ab-angle->ABCF B

Percentage Accurate: 54.1% → 67.0%

Time: 6.8s

Alternatives: 16

Speedup: 13.7×

• 32.8% 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: 54.1% 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: 67.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{2}\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 1.6 \cdot 10^{+142}:\\ \;\;\;\;\left(2 \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle\_m, \mathsf{PI}\left(\right), t\_0\right)\right)\right) \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle\_m}{180}\right) \cdot \sin \left(\left(-\mathsf{PI}\left(\right)\right) \cdot \frac{angle\_m}{180} + t\_0\right)\right)\\ \end{array} \end{array} \end{array} \]

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 (/ (PI) 2.0)))
   (*
    angle_s
    (if (<= angle_m 1.6e+142)
      (*
       (* 2.0 (sin (fma (* 0.005555555555555556 angle_m) (PI) t_0)))
       (* (* (sin (* (* 0.005555555555555556 angle_m) (PI))) (+ a b)) (- b a)))
      (*
       (* (* (+ b a) (- b a)) 2.0)
       (*
        (sin (* (PI) (/ angle_m 180.0)))
        (sin (+ (* (- (PI)) (/ angle_m 180.0)) t_0))))))))

Derivation

1. Split input into 2 regimes

2. if angle < 1.60000000000000003e142

   1. Initial program 62.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 65.5%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. lower-*.f64 N/A

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

   7. Applied rewrites 73.7%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-sin.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-PI.f64 75.0

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

   9. Applied rewrites 75.0%

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

   if 1.60000000000000003e142 < angle

   1. Initial program 50.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

   4. Applied rewrites 50.8%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-PI.f64 39.1

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

   6. Applied rewrites 39.1%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 1.6 \cdot 10^{+142}:\\ \;\;\;\;\left(2 \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \mathsf{PI}\left(\right), \frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \sin \left(\left(-\mathsf{PI}\left(\right)\right) \cdot \frac{angle}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 57.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \frac{angle\_m}{180}\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0 \leq -2 \cdot 10^{+59}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle\_m \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.011111111111111112 \cdot angle\_m\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \end{array} \end{array} \end{array} \]

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 (* (PI) (/ angle_m 180.0))))
   (*
    angle_s
    (if (<=
         (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin t_0)) (cos t_0))
         -2e+59)
      (* (* -0.011111111111111112 a) (* (* angle_m (PI)) a))
      (* (* 0.011111111111111112 angle_m) (* (* (PI) (+ a b)) (- b a)))))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) < -1.99999999999999994e59

   1. Initial program 57.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 47.8

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

   5. Applied rewrites 47.8%

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

   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. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-PI.f64 29.7

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

   8. Applied rewrites 29.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 29.7

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

   10. Applied rewrites 29.7%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-PI.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 N/A

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

       13. lift-PI.f64 32.5

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

   12. Applied rewrites 32.5%

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

   if -1.99999999999999994e59 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64)))))

   1. Initial program 62.6%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      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 62.1

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

   5. Applied rewrites 62.1%

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

      1. lift-*.f64 N/A

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

      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-PI.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. *-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

   7. Applied rewrites 62.1%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \leq -2 \cdot 10^{+59}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.011111111111111112 \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \frac{angle\_m}{180}\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 3.1 \cdot 10^{+137}:\\ \;\;\;\;\left(2 \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle\_m, \mathsf{PI}\left(\right), \frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin t\_0 \cdot \cos t\_0\right)\\ \end{array} \end{array} \end{array} \]

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 (* (PI) (/ angle_m 180.0))))
   (*
    angle_s
    (if (<= angle_m 3.1e+137)
      (*
       (* 2.0 (sin (fma (* 0.005555555555555556 angle_m) (PI) (/ (PI) 2.0))))
       (* (* (sin (* (* 0.005555555555555556 angle_m) (PI))) (+ a b)) (- b a)))
      (* (* (* (+ b a) (- b a)) 2.0) (* (sin t_0) (cos t_0)))))))

Derivation

1. Split input into 2 regimes

2. if angle < 3.0999999999999999e137

   1. Initial program 62.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 65.7%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. lower-*.f64 N/A

