ab-angle->ABCF B

Percentage Accurate: 53.5% → 67.0%

Time: 12.6s

Alternatives: 16

Speedup: 10.3×

• Could not converge on a ground truth

  1. a = 1.162737645138401e+156
  2. b = 1.3123285653020884e-58
  3. angle = 6.124715124841019e+23

• 32.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

Initial Program: 53.5% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 67.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+257}:\\ \;\;\;\;\left(\sin \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\_m\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{{\left(a + b\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \left(\left(2 \cdot \sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}} \cdot angle\_m}{-180}\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 2e+257)
    (*
     (* (sin (* 0.011111111111111112 (* (PI) angle_m))) (- b a))
     (/ 1.0 (pow (+ a b) -1.0)))
    (*
     (* (- b a) (+ a b))
     (*
      (* 2.0 (sin (* (* 0.005555555555555556 angle_m) (PI))))
      (cos (/ (* (cbrt (pow (PI) 3.0)) angle_m) -180.0)))))))

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2.00000000000000006e257

   1. Initial program 54.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 68.8%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lift-*.f64 N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. clear-num N/A

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

      7. lift-*.f64 N/A

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

      8. flip-+ N/A

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

      9. lift-+.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 68.8

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 68.8

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

   6. Applied rewrites 68.8%

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

   if 2.00000000000000006e257 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 12.1%

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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 21.7%

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

      1. lift-PI.f64 N/A

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

      2. add-cbrt-cube N/A

         \[\leadsto \left(\cos \left(\frac{angle \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}}{-180}\right) \cdot \left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right) \]

      3. lower-cbrt.f64 N/A

         \[\leadsto \left(\cos \left(\frac{angle \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}}{-180}\right) \cdot \left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right) \]

      4. rem-cube-cbrt N/A

         \[\leadsto \left(\cos \left(\frac{angle \cdot \sqrt[3]{\color{blue}{{\left(\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}\right)}^{3}}}}{-180}\right) \cdot \left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right) \]

      5. add-cbrt-cube N/A

         \[\leadsto \left(\cos \left(\frac{angle \cdot \sqrt[3]{{\color{blue}{\mathsf{PI}\left(\right)}}^{3}}}{-180}\right) \cdot \left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right) \]

      6. lift-PI.f64 N/A

         \[\leadsto \left(\cos \left(\frac{angle \cdot \sqrt[3]{{\color{blue}{\mathsf{PI}\left(\right)}}^{3}}}{-180}\right) \cdot \left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right) \]

      7. lower-pow.f64 35.6

         \[\leadsto \left(\cos \left(\frac{angle \cdot \sqrt[3]{\color{blue}{{\mathsf{PI}\left(\right)}^{3}}}}{-180}\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right) \]

   6. Applied rewrites 35.6%

      \[\leadsto \left(\cos \left(\frac{angle \cdot \color{blue}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}}}{-180}\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+257}:\\ \;\;\;\;\left(\sin \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{{\left(a + b\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \left(\left(2 \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}} \cdot angle}{-180}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 65.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := {b}^{2} - {a}^{2}\\ t_1 := \sin \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\_m\right)\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+307}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot a\right) \cdot angle\_m\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-230}:\\ \;\;\;\;\left(\left(-a\right) \cdot a\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 \cdot b\right) \cdot b\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (- (pow b 2.0) (pow a 2.0)))
        (t_1 (sin (* 0.011111111111111112 (* (PI) angle_m)))))
   (*
    angle_s
    (if (<= t_0 -2e+307)
      (* (* (* -0.011111111111111112 a) angle_m) (* (PI) a))
      (if (<= t_0 5e-230) (* (* (- a) a) t_1) (* (* t_1 b) b))))))

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -1.99999999999999997e307

   1. Initial program 54.0%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 59.3

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

   5. Applied rewrites 59.3%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 59.3%

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

   10. Applied rewrites 76.3%

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

   12. Applied rewrites 76.4%

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

   if -1.99999999999999997e307 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < 5.00000000000000035e-230

