ab-angle->ABCF B

Percentage Accurate: 54.2% → 67.9%

Time: 11.1s

Alternatives: 15

Speedup: 16.8×

• Could not converge on a ground truth

  1. a = 4.488280663459262e-66
  2. b = 1.2019664097666402e-91
  3. angle = 3.020214753977713e+97

• 32.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

Initial Program: 54.2% 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: 67.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} t_0 := \sqrt[3]{{\mathsf{PI}\left(\right)}^{1.5}}\\ \mathbf{if}\;a\_m \leq 2.2 \cdot 10^{+269}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(0.011111111111111112 \cdot \left(\left(t\_0 \cdot t\_0\right) \cdot angle\right)\right) \cdot \left(b\_m - a\_m\right)\right) \cdot \left(b\_m + a\_m\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (let* ((t_0 (cbrt (pow (PI) 1.5))))
   (if (<= a_m 2.2e+269)
     (* (+ b_m a_m) (* (- b_m a_m) (sin (* (* (PI) (/ angle 180.0)) 2.0))))
     (*
      (* (sin (* 0.011111111111111112 (* (* t_0 t_0) angle))) (- b_m a_m))
      (+ b_m a_m)))))

Derivation

1. Split input into 2 regimes

2. if a < 2.1999999999999999e269

   1. Initial program 59.6%

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

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift--.f64 N/A

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

      6. flip-- N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 60.9%

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

   6. Applied rewrites 71.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. lift-/.f64 73.0

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

   8. Applied rewrites 73.0%

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

   if 2.1999999999999999e269 < a

   1. Initial program 25.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 74.8%

      \[\leadsto \color{blue}{\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot 0.011111111111111112\right)\right)} \]
   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-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 99.4%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\left(\sqrt[3]{{\mathsf{PI}\left(\right)}^{1.5}} \cdot \sqrt[3]{{\mathsf{PI}\left(\right)}^{1.5}}\right)}\right) \cdot 0.011111111111111112\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.2 \cdot 10^{+269}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(0.011111111111111112 \cdot \left(\left(\sqrt[3]{{\mathsf{PI}\left(\right)}^{1.5}} \cdot \sqrt[3]{{\mathsf{PI}\left(\right)}^{1.5}}\right) \cdot angle\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 57.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} t_0 := {b\_m}^{2} - {a\_m}^{2}\\ t_1 := \mathsf{PI}\left(\right) \cdot angle\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-278}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot a\_m\right) \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\_m\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(t\_1 \cdot 0.011111111111111112, b\_m \cdot b\_m, 0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot \left(t\_1 \cdot a\_m\right)\right) \cdot a\_m\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (let* ((t_0 (- (pow b_m 2.0) (pow a_m 2.0))) (t_1 (* (PI) angle)))
   (if (<= t_0 -5e-278)
     (* (* (* -0.011111111111111112 a_m) angle) (* (PI) a_m))
     (if (<= t_0 INFINITY)
       (fma (* t_1 0.011111111111111112) (* b_m b_m) 0.0)
       (* (* -0.011111111111111112 (* t_1 a_m)) a_m)))))

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -4.99999999999999985e-278

   1. Initial program 57.5%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      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 52.0

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

   5. Applied rewrites 52.0%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 51.9%

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

   10. Applied rewrites 61.6%

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

   12. Applied rewrites 61.7%

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

   if -4.99999999999999985e-278 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < +inf.0

   1. Initial program 66.3%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      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. 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 15.3%

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

   10. Applied rewrites 14.6%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 62.4%

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

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

   1. Initial program 0.0%

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

   3. 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 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 56.7%

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

   10. Applied rewrites 62.5%

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

   12. Applied rewrites 62.6%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -5 \cdot 10^{-278}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot a\right) \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\right)\\ \mathbf{elif}\;{b}^{2} - {a}^{2} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112, b \cdot b, 0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} t_0 := {b\_m}^{2} - {a\_m}^{2}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-278}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot a\_m\right) \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\_m\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(\left(\left(b\_m \cdot b\_m\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\_m\right)\right) \cdot a\_m\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (let* ((t_0 (- (pow b_m 2.0) (pow a_m 2.0))))
   (if (<= t_0 -5e-278)
     (* (* (* -0.011111111111111112 a_m) angle) (* (PI) a_m))
     (if (<= t_0 INFINITY)
       (* (* (* (* b_m b_m) (PI)) angle) 0.011111111111111112)
       (* (* -0.011111111111111112 (* (* (PI) angle) a_m)) a_m)))))

