ab-angle->ABCF B

Percentage Accurate: 54.8% → 67.6%

Time: 11.5s

Alternatives: 15

Speedup: 10.3×

• 32.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

Initial Program: 54.8% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 67.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} t_0 := -\left(\frac{a\_m}{b\_m} - 1\right)\\ t_1 := \sqrt[3]{{\mathsf{PI}\left(\right)}^{1.5}}\\ t_2 := \sqrt{\mathsf{PI}\left(\right)}\\ t_3 := t\_2 \cdot angle\\ \mathbf{if}\;b\_m \leq 5.4 \cdot 10^{-24}:\\ \;\;\;\;\left(a\_m + b\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(\left(angle \cdot \left(t\_1 \cdot t\_1\right)\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{elif}\;b\_m \leq 3.2 \cdot 10^{+166}:\\ \;\;\;\;\left(a\_m + b\_m\right) \cdot \left(\left(\sin \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(t\_3 \cdot \left(0.011111111111111112 \cdot {\mathsf{PI}\left(\right)}^{0.16666666666666666}\right)\right)\right) \cdot t\_0\right) \cdot b\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a\_m + b\_m\right) \cdot \left(\left(\sin \left(t\_3 \cdot \left(t\_2 \cdot 0.011111111111111112\right)\right) \cdot t\_0\right) \cdot b\_m\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (let* ((t_0 (- (- (/ a_m b_m) 1.0)))
        (t_1 (cbrt (pow (PI) 1.5)))
        (t_2 (sqrt (PI)))
        (t_3 (* t_2 angle)))
   (if (<= b_m 5.4e-24)
     (*
      (+ a_m b_m)
      (* (- b_m a_m) (sin (* (* angle (* t_1 t_1)) 0.011111111111111112))))
     (if (<= b_m 3.2e+166)
       (*
        (+ a_m b_m)
        (*
         (*
          (sin
           (*
            (cbrt (PI))
            (* t_3 (* 0.011111111111111112 (pow (PI) 0.16666666666666666)))))
          t_0)
         b_m))
       (*
        (+ a_m b_m)
        (* (* (sin (* t_3 (* t_2 0.011111111111111112))) t_0) b_m))))))

Derivation

1. Split input into 3 regimes

2. if b < 5.40000000000000014e-24

   1. Initial program 53.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. 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 67.2%

      \[\leadsto \color{blue}{\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot 0.011111111111111112\right)\right)} \]
   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. lift-PI.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. add-sqr-sqrt N/A

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

      6. unswap-sqr N/A

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

      7. cbrt-prod N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. unpow1 N/A

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

      11. lift-PI.f64 N/A

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

      12. pow1/2 N/A

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

      13. pow-prod-up N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-pow.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-cbrt.f64 N/A

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

      19. unpow1 N/A

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

      20. lift-PI.f64 N/A

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

   6. Applied rewrites 66.5%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\left(\sqrt[3]{{\mathsf{PI}\left(\right)}^{1.5}} \cdot \sqrt[3]{{\mathsf{PI}\left(\right)}^{1.5}}\right)}\right) \cdot 0.011111111111111112\right)\right) \]

   if 5.40000000000000014e-24 < b < 3.19999999999999968e166

   1. Initial program 53.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 64.4%

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

   5. Taylor expanded in b around inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. remove-double-neg N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-neg-out N/A

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

      6. mul-1-neg N/A

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

      7. neg-mul-1 N/A

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

      8. associate-*l* N/A

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

      9. distribute-lft-in N/A

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

      10. distribute-rgt-in N/A

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

   7. Applied rewrites 64.4%

      \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(\left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(\frac{a}{b} - 1\right)\right) \cdot \left(-b\right)\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 78.6%

      \[\leadsto \left(a + b\right) \cdot \left(\left(\sin \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot angle\right) \cdot \left(0.011111111111111112 \cdot {\mathsf{PI}\left(\right)}^{0.16666666666666666}\right)\right)\right) \cdot \left(\frac{a}{b} - 1\right)\right) \cdot \left(-b\right)\right) \]

   if 3.19999999999999968e166 < b

   1. Initial program 55.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. 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 84.0%

