ab-angle->ABCF B

Percentage Accurate: 53.6% → 66.7%

Time: 10.8s

Alternatives: 18

Speedup: 13.7×

• 33.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

Initial Program: 53.6% 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: 66.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;2 \cdot \left({b}^{2} - {a}^{2}\right) \leq -5 \cdot 10^{+268}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), t\_0, 0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \sqrt[3]{t\_0}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* (PI) (PI)) (PI))))
   (if (<= (* 2.0 (- (pow b 2.0) (pow a 2.0))) -5e+268)
     (*
      (+ a b)
      (*
       (- b a)
       (*
        (fma
         (* -2.2862368541380886e-7 (* angle angle))
         t_0
         (* 0.011111111111111112 (PI)))
        angle)))
     (* (+ a b) (* (- b a) (sin (* 2.0 (* (/ angle 180.0) (cbrt t_0)))))))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -5.0000000000000002e268

   1. Initial program 45.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* 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. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   4. Applied rewrites 74.0%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 83.9

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

   7. Applied rewrites 83.9%

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

   9. Applied rewrites 83.9%

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

   if -5.0000000000000002e268 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 62.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. 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-*r* 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. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   4. Applied rewrites 70.7%

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

      1. lift-PI.f64 N/A

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

      2. add-cbrt-cube N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right)\right)\right) \]

      3. lower-cbrt.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right)\right)\right) \]

      4. rem-cube-cbrt N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \sqrt[3]{\color{blue}{{\left(\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}\right)}^{3}}}\right)\right)\right) \]

      5. add-cbrt-cube N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \sqrt[3]{{\color{blue}{\mathsf{PI}\left(\right)}}^{3}}\right)\right)\right) \]

      6. lift-PI.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \sqrt[3]{{\color{blue}{\mathsf{PI}\left(\right)}}^{3}}\right)\right)\right) \]

      7. lower-pow.f64 76.0

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \sqrt[3]{\color{blue}{{\mathsf{PI}\left(\right)}^{3}}}\right)\right)\right) \]

   6. Applied rewrites 76.0%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \color{blue}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}}\right)\right)\right) \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \sqrt[3]{\color{blue}{{\mathsf{PI}\left(\right)}^{3}}}\right)\right)\right) \]

      2. unpow3 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \sqrt[3]{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right)\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \sqrt[3]{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right)\right)\right) \]

      4. lower-*.f64 76.0

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \sqrt[3]{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

   8. Applied rewrites 76.0%

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

Alternative 2: 66.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;2 \cdot \left({b}^{2} - {a}^{2}\right) \leq -5 \cdot 10^{+268}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), 0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \left(t\_0 \cdot t\_0\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (sqrt (PI))))
   (if (<= (* 2.0 (- (pow b 2.0) (pow a 2.0))) -5e+268)
     (*
      (+ a b)
      (*
       (- b a)
       (*
        (fma
         (* -2.2862368541380886e-7 (* angle angle))
         (* (* (PI) (PI)) (PI))
         (* 0.011111111111111112 (PI)))
        angle)))
     (* (+ a b) (* (- b a) (sin (* 2.0 (* (/ angle 180.0) (* t_0 t_0)))))))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -5.0000000000000002e268

   1. Initial program 45.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* 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. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   4. Applied rewrites 74.0%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 83.9

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

   7. Applied rewrites 83.9%

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

   9. Applied rewrites 83.9%

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

   if -5.0000000000000002e268 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 62.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. 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-*r* 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. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   4. Applied rewrites 70.7%

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

      1. lift-PI.f64 N/A

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

      2. add-sqr-sqrt N/A

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

      3. lower-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lower-sqrt.f64 74.9

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

   6. Applied rewrites 74.9%

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

Alternative 3: 65.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 2 \cdot 10^{+32}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{angle}{180} \cdot \left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), 0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (pow a 2.0) 2e+32)
   (* (+ a b) (* (- b a) (sin (* (/ angle 180.0) (+ (PI) (PI))))))
   (*
    (+ a b)
    (*
     (- b a)
     (*
      (fma
       (* -2.2862368541380886e-7 (* angle angle))
       (* (* (PI) (PI)) (PI))
       (* 0.011111111111111112 (PI)))
      angle)))))