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

   7. Applied rewrites 73.9%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-sin.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-PI.f64 75.3

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

   9. Applied rewrites 75.3%

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

   if 3.0999999999999999e137 < angle

   1. Initial program 49.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

   4. Applied rewrites 49.7%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 3.1 \cdot 10^{+137}:\\ \;\;\;\;\left(2 \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \mathsf{PI}\left(\right), \frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 3.1 \cdot 10^{+137}:\\ \;\;\;\;\left(2 \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle\_m, \mathsf{PI}\left(\right), \frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{angle\_m}{180}\right)\right)\\ \end{array} \end{array} \]

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 3.1e+137)
    (*
     (* 2.0 (sin (fma (* 0.005555555555555556 angle_m) (PI) (/ (PI) 2.0))))
     (* (* (sin (* (* 0.005555555555555556 angle_m) (PI))) (+ a b)) (- b a)))
    (* (* (- b a) (+ a b)) (sin (* 2.0 (* (PI) (/ angle_m 180.0))))))))

Derivation

1. Split input into 2 regimes

2. if angle < 3.0999999999999999e137

   1. Initial program 62.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 65.7%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. lower-*.f64 N/A

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

   7. Applied rewrites 73.9%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-sin.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-PI.f64 75.3

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

   9. Applied rewrites 75.3%

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

   if 3.0999999999999999e137 < angle

   1. Initial program 49.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

   4. Applied rewrites 49.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

   6. Applied rewrites 49.7%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 3.1 \cdot 10^{+137}:\\ \;\;\;\;\left(2 \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \mathsf{PI}\left(\right), \frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(angle\_m \cdot \mathsf{PI}\left(\right)\right) \cdot 0.005555555555555556\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 4 \cdot 10^{+256}:\\ \;\;\;\;\left(\left(\cos t\_0 \cdot 2\right) \cdot \left(\left(a + b\right) \cdot \sin t\_0\right)\right) \cdot \left(b - a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\_m\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\left(\left(\left(a + b\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\_m\right) \cdot 0.005555555555555556\right) \cdot \left(b - a\right)\right)\\ \end{array} \end{array} \end{array} \]

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 (* (* angle_m (PI)) 0.005555555555555556)))
   (*
    angle_s
    (if (<= a 4e+256)
      (* (* (* (cos t_0) 2.0) (* (+ a b) (sin t_0))) (- b a))
      (*
       (* 2.0 (cos (* (* (PI) angle_m) 0.005555555555555556)))
       (* (* (* (* (+ a b) (PI)) angle_m) 0.005555555555555556) (- b a)))))))

Derivation

1. Split input into 2 regimes

2. if a < 4.0000000000000001e256

   1. Initial program 61.3%

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

   3. Taylor expanded in angle around 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 63.8%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. lower-*.f64 N/A

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

   7. Applied rewrites 70.0%

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

   9. Applied rewrites 71.3%

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

   if 4.0000000000000001e256 < a

   1. Initial program 57.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 58.4%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. lower-*.f64 N/A

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

   7. Applied rewrites 71.2%

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

   8. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-PI.f64 100.0

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

   10. Applied rewrites 100.0%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4 \cdot 10^{+256}:\\ \;\;\;\;\left(\left(\cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot 0.005555555555555556\right) \cdot 2\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b - a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\left(\left(\left(a + b\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(b - a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 3 \cdot 10^{+24}:\\ \;\;\;\;\left(2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\_m\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(\cos \left(-0.5 \cdot \mathsf{PI}\left(\right)\right) - \cos \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \left(angle\_m \cdot \mathsf{PI}\left(\right)\right) \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

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 3e+24)
    (*
     (* 2.0 (cos (* (* (PI) angle_m) 0.005555555555555556)))
     (* (* (sin (* (* 0.005555555555555556 angle_m) (PI))) (+ a b)) (- b a)))
    (*
     (* (+ a b) (- b a))
     (-
      (cos (* -0.5 (PI)))
      (cos (fma 0.5 (PI) (* (* angle_m (PI)) 0.011111111111111112))))))))