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

   5. Taylor expanded in b around inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. remove-double-neg N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-neg-out N/A

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

      6. mul-1-neg N/A

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

      7. neg-mul-1 N/A

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

      8. associate-*l* N/A

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

      9. distribute-lft-in N/A

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

      10. distribute-rgt-in N/A

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

   7. Applied rewrites 84.0%

      \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(\left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(\frac{a}{b} - 1\right)\right) \cdot \left(-b\right)\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 91.9%

      \[\leadsto \left(a + b\right) \cdot \left(\left(\sin \left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot angle\right) \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot 0.011111111111111112\right)\right) \cdot \left(\frac{a}{b} - 1\right)\right) \cdot \left(-b\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.4 \cdot 10^{-24}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \left(\sqrt[3]{{\mathsf{PI}\left(\right)}^{1.5}} \cdot \sqrt[3]{{\mathsf{PI}\left(\right)}^{1.5}}\right)\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{+166}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(\sin \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot angle\right) \cdot \left(0.011111111111111112 \cdot {\mathsf{PI}\left(\right)}^{0.16666666666666666}\right)\right)\right) \cdot \left(-\left(\frac{a}{b} - 1\right)\right)\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(\sin \left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot angle\right) \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot 0.011111111111111112\right)\right) \cdot \left(-\left(\frac{a}{b} - 1\right)\right)\right) \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 67.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \frac{angle}{180}\\ \mathbf{if}\;\left(\left(2 \cdot \left({b\_m}^{2} - {a\_m}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0 \leq -2 \cdot 10^{+251}:\\ \;\;\;\;\left(a\_m + b\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \left(\mathsf{fma}\left(\left({\mathsf{PI}\left(\right)}^{3} \cdot -2.2862368541380886 \cdot 10^{-7}\right) \cdot angle, angle, 0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a\_m + b\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(\left(\left(\left(angle \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot {\mathsf{PI}\left(\right)}^{0.16666666666666666}\right) \cdot 0.011111111111111112\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (let* ((t_0 (* (PI) (/ angle 180.0))))
   (if (<=
        (* (* (* 2.0 (- (pow b_m 2.0) (pow a_m 2.0))) (sin t_0)) (cos t_0))
        -2e+251)
     (*
      (+ a_m b_m)
      (*
       (- b_m a_m)
       (*
        (fma
         (* (* (pow (PI) 3.0) -2.2862368541380886e-7) angle)
         angle
         (* 0.011111111111111112 (PI)))
        angle)))
     (*
      (+ a_m b_m)
      (*
       (- b_m a_m)
       (sin
        (*
         (*
          (* (* angle (sqrt (PI))) (cbrt (PI)))
          (pow (PI) 0.16666666666666666))
         0.011111111111111112)))))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) < -2.0000000000000001e251

   1. Initial program 48.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 66.1%

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

   5. Taylor expanded in angle around 0

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

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 63.6%

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

   if -2.0000000000000001e251 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64)))))

   1. Initial program 55.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. 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 70.3%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. add-sqr-sqrt N/A

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

      4. associate-*r* N/A

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

      5. add-cube-cbrt N/A

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

      6. sqrt-prod N/A

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

      7. pow2 N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. unpow1 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

   6. Applied rewrites 71.0%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\color{blue}{\left(\left(\left(angle \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot {\mathsf{PI}\left(\right)}^{0.16666666666666666}\right)} \cdot 0.011111111111111112\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 67.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} t_0 := 0.011111111111111112 \cdot \mathsf{PI}\left(\right)\\ t_1 := \mathsf{PI}\left(\right) \cdot \frac{angle}{180}\\ \mathbf{if}\;\left(\left(2 \cdot \left({b\_m}^{2} - {a\_m}^{2}\right)\right) \cdot \sin t\_1\right) \cdot \cos t\_1 \leq -4 \cdot 10^{+286}:\\ \;\;\;\;\left(a\_m + b\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \left(\mathsf{fma}\left(\left({\mathsf{PI}\left(\right)}^{3} \cdot -2.2862368541380886 \cdot 10^{-7}\right) \cdot angle, angle, t\_0\right) \cdot angle\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b\_m - a\_m\right) \cdot \left(\left(b\_m + a\_m\right) \cdot \sin \left(t\_0 \cdot angle\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (let* ((t_0 (* 0.011111111111111112 (PI))) (t_1 (* (PI) (/ angle 180.0))))
   (if (<=
        (* (* (* 2.0 (- (pow b_m 2.0) (pow a_m 2.0))) (sin t_1)) (cos t_1))
        -4e+286)
     (*
      (+ a_m b_m)
      (*
       (- b_m a_m)
       (*
        (fma (* (* (pow (PI) 3.0) -2.2862368541380886e-7) angle) angle t_0)
        angle)))
     (* (- b_m a_m) (* (+ b_m a_m) (sin (* t_0 angle)))))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) < -4.00000000000000013e286