Derivation

1. Split input into 2 regimes

2. if (pow.f64 a #s(literal 2 binary64)) < 2.00000000000000011e32

   1. Initial program 66.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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-*r* 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. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   4. Applied rewrites 73.1%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 73.1

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

   6. Applied rewrites 73.1%

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

   if 2.00000000000000011e32 < (pow.f64 a #s(literal 2 binary64))

   1. Initial program 50.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* 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. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   4. Applied rewrites 69.8%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 76.1

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

   7. Applied rewrites 76.1%

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

   9. Applied rewrites 76.1%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), 0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 57.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a}^{2}\right) \leq -4 \cdot 10^{-299}:\\ \;\;\;\;\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(-0.011111111111111112 \cdot a\right)\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (* 2.0 (- (pow b 2.0) (pow a 2.0))) -4e-299)
   (* (* (* angle (PI)) (* -0.011111111111111112 a)) a)
   (* (* (* (PI) (* b b)) angle) 0.011111111111111112)))

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -3.99999999999999997e-299

   1. Initial program 55.0%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 51.9

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

   5. Applied rewrites 51.9%

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

   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 51.7%

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

   10. Applied rewrites 66.8%

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

   if -3.99999999999999997e-299 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 61.9%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 61.4

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

   5. Applied rewrites 61.4%

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

   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 61.9%

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

Alternative 5: 64.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 5 \cdot 10^{-303}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), 0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (pow a 2.0) 5e-303)
   (* (+ a b) (* (sin (* (* (PI) angle) 0.011111111111111112)) b))
   (*
    (+ a b)
    (*
     (- b a)
     (*
      (fma
       (* -2.2862368541380886e-7 (* angle angle))
       (* (* (PI) (PI)) (PI))
       (* 0.011111111111111112 (PI)))
      angle)))))

Derivation

1. Split input into 2 regimes

2. if (pow.f64 a #s(literal 2 binary64)) < 4.9999999999999998e-303

   1. Initial program 71.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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-*r* 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. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   4. Applied rewrites 76.7%

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

   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 78.0

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

   7. Applied rewrites 78.0%

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

   if 4.9999999999999998e-303 < (pow.f64 a #s(literal 2 binary64))

   1. Initial program 54.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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-*r* 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. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   4. Applied rewrites 69.7%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 72.6

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

   7. Applied rewrites 72.6%

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

   9. Applied rewrites 72.6%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), 0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 67.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 1.35 \cdot 10^{+233}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(angle \cdot \frac{\mathsf{PI}\left(\right)}{180}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 1.35e+233)
   (* (+ a b) (* (- b a) (sin (* 2.0 (* angle (/ (PI) 180.0))))))
   (- (* (+ a b) (* (- b a) (sin (* 2.0 (* (/ angle 180.0) (PI)))))))))

Derivation

1. Split input into 2 regimes

2. if angle < 1.35000000000000004e233

   1. Initial program 60.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* 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. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   4. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 77.3

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

   6. Applied rewrites 77.3%

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

   if 1.35000000000000004e233 < angle

   1. Initial program 23.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* 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. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   4. Applied rewrites 23.8%

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

      1. lift-PI.f64 N/A

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

      2. add-cbrt-cube N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right)\right)\right) \]

      3. lower-cbrt.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right)\right)\right) \]

      4. rem-cube-cbrt N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \sqrt[3]{\color{blue}{{\left(\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}\right)}^{3}}}\right)\right)\right) \]

      5. add-cbrt-cube N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \sqrt[3]{{\color{blue}{\mathsf{PI}\left(\right)}}^{3}}\right)\right)\right) \]

      6. lift-PI.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \sqrt[3]{{\color{blue}{\mathsf{PI}\left(\right)}}^{3}}\right)\right)\right) \]

      7. lower-pow.f64 31.8

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \sqrt[3]{\color{blue}{{\mathsf{PI}\left(\right)}^{3}}}\right)\right)\right) \]

   6. Applied rewrites 31.8%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \color{blue}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}}\right)\right)\right) \]
   7. Step-by-step derivation

      1. lift-cbrt.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \color{blue}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}}\right)\right)\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \sqrt[3]{\color{blue}{{\mathsf{PI}\left(\right)}^{3}}}\right)\right)\right) \]

