ab-angle->ABCF B

Percentage Accurate: 54.1% → 67.1%

Time: 13.1s

Alternatives: 17

Speedup: 13.7×

• Could not converge on a ground truth

  1. a = 8.342993830872738e+140
  2. b = 2.0742647783679128e-22
  3. angle = 1.4109533196537183e+137

• 33.6% 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: 67.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 6.5 \cdot 10^{+218}:\\ \;\;\;\;\left(\sin \left(\left(\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)} \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(b\_m - a\_m\right)\right) \cdot \left(b\_m + a\_m\right)\\ \mathbf{elif}\;b\_m \leq 2.6 \cdot 10^{+238}:\\ \;\;\;\;\left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(b\_m - a\_m\right)\right) \cdot 0.011111111111111112\right) \cdot angle\right) \cdot \left(b\_m + a\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -2.2862368541380886 \cdot 10^{-7}, {\mathsf{PI}\left(\right)}^{3}, \mathsf{PI}\left(\right) \cdot 0.011111111111111112\right) \cdot angle\right) \cdot \left(b\_m - a\_m\right)\right) \cdot \left(b\_m + a\_m\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= b_m 6.5e+218)
   (*
    (*
     (sin (* (* (cbrt (* (* (PI) (PI)) (PI))) angle) 0.011111111111111112))
     (- b_m a_m))
    (+ b_m a_m))
   (if (<= b_m 2.6e+238)
     (* (* (* (* (PI) (- b_m a_m)) 0.011111111111111112) angle) (+ b_m a_m))
     (*
      (*
       (*
        (fma
         (* (* angle angle) -2.2862368541380886e-7)
         (pow (PI) 3.0)
         (* (PI) 0.011111111111111112))
        angle)
       (- b_m a_m))
      (+ b_m a_m)))))

Derivation

1. Split input into 3 regimes

2. if b < 6.4999999999999999e218

   1. Initial program 49.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 62.9%

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

      1. lift-PI.f64 N/A

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

      2. add-cbrt-cube N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{90}\right)\right) \]

      3. lower-cbrt.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{90}\right)\right) \]

      4. rem-cube-cbrt N/A

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

      5. add-cbrt-cube N/A

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

      6. lift-PI.f64 N/A

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

      7. lower-pow.f64 66.0

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{\color{blue}{{\mathsf{PI}\left(\right)}^{3}}}\right) \cdot 0.011111111111111112\right)\right) \]

   6. Applied rewrites 66.0%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}}\right) \cdot 0.011111111111111112\right)\right) \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{90}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{90}\right)\right) \]

      4. lower-*.f64 66.0

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \mathsf{PI}\left(\right)}\right) \cdot 0.011111111111111112\right)\right) \]

   8. Applied rewrites 66.0%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right) \cdot 0.011111111111111112\right)\right) \]

   if 6.4999999999999999e218 < b < 2.6e238

   1. Initial program 25.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 49.6%

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

      1. lift-PI.f64 N/A

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

      2. add-cbrt-cube N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{90}\right)\right) \]

      3. lower-cbrt.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{90}\right)\right) \]

      4. rem-cube-cbrt N/A

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

      5. add-cbrt-cube N/A

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

      6. lift-PI.f64 N/A

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

      7. lower-pow.f64 49.6

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{\color{blue}{{\mathsf{PI}\left(\right)}^{3}}}\right) \cdot 0.011111111111111112\right)\right) \]

   6. Applied rewrites 49.6%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}}\right) \cdot 0.011111111111111112\right)\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. rem-cbrt-cube N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lift-PI.f64 N/A

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

      10. add-cube-cbrt N/A

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \frac{1}{90}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\left(\left(angle \cdot \frac{1}{90}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\left(\left(angle \cdot \frac{1}{90}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\color{blue}{\left(\left(angle \cdot \frac{1}{90}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\color{blue}{\left(\frac{1}{90} \cdot angle\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\color{blue}{\left(\frac{1}{90} \cdot angle\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \]

      16. pow2 N/A

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

      17. lower-pow.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. lower-cbrt.f64 N/A

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

      20. lift-PI.f64 N/A

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

      21. lower-cbrt.f64 74.4

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(0.011111111111111112 \cdot angle\right) \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}\right) \cdot \color{blue}{\sqrt[3]{\mathsf{PI}\left(\right)}}\right)\right) \]

   8. Applied rewrites 74.4%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\left(\left(0.011111111111111112 \cdot angle\right) \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right) \]

   9. Taylor expanded in angle around 0

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. lower--.f64 N/A

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

       9. lower-PI.f64 100.0

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

   11. Applied rewrites 100.0%

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

   if 2.6e238 < b

   1. Initial program 50.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 79.8%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(\left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(\left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\left(\color{blue}{\left(\frac{-1}{4374000} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3}} + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{-1}{4374000} \cdot {angle}^{2}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)} \cdot angle\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{4374000} \cdot {angle}^{2}}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{4374000} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{4374000} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{{\mathsf{PI}\left(\right)}^{3}}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right) \]

      9. lower-PI.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), {\color{blue}{\mathsf{PI}\left(\right)}}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), {\mathsf{PI}\left(\right)}^{3}, \color{blue}{\frac{1}{90} \cdot \mathsf{PI}\left(\right)}\right) \cdot angle\right)\right) \]

      11. lower-PI.f64 99.8

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), {\mathsf{PI}\left(\right)}^{3}, 0.011111111111111112 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot angle\right)\right) \]