Derivation

1. Split input into 2 regimes

2. if angle < 2.99999999999999995e24

   1. Initial program 65.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 69.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. lower-*.f64 N/A

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

   7. Applied rewrites 78.5%

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

   if 2.99999999999999995e24 < angle

   1. Initial program 41.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

   4. Applied rewrites 41.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-cos.f64 N/A

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

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

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

      5. sin-mult N/A

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

      6. lower-/.f64 N/A

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

   6. Applied rewrites 30.6%

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

   7. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

   9. Applied rewrites 35.7%

      \[\leadsto \color{blue}{\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(\cos \left(-0.5 \cdot \mathsf{PI}\left(\right)\right) - \cos \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot 0.011111111111111112\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 3 \cdot 10^{+24}:\\ \;\;\;\;\left(2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(\cos \left(-0.5 \cdot \mathsf{PI}\left(\right)\right) - \cos \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 0.000215:\\ \;\;\;\;\left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\_m\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(\cos \left(-0.5 \cdot \mathsf{PI}\left(\right)\right) - \cos \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \left(angle\_m \cdot \mathsf{PI}\left(\right)\right) \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

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 0.000215)
    (* (* (* (* (PI) angle_m) (+ a b)) (- b a)) 0.011111111111111112)
    (*
     (* (+ a b) (- b a))
     (-
      (cos (* -0.5 (PI)))
      (cos (fma 0.5 (PI) (* (* angle_m (PI)) 0.011111111111111112))))))))

Derivation

1. Split input into 2 regimes

2. if angle < 2.14999999999999995e-4

   1. Initial program 66.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 66.9

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

   5. Applied rewrites 66.9%

      \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]
   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. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift--.f64 76.2

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

   7. Applied rewrites 76.2%

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

   if 2.14999999999999995e-4 < angle

   1. Initial program 42.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

   4. Applied rewrites 42.2%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-cos.f64 N/A

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

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

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

      5. sin-mult N/A

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

      6. lower-/.f64 N/A

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

   6. Applied rewrites 32.6%

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

   7. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

   9. Applied rewrites 36.8%

      \[\leadsto \color{blue}{\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(\cos \left(-0.5 \cdot \mathsf{PI}\left(\right)\right) - \cos \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot 0.011111111111111112\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 0.000215:\\ \;\;\;\;\left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(\cos \left(-0.5 \cdot \mathsf{PI}\left(\right)\right) - \cos \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a}^{2}\right) \leq -5 \cdot 10^{-199}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle\_m \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot angle\_m\right) \cdot \left(b \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

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))) -5e-199)
    (* (* -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)))) < -4.9999999999999996e-199

   1. Initial program 59.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 57.6

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

   5. Applied rewrites 57.6%

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

   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. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-PI.f64 56.9

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

   8. Applied rewrites 56.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 56.9

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

   10. Applied rewrites 56.9%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-PI.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 N/A

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

       13. lift-PI.f64 63.1

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

   12. Applied rewrites 63.1%

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

   if -4.9999999999999996e-199 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 62.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 58.8

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

   5. Applied rewrites 58.8%

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

   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 56.5%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a}^{2}\right) \leq -5 \cdot 10^{-199}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(b \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a}^{2}\right) \leq -5 \cdot 10^{-199}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle\_m \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle\_m \cdot \left(b \cdot b\right)\right) \cdot \left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

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 (<= (* 2.0 (- (pow b 2.0) (pow a 2.0))) -5e-199)
    (* (* -0.011111111111111112 a) (* (* angle_m (PI)) a))
    (* (* angle_m (* b b)) (* 0.011111111111111112 (PI))))))

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)))) < -4.9999999999999996e-199

   1. Initial program 59.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 57.6

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

   5. Applied rewrites 57.6%

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

   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. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-PI.f64 56.9

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

   8. Applied rewrites 56.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 56.9

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

   10. Applied rewrites 56.9%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-PI.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 N/A