   1. Initial program 54.8%

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

   4. Applied rewrites 24.0%

      \[\leadsto \color{blue}{\frac{\left(\sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot 0.011111111111111112\right) + 0\right) \cdot \left(\left({b}^{4} - {a}^{4}\right) \cdot 2\right)}{2 \cdot \mathsf{fma}\left(a, a, b \cdot b\right)}} \]

   5. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-PI.f64 55.4

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

   7. Applied rewrites 55.4%

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

   if 5.00000000000000035e-230 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 49.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 69.1%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lift-*.f64 N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. clear-num N/A

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

      7. lift-*.f64 N/A

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

      8. flip-+ N/A

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

      9. lift-+.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 69.1

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 69.1

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

   6. Applied rewrites 69.1%

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

      1. lift-PI.f64 N/A

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

      2. add-sqr-sqrt N/A

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

      3. lower-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lower-sqrt.f64 62.8

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

   8. Applied rewrites 62.8%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. distribute-rgt1-in N/A

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

       6. metadata-eval N/A

          \[\leadsto b \cdot \left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \left(\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{0} \cdot b\right)\right) \cdot a \]

       7. mul0-lft N/A

          \[\leadsto b \cdot \left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \left(\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{0}\right) \cdot a \]

       8. associate-*r* N/A

          \[\leadsto b \cdot \left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \color{blue}{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(0 \cdot a\right)} \]

       9. mul0-lft N/A

          \[\leadsto b \cdot \left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{0} \]

       10. mul0-lft N/A

           \[\leadsto b \cdot \left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(0 \cdot b\right)} \]

       11. associate-*r* N/A

           \[\leadsto b \cdot \left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \color{blue}{\left(\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 0\right) \cdot b} \]

       12. mul0-lft N/A

           \[\leadsto b \cdot \left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \left(\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(0 \cdot a\right)}\right) \cdot b \]

       13. metadata-eval N/A

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

       14. distribute-rgt1-in N/A

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

       15. *-commutative N/A

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

   11. Applied rewrites 65.0%

       \[\leadsto \color{blue}{\left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(0 + b\right)\right) \cdot b} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -2 \cdot 10^{+307}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot a\right) \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\right)\\ \mathbf{elif}\;{b}^{2} - {a}^{2} \leq 5 \cdot 10^{-230}:\\ \;\;\;\;\left(\left(-a\right) \cdot a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot b\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot angle\_m\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 6.3 \cdot 10^{+232}:\\ \;\;\;\;\left(\sin \left(0.011111111111111112 \cdot t\_0\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{{\left(a + b\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}\right) \cdot 2\right) \cdot \cos \left(\frac{t\_0}{-180}\right)\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* (PI) angle_m)))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 6.3e+232)
      (*
       (* (sin (* 0.011111111111111112 t_0)) (- b a))
       (/ 1.0 (pow (+ a b) -1.0)))
      (*
       (*
        (*
         (sin (* (* 0.005555555555555556 angle_m) (cbrt (pow (PI) 3.0))))
         2.0)
        (cos (/ t_0 -180.0)))
       (* (- b a) (+ a b)))))))

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 6.2999999999999995e232

   1. Initial program 54.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 70.2%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lift-*.f64 N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. clear-num N/A

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

      7. lift-*.f64 N/A

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

      8. flip-+ N/A

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

      9. lift-+.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 70.2

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 70.2

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

   6. Applied rewrites 70.2%

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

   if 6.2999999999999995e232 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 19.9%

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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 25.0%

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

      1. lift-PI.f64 N/A

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

      2. add-cbrt-cube N/A

         \[\leadsto \left(\cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot \left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right) \cdot 2\right)\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right) \]

      3. lower-cbrt.f64 N/A

         \[\leadsto \left(\cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot \left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right) \cdot 2\right)\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right) \]

      4. rem-cube-cbrt N/A

         \[\leadsto \left(\cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot \left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\color{blue}{{\left(\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}\right)}^{3}}}\right) \cdot 2\right)\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right) \]