   7. Applied rewrites 99.8%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(\mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), {\mathsf{PI}\left(\right)}^{3}, 0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{+218}:\\ \;\;\;\;\left(\sin \left(\left(\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)} \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{+238}:\\ \;\;\;\;\left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\right) \cdot angle\right) \cdot \left(b + a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -2.2862368541380886 \cdot 10^{-7}, {\mathsf{PI}\left(\right)}^{3}, \mathsf{PI}\left(\right) \cdot 0.011111111111111112\right) \cdot angle\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 67.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+146}:\\ \;\;\;\;\left(\sin \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left({\left({\left({\mathsf{PI}\left(\right)}^{0.16666666666666666}\right)}^{2}\right)}^{2} \cdot \left(0.011111111111111112 \cdot angle\right)\right)\right) \cdot \left(b\_m - a\_m\right)\right) \cdot \left(b\_m + a\_m\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{2}{{\left(b\_m + a\_m\right)}^{-1}} \cdot \frac{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}{{\left(b\_m - a\_m\right)}^{-1}}\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= (/ angle 180.0) 2e+146)
   (*
    (*
     (sin
      (*
       (cbrt (PI))
       (*
        (pow (pow (pow (PI) 0.16666666666666666) 2.0) 2.0)
        (* 0.011111111111111112 angle))))
     (- b_m a_m))
    (+ b_m a_m))
   (*
    1.0
    (*
     (/ 2.0 (pow (+ b_m a_m) -1.0))
     (/
      (sin (* (* (PI) angle) 0.005555555555555556))
      (pow (- b_m a_m) -1.0))))))

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 1.99999999999999987e146

   1. Initial program 55.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 71.1%

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

      1. lift-PI.f64 N/A

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

      2. add-cbrt-cube N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{90}\right)\right) \]

      3. lower-cbrt.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{90}\right)\right) \]

      4. rem-cube-cbrt N/A

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

      5. add-cbrt-cube N/A

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

      6. lift-PI.f64 N/A

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

      7. lower-pow.f64 73.9

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{\color{blue}{{\mathsf{PI}\left(\right)}^{3}}}\right) \cdot 0.011111111111111112\right)\right) \]

   6. Applied rewrites 73.9%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}}\right) \cdot 0.011111111111111112\right)\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. rem-cbrt-cube N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lift-PI.f64 N/A

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

      10. add-cube-cbrt N/A

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \frac{1}{90}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\left(\left(angle \cdot \frac{1}{90}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\left(\left(angle \cdot \frac{1}{90}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\color{blue}{\left(\left(angle \cdot \frac{1}{90}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\color{blue}{\left(\frac{1}{90} \cdot angle\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\color{blue}{\left(\frac{1}{90} \cdot angle\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \]

      16. pow2 N/A

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

      17. lower-pow.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. lower-cbrt.f64 N/A

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

      20. lift-PI.f64 N/A

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

      21. lower-cbrt.f64 73.0

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(0.011111111111111112 \cdot angle\right) \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}\right) \cdot \color{blue}{\sqrt[3]{\mathsf{PI}\left(\right)}}\right)\right) \]

   8. Applied rewrites 73.0%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\left(\left(0.011111111111111112 \cdot angle\right) \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right) \]
   9. Step-by-step derivation

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-cbrt.f64 N/A

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

      7. pow1/3 N/A

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

      8. pow-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 74.0

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(0.011111111111111112 \cdot angle\right) \cdot {\left({\left({\mathsf{PI}\left(\right)}^{\color{blue}{0.16666666666666666}}\right)}^{2}\right)}^{2}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \]

   10. Applied rewrites 74.0%

       \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(0.011111111111111112 \cdot angle\right) \cdot {\color{blue}{\left({\left({\mathsf{PI}\left(\right)}^{0.16666666666666666}\right)}^{2}\right)}}^{2}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \]

   if 1.99999999999999987e146 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 14.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift--.f64 N/A

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

      6. flip-- N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 20.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. div-inv N/A

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

      10. lift-/.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. unpow-prod-down N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 19.3%

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

   7. Taylor expanded in angle around 0

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

   9. Applied rewrites 38.4%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+146}:\\ \;\;\;\;\left(\sin \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left({\left({\left({\mathsf{PI}\left(\right)}^{0.16666666666666666}\right)}^{2}\right)}^{2} \cdot \left(0.011111111111111112 \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{2}{{\left(b + a\right)}^{-1}} \cdot \frac{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}{{\left(b - a\right)}^{-1}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} t_0 := {b\_m}^{2} - {a\_m}^{2}\\ t_1 := \left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\_m\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-105}:\\ \;\;\;\;\left(t\_1 \cdot a\_m\right) \cdot -0.011111111111111112\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(\left(\left(b\_m \cdot b\_m\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\_m\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (let* ((t_0 (- (pow b_m 2.0) (pow a_m 2.0))) (t_1 (* (* (PI) angle) a_m)))
   (if (<= t_0 -1e-105)
     (* (* t_1 a_m) -0.011111111111111112)
     (if (<= t_0 INFINITY)
       (* (* (* (* b_m b_m) (PI)) angle) 0.011111111111111112)
       (* (* -0.011111111111111112 a_m) t_1)))))