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

       13. lift-PI.f64 63.1

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

   12. Applied rewrites 63.1%

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

   if -4.9999999999999996e-199 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 62.3%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 63.6%

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

   6. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   8. Applied rewrites 43.9%

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

   9. Taylor expanded in a around 0

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

       1. lift-*.f64 N/A

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

       2. lift-PI.f64 56.7

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

   11. Applied rewrites 56.7%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a}^{2}\right) \leq -5 \cdot 10^{-199}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot \left(b \cdot b\right)\right) \cdot \left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot angle\_m\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 2 \cdot 10^{-310}:\\ \;\;\;\;\sin \left(t\_0 \cdot 0.011111111111111112\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_0 \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \end{array} \]

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 (* (PI) angle_m)))
   (*
    angle_s
    (if (<= (pow a 2.0) 2e-310)
      (* (sin (* t_0 0.011111111111111112)) (* b b))
      (* (* (* t_0 (+ a b)) (- b a)) 0.011111111111111112)))))

Derivation

1. Split input into 2 regimes

2. if (pow.f64 a #s(literal 2 binary64)) < 1.999999999999994e-310

   1. Initial program 74.5%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 74.4%

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

   6. Taylor expanded in a around 0

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

      1. lower-sin.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 74.4

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

   8. Applied rewrites 74.4%

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

   if 1.999999999999994e-310 < (pow.f64 a #s(literal 2 binary64))

   1. Initial program 56.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 56.4

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

   5. Applied rewrites 56.4%

      \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]
   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. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift--.f64 65.6

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

   7. Applied rewrites 65.6%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 2 \cdot 10^{-310}:\\ \;\;\;\;\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot angle\_m\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 2.25 \cdot 10^{-222}:\\ \;\;\;\;\sin \left(t\_0 \cdot 0.011111111111111112\right) \cdot \left(b \cdot b\right)\\ \mathbf{elif}\;a \leq 2.55 \cdot 10^{+153}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_0 \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \end{array} \]

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 (* (PI) angle_m)))
   (*
    angle_s
    (if (<= a 2.25e-222)
      (* (sin (* t_0 0.011111111111111112)) (* b b))
      (if (<= a 2.55e+153)
        (*
         2.0
         (*
          (* (sin (* (* 0.005555555555555556 angle_m) (PI))) (+ a b))
          (- b a)))
        (* (* (* t_0 (+ a b)) (- b a)) 0.011111111111111112))))))

Derivation

1. Split input into 3 regimes

2. if a < 2.25000000000000007e-222

   1. Initial program 62.4%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 56.4%

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

   6. Taylor expanded in a around 0

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

      1. lower-sin.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 44.5

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

   8. Applied rewrites 44.5%

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

   if 2.25000000000000007e-222 < a < 2.55000000000000018e153

   1. Initial program 61.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 60.4%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. lower-*.f64 N/A

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

   7. Applied rewrites 65.1%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 65.6%

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

   if 2.55000000000000018e153 < a

   1. Initial program 52.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 65.1

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

   5. Applied rewrites 65.1%

      \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]
   6. 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. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift--.f64 83.9

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

   7. Applied rewrites 83.9%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.25 \cdot 10^{-222}:\\ \;\;\;\;\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(b \cdot b\right)\\ \mathbf{elif}\;a \leq 2.55 \cdot 10^{+153}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 4.7 \cdot 10^{-87}:\\ \;\;\;\;\left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\_m\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{angle\_m}{180}\right)\right)\\ \end{array} \end{array} \]

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 4.7e-87)
    (* (* (* (* (PI) angle_m) (+ a b)) (- b a)) 0.011111111111111112)
    (* (* (- b a) (+ a b)) (sin (* 2.0 (* (PI) (/ angle_m 180.0))))))))

Derivation

1. Split input into 2 regimes

2. if angle < 4.7000000000000001e-87

   1. Initial program 63.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 63.9

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

   5. Applied rewrites 63.9%

      \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]
   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. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift--.f64 74.0

