ab-angle->ABCF B

Percentage Accurate: 53.8% → 67.7%

Time: 9.6s

Alternatives: 14

Speedup: 13.7×

• 32.9% 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: 53.8% 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: 67.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 6 \cdot 10^{+138}:\\ \;\;\;\;\left(\left(\left(a + b\right) \cdot \sin \left(\left(angle\_m \cdot \mathsf{PI}\left(\right)\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \left(-\mathsf{PI}\left(\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\_m\right) \cdot 0.005555555555555556\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 6e+138)
    (*
     (*
      (* (+ a b) (sin (* (* angle_m (PI)) 0.005555555555555556)))
      (*
       (- b a)
       (sin (+ (* (* 0.005555555555555556 angle_m) (- (PI))) (/ (PI) 2.0)))))
     2.0)
    (*
     2.0
     (*
      (sin (* (* (PI) angle_m) 0.005555555555555556))
      (* (+ b a) (- b a)))))))

Derivation

1. Split input into 2 regimes

2. if angle < 6.0000000000000002e138

   1. Initial program 62.9%

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

   3. Taylor expanded in angle around 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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 67.4%

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

   7. Applied rewrites 77.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lift-*.f64 N/A

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

   9. Applied rewrites 77.8%

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

       1. lift-cos.f64 N/A

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

       2. cos-neg-rev N/A

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

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

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

       4. lower-sin.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-neg.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-PI.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. associate-*r* N/A

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

       12. lower-*.f64 N/A

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

       13. lower-*.f64 N/A

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

       14. lift-PI.f64 N/A

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

       15. lower-/.f64 N/A

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

       16. lift-PI.f64 76.8

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

   11. Applied rewrites 76.8%

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

   if 6.0000000000000002e138 < angle

   1. Initial program 30.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 35.1%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 36.4%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 6 \cdot 10^{+138}:\\ \;\;\;\;\left(\left(\left(a + b\right) \cdot \sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \left(-\mathsf{PI}\left(\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 68.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(0.005555555555555556 \cdot angle\_m\right) \cdot \mathsf{PI}\left(\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 5.8 \cdot 10^{+138}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(\sin t\_0 \cdot \left(\cos t\_0 \cdot \left(b - a\right)\right)\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\_m\right) \cdot 0.005555555555555556\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* (* 0.005555555555555556 angle_m) (PI))))
   (*
    angle_s
    (if (<= angle_m 5.8e+138)
      (* (* (+ a b) (* (sin t_0) (* (cos t_0) (- b a)))) 2.0)
      (*
       2.0
       (*
        (sin (* (* (PI) angle_m) 0.005555555555555556))
        (* (+ b a) (- b a))))))))

Derivation

1. Split input into 2 regimes

2. if angle < 5.80000000000000019e138

   1. Initial program 62.9%

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

   3. Taylor expanded in angle around 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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 67.4%

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

   7. Applied rewrites 77.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lift-*.f64 N/A

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

   9. Applied rewrites 77.8%

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

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-sin.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-PI.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift--.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lift-cos.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. lift-PI.f64 N/A

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

       13. lift-*.f64 N/A

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

   11. Applied rewrites 78.1%

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

   if 5.80000000000000019e138 < angle

   1. Initial program 30.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 35.1%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 36.4%

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

Alternative 3: 67.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(a + b\right) \cdot \sin \left(\left(angle\_m \cdot \mathsf{PI}\left(\right)\right) \cdot 0.005555555555555556\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq 4.1 \cdot 10^{+68}:\\ \;\;\;\;\left(t\_0 \cdot \left(\left(b - a\right) \cdot \cos \left(angle\_m \cdot \left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot \left(b - a\right)\right) \cdot 2\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* (+ a b) (sin (* (* angle_m (PI)) 0.005555555555555556)))))
   (*
    angle_s
    (if (<= b 4.1e+68)
      (*
       (* t_0 (* (- b a) (cos (* angle_m (* 0.005555555555555556 (PI))))))
       2.0)
      (* (* t_0 (- b a)) 2.0)))))

Derivation

1. Split input into 2 regimes

2. if b < 4.0999999999999999e68

   1. Initial program 60.6%

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

   3. Taylor expanded in angle around 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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 63.9%