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -9.99999999999999965e-106

   1. Initial program 49.2%

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 48.4

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

   5. Applied rewrites 48.4%

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

   6. Taylor expanded in 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

   8. Applied rewrites 48.4%

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

   10. Applied rewrites 64.1%

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

   11. Taylor expanded in a around inf

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

   13. Applied rewrites 64.1%

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

   if -9.99999999999999965e-106 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < +inf.0

   1. Initial program 55.4%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 55.0

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

   5. Applied rewrites 55.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 55.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 46.5

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

   5. Applied rewrites 46.5%

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

   6. Taylor expanded in 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

   8. Applied rewrites 44.2%

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

   10. Applied rewrites 50.1%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -1 \cdot 10^{-105}:\\ \;\;\;\;\left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right) \cdot a\right) \cdot -0.011111111111111112\\ \mathbf{elif}\;{b}^{2} - {a}^{2} \leq \infty:\\ \;\;\;\;\left(\left(\left(b \cdot b\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} t_0 := {b\_m}^{2} - {a\_m}^{2}\\ t_1 := \left(-0.011111111111111112 \cdot a\_m\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\_m\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-105}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(\left(\left(b\_m \cdot b\_m\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (let* ((t_0 (- (pow b_m 2.0) (pow a_m 2.0)))
        (t_1 (* (* -0.011111111111111112 a_m) (* (* (PI) angle) a_m))))
   (if (<= t_0 -1e-105)
     t_1
     (if (<= t_0 INFINITY)
       (* (* (* (* b_m b_m) (PI)) angle) 0.011111111111111112)
       t_1))))

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -9.99999999999999965e-106 or +inf.0 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 42.5%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 48.2

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

   5. Applied rewrites 48.2%

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

   6. Taylor expanded in 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

   8. Applied rewrites 47.9%

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

   10. Applied rewrites 62.2%

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

   if -9.99999999999999965e-106 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < +inf.0

   1. Initial program 55.4%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 55.0

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

   5. Applied rewrites 55.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 55.1%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -1 \cdot 10^{-105}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\\ \mathbf{elif}\;{b}^{2} - {a}^{2} \leq \infty:\\ \;\;\;\;\left(\left(\left(b \cdot b\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+142}:\\ \;\;\;\;\left(\sin \left(\left(\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}} \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(b\_m - a\_m\right)\right) \cdot \frac{-1}{\frac{-1}{b\_m + a\_m}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{2}{{\left(b\_m + a\_m\right)}^{-1}} \cdot \frac{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}{{\left(b\_m - a\_m\right)}^{-1}}\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= (/ angle 180.0) 2e+142)
   (*
    (*
     (sin (* (* (cbrt (pow (PI) 3.0)) angle) 0.011111111111111112))
     (- b_m a_m))
    (/ -1.0 (/ -1.0 (+ b_m a_m))))
   (*
    1.0
    (*
     (/ 2.0 (pow (+ b_m a_m) -1.0))
     (/
      (sin (* (* (PI) angle) 0.005555555555555556))
      (pow (- b_m a_m) -1.0))))))

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2.0000000000000001e142

   1. Initial program 55.6%

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

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 71.7%

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

      1. lift-PI.f64 N/A

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

      2. add-cbrt-cube N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{90}\right)\right) \]

      3. lower-cbrt.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{90}\right)\right) \]

      4. rem-cube-cbrt N/A

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

      5. add-cbrt-cube N/A

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

      6. lift-PI.f64 N/A

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

      7. lower-pow.f64 74.6

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{\color{blue}{{\mathsf{PI}\left(\right)}^{3}}}\right) \cdot 0.011111111111111112\right)\right) \]

   6. Applied rewrites 74.6%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}}\right) \cdot 0.011111111111111112\right)\right) \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 74.6

         \[\leadsto \color{blue}{\left(b + a\right)} \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}\right) \cdot 0.011111111111111112\right)\right) \]

      4. remove-double-div N/A

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

      5. unpow-1 N/A

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

      6. lift-pow.f64 N/A

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

      7. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left({\left(b + a\right)}^{-1}\right)}} \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}\right) \cdot \frac{1}{90}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1}}{\mathsf{neg}\left({\left(b + a\right)}^{-1}\right)} \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}\right) \cdot \frac{1}{90}\right)\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{\mathsf{neg}\left({\left(b + a\right)}^{-1}\right)}} \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}\right) \cdot \frac{1}{90}\right)\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{-1}{\mathsf{neg}\left(\color{blue}{{\left(b + a\right)}^{-1}}\right)} \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}\right) \cdot \frac{1}{90}\right)\right) \]

      11. unpow-1 N/A

          \[\leadsto \frac{-1}{\mathsf{neg}\left(\color{blue}{\frac{1}{b + a}}\right)} \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}\right) \cdot \frac{1}{90}\right)\right) \]

      12. distribute-neg-frac N/A

          \[\leadsto \frac{-1}{\color{blue}{\frac{\mathsf{neg}\left(1\right)}{b + a}}} \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}\right) \cdot \frac{1}{90}\right)\right) \]

      13. metadata-eval N/A

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

      14. lower-/.f64 74.6

          \[\leadsto \frac{-1}{\color{blue}{\frac{-1}{b + a}}} \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}\right) \cdot 0.011111111111111112\right)\right) \]

   8. Applied rewrites 74.6%

      \[\leadsto \color{blue}{\frac{-1}{\frac{-1}{b + a}}} \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}\right) \cdot 0.011111111111111112\right)\right) \]

   if 2.0000000000000001e142 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 16.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift--.f64 N/A