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -4.99999999999999985e-278

   1. Initial program 57.5%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      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 52.0

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

   5. Applied rewrites 52.0%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 51.9%

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

   10. Applied rewrites 61.6%

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

   12. Applied rewrites 61.7%

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

   if -4.99999999999999985e-278 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < +inf.0

   1. Initial program 66.3%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      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. 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 62.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 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 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 56.7%

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

   10. Applied rewrites 62.5%

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

   12. Applied rewrites 62.6%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -5 \cdot 10^{-278}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot a\right) \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\right)\\ \mathbf{elif}\;{b}^{2} - {a}^{2} \leq \infty:\\ \;\;\;\;\left(\left(\left(b \cdot b\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} t_0 := {b\_m}^{2} - {a\_m}^{2}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-278}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot a\_m\right) \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\_m\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(\left(\left(b\_m \cdot b\_m\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\_m\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\_m\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (let* ((t_0 (- (pow b_m 2.0) (pow a_m 2.0))))
   (if (<= t_0 -5e-278)
     (* (* (* -0.011111111111111112 a_m) angle) (* (PI) a_m))
     (if (<= t_0 INFINITY)
       (* (* (* (* b_m b_m) (PI)) angle) 0.011111111111111112)
       (* (* -0.011111111111111112 a_m) (* (* (PI) angle) a_m))))))

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -4.99999999999999985e-278

   1. Initial program 57.5%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      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 52.0

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

   5. Applied rewrites 52.0%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 51.9%

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

   10. Applied rewrites 61.6%

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

   12. Applied rewrites 61.7%

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

   if -4.99999999999999985e-278 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < +inf.0

   1. Initial program 66.3%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      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. 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 62.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 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 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 56.7%

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

   10. Applied rewrites 62.5%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -5 \cdot 10^{-278}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot a\right) \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\right)\\ \mathbf{elif}\;{b}^{2} - {a}^{2} \leq \infty:\\ \;\;\;\;\left(\left(\left(b \cdot b\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.2% accurate, 1.3× speedup

Math

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

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (let* ((t_0 (sin (* (* (PI) angle) 0.011111111111111112))))
   (if (<= (- (pow b_m 2.0) (pow a_m 2.0)) 1e-310)
     (* (* (- a_m) t_0) (+ b_m a_m))
     (* (* t_0 b_m) (+ b_m a_m)))))

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < 9.999999999999969e-311

   1. Initial program 61.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. 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 67.7%

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

   5. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 67.6

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

   7. Applied rewrites 67.6%

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

   if 9.999999999999969e-311 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 56.0%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. 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 75.3%

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

   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 73.9

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

   7. Applied rewrites 73.9%

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

4. Final simplification 70.9%

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

Alternative 6: 64.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;{a\_m}^{2} \leq 2 \cdot 10^{-300}:\\ \;\;\;\;\left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot b\_m\right) \cdot \left(b\_m + a\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(b\_m - a\_m\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(b\_m + a\_m\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= (pow a_m 2.0) 2e-300)
   (* (* (sin (* (* (PI) angle) 0.011111111111111112)) b_m) (+ b_m a_m))
   (* (* (* (* (- b_m a_m) (PI)) angle) 0.011111111111111112) (+ b_m a_m))))

Derivation

1. Split input into 2 regimes

2. if (pow.f64 a #s(literal 2 binary64)) < 2.00000000000000005e-300

   1. Initial program 65.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 72.4%

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

   5. Taylor expanded in 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 72.3

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

   7. Applied rewrites 72.3%

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

   if 2.00000000000000005e-300 < (pow.f64 a #s(literal 2 binary64))

   1. Initial program 56.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 71.4%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-PI.f64 68.7

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

   7. Applied rewrites 68.7%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 2 \cdot 10^{-300}:\\ \;\;\;\;\left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot b\right) \cdot \left(b + a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(b + a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \left(b\_m + a\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right)\right) \end{array} \]

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 (* (* (PI) (/ angle 180.0)) 2.0)))))

Derivation

1. Initial program 58.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift--.f64 N/A

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

   6. flip-- N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 60.6%

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

6. Applied rewrites 71.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. div-inv N/A

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

   5. lift-/.f64 73.0

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

8. Applied rewrites 73.0%

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

9. Final simplification 73.0%

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

Alternative 8: 66.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 1.52 \cdot 10^{+180}:\\ \;\;\;\;\left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(b\_m - a\_m\right)\right) \cdot \left(b\_m + a\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(b\_m - a\_m\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(b\_m + a\_m\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= a_m 1.52e+180)
   (*
    (* (sin (* (* (PI) angle) 0.011111111111111112)) (- b_m a_m))
    (+ b_m a_m))
   (* (* (* (* (- b_m a_m) (PI)) angle) 0.011111111111111112) (+ b_m a_m))))