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

   7. Applied rewrites 72.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lift-*.f64 N/A

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

   9. Applied rewrites 72.3%

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

       1. lift-*.f64 N/A

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

       2. lift-PI.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-PI.f64 72.1

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

   11. Applied rewrites 72.1%

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

   if 4.0999999999999999e68 < b

   1. Initial program 44.0%

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

   3. Taylor expanded in angle around 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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 54.3%

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

   7. Applied rewrites 64.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lift-*.f64 N/A

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

   9. Applied rewrites 64.3%

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

   10. Taylor expanded in angle around 0

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

       1. lift--.f64 80.0

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

   12. Applied rewrites 80.0%

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

Alternative 4: 67.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \sin \left(\left(angle\_m \cdot \mathsf{PI}\left(\right)\right) \cdot 0.005555555555555556\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq 4.1 \cdot 10^{+68}:\\ \;\;\;\;\left(\left(\left(t\_0 \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \cos \left(angle\_m \cdot \left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a + b\right) \cdot t\_0\right) \cdot \left(b - a\right)\right) \cdot 2\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (sin (* (* angle_m (PI)) 0.005555555555555556))))
   (*
    angle_s
    (if (<= b 4.1e+68)
      (*
       (*
        (* (* t_0 (+ a b)) (- b a))
        (cos (* angle_m (* 0.005555555555555556 (PI)))))
       2.0)
      (* (* (* (+ a b) t_0) (- b a)) 2.0)))))

Derivation

1. Split input into 2 regimes

2. if b < 4.0999999999999999e68

   1. Initial program 60.6%

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

   3. Taylor expanded in angle around 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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 63.9%

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

   7. Applied rewrites 72.3%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-PI.f64 72.1

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

   9. Applied rewrites 72.1%

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

   if 4.0999999999999999e68 < b

   1. Initial program 44.0%

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

   3. Taylor expanded in angle around 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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 54.3%

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

   7. Applied rewrites 64.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lift-*.f64 N/A

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

   9. Applied rewrites 64.3%

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

   10. Taylor expanded in angle around 0

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

       1. lift--.f64 80.0

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

   12. Applied rewrites 80.0%

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

Alternative 5: 68.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(angle\_m \cdot \mathsf{PI}\left(\right)\right) \cdot 0.005555555555555556\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq 4.1 \cdot 10^{+68}:\\ \;\;\;\;\left(\left(\left(\sin \left(angle\_m \cdot \left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \cos t\_0\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a + b\right) \cdot \sin t\_0\right) \cdot \left(b - a\right)\right) \cdot 2\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* (* angle_m (PI)) 0.005555555555555556)))
   (*
    angle_s
    (if (<= b 4.1e+68)
      (*
       (*
        (* (* (sin (* angle_m (* 0.005555555555555556 (PI)))) (+ a b)) (- b a))
        (cos t_0))
       2.0)
      (* (* (* (+ a b) (sin t_0)) (- b a)) 2.0)))))

Derivation

1. Split input into 2 regimes

2. if b < 4.0999999999999999e68

   1. Initial program 60.6%

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

   3. Taylor expanded in angle around 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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 63.9%

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

   7. Applied rewrites 72.3%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-PI.f64 72.0

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

   9. Applied rewrites 72.0%

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

   if 4.0999999999999999e68 < b

   1. Initial program 44.0%

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

   3. Taylor expanded in angle around 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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 54.3%

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

   7. Applied rewrites 64.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lift-*.f64 N/A

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

   9. Applied rewrites 64.3%

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

   10. Taylor expanded in angle around 0

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

       1. lift--.f64 80.0

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

   12. Applied rewrites 80.0%

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

Alternative 6: 58.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a}^{2}\right) \leq -4 \cdot 10^{-262}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle\_m \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot angle\_m\right) \cdot \left(b \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (* 2.0 (- (pow b 2.0) (pow a 2.0))) -4e-262)
    (* (* -0.011111111111111112 a) (* (* angle_m (PI)) a))
    (* (* (* (PI) angle_m) (* b (- b a))) 0.011111111111111112))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -4.00000000000000005e-262

   1. Initial program 62.9%

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

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 56.3

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

   5. Applied rewrites 56.3%

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

   6. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 56.0

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

   8. Applied rewrites 56.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 56.0

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

   10. Applied rewrites 56.0%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-PI.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*l* N/A

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

       7. lower-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. lift-PI.f64 64.8

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

   12. Applied rewrites 64.8%

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

   if -4.00000000000000005e-262 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 53.3%