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

      6. flip-- N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 20.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. div-inv N/A

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

      10. lift-/.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. unpow-prod-down N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 21.4%

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

   7. Taylor expanded in angle around 0

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

   9. Applied rewrites 36.5%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+142}:\\ \;\;\;\;\left(\sin \left(\left(\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}} \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(b - a\right)\right) \cdot \frac{-1}{\frac{-1}{b + a}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{2}{{\left(b + a\right)}^{-1}} \cdot \frac{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}{{\left(b - a\right)}^{-1}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+145}:\\ \;\;\;\;\left(\sin \left(\left(\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}} \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(b\_m - a\_m\right)\right) \cdot \frac{-1}{\frac{-1}{b\_m + a\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2}{{\left(\left(b\_m - a\_m\right) \cdot \left(b\_m + a\_m\right)\right)}^{-1}} \cdot 1\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= (/ angle 180.0) 5e+145)
   (*
    (*
     (sin (* (* (cbrt (pow (PI) 3.0)) angle) 0.011111111111111112))
     (- b_m a_m))
    (/ -1.0 (/ -1.0 (+ b_m a_m))))
   (*
    (/
     (* (sin (* (* 0.005555555555555556 angle) (PI))) 2.0)
     (pow (* (- b_m a_m) (+ b_m a_m)) -1.0))
    1.0)))

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.99999999999999967e145

   1. Initial program 55.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 71.4%

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

      1. lift-PI.f64 N/A

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

      2. add-cbrt-cube N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{90}\right)\right) \]

      3. lower-cbrt.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{90}\right)\right) \]

      4. rem-cube-cbrt N/A

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

      5. add-cbrt-cube N/A

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

      6. lift-PI.f64 N/A

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

      7. lower-pow.f64 74.3

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{\color{blue}{{\mathsf{PI}\left(\right)}^{3}}}\right) \cdot 0.011111111111111112\right)\right) \]

   6. Applied rewrites 74.3%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}}\right) \cdot 0.011111111111111112\right)\right) \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 74.3

         \[\leadsto \color{blue}{\left(b + a\right)} \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}\right) \cdot 0.011111111111111112\right)\right) \]

      4. remove-double-div N/A

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

      5. unpow-1 N/A

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

      6. lift-pow.f64 N/A

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

      7. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left({\left(b + a\right)}^{-1}\right)}} \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}\right) \cdot \frac{1}{90}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1}}{\mathsf{neg}\left({\left(b + a\right)}^{-1}\right)} \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}\right) \cdot \frac{1}{90}\right)\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{\mathsf{neg}\left({\left(b + a\right)}^{-1}\right)}} \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}\right) \cdot \frac{1}{90}\right)\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{-1}{\mathsf{neg}\left(\color{blue}{{\left(b + a\right)}^{-1}}\right)} \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}\right) \cdot \frac{1}{90}\right)\right) \]

      11. unpow-1 N/A

          \[\leadsto \frac{-1}{\mathsf{neg}\left(\color{blue}{\frac{1}{b + a}}\right)} \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}\right) \cdot \frac{1}{90}\right)\right) \]

      12. distribute-neg-frac N/A

          \[\leadsto \frac{-1}{\color{blue}{\frac{\mathsf{neg}\left(1\right)}{b + a}}} \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}\right) \cdot \frac{1}{90}\right)\right) \]

      13. metadata-eval N/A

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

      14. lower-/.f64 74.3

          \[\leadsto \frac{-1}{\color{blue}{\frac{-1}{b + a}}} \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}\right) \cdot 0.011111111111111112\right)\right) \]

   8. Applied rewrites 74.3%

      \[\leadsto \color{blue}{\frac{-1}{\frac{-1}{b + a}}} \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}\right) \cdot 0.011111111111111112\right)\right) \]

   if 4.99999999999999967e145 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 16.2%

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

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift--.f64 N/A

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

      6. flip-- N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 19.9%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 40.8%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+145}:\\ \;\;\;\;\left(\sin \left(\left(\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}} \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(b - a\right)\right) \cdot \frac{-1}{\frac{-1}{b + a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2}{{\left(\left(b - a\right) \cdot \left(b + a\right)\right)}^{-1}} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+145}:\\ \;\;\;\;\left(\sin \left(\left(\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)} \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(b\_m - a\_m\right)\right) \cdot \left(b\_m + a\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2}{{\left(\left(b\_m - a\_m\right) \cdot \left(b\_m + a\_m\right)\right)}^{-1}} \cdot 1\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= (/ angle 180.0) 5e+145)
   (*
    (*
     (sin (* (* (cbrt (* (* (PI) (PI)) (PI))) angle) 0.011111111111111112))
     (- b_m a_m))
    (+ b_m a_m))
   (*
    (/
     (* (sin (* (* 0.005555555555555556 angle) (PI))) 2.0)
     (pow (* (- b_m a_m) (+ b_m a_m)) -1.0))
    1.0)))

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.99999999999999967e145

   1. Initial program 55.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 71.4%

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

      1. lift-PI.f64 N/A

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

      2. add-cbrt-cube N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{90}\right)\right) \]

      3. lower-cbrt.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{90}\right)\right) \]