      3. rem-cbrt-cube 23.8

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

      4. rem-square-sqrt N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. sqr-neg-rev N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. unpow1 N/A

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

      10. metadata-eval N/A

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

      11. pow-pow N/A

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

      12. pow2 N/A

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

      13. sqr-neg-rev N/A

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

      14. pow2 N/A

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

      15. pow-pow N/A

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

      16. metadata-eval N/A

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

      17. unpow1 N/A

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

      18. lift-sqrt.f64 N/A

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

      19. lift-sqrt.f64 N/A

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

      20. rem-square-sqrt N/A

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

      21. lower-neg.f64 48.9

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

   8. Applied rewrites 48.9%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 1.35 \cdot 10^{+233}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(angle \cdot \frac{\mathsf{PI}\left(\right)}{180}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 8.2 \cdot 10^{+26}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(angle \cdot \frac{\mathsf{PI}\left(\right)}{180}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), 0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 8.2e+26)
   (* (+ a b) (* (- b a) (sin (* 2.0 (* angle (/ (PI) 180.0))))))
   (*
    (+ a b)
    (*
     (- b a)
     (*
      (fma
       (* -2.2862368541380886e-7 (* angle angle))
       (* (* (PI) (PI)) (PI))
       (* 0.011111111111111112 (PI)))
      angle)))))

Derivation

1. Split input into 2 regimes

2. if a < 8.19999999999999967e26

   1. Initial program 62.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. 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-*r* 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. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   4. Applied rewrites 73.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 76.7

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

   6. Applied rewrites 76.7%

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

   if 8.19999999999999967e26 < a

   1. Initial program 46.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* 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. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   4. Applied rewrites 65.9%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 77.8

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

   7. Applied rewrites 77.8%

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

   9. Applied rewrites 77.8%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), 0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 62.4% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-a, a, b \cdot b\right) \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;angle \leq 7.2 \cdot 10^{+42}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), 0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right)\\ \mathbf{elif}\;angle \leq 2.7 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(-0.011111111111111112 \cdot angle, \mathsf{PI}\left(\right), \frac{-0.011111111111111112 \cdot \frac{\left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b\right)\right) \cdot angle}{a}}{-a}\right) \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;angle \leq 1.85 \cdot 10^{+197} \lor \neg \left(angle \leq 10^{+275}\right):\\ \;\;\;\;\left(t\_0 \cdot angle\right) \cdot -0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (fma (- a) a (* b b)) (PI))))
   (if (<= angle 7.2e+42)
     (*
      (+ a b)
      (*
       (- b a)
       (*
        (fma
         (* -2.2862368541380886e-7 (* angle angle))
         (* (* (PI) (PI)) (PI))
         (* 0.011111111111111112 (PI)))
        angle)))
     (if (<= angle 2.7e+70)
       (*
        (fma
         (* -0.011111111111111112 angle)
         (PI)
         (/ (* -0.011111111111111112 (/ (* (* (PI) (* b b)) angle) a)) (- a)))
        (* a a))
       (if (or (<= angle 1.85e+197) (not (<= angle 1e+275)))
         (* (* t_0 angle) -0.011111111111111112)
         (* 0.011111111111111112 (* angle t_0)))))))

Derivation

1. Split input into 4 regimes

2. if angle < 7.2000000000000002e42

   1. Initial program 66.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* 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. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   4. Applied rewrites 81.4%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 79.4

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

   7. Applied rewrites 79.4%

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

   9. Applied rewrites 79.4%

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

   if 7.2000000000000002e42 < angle < 2.7e70

   1. Initial program 34.6%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 59.2

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

   5. Applied rewrites 59.2%

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

   6. Taylor expanded in a around -inf

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

   8. Applied rewrites 59.2%

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

   if 2.7e70 < angle < 1.8500000000000002e197 or 9.9999999999999996e274 < angle

   1. Initial program 27.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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-*r* 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. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   4. Applied rewrites 27.1%

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

      1. lift-PI.f64 N/A

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

      2. add-cbrt-cube N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right)\right)\right) \]

      3. lower-cbrt.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right)\right)\right) \]

      4. rem-cube-cbrt N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \sqrt[3]{\color{blue}{{\left(\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}\right)}^{3}}}\right)\right)\right) \]