Derivation

1. Split input into 2 regimes

2. if a < 1.52e180

   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. 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 70.5%

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

   if 1.52e180 < a

   1. Initial program 43.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 83.1%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-PI.f64 83.2

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

   7. Applied rewrites 83.2%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.52 \cdot 10^{+180}:\\ \;\;\;\;\left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(b + a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.0% accurate, 3.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 58.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lift--.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. lift-pow.f64 N/A

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

   11. unpow2 N/A

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

   12. difference-of-squares N/A

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

   13. associate-*l* N/A

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

   14. lower-*.f64 N/A

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

4. Applied rewrites 71.7%

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

   1. lift-*.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. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 71.7

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-+.f64 71.7

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

   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 71.8

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

6. Applied rewrites 71.8%

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

7. Final simplification 71.8%

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

Alternative 10: 67.7% accurate, 3.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 58.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift--.f64 N/A

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

   6. flip-- N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 60.6%

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

6. Applied rewrites 71.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. metadata-eval N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 71.2

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

8. Applied rewrites 71.2%

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

Alternative 11: 57.4% accurate, 13.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 5 \cdot 10^{+155}:\\ \;\;\;\;\left(\left(b\_m + a\_m\right) \cdot \left(b\_m - a\_m\right)\right) \cdot \left(\left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\_m\right) \cdot a\_m\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= a_m 5e+155)
   (* (* (+ b_m a_m) (- b_m a_m)) (* (* 0.011111111111111112 (PI)) angle))
   (* -0.011111111111111112 (* (* (* (PI) angle) a_m) a_m))))

Derivation

1. Split input into 2 regimes

2. if a < 4.9999999999999999e155

   1. Initial program 60.9%

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

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

   5. Applied rewrites 58.8%

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

   if 4.9999999999999999e155 < a

   1. Initial program 38.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 43.6

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

   5. Applied rewrites 43.6%

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

   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 39.0%

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

   10. Applied rewrites 66.4%

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

   12. Applied rewrites 66.7%

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

4. Final simplification 59.7%

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

Alternative 12: 62.1% accurate, 16.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 58.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lift--.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. lift-pow.f64 N/A

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

   11. unpow2 N/A

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

   12. difference-of-squares N/A

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

   13. associate-*l* N/A

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

   14. lower-*.f64 N/A

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

4. Applied rewrites 71.7%

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

5. Taylor expanded in angle around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-PI.f64 67.4

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

7. Applied rewrites 67.4%

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

8. Final simplification 67.4%

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

Alternative 13: 62.1% accurate, 16.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 58.5%

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

3. Taylor expanded in angle around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-PI.f64 N/A

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

   10. unpow2 N/A

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

   11. unpow2 N/A

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

   12. difference-of-squares N/A

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

   13. lower-*.f64 N/A

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

   14. lower-+.f64 N/A

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

   15. lower--.f64 57.2

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

5. Applied rewrites 57.2%

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

7. Applied rewrites 67.3%

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

8. Final simplification 67.3%

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

Alternative 14: 38.1% accurate, 21.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 58.5%

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

3. Taylor expanded in angle around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-PI.f64 N/A

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

   10. unpow2 N/A

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

   11. unpow2 N/A

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

   12. difference-of-squares N/A

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

   13. lower-*.f64 N/A

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

   14. lower-+.f64 N/A

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

   15. lower--.f64 57.2

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

5. Applied rewrites 57.2%

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

6. Taylor expanded in a around inf

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

8. Applied rewrites 33.1%

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

10. Applied rewrites 37.1%

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

12. Applied rewrites 37.1%

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

13. Final simplification 37.1%

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

Alternative 15: 38.1% accurate, 21.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 58.5%

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

3. Taylor expanded in angle around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-PI.f64 N/A

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

   10. unpow2 N/A

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

   11. unpow2 N/A

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

   12. difference-of-squares N/A

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

   13. lower-*.f64 N/A

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

   14. lower-+.f64 N/A

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

   15. lower--.f64 57.2

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

5. Applied rewrites 57.2%

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

6. Taylor expanded in a around inf

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

8. Applied rewrites 33.1%

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

10. Applied rewrites 37.1%

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

11. Final simplification 37.1%

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

Reproduce

herbie shell --seed 2024331 
(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)))))