      4. rem-cube-cbrt N/A

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

      5. add-cbrt-cube N/A

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

      6. lift-PI.f64 N/A

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

      7. lower-pow.f64 74.3

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{\color{blue}{{\mathsf{PI}\left(\right)}^{3}}}\right) \cdot 0.011111111111111112\right)\right) \]

   6. Applied rewrites 74.3%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}}\right) \cdot 0.011111111111111112\right)\right) \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{90}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{90}\right)\right) \]

      4. lower-*.f64 74.3

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \mathsf{PI}\left(\right)}\right) \cdot 0.011111111111111112\right)\right) \]

   8. Applied rewrites 74.3%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right) \cdot 0.011111111111111112\right)\right) \]

   if 4.99999999999999967e145 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 16.2%

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

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift--.f64 N/A

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

      6. flip-- N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 19.9%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 40.8%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+145}:\\ \;\;\;\;\left(\sin \left(\left(\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)} \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2}{{\left(\left(b - a\right) \cdot \left(b + a\right)\right)}^{-1}} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot 0.011111111111111112\\ t_1 := \sqrt{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;a\_m \leq 4.3 \cdot 10^{+145}:\\ \;\;\;\;\left(\sin \left(\left(t\_1 \cdot \left(0.011111111111111112 \cdot angle\right)\right) \cdot t\_1\right) \cdot \left(b\_m - a\_m\right)\right) \cdot \left(b\_m + a\_m\right)\\ \mathbf{elif}\;a\_m \leq 2 \cdot 10^{+236}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -2.2862368541380886 \cdot 10^{-7}, {\mathsf{PI}\left(\right)}^{3}, t\_0\right) \cdot angle\right) \cdot \left(b\_m - a\_m\right)\right) \cdot \left(b\_m + a\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_0 \cdot angle\right) \cdot \left(b\_m - a\_m\right)\right) \cdot \left(b\_m + a\_m\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (let* ((t_0 (* (PI) 0.011111111111111112)) (t_1 (sqrt (PI))))
   (if (<= a_m 4.3e+145)
     (*
      (* (sin (* (* t_1 (* 0.011111111111111112 angle)) t_1)) (- b_m a_m))
      (+ b_m a_m))
     (if (<= a_m 2e+236)
       (*
        (*
         (*
          (fma (* (* angle angle) -2.2862368541380886e-7) (pow (PI) 3.0) t_0)
          angle)
         (- b_m a_m))
        (+ b_m a_m))
       (* (* (* t_0 angle) (- b_m a_m)) (+ b_m a_m))))))

Derivation

1. Split input into 3 regimes

2. if a < 4.29999999999999998e145

   1. Initial program 51.6%

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

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 60.4%

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

      1. lift-PI.f64 N/A

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

      2. add-cbrt-cube N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{90}\right)\right) \]

      3. lower-cbrt.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{90}\right)\right) \]

      4. rem-cube-cbrt N/A

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

      5. add-cbrt-cube N/A

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

      6. lift-PI.f64 N/A

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

      7. lower-pow.f64 62.9

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{\color{blue}{{\mathsf{PI}\left(\right)}^{3}}}\right) \cdot 0.011111111111111112\right)\right) \]

   6. Applied rewrites 62.9%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}}\right) \cdot 0.011111111111111112\right)\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. rem-cbrt-cube N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lift-PI.f64 N/A

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

      10. add-sqr-sqrt N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-PI.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. lower-sqrt.f64 61.0

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

   8. Applied rewrites 61.0%

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

   if 4.29999999999999998e145 < a < 2.00000000000000011e236

   1. Initial program 27.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 82.2%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(\left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(\left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\left(\color{blue}{\left(\frac{-1}{4374000} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3}} + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{-1}{4374000} \cdot {angle}^{2}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)} \cdot angle\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{4374000} \cdot {angle}^{2}}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{4374000} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{4374000} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{{\mathsf{PI}\left(\right)}^{3}}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right) \]

      9. lower-PI.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), {\color{blue}{\mathsf{PI}\left(\right)}}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), {\mathsf{PI}\left(\right)}^{3}, \color{blue}{\frac{1}{90} \cdot \mathsf{PI}\left(\right)}\right) \cdot angle\right)\right) \]

      11. lower-PI.f64 86.7

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), {\mathsf{PI}\left(\right)}^{3}, 0.011111111111111112 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot angle\right)\right) \]

   7. Applied rewrites 86.7%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(\mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), {\mathsf{PI}\left(\right)}^{3}, 0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}\right) \]