      5. add-cbrt-cube N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \sqrt[3]{{\color{blue}{\mathsf{PI}\left(\right)}}^{3}}\right)\right)\right) \]

      6. lift-PI.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \sqrt[3]{{\color{blue}{\mathsf{PI}\left(\right)}}^{3}}\right)\right)\right) \]

      7. lower-pow.f64 36.1

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \sqrt[3]{\color{blue}{{\mathsf{PI}\left(\right)}^{3}}}\right)\right)\right) \]

   6. Applied rewrites 36.1%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \color{blue}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}}\right)\right)\right) \]
   7. Step-by-step derivation

      1. lift-cbrt.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \color{blue}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}}\right)\right)\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \sqrt[3]{\color{blue}{{\mathsf{PI}\left(\right)}^{3}}}\right)\right)\right) \]

      3. rem-cbrt-cube 27.1

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

      4. rem-square-sqrt N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. sqr-neg-rev N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. unpow1 N/A

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

      10. metadata-eval N/A

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

      11. pow-pow N/A

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

      12. pow2 N/A

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

      13. sqr-neg-rev N/A

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

      14. pow2 N/A

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

      15. pow-pow N/A

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

      16. metadata-eval N/A

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

      17. unpow1 N/A

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

      18. lift-sqrt.f64 N/A

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

      19. lift-sqrt.f64 N/A

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

      20. rem-square-sqrt N/A

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

      21. lower-neg.f64 42.3

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

   8. Applied rewrites 42.3%

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

   9. Taylor expanded in angle around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 47.0%

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

   if 1.8500000000000002e197 < angle < 9.9999999999999996e274

   1. Initial program 18.7%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 51.6

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

   5. Applied rewrites 51.6%

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

   7. Applied rewrites 51.6%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 7.2 \cdot 10^{+42}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), 0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right)\\ \mathbf{elif}\;angle \leq 2.7 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(-0.011111111111111112 \cdot angle, \mathsf{PI}\left(\right), \frac{-0.011111111111111112 \cdot \frac{\left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b\right)\right) \cdot angle}{a}}{-a}\right) \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;angle \leq 1.85 \cdot 10^{+197} \lor \neg \left(angle \leq 10^{+275}\right):\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-a, a, b \cdot b\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot -0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\mathsf{fma}\left(-a, a, b \cdot b\right) \cdot \mathsf{PI}\left(\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.6% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-a, a, b \cdot b\right) \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;angle \leq 100:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), 0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right)\\ \mathbf{elif}\;angle \leq 1.85 \cdot 10^{+197} \lor \neg \left(angle \leq 10^{+275}\right):\\ \;\;\;\;\left(t\_0 \cdot angle\right) \cdot -0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (fma (- a) a (* b b)) (PI))))
   (if (<= angle 100.0)
     (*
      (+ a b)
      (*
       (- b a)
       (*
        (fma
         (* -2.2862368541380886e-7 (* angle angle))
         (* (* (PI) (PI)) (PI))
         (* 0.011111111111111112 (PI)))
        angle)))
     (if (or (<= angle 1.85e+197) (not (<= angle 1e+275)))
       (* (* t_0 angle) -0.011111111111111112)
       (* 0.011111111111111112 (* angle t_0))))))

Derivation

1. Split input into 3 regimes

2. if angle < 100

   1. Initial program 66.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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-*r* 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. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   4. Applied rewrites 82.8%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 80.5

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

   7. Applied rewrites 80.5%

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

   9. Applied rewrites 80.5%

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

   if 100 < angle < 1.8500000000000002e197 or 9.9999999999999996e274 < angle

   1. Initial program 31.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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-*r* 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. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   4. Applied rewrites 31.9%

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

      1. lift-PI.f64 N/A

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

      2. add-cbrt-cube N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right)\right)\right) \]

      3. lower-cbrt.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right)\right)\right) \]

      4. rem-cube-cbrt N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \sqrt[3]{\color{blue}{{\left(\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}\right)}^{3}}}\right)\right)\right) \]

      5. add-cbrt-cube N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \sqrt[3]{{\color{blue}{\mathsf{PI}\left(\right)}}^{3}}\right)\right)\right) \]

      6. lift-PI.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \sqrt[3]{{\color{blue}{\mathsf{PI}\left(\right)}}^{3}}\right)\right)\right) \]