   1. Initial program 50.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. 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 69.0%

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

   5. 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. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 67.1%

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

   if -4.00000000000000013e286 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64)))))

   1. Initial program 54.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 69.4%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 69.4

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 69.5

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

   6. Applied rewrites 69.5%

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

Alternative 4: 67.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ t_1 := {\left(\sqrt[3]{t\_0}\right)}^{3}\\ \mathbf{if}\;b\_m \leq 3.2 \cdot 10^{+166}:\\ \;\;\;\;\left(a\_m + b\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(\left(angle \cdot \left(t\_1 \cdot t\_1\right)\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a\_m + b\_m\right) \cdot \left(\left(\sin \left(\left(t\_0 \cdot angle\right) \cdot \left(t\_0 \cdot 0.011111111111111112\right)\right) \cdot \left(-\left(\frac{a\_m}{b\_m} - 1\right)\right)\right) \cdot b\_m\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (let* ((t_0 (sqrt (PI))) (t_1 (pow (cbrt t_0) 3.0)))
   (if (<= b_m 3.2e+166)
     (*
      (+ a_m b_m)
      (* (- b_m a_m) (sin (* (* angle (* t_1 t_1)) 0.011111111111111112))))
     (*
      (+ a_m b_m)
      (*
       (*
        (sin (* (* t_0 angle) (* t_0 0.011111111111111112)))
        (- (- (/ a_m b_m) 1.0)))
       b_m)))))

Derivation

1. Split input into 2 regimes

2. if b < 3.19999999999999968e166

   1. Initial program 53.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 66.8%

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

      1. rem-cube-cbrt N/A

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

      2. lift-PI.f64 N/A

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

      3. add-sqr-sqrt N/A

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

      4. cbrt-prod N/A

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

      5. unpow-prod-down N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cbrt.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-cbrt.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. lower-sqrt.f64 68.4

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

   6. Applied rewrites 68.4%

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

   if 3.19999999999999968e166 < b

   1. Initial program 55.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. 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 84.0%

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

   5. Taylor expanded in b around inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. remove-double-neg N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-neg-out N/A

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

      6. mul-1-neg N/A

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

      7. neg-mul-1 N/A

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

      8. associate-*l* N/A

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

      9. distribute-lft-in N/A

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

      10. distribute-rgt-in N/A

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

   7. Applied rewrites 84.0%

      \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(\left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(\frac{a}{b} - 1\right)\right) \cdot \left(-b\right)\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 91.9%

      \[\leadsto \left(a + b\right) \cdot \left(\left(\sin \left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot angle\right) \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot 0.011111111111111112\right)\right) \cdot \left(\frac{a}{b} - 1\right)\right) \cdot \left(-b\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.2 \cdot 10^{+166}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \left({\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3} \cdot {\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}\right)\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(\sin \left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot angle\right) \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot 0.011111111111111112\right)\right) \cdot \left(-\left(\frac{a}{b} - 1\right)\right)\right) \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} t_0 := {b\_m}^{2} - {a\_m}^{2}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-194} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\_m\right) \cdot a\_m\right) \cdot -0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \left(\left(\left(angle \cdot b\_m\right) \cdot b\_m\right) \cdot 0.011111111111111112\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (let* ((t_0 (- (pow b_m 2.0) (pow a_m 2.0))))
   (if (or (<= t_0 2e-194) (not (<= t_0 INFINITY)))
     (* (* (* (* (PI) angle) a_m) a_m) -0.011111111111111112)
     (* (PI) (* (* (* angle b_m) b_m) 0.011111111111111112)))))