   if 2.00000000000000011e236 < a

   1. Initial program 50.9%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 69.0

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

   5. Applied rewrites 69.0%

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

   7. Applied rewrites 99.7%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.3 \cdot 10^{+145}:\\ \;\;\;\;\left(\sin \left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(0.011111111111111112 \cdot angle\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\\ \mathbf{elif}\;a \leq 2 \cdot 10^{+236}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -2.2862368541380886 \cdot 10^{-7}, {\mathsf{PI}\left(\right)}^{3}, \mathsf{PI}\left(\right) \cdot 0.011111111111111112\right) \cdot angle\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{PI}\left(\right) \cdot 0.011111111111111112\right) \cdot angle\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 1.32 \cdot 10^{+125}:\\ \;\;\;\;\left(\sin \left(\mathsf{PI}\left(\right) \cdot \left(0.011111111111111112 \cdot angle\right)\right) \cdot \left(b\_m + a\_m\right)\right) \cdot \left(b\_m - a\_m\right)\\ \mathbf{elif}\;b\_m \leq 2.6 \cdot 10^{+238}:\\ \;\;\;\;\left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(b\_m - a\_m\right)\right) \cdot 0.011111111111111112\right) \cdot angle\right) \cdot \left(b\_m + a\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -2.2862368541380886 \cdot 10^{-7}, {\mathsf{PI}\left(\right)}^{3}, \mathsf{PI}\left(\right) \cdot 0.011111111111111112\right) \cdot angle\right) \cdot \left(b\_m - a\_m\right)\right) \cdot \left(b\_m + a\_m\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= b_m 1.32e+125)
   (*
    (* (sin (* (PI) (* 0.011111111111111112 angle))) (+ b_m a_m))
    (- b_m a_m))
   (if (<= b_m 2.6e+238)
     (* (* (* (* (PI) (- b_m a_m)) 0.011111111111111112) angle) (+ b_m a_m))
     (*
      (*
       (*
        (fma
         (* (* angle angle) -2.2862368541380886e-7)
         (pow (PI) 3.0)
         (* (PI) 0.011111111111111112))
        angle)
       (- b_m a_m))
      (+ b_m a_m)))))

Derivation

1. Split input into 3 regimes

2. if b < 1.32000000000000006e125

   1. Initial program 50.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 62.2%

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

      1. lift-PI.f64 N/A

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

      2. add-cbrt-cube N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{90}\right)\right) \]

      3. lower-cbrt.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{90}\right)\right) \]

      4. rem-cube-cbrt N/A

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

      5. add-cbrt-cube N/A

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

      6. lift-PI.f64 N/A

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

      7. lower-pow.f64 64.6

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{\color{blue}{{\mathsf{PI}\left(\right)}^{3}}}\right) \cdot 0.011111111111111112\right)\right) \]

   6. Applied rewrites 64.6%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}}\right) \cdot 0.011111111111111112\right)\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cbrt.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. rem-cbrt-cube N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. associate-*r* N/A

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

   8. Applied rewrites 61.8%

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

   if 1.32000000000000006e125 < b < 2.6e238

   1. Initial program 42.6%

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

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 68.0%

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

      1. lift-PI.f64 N/A

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

      2. add-cbrt-cube N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{90}\right)\right) \]

      3. lower-cbrt.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{90}\right)\right) \]

      4. rem-cube-cbrt N/A

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

      5. add-cbrt-cube N/A

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

      6. lift-PI.f64 N/A

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

      7. lower-pow.f64 78.4

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{\color{blue}{{\mathsf{PI}\left(\right)}^{3}}}\right) \cdot 0.011111111111111112\right)\right) \]

   6. Applied rewrites 78.4%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}}\right) \cdot 0.011111111111111112\right)\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. rem-cbrt-cube N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lift-PI.f64 N/A

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

      10. add-cube-cbrt N/A

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \frac{1}{90}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\left(\left(angle \cdot \frac{1}{90}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\left(\left(angle \cdot \frac{1}{90}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\color{blue}{\left(\left(angle \cdot \frac{1}{90}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\color{blue}{\left(\frac{1}{90} \cdot angle\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\color{blue}{\left(\frac{1}{90} \cdot angle\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \]

      16. pow2 N/A

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

      17. lower-pow.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. lower-cbrt.f64 N/A

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

      20. lift-PI.f64 N/A

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

      21. lower-cbrt.f64 73.8

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(0.011111111111111112 \cdot angle\right) \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}\right) \cdot \color{blue}{\sqrt[3]{\mathsf{PI}\left(\right)}}\right)\right) \]

   8. Applied rewrites 73.8%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\left(\left(0.011111111111111112 \cdot angle\right) \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right) \]

   9. Taylor expanded in angle around 0

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. lower--.f64 N/A

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

       9. lower-PI.f64 84.5

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

   11. Applied rewrites 84.5%

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

   if 2.6e238 < b

   1. Initial program 50.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 79.8%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(\left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(\left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\left(\color{blue}{\left(\frac{-1}{4374000} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3}} + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{-1}{4374000} \cdot {angle}^{2}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)} \cdot angle\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{4374000} \cdot {angle}^{2}}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{4374000} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{4374000} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{{\mathsf{PI}\left(\right)}^{3}}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right) \]

      9. lower-PI.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), {\color{blue}{\mathsf{PI}\left(\right)}}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), {\mathsf{PI}\left(\right)}^{3}, \color{blue}{\frac{1}{90} \cdot \mathsf{PI}\left(\right)}\right) \cdot angle\right)\right) \]

      11. lower-PI.f64 99.8

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), {\mathsf{PI}\left(\right)}^{3}, 0.011111111111111112 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot angle\right)\right) \]

   7. Applied rewrites 99.8%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(\mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), {\mathsf{PI}\left(\right)}^{3}, 0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.32 \cdot 10^{+125}:\\ \;\;\;\;\left(\sin \left(\mathsf{PI}\left(\right) \cdot \left(0.011111111111111112 \cdot angle\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{+238}:\\ \;\;\;\;\left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\right) \cdot angle\right) \cdot \left(b + a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -2.2862368541380886 \cdot 10^{-7}, {\mathsf{PI}\left(\right)}^{3}, \mathsf{PI}\left(\right) \cdot 0.011111111111111112\right) \cdot angle\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+145}:\\ \;\;\;\;\left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(b\_m - a\_m\right)\right) \cdot \left(b\_m + a\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-a\_m\right) \cdot \left(b\_m + a\_m\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 0.011111111111111112\right) \cdot angle\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= (/ angle 180.0) 5e+145)
   (*
    (* (sin (* (* (PI) angle) 0.011111111111111112)) (- b_m a_m))
    (+ b_m a_m))
   (* (* (- a_m) (+ b_m a_m)) (* (* (PI) 0.011111111111111112) angle))))