      7. lower-pow.f64 42.1

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \sqrt[3]{\color{blue}{{\mathsf{PI}\left(\right)}^{3}}}\right)\right)\right) \]

   6. Applied rewrites 42.1%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \color{blue}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}}\right)\right)\right) \]
   7. Step-by-step derivation

      1. lift-cbrt.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \color{blue}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}}\right)\right)\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \sqrt[3]{\color{blue}{{\mathsf{PI}\left(\right)}^{3}}}\right)\right)\right) \]

      3. rem-cbrt-cube 31.9

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

      4. rem-square-sqrt N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. sqr-neg-rev N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. unpow1 N/A

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

      10. metadata-eval N/A

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

      11. pow-pow N/A

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

      12. pow2 N/A

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

      13. sqr-neg-rev N/A

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

      14. pow2 N/A

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

      15. pow-pow N/A

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

      16. metadata-eval N/A

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

      17. unpow1 N/A

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

      18. lift-sqrt.f64 N/A

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

      19. lift-sqrt.f64 N/A

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

      20. rem-square-sqrt N/A

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

      21. lower-neg.f64 40.8

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

   8. Applied rewrites 40.8%

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

   9. Taylor expanded in angle around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 43.3%

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

   if 1.8500000000000002e197 < angle < 9.9999999999999996e274

   1. Initial program 18.7%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 51.6

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

   5. Applied rewrites 51.6%

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

   7. Applied rewrites 51.6%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 100:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), 0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right)\\ \mathbf{elif}\;angle \leq 1.85 \cdot 10^{+197} \lor \neg \left(angle \leq 10^{+275}\right):\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-a, a, b \cdot b\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot -0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\mathsf{fma}\left(-a, a, b \cdot b\right) \cdot \mathsf{PI}\left(\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.9% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-a, a, b \cdot b\right) \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;angle \leq 1000:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(\left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right)\\ \mathbf{elif}\;angle \leq 1.85 \cdot 10^{+197} \lor \neg \left(angle \leq 10^{+275}\right):\\ \;\;\;\;\left(t\_0 \cdot angle\right) \cdot -0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (fma (- a) a (* b b)) (PI))))
   (if (<= angle 1000.0)
     (* (+ a b) (* (* (* (- b a) (PI)) angle) 0.011111111111111112))
     (if (or (<= angle 1.85e+197) (not (<= angle 1e+275)))
       (* (* t_0 angle) -0.011111111111111112)
       (* 0.011111111111111112 (* angle t_0))))))

Derivation

1. Split input into 3 regimes

2. if angle < 1e3

   1. Initial program 66.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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-*r* 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. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   4. Applied rewrites 82.8%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-PI.f64 79.1

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

   7. Applied rewrites 79.1%

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

   if 1e3 < angle < 1.8500000000000002e197 or 9.9999999999999996e274 < angle

   1. Initial program 31.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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-*r* 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. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   4. Applied rewrites 31.9%

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

      1. lift-PI.f64 N/A

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

      2. add-cbrt-cube N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right)\right)\right) \]

      3. lower-cbrt.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right)\right)\right) \]

      4. rem-cube-cbrt N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \sqrt[3]{\color{blue}{{\left(\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}\right)}^{3}}}\right)\right)\right) \]

      5. add-cbrt-cube N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \sqrt[3]{{\color{blue}{\mathsf{PI}\left(\right)}}^{3}}\right)\right)\right) \]

      6. lift-PI.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \sqrt[3]{{\color{blue}{\mathsf{PI}\left(\right)}}^{3}}\right)\right)\right) \]

      7. lower-pow.f64 42.1

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \sqrt[3]{\color{blue}{{\mathsf{PI}\left(\right)}^{3}}}\right)\right)\right) \]

   6. Applied rewrites 42.1%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \color{blue}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}}\right)\right)\right) \]
   7. Step-by-step derivation

      1. lift-cbrt.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \color{blue}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}}\right)\right)\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \sqrt[3]{\color{blue}{{\mathsf{PI}\left(\right)}^{3}}}\right)\right)\right) \]

      3. rem-cbrt-cube 31.9

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

      4. rem-square-sqrt N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. sqr-neg-rev N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. unpow1 N/A