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < 2.00000000000000004e-194 or +inf.0 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 48.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 56.6

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

   5. Applied rewrites 56.6%

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

   6. Taylor expanded in 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 68.4%

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

   if 2.00000000000000004e-194 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < +inf.0

   1. Initial program 60.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 52.5

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

   5. Applied rewrites 52.5%

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

   6. Taylor expanded in a around 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 51.9%

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

   10. Applied rewrites 51.9%

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

   12. Applied rewrites 59.0%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq 2 \cdot 10^{-194} \lor \neg \left({b}^{2} - {a}^{2} \leq \infty\right):\\ \;\;\;\;\left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right) \cdot a\right) \cdot -0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \left(\left(\left(angle \cdot b\right) \cdot b\right) \cdot 0.011111111111111112\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} t_0 := {b\_m}^{2} - {a\_m}^{2}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-194} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(-0.011111111111111112 \cdot \left(a\_m \cdot a\_m\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \left(\left(\left(angle \cdot b\_m\right) \cdot b\_m\right) \cdot 0.011111111111111112\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (let* ((t_0 (- (pow b_m 2.0) (pow a_m 2.0))))
   (if (or (<= t_0 2e-194) (not (<= t_0 INFINITY)))
     (* (* -0.011111111111111112 (* a_m a_m)) (* (PI) angle))
     (* (PI) (* (* (* angle b_m) b_m) 0.011111111111111112)))))

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < 2.00000000000000004e-194 or +inf.0 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 48.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 56.6

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

   5. Applied rewrites 56.6%

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

   6. Taylor expanded in 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 19.0%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 57.8%

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

   if 2.00000000000000004e-194 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < +inf.0

   1. Initial program 60.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 52.5

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

   5. Applied rewrites 52.5%

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

   6. Taylor expanded in a around 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 51.9%

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

   10. Applied rewrites 51.9%

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

   12. Applied rewrites 59.0%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq 2 \cdot 10^{-194} \lor \neg \left({b}^{2} - {a}^{2} \leq \infty\right):\\ \;\;\;\;\left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \left(\left(\left(angle \cdot b\right) \cdot b\right) \cdot 0.011111111111111112\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} t_0 := \sqrt[3]{\mathsf{PI}\left(\right)}\\ t_1 := \sqrt{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;b\_m \leq 2.4 \cdot 10^{+166}:\\ \;\;\;\;\left(a\_m + b\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(\left(\left(angle \cdot {t\_0}^{2}\right) \cdot t\_0\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a\_m + b\_m\right) \cdot \left(\left(\sin \left(\left(t\_1 \cdot angle\right) \cdot \left(t\_1 \cdot 0.011111111111111112\right)\right) \cdot \left(-\left(\frac{a\_m}{b\_m} - 1\right)\right)\right) \cdot b\_m\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (let* ((t_0 (cbrt (PI))) (t_1 (sqrt (PI))))
   (if (<= b_m 2.4e+166)
     (*
      (+ a_m b_m)
      (*
       (- b_m a_m)
       (sin (* (* (* angle (pow t_0 2.0)) t_0) 0.011111111111111112))))
     (*
      (+ a_m b_m)
      (*
       (*
        (sin (* (* t_1 angle) (* t_1 0.011111111111111112)))
        (- (- (/ a_m b_m) 1.0)))
       b_m)))))

Derivation

1. Split input into 2 regimes

2. if b < 2.39999999999999992e166

   1. Initial program 53.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 66.8%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. add-cube-cbrt N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-pow.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-cbrt.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-cbrt.f64 70.9

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

   6. Applied rewrites 70.9%

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

   if 2.39999999999999992e166 < b

   1. Initial program 55.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. 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 84.0%

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

   5. Taylor expanded in b around inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. remove-double-neg N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-neg-out N/A