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.99999999999999967e145

   1. Initial program 55.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 71.4%

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

   if 4.99999999999999967e145 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 16.2%

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 22.5

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

   5. Applied rewrites 22.5%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 17.8%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+145}:\\ \;\;\;\;\left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-a\right) \cdot \left(b + a\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 0.011111111111111112\right) \cdot angle\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 1.32 \cdot 10^{+125}:\\ \;\;\;\;\left(\sin \left(\mathsf{PI}\left(\right) \cdot \left(0.011111111111111112 \cdot angle\right)\right) \cdot \left(b\_m + a\_m\right)\right) \cdot \left(b\_m - a\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(b\_m - a\_m\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(b\_m + a\_m\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= b_m 1.32e+125)
   (*
    (* (sin (* (PI) (* 0.011111111111111112 angle))) (+ b_m a_m))
    (- b_m a_m))
   (* (* (* (* (PI) (- b_m a_m)) angle) 0.011111111111111112) (+ b_m a_m))))

Derivation

1. Split input into 2 regimes

2. if b < 1.32000000000000006e125

   1. Initial program 50.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 62.2%

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

      1. lift-PI.f64 N/A

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

      2. add-cbrt-cube N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{90}\right)\right) \]

      3. lower-cbrt.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{90}\right)\right) \]

      4. rem-cube-cbrt N/A

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

      5. add-cbrt-cube N/A

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

      6. lift-PI.f64 N/A

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

      7. lower-pow.f64 64.6

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{\color{blue}{{\mathsf{PI}\left(\right)}^{3}}}\right) \cdot 0.011111111111111112\right)\right) \]

   6. Applied rewrites 64.6%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}}\right) \cdot 0.011111111111111112\right)\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cbrt.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. rem-cbrt-cube N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. associate-*r* N/A

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

   8. Applied rewrites 61.8%

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

   if 1.32000000000000006e125 < b

   1. Initial program 45.2%

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

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 72.1%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-PI.f64 79.3

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

   7. Applied rewrites 79.3%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.32 \cdot 10^{+125}:\\ \;\;\;\;\left(\sin \left(\mathsf{PI}\left(\right) \cdot \left(0.011111111111111112 \cdot angle\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(b + a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 39.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;{a\_m}^{2} \leq 10^{+251}:\\ \;\;\;\;\left(\left(a\_m \cdot a\_m\right) \cdot -0.011111111111111112\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a\_m \cdot angle\right) \cdot \left(-0.011111111111111112 \cdot a\_m\right)\right) \cdot \mathsf{PI}\left(\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= (pow a_m 2.0) 1e+251)
   (* (* (* a_m a_m) -0.011111111111111112) (* (PI) angle))
   (* (* (* a_m angle) (* -0.011111111111111112 a_m)) (PI))))

Derivation

1. Split input into 2 regimes

2. if (pow.f64 a #s(literal 2 binary64)) < 1e251

   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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 53.5

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

   5. Applied rewrites 53.5%

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

   6. Taylor expanded in 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

   8. Applied rewrites 29.4%

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

   if 1e251 < (pow.f64 a #s(literal 2 binary64))

   1. Initial program 33.4%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 47.7

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

   5. Applied rewrites 47.7%

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

   6. Taylor expanded in 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

   8. Applied rewrites 43.2%

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

   10. Applied rewrites 65.7%

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

   12. Applied rewrites 65.7%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 10^{+251}:\\ \;\;\;\;\left(\left(a \cdot a\right) \cdot -0.011111111111111112\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot angle\right) \cdot \left(-0.011111111111111112 \cdot a\right)\right) \cdot \mathsf{PI}\left(\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 39.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot angle\\ \mathbf{if}\;{a\_m}^{2} \leq 5 \cdot 10^{+246}:\\ \;\;\;\;\left(\left(a\_m \cdot a\_m\right) \cdot -0.011111111111111112\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\_m\right) \cdot \left(t\_0 \cdot a\_m\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (let* ((t_0 (* (PI) angle)))
   (if (<= (pow a_m 2.0) 5e+246)
     (* (* (* a_m a_m) -0.011111111111111112) t_0)
     (* (* -0.011111111111111112 a_m) (* t_0 a_m)))))

Derivation

1. Split input into 2 regimes

2. if (pow.f64 a #s(literal 2 binary64)) < 4.99999999999999976e246

   1. Initial program 55.9%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 53.3

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

   5. Applied rewrites 53.3%

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

   6. Taylor expanded in 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

   8. Applied rewrites 29.0%

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

   if 4.99999999999999976e246 < (pow.f64 a #s(literal 2 binary64))