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

      10. metadata-eval N/A

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

      11. pow-pow N/A

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

      12. pow2 N/A

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

      13. sqr-neg-rev N/A

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

      14. pow2 N/A

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

      15. pow-pow N/A

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

      16. metadata-eval N/A

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

      17. unpow1 N/A

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

      18. lift-sqrt.f64 N/A

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

      19. lift-sqrt.f64 N/A

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

      20. rem-square-sqrt N/A

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

      21. lower-neg.f64 40.8

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

   8. Applied rewrites 40.8%

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

   9. Taylor expanded in angle around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 43.3%

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

   if 1.8500000000000002e197 < angle < 9.9999999999999996e274

   1. Initial program 18.7%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 51.6

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

   5. Applied rewrites 51.6%

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

   7. Applied rewrites 51.6%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 1000:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(\left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right)\\ \mathbf{elif}\;angle \leq 1.85 \cdot 10^{+197} \lor \neg \left(angle \leq 10^{+275}\right):\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-a, a, b \cdot b\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot -0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\mathsf{fma}\left(-a, a, b \cdot b\right) \cdot \mathsf{PI}\left(\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.6% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\\ \mathbf{if}\;angle \leq 1000:\\ \;\;\;\;\left(a + b\right) \cdot \left(t\_0 \cdot 0.011111111111111112\right)\\ \mathbf{elif}\;angle \leq 1.85 \cdot 10^{+197} \lor \neg \left(angle \leq 1.08 \cdot 10^{+290}\right):\\ \;\;\;\;\left(a + b\right) \cdot \left(t\_0 \cdot -0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\mathsf{fma}\left(-a, a, b \cdot b\right) \cdot \mathsf{PI}\left(\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* (- b a) (PI)) angle)))
   (if (<= angle 1000.0)
     (* (+ a b) (* t_0 0.011111111111111112))
     (if (or (<= angle 1.85e+197) (not (<= angle 1.08e+290)))
       (* (+ a b) (* t_0 -0.011111111111111112))
       (* 0.011111111111111112 (* angle (* (fma (- a) a (* b b)) (PI))))))))

Derivation

1. Split input into 3 regimes

2. if angle < 1e3

   1. Initial program 66.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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-*r* 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. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   4. Applied rewrites 82.8%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-PI.f64 79.1

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

   7. Applied rewrites 79.1%

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

   if 1e3 < angle < 1.8500000000000002e197 or 1.08e290 < angle

   1. Initial program 30.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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-*r* 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. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   4. Applied rewrites 30.9%

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

      1. lift-PI.f64 N/A

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

      2. add-cbrt-cube N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right)\right)\right) \]

      3. lower-cbrt.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right)\right)\right) \]

      4. rem-cube-cbrt N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \sqrt[3]{\color{blue}{{\left(\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}\right)}^{3}}}\right)\right)\right) \]

      5. add-cbrt-cube N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \sqrt[3]{{\color{blue}{\mathsf{PI}\left(\right)}}^{3}}\right)\right)\right) \]

      6. lift-PI.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \sqrt[3]{{\color{blue}{\mathsf{PI}\left(\right)}}^{3}}\right)\right)\right) \]

      7. lower-pow.f64 41.2

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \sqrt[3]{\color{blue}{{\mathsf{PI}\left(\right)}^{3}}}\right)\right)\right) \]

   6. Applied rewrites 41.2%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \color{blue}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}}\right)\right)\right) \]
   7. Step-by-step derivation

      1. lift-cbrt.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \color{blue}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}}\right)\right)\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \sqrt[3]{\color{blue}{{\mathsf{PI}\left(\right)}^{3}}}\right)\right)\right) \]

      3. rem-cbrt-cube 30.9

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

      4. rem-square-sqrt N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. sqr-neg-rev N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. unpow1 N/A