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

   7. Applied rewrites 74.0%

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

   if 4.7000000000000001e-87 < angle

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

   4. Applied rewrites 56.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

   6. Applied rewrites 56.0%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 4.7 \cdot 10^{-87}:\\ \;\;\;\;\left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 66.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot angle\_m\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 1.46 \cdot 10^{-88}:\\ \;\;\;\;\left(\left(t\_0 \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot t\_0\right) \cdot 0.005555555555555556\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\\ \end{array} \end{array} \end{array} \]

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 (* (PI) angle_m)))
   (*
    angle_s
    (if (<= angle_m 1.46e-88)
      (* (* (* t_0 (+ a b)) (- b a)) 0.011111111111111112)
      (* (sin (* (* 2.0 t_0) 0.005555555555555556)) (* (- b a) (+ a b)))))))

Derivation

1. Split input into 2 regimes

2. if angle < 1.4600000000000001e-88

   1. Initial program 63.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 63.9

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

   5. Applied rewrites 63.9%

      \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]
   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. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift--.f64 74.0

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

   7. Applied rewrites 74.0%

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

   if 1.4600000000000001e-88 < angle

   1. Initial program 54.6%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 55.3%

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

   7. Applied rewrites 55.3%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 1.46 \cdot 10^{-88}:\\ \;\;\;\;\left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot 0.005555555555555556\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 65.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot angle\_m\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 1.2 \cdot 10^{+142}:\\ \;\;\;\;\left(\left(t\_0 \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(t\_0 \cdot 0.005555555555555556\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

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 (* (PI) angle_m)))
   (*
    angle_s
    (if (<= angle_m 1.2e+142)
      (* (* (* t_0 (+ a b)) (- b a)) 0.011111111111111112)
      (* 2.0 (* (sin (* t_0 0.005555555555555556)) (* (+ b a) (- b a))))))))

Derivation

1. Split input into 2 regimes

2. if angle < 1.2e142

   1. Initial program 62.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 62.1

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

   5. Applied rewrites 62.1%

      \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]
   6. 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. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift--.f64 70.3

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

   7. Applied rewrites 70.3%

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

   if 1.2e142 < angle

   1. Initial program 50.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 48.6%

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

   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 40.8%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 1.2 \cdot 10^{+142}:\\ \;\;\;\;\left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 63.5% accurate, 13.7× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot angle\_m\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 1.25 \cdot 10^{+155}:\\ \;\;\;\;\left(\left(t\_0 \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot \left(\left(b + a\right) \cdot \left(-a\right)\right)\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \end{array} \]

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 (* (PI) angle_m)))
   (*
    angle_s
    (if (<= angle_m 1.25e+155)
      (* (* (* t_0 (+ a b)) (- b a)) 0.011111111111111112)
      (* (* t_0 (* (+ b a) (- a))) 0.011111111111111112)))))

Derivation

1. Split input into 2 regimes

2. if angle < 1.25e155

   1. Initial program 62.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 61.8

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

   5. Applied rewrites 61.8%

      \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]
   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. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift--.f64 70.0

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

   7. Applied rewrites 70.0%

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

   if 1.25e155 < angle

   1. Initial program 52.0%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 29.7

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

   5. Applied rewrites 29.7%

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

   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 27.1

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

   8. Applied rewrites 27.1%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 1.25 \cdot 10^{+155}:\\ \;\;\;\;\left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(-a\right)\right)\right) \cdot 0.011111111111111112\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 38.6% 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 61.2%

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

3. Taylor expanded in angle around 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 58.3

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

5. Applied rewrites 58.3%

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

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. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lift-PI.f64 36.1

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

8. Applied rewrites 36.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 36.1

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

10. Applied rewrites 36.1%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-PI.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. associate-*l* N/A

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

    7. *-commutative N/A

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

    8. lower-*.f64 N/A

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

    9. lift-*.f64 N/A

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

    10. *-commutative N/A

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

    11. lower-*.f64 N/A

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

    12. lower-*.f64 N/A

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

    13. lift-PI.f64 38.0

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

12. Applied rewrites 38.0%

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

13. Final simplification 38.0%

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

Reproduce

herbie shell --seed 2025051 
(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)))))