   1. Initial program 34.2%

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 48.4

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

   5. Applied rewrites 48.4%

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

   6. Taylor expanded in 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

   8. Applied rewrites 43.9%

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

   10. Applied rewrites 66.2%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 5 \cdot 10^{+246}:\\ \;\;\;\;\left(\left(a \cdot a\right) \cdot -0.011111111111111112\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 63.2% accurate, 10.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} t_0 := \left(\mathsf{PI}\left(\right) \cdot 0.011111111111111112\right) \cdot angle\\ \mathbf{if}\;\frac{angle}{180} \leq 10^{+143}:\\ \;\;\;\;\left(t\_0 \cdot \left(b\_m - a\_m\right)\right) \cdot \left(b\_m + a\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-a\_m\right) \cdot \left(b\_m + a\_m\right)\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (let* ((t_0 (* (* (PI) 0.011111111111111112) angle)))
   (if (<= (/ angle 180.0) 1e+143)
     (* (* t_0 (- b_m a_m)) (+ b_m a_m))
     (* (* (- a_m) (+ b_m a_m)) t_0))))

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 1e143

   1. Initial program 55.6%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 57.4

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

   5. Applied rewrites 57.4%

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

   7. Applied rewrites 68.3%

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

   if 1e143 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 16.4%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 22.0

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

   5. Applied rewrites 22.0%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 17.4%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{+143}:\\ \;\;\;\;\left(\left(\left(\mathsf{PI}\left(\right) \cdot 0.011111111111111112\right) \cdot angle\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-a\right) \cdot \left(b + a\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 0.011111111111111112\right) \cdot angle\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 58.1% accurate, 13.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 2.4 \cdot 10^{+148}:\\ \;\;\;\;\left(\left(\left(b\_m - a\_m\right) \cdot \left(b\_m + a\_m\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot 0.011111111111111112\right)\right) \cdot angle\\ \mathbf{else}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\_m\right)\right) \cdot a\_m\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= a_m 2.4e+148)
   (* (* (* (- b_m a_m) (+ b_m a_m)) (* (PI) 0.011111111111111112)) angle)
   (* (* -0.011111111111111112 (* (* (PI) angle) a_m)) a_m)))

Derivation

1. Split input into 2 regimes

2. if a < 2.39999999999999995e148

   1. Initial program 51.6%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 52.7

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

   5. Applied rewrites 52.7%

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

   7. Applied rewrites 52.8%

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

   if 2.39999999999999995e148 < a

   1. Initial program 32.6%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 44.8

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

   5. Applied rewrites 44.8%

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

   6. Taylor expanded in 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

   8. Applied rewrites 40.4%

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

   10. Applied rewrites 72.6%

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

   12. Applied rewrites 72.6%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.4 \cdot 10^{+148}:\\ \;\;\;\;\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot 0.011111111111111112\right)\right) \cdot angle\\ \mathbf{else}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 58.2% accurate, 13.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(\left(b\_m - a\_m\right) \cdot \left(b\_m + a\_m\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 0.011111111111111112\right) \cdot angle\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\_m\right) \cdot a\_m\right) \cdot -0.011111111111111112\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= a_m 1.35e+154)
   (* (* (- b_m a_m) (+ b_m a_m)) (* (* (PI) 0.011111111111111112) angle))
   (* (* (* (* (PI) angle) a_m) a_m) -0.011111111111111112)))

Derivation

1. Split input into 2 regimes

2. if a < 1.35000000000000003e154

   1. Initial program 51.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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

         \[\leadsto \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      9. lower-PI.f64 N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      11. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      12. difference-of-squares N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      14. lower-+.f64 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      15. lower--.f64 53.0

          \[\leadsto \left(angle \cdot \left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

   5. Applied rewrites 53.0%

      \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   if 1.35000000000000003e154 < a

   1. Initial program 28.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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

         \[\leadsto \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      9. lower-PI.f64 N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      11. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      12. difference-of-squares N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      14. lower-+.f64 N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      15. lower--.f64 41.7

          \[\leadsto \left(angle \cdot \left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

   5. Applied rewrites 41.7%

      \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   6. Taylor expanded in 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

   8. Applied rewrites 36.8%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 72.8%

       \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \]

   11. Taylor expanded in a around inf

       \[\leadsto {a}^{2} \cdot \color{blue}{\left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{90} \cdot \frac{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)}{a}\right)} \]

   13. Applied rewrites 72.8%

       \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right) \cdot a\right) \cdot \color{blue}{-0.011111111111111112} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 0.011111111111111112\right) \cdot angle\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right) \cdot a\right) \cdot -0.011111111111111112\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 38.4% accurate, 21.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \left(-0.011111111111111112 \cdot a\_m\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\_m\right) \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (* (* -0.011111111111111112 a_m) (* (* (PI) angle) a_m)))

Derivation

1. Initial program 49.5%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
2. Add Preprocessing

3. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \frac{1}{90}} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

   3. *-commutative N/A

      \[\leadsto angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

   4. associate-*r* N/A

      \[\leadsto angle \cdot \color{blue}{\left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

   5. associate-*r* N/A

      \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \left(angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

   9. lower-PI.f64 N/A

      \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

   10. unpow2 N/A

       \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

   11. unpow2 N/A

       \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

   12. difference-of-squares N/A

       \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   13. lower-*.f64 N/A

       \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   14. lower-+.f64 N/A

       \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

   15. lower--.f64 51.8

       \[\leadsto \left(angle \cdot \left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

5. Applied rewrites 51.8%

   \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

6. Taylor expanded in 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

8. Applied rewrites 33.4%

   \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \]
9. Step-by-step derivation

10. Applied rewrites 37.6%

    \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \]

11. Final simplification 37.6%

    \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024296 
(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)))))