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

      10. metadata-eval N/A

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

      11. pow-pow N/A

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

      12. pow2 N/A

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

      13. sqr-neg-rev N/A

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

      14. pow2 N/A

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

      15. pow-pow N/A

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

      16. metadata-eval N/A

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

      17. unpow1 N/A

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

      18. lift-sqrt.f64 N/A

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

      19. lift-sqrt.f64 N/A

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

      20. rem-square-sqrt N/A

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

      21. lower-neg.f64 39.9

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

   8. Applied rewrites 39.9%

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

   9. Taylor expanded in angle around 0

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

       1. *-commutative N/A

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

       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 38.8

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

   11. Applied rewrites 38.8%

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

   if 1.8500000000000002e197 < angle < 1.08e290

   1. Initial program 24.4%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 51.5

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

   5. Applied rewrites 51.5%

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

   7. Applied rewrites 51.5%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 1000:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(\left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right)\\ \mathbf{elif}\;angle \leq 1.85 \cdot 10^{+197} \lor \neg \left(angle \leq 1.08 \cdot 10^{+290}\right):\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(\left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot -0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\mathsf{fma}\left(-a, a, b \cdot b\right) \cdot \mathsf{PI}\left(\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.1% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\\ \mathbf{if}\;angle \leq 1000:\\ \;\;\;\;\left(a + b\right) \cdot \left(t\_0 \cdot 0.011111111111111112\right)\\ \mathbf{elif}\;angle \leq 1.85 \cdot 10^{+197}:\\ \;\;\;\;\left(a + b\right) \cdot \left(t\_0 \cdot -0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* (- b a) (PI)) angle)))
   (if (<= angle 1000.0)
     (* (+ a b) (* t_0 0.011111111111111112))
     (if (<= angle 1.85e+197)
       (* (+ a b) (* t_0 -0.011111111111111112))
       (* (* (* (PI) (* b b)) angle) 0.011111111111111112)))))

Derivation

1. Split input into 3 regimes

2. if angle < 1e3

   1. Initial program 66.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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-*r* 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. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   4. Applied rewrites 82.8%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-PI.f64 79.1

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

   7. Applied rewrites 79.1%

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

   if 1e3 < angle < 1.8500000000000002e197

   1. Initial program 33.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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-*r* 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. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   4. Applied rewrites 33.5%

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

      1. lift-PI.f64 N/A

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

      2. add-cbrt-cube N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right)\right)\right) \]

      3. lower-cbrt.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right)\right)\right) \]

      4. rem-cube-cbrt N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \sqrt[3]{\color{blue}{{\left(\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}\right)}^{3}}}\right)\right)\right) \]

      5. add-cbrt-cube N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \sqrt[3]{{\color{blue}{\mathsf{PI}\left(\right)}}^{3}}\right)\right)\right) \]

      6. lift-PI.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \sqrt[3]{{\color{blue}{\mathsf{PI}\left(\right)}}^{3}}\right)\right)\right) \]

      7. lower-pow.f64 46.2

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \sqrt[3]{\color{blue}{{\mathsf{PI}\left(\right)}^{3}}}\right)\right)\right) \]

   6. Applied rewrites 46.2%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \color{blue}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}}\right)\right)\right) \]
   7. Step-by-step derivation

      1. lift-cbrt.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \color{blue}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}}\right)\right)\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \sqrt[3]{\color{blue}{{\mathsf{PI}\left(\right)}^{3}}}\right)\right)\right) \]

      3. rem-cbrt-cube 33.5

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

      4. rem-square-sqrt N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. sqr-neg-rev N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. unpow1 N/A

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

      10. metadata-eval N/A

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

      11. pow-pow N/A

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

      12. pow2 N/A

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

      13. sqr-neg-rev N/A

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

      14. pow2 N/A

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

      15. pow-pow N/A

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

      16. metadata-eval N/A

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

      17. unpow1 N/A

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

      18. lift-sqrt.f64 N/A

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

      19. lift-sqrt.f64 N/A

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

      20. rem-square-sqrt N/A

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

      21. lower-neg.f64 38.7

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

   8. Applied rewrites 38.7%

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

   9. Taylor expanded in angle around 0

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

       1. *-commutative N/A

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

       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 35.0

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

   11. Applied rewrites 35.0%

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

   if 1.8500000000000002e197 < angle

   1. Initial program 21.7%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 32.7

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

   5. Applied rewrites 32.7%

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

   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 27.6%

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

Alternative 13: 62.9% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 1000000000:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(\left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.011111111111111112 \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 1000000000.0)
   (* (+ a b) (* (* (* (- b a) (PI)) angle) 0.011111111111111112))
   (* (* 0.011111111111111112 angle) (* (* (PI) (+ b a)) (- b a)))))