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

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 56.0

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

   5. Applied rewrites 56.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 55.9%

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(b \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 57.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a}^{2}\right) \leq -4 \cdot 10^{-262}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle\_m \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot angle\_m\right) \cdot \left(b \cdot b\right)\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (* 2.0 (- (pow b 2.0) (pow a 2.0))) -4e-262)
    (* (* -0.011111111111111112 a) (* (* angle_m (PI)) a))
    (* (* (* (PI) angle_m) (* b b)) 0.011111111111111112))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -4.00000000000000005e-262

   1. Initial program 62.9%

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

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 56.3

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

   5. Applied rewrites 56.3%

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

   6. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 56.0

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

   8. Applied rewrites 56.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 56.0

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

   10. Applied rewrites 56.0%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-PI.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*l* N/A

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

       7. lower-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. lift-PI.f64 64.8

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

   12. Applied rewrites 64.8%

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

   if -4.00000000000000005e-262 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 53.3%

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

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 56.0

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

   5. Applied rewrites 56.0%

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

   6. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 54.7

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(b \cdot b\right)\right) \cdot 0.011111111111111112 \]

   8. Applied rewrites 54.7%

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

Alternative 8: 58.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a}^{2}\right) \leq -4 \cdot 10^{-262}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle\_m \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b\right)\right) \cdot angle\_m\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (* 2.0 (- (pow b 2.0) (pow a 2.0))) -4e-262)
    (* (* -0.011111111111111112 a) (* (* angle_m (PI)) a))
    (* (* (* (PI) (* b b)) angle_m) 0.011111111111111112))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -4.00000000000000005e-262

   1. Initial program 62.9%

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

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 56.3

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

   5. Applied rewrites 56.3%

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

   6. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 56.0

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

   8. Applied rewrites 56.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 56.0

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

   10. Applied rewrites 56.0%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-PI.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*l* N/A

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

       7. lower-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. lift-PI.f64 64.8

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

   12. Applied rewrites 64.8%

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

   if -4.00000000000000005e-262 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 53.3%

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

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 56.0

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

   5. Applied rewrites 56.0%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 54.7

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112 \]

   8. Applied rewrites 54.7%

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

Alternative 9: 68.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 6.2 \cdot 10^{-8}:\\ \;\;\;\;\left(\left(\left(angle\_m \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;angle\_m \leq 5.8 \cdot 10^{+138}:\\ \;\;\;\;\sin \left(2 \cdot \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\_m\right) \cdot 0.005555555555555556\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 6.2e-8)
    (* (* (* (* angle_m (PI)) (+ a b)) (- b a)) 0.011111111111111112)
    (if (<= angle_m 5.8e+138)
      (*
       (sin (* 2.0 (* (* 0.005555555555555556 angle_m) (PI))))
       (* (+ a b) (- b a)))
      (*
       2.0
       (*
        (sin (* (* (PI) angle_m) 0.005555555555555556))
        (* (+ b a) (- b a))))))))

Derivation

1. Split input into 3 regimes

2. if angle < 6.2e-8

   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. 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

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 65.2

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

   5. Applied rewrites 65.2%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lift--.f64 75.5

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

   7. Applied rewrites 75.5%

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

   if 6.2e-8 < angle < 5.80000000000000019e138

   1. Initial program 47.0%

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

   3. Taylor expanded in angle around 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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 44.4%

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

   7. Applied rewrites 44.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lift-*.f64 N/A

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

   9. Applied rewrites 44.4%

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

   11. Applied rewrites 50.4%

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

   if 5.80000000000000019e138 < angle

   1. Initial program 30.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 35.1%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 36.4%

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

Alternative 10: 66.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 75000000000:\\ \;\;\;\;\left(\left(\left(angle\_m \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\_m\right) \cdot 0.005555555555555556\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 75000000000.0)
    (* (* (* (* angle_m (PI)) (+ a b)) (- b a)) 0.011111111111111112)
    (*
     2.0
     (*
      (sin (* (* (PI) angle_m) 0.005555555555555556))
      (* (+ b a) (- b a)))))))

Derivation

1. Split input into 2 regimes

2. if angle < 7.5e10

   1. Initial program 65.4%

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

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 65.2

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

   5. Applied rewrites 65.2%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lift--.f64 75.3

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

   7. Applied rewrites 75.3%

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

   if 7.5e10 < angle

   1. Initial program 31.7%

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

   3. 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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 34.2%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 34.6%