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

      6. mul-1-neg N/A

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

      7. neg-mul-1 N/A

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

      8. associate-*l* N/A

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

      9. distribute-lft-in N/A

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

      10. distribute-rgt-in N/A

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

   7. Applied rewrites 84.0%

      \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(\left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(\frac{a}{b} - 1\right)\right) \cdot \left(-b\right)\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 91.9%

      \[\leadsto \left(a + b\right) \cdot \left(\left(\sin \left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot angle\right) \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot 0.011111111111111112\right)\right) \cdot \left(\frac{a}{b} - 1\right)\right) \cdot \left(-b\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{+166}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(\sin \left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot angle\right) \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot 0.011111111111111112\right)\right) \cdot \left(-\left(\frac{a}{b} - 1\right)\right)\right) \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} t_0 := \sqrt[3]{\mathsf{PI}\left(\right)}\\ t_1 := \sqrt{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;b\_m \leq 2.4 \cdot 10^{+166}:\\ \;\;\;\;\left(a\_m + b\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(\left(angle \cdot \left({t\_0}^{2} \cdot t\_0\right)\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a\_m + b\_m\right) \cdot \left(\left(\sin \left(\left(t\_1 \cdot angle\right) \cdot \left(t\_1 \cdot 0.011111111111111112\right)\right) \cdot \left(-\left(\frac{a\_m}{b\_m} - 1\right)\right)\right) \cdot b\_m\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (let* ((t_0 (cbrt (PI))) (t_1 (sqrt (PI))))
   (if (<= b_m 2.4e+166)
     (*
      (+ a_m b_m)
      (*
       (- b_m a_m)
       (sin (* (* angle (* (pow t_0 2.0) t_0)) 0.011111111111111112))))
     (*
      (+ a_m b_m)
      (*
       (*
        (sin (* (* t_1 angle) (* t_1 0.011111111111111112)))
        (- (- (/ a_m b_m) 1.0)))
       b_m)))))

Derivation

1. Split input into 2 regimes

2. if b < 2.39999999999999992e166

   1. Initial program 53.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 66.8%

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

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lower-cbrt.f64 69.6

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

   6. Applied rewrites 69.6%

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

   if 2.39999999999999992e166 < b

   1. Initial program 55.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. 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 84.0%

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

   5. Taylor expanded in b around inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. remove-double-neg N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-neg-out N/A

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

      6. mul-1-neg N/A

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

      7. neg-mul-1 N/A

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

      8. associate-*l* N/A

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

      9. distribute-lft-in N/A

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

      10. distribute-rgt-in N/A

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

   7. Applied rewrites 84.0%

      \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(\left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(\frac{a}{b} - 1\right)\right) \cdot \left(-b\right)\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 91.9%

      \[\leadsto \left(a + b\right) \cdot \left(\left(\sin \left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot angle\right) \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot 0.011111111111111112\right)\right) \cdot \left(\frac{a}{b} - 1\right)\right) \cdot \left(-b\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{+166}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(\sin \left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot angle\right) \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot 0.011111111111111112\right)\right) \cdot \left(-\left(\frac{a}{b} - 1\right)\right)\right) \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;{a\_m}^{2} \leq 5 \cdot 10^{-275}:\\ \;\;\;\;\left(a\_m + b\_m\right) \cdot \left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot b\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot \left(\left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(b\_m - a\_m\right)\right)\right) \cdot \left(b\_m + a\_m\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= (pow a_m 2.0) 5e-275)
   (* (+ a_m b_m) (* (sin (* (* (PI) angle) 0.011111111111111112)) b_m))
   (* (* angle (* (* 0.011111111111111112 (PI)) (- b_m a_m))) (+ b_m a_m))))

Derivation

1. Split input into 2 regimes

2. if (pow.f64 a #s(literal 2 binary64)) < 4.99999999999999983e-275

   1. Initial program 64.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 71.3%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 70.6

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

   7. Applied rewrites 70.6%

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

   if 4.99999999999999983e-275 < (pow.f64 a #s(literal 2 binary64))

   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 55.2

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

   5. Applied rewrites 55.2%

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

   7. Applied rewrites 66.5%

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

Alternative 10: 65.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} t_0 := 0.011111111111111112 \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\left(angle \cdot \left(t\_0 \cdot \left(b\_m - a\_m\right)\right)\right) \cdot \left(b\_m + a\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b\_m + a\_m\right) \cdot \left(b\_m - a\_m\right)\right) \cdot \sin \left(t\_0 \cdot angle\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (let* ((t_0 (* 0.011111111111111112 (PI))))
   (if (<= (/ angle 180.0) 5e-13)
     (* (* angle (* t_0 (- b_m a_m))) (+ b_m a_m))
     (* (* (+ b_m a_m) (- b_m a_m)) (sin (* t_0 angle))))))