Derivation

1. Split input into 2 regimes

2. if angle < 1e9

   1. Initial program 66.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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-*r* 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. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   4. Applied rewrites 82.7%

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

   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 78.8

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

   7. Applied rewrites 78.8%

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

   if 1e9 < angle

   1. Initial program 28.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. 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-*r* 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. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   4. Applied rewrites 30.7%

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

   5. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. difference-of-squares-rev N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares-rev N/A

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

      11. +-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-PI.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lower--.f64 26.8

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

   7. Applied rewrites 26.8%

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

Alternative 14: 62.8% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 7.5 \cdot 10^{-40}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(\left(0.011111111111111112 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.011111111111111112 \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 7.5e-40)
   (* (- b a) (* (+ b a) (* (* 0.011111111111111112 angle) (PI))))
   (* (* 0.011111111111111112 angle) (* (* (PI) (+ b a)) (- b a)))))

Derivation

1. Split input into 2 regimes

2. if angle < 7.50000000000000069e-40

   1. Initial program 66.1%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 64.1

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

   5. Applied rewrites 64.1%

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

   7. Applied rewrites 78.1%

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

   if 7.50000000000000069e-40 < angle

   1. Initial program 36.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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-*r* 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. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   4. Applied rewrites 39.8%

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

   5. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. difference-of-squares-rev N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares-rev N/A

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

      11. +-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-PI.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lower--.f64 35.6

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

   7. Applied rewrites 35.6%

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

Alternative 15: 62.8% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.011111111111111112 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;angle \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* 0.011111111111111112 angle) (PI))))
   (if (<= angle 2e-18)
     (* (- b a) (* (+ b a) t_0))
     (* t_0 (* (+ b a) (- b a))))))

Derivation

1. Split input into 2 regimes

2. if angle < 2.0000000000000001e-18

   1. Initial program 66.9%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 65.0

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

   5. Applied rewrites 65.0%

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

   7. Applied rewrites 78.6%

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

   if 2.0000000000000001e-18 < angle

   1. Initial program 31.4%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 30.3

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

   5. Applied rewrites 30.3%

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

Alternative 16: 38.9% accurate, 16.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 2 \cdot 10^{-26}:\\ \;\;\;\;\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(-0.011111111111111112 \cdot a\right)\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 2e-26)
   (* (* (* angle (PI)) (* -0.011111111111111112 a)) a)
   (* (* -0.011111111111111112 (* a a)) (* (PI) angle))))

Derivation

1. Split input into 2 regimes

2. if angle < 2.0000000000000001e-26

   1. Initial program 66.6%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 64.6

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

   5. Applied rewrites 64.6%

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

   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 38.4%

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

   10. Applied rewrites 47.1%

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

   if 2.0000000000000001e-26 < angle

   1. Initial program 33.6%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 32.5

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

   5. Applied rewrites 32.5%

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

   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 21.1%

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

Alternative 17: 37.1% accurate, 16.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 8 \cdot 10^{-7}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 8e-7)
   (* (* -0.011111111111111112 (* a a)) (* (PI) angle))
   (* (* -0.011111111111111112 a) (* a (* angle (PI))))))

Derivation

1. Split input into 2 regimes

2. if a < 7.9999999999999996e-7

   1. Initial program 62.7%

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

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 61.6

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

   5. Applied rewrites 61.6%

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

   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 34.2%

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

   if 7.9999999999999996e-7 < a

   1. Initial program 47.8%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 44.3

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

   5. Applied rewrites 44.3%

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

   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 34.4%

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

   10. Applied rewrites 49.4%

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

Alternative 18: 38.1% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (* (* -0.011111111111111112 a) (* a (* angle (PI)))))

Derivation

1. Initial program 58.7%

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

3. Taylor expanded in angle around 0

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. unpow2 N/A

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

   8. unpow2 N/A

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

   9. difference-of-squares N/A

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

   10. lower-*.f64 N/A

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

   11. lower-+.f64 N/A

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

   12. lower--.f64 57.0

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

5. Applied rewrites 57.0%

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

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 34.3%

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

10. Applied rewrites 40.0%

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

Reproduce

herbie shell --seed 2024363 
(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)))))