right) \]

   if 1.89999999999999986e241 < a

   1. Initial program 46.2%

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

   3. Taylor expanded in angle around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 64.4%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \left(angle \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + {angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot 2\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 91.6%

      \[\leadsto \left(\mathsf{fma}\left({\mathsf{PI}\left(\right)}^{3} \cdot -1.1431184270690443 \cdot 10^{-7}, angle \cdot angle, \mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot 2\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 91.6%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right) \cdot \left(\left(angle \cdot angle\right) \cdot -1.1431184270690443 \cdot 10^{-7}\right), 0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \left(\left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right) \cdot 2\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(\left(a \cdot a\right) \cdot -0.011111111111111112, \mathsf{PI}\left(\right) \cdot angle, \left(\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot b\right)\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{+241}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot a\right) \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \left(-1.1431184270690443 \cdot 10^{-7} \cdot \left(angle \cdot angle\right)\right) \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 37.8% accurate, 21.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right) \cdot \left(-0.011111111111111112 \cdot a\right) \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (* (* (* (PI) angle) a) (* -0.011111111111111112 a)))

Derivation

1. Initial program 53.4%

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

3. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \frac{1}{90}} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

   3. *-commutative N/A

      \[\leadsto angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

   4. associate-*r* N/A

      \[\leadsto angle \cdot \color{blue}{\left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

   5. associate-*r* N/A

      \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \left(angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

   9. lower-PI.f64 N/A

      \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

   10. unpow2 N/A

       \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

   11. unpow2 N/A

       \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

   12. difference-of-squares N/A

       \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   13. lower-*.f64 N/A

       \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   14. lower-+.f64 N/A

       \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

   15. lower--.f64 52.7

       \[\leadsto \left(angle \cdot \left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

5. Applied rewrites 52.7%

   \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

6. Taylor expanded in b around 0

   \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
7. Step-by-step derivation

8. Applied rewrites 35.5%

   \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \]
9. Step-by-step derivation

10. Applied rewrites 39.1%

    \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]

11. Final simplification 39.1%

    \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right) \cdot \left(-0.011111111111111112 \cdot a\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024249 
(FPCore (a b angle)
  :name "ab-angle->ABCF B"
  :precision binary64
  (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* (PI) (/ angle 180.0)))) (cos (* (PI) (/ angle 180.0)))))