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.9999999999999999e-13

   1. Initial program 60.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 62.5

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

   5. Applied rewrites 62.5%

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

   7. Applied rewrites 75.1%

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

   if 4.9999999999999999e-13 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 33.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 37.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift--.f64 N/A

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

      7. difference-of-squares N/A

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

      8. pow2 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lower-*.f64 34.2

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

      14. lift--.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. pow2 N/A

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

      17. lift-pow.f64 N/A

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

      18. unpow2 N/A

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

      19. difference-of-squares N/A

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

      20. lift-+.f64 N/A

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

      21. lift--.f64 N/A

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

      22. lift-*.f64 37.3

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

      23. lift-*.f64 N/A

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

      24. lift-*.f64 N/A

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

      25. associate-*l* N/A

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

   6. Applied rewrites 38.7%

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

Alternative 11: 62.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{+118}:\\ \;\;\;\;\left(angle \cdot \left(\left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(b\_m - a\_m\right)\right)\right) \cdot \left(b\_m + a\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(b\_m \cdot b\_m\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= (/ angle 180.0) 1e+118)
   (* (* angle (* (* 0.011111111111111112 (PI)) (- b_m a_m))) (+ b_m a_m))
   (* (sin (* (* (PI) angle) 0.011111111111111112)) (* b_m b_m))))

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 9.99999999999999967e117

   1. Initial program 58.9%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 62.0

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

   5. Applied rewrites 62.0%

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

   7. Applied rewrites 73.1%

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

   if 9.99999999999999967e117 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 22.0%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 22.8%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 21.0

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

   7. Applied rewrites 21.0%

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

Alternative 12: 68.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \left(b\_m - a\_m\right) \cdot \left(\left(b\_m + a\_m\right) \cdot \sin \left(\left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right) \end{array} \]

FPCore

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

Derivation

1. Initial program 53.6%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
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 69.3%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-+.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 69.3

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*l* N/A

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

   14. *-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 70.6

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

6. Applied rewrites 70.6%

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

Alternative 13: 68.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \left(a\_m + b\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot 0.011111111111111112\right)\right) \end{array} \]

FPCore

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

Derivation

1. Initial program 53.6%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
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 69.3%

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

Alternative 14: 63.1% accurate, 10.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} t_0 := 0.011111111111111112 \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+126}:\\ \;\;\;\;\left(angle \cdot \left(t\_0 \cdot \left(b\_m - a\_m\right)\right)\right) \cdot \left(b\_m + a\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot t\_0\right) \cdot \left(\left(b\_m + a\_m\right) \cdot \left(-a\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (let* ((t_0 (* 0.011111111111111112 (PI))))
   (if (<= (/ angle 180.0) 5e+126)
     (* (* angle (* t_0 (- b_m a_m))) (+ b_m a_m))
     (* (* angle t_0) (* (+ b_m a_m) (- a_m))))))

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.99999999999999977e126

   1. Initial program 58.7%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      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 61.8

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

   5. Applied rewrites 61.8%

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

   7. Applied rewrites 72.8%

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

   if 4.99999999999999977e126 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 22.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 12.9

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

   5. Applied rewrites 12.9%

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

   6. Taylor expanded in 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 11.0%

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

Alternative 15: 38.2% accurate, 21.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \mathsf{PI}\left(\right) \cdot \left(\left(\left(angle \cdot b\_m\right) \cdot b\_m\right) \cdot 0.011111111111111112\right) \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (* (PI) (* (* (* angle b_m) b_m) 0.011111111111111112)))

Derivation

1. Initial program 53.6%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
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 54.9

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

5. Applied rewrites 54.9%

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

6. Taylor expanded in 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 32.7%

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

10. Applied rewrites 32.8%

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

12. Applied rewrites 35.0%

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

Reproduce

herbie shell --seed 2024313 
(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)))))