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

Alternative 11: 66.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(\left(\left(a + b\right) \cdot \sin \left(\left(angle\_m \cdot \mathsf{PI}\left(\right)\right) \cdot 0.005555555555555556\right)\right) \cdot \left(b - a\right)\right) \cdot 2\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (*
   (* (* (+ a b) (sin (* (* angle_m (PI)) 0.005555555555555556))) (- b a))
   2.0)))

Derivation

1. Initial program 57.5%

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

3. Taylor expanded in angle around 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. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lower-*.f64 N/A

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

5. Applied rewrites 62.1%

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

7. Applied rewrites 70.8%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. lift-*.f64 N/A

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

9. Applied rewrites 70.8%

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

10. Taylor expanded in angle around 0

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

    1. lift--.f64 70.4

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

12. Applied rewrites 70.4%

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

Alternative 12: 63.5% accurate, 13.7× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 2.3 \cdot 10^{+235}:\\ \;\;\;\;\left(\left(\left(angle\_m \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot angle\_m\right) \cdot \left(\left(b + a\right) \cdot \left(-a\right)\right)\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 2.3e+235)
    (* (* (* (* angle_m (PI)) (+ a b)) (- b a)) 0.011111111111111112)
    (* (* (* (PI) angle_m) (* (+ b a) (- a))) 0.011111111111111112))))

Derivation

1. Split input into 2 regimes

2. if angle < 2.3e235

   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

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 58.2

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

   5. Applied rewrites 58.2%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lift--.f64 66.5

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

   7. Applied rewrites 66.5%

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

   if 2.3e235 < angle

   1. Initial program 37.9%

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

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 28.5

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

   5. Applied rewrites 28.5%

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

   6. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 34.5

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

   8. Applied rewrites 34.5%

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(-a\right)\right)\right) \cdot 0.011111111111111112 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 63.5% accurate, 13.7× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 2.3 \cdot 10^{+235}:\\ \;\;\;\;\left(\left(\left(\left(a + b\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\_m\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot angle\_m\right) \cdot \left(\left(b + a\right) \cdot \left(-a\right)\right)\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 2.3e+235)
    (* (* (* (* (+ a b) (PI)) angle_m) (- b a)) 0.011111111111111112)
    (* (* (* (PI) angle_m) (* (+ b a) (- a))) 0.011111111111111112))))

Derivation

1. Split input into 2 regimes

2. if angle < 2.3e235

   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

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 58.2

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

   5. Applied rewrites 58.2%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   8. Applied rewrites 59.3%

      \[\leadsto \mathsf{fma}\left(angle \cdot \mathsf{fma}\left(\mathsf{PI}\left(\right), b, \left(0 \cdot a\right) \cdot \mathsf{PI}\left(\right)\right), b, \left(-a \cdot a\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 0.011111111111111112 \]

   9. Taylor expanded in angle around 0

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

       1. associate-*r* N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. +-commutative N/A

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

       7. associate-*r* N/A

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

       8. lower-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. lift-+.f64 N/A

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

       14. lift-PI.f64 N/A

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

       15. lift--.f64 66.5

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

   11. Applied rewrites 66.5%

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

   if 2.3e235 < angle

   1. Initial program 37.9%

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

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 28.5

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

   5. Applied rewrites 28.5%

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

   6. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 34.5

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

   8. Applied rewrites 34.5%

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(-a\right)\right)\right) \cdot 0.011111111111111112 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 38.0% accurate, 21.6× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle\_m \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (* angle_s (* (* -0.011111111111111112 a) (* (* angle_m (PI)) a))))

Derivation

1. Initial program 57.5%

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

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

   2. lower-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. unpow2 N/A

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

   9. unpow2 N/A

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

   10. difference-of-squares N/A

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

   11. lower-*.f64 N/A

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

   12. lower-+.f64 N/A

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

   13. lower--.f64 56.1

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

5. Applied rewrites 56.1%

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

6. Taylor expanded in a around inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-PI.f64 38.9

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

8. Applied rewrites 38.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 38.9

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

10. Applied rewrites 38.9%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-PI.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. associate-*l* N/A

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

    7. lower-*.f64 N/A

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

    8. lift-*.f64 N/A

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

    9. *-commutative N/A

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

    10. lower-*.f64 N/A

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

    11. lift-*.f64 N/A

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

    12. lift-PI.f64 42.9

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

12. Applied rewrites 42.9%

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

Reproduce

herbie shell --seed 2025046 
(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)))))