      5. add-cbrt-cube N/A

         \[\leadsto \left(\cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot \left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{{\color{blue}{\mathsf{PI}\left(\right)}}^{3}}\right) \cdot 2\right)\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right) \]

      6. lift-PI.f64 N/A

         \[\leadsto \left(\cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot \left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{{\color{blue}{\mathsf{PI}\left(\right)}}^{3}}\right) \cdot 2\right)\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right) \]

      7. lower-pow.f64 29.0

         \[\leadsto \left(\cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt[3]{\color{blue}{{\mathsf{PI}\left(\right)}^{3}}}\right) \cdot 2\right)\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right) \]

   6. Applied rewrites 29.0%

      \[\leadsto \left(\cos \left(\frac{angle \cdot \mathsf{PI}\left(\right)}{-180}\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \color{blue}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}}\right) \cdot 2\right)\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 6.3 \cdot 10^{+232}:\\ \;\;\;\;\left(\sin \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{{\left(a + b\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}\right) \cdot 2\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{-180}\right)\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \sin \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\_m\right)\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq 5 \cdot 10^{-230}:\\ \;\;\;\;a \cdot \left(t\_0 \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot b\right) \cdot b\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (sin (* 0.011111111111111112 (* (PI) angle_m)))))
   (*
    angle_s
    (if (<= (- (pow b 2.0) (pow a 2.0)) 5e-230)
      (* a (* t_0 (- b a)))
      (* (* t_0 b) b)))))

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < 5.00000000000000035e-230

   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-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 64.5%

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

      1. lift-+.f64 N/A

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

      2. unpow1 N/A

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

      3. sqr-pow N/A

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

      4. lower-fma.f64 N/A

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

      5. metadata-eval N/A

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

      6. unpow1/2 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. metadata-eval N/A

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

      9. unpow1/2 N/A

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

      10. lower-sqrt.f64 31.2

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

   6. Applied rewrites 31.2%

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

   7. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity 64.5

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

   9. Applied rewrites 64.5%

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

   if 5.00000000000000035e-230 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 49.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 69.1%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lift-*.f64 N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. clear-num N/A

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

      7. lift-*.f64 N/A

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

      8. flip-+ N/A

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

      9. lift-+.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 69.1

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 69.1

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

   6. Applied rewrites 69.1%

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

      1. lift-PI.f64 N/A

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

      2. add-sqr-sqrt N/A

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

      3. lower-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lower-sqrt.f64 62.8

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

   8. Applied rewrites 62.8%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. distribute-rgt1-in N/A

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

       6. metadata-eval N/A

          \[\leadsto b \cdot \left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \left(\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{0} \cdot b\right)\right) \cdot a \]

       7. mul0-lft N/A

          \[\leadsto b \cdot \left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \left(\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{0}\right) \cdot a \]

       8. associate-*r* N/A

          \[\leadsto b \cdot \left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \color{blue}{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(0 \cdot a\right)} \]

       9. mul0-lft N/A

          \[\leadsto b \cdot \left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{0} \]

       10. mul0-lft N/A

           \[\leadsto b \cdot \left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(0 \cdot b\right)} \]

       11. associate-*r* N/A

           \[\leadsto b \cdot \left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \color{blue}{\left(\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 0\right) \cdot b} \]

       12. mul0-lft N/A

           \[\leadsto b \cdot \left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \left(\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(0 \cdot a\right)}\right) \cdot b \]

       13. metadata-eval N/A

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

       14. distribute-rgt1-in N/A

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

       15. *-commutative N/A

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

   11. Applied rewrites 65.0%

       \[\leadsto \color{blue}{\left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(0 + b\right)\right) \cdot b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq 5 \cdot 10^{-230}:\\ \;\;\;\;a \cdot \left(\sin \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot b\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \sin \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\_m\right)\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq 5 \cdot 10^{-230}:\\ \;\;\;\;\left(\left(-a\right) \cdot t\_0\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot b\right) \cdot b\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (sin (* 0.011111111111111112 (* (PI) angle_m)))))
   (*
    angle_s
    (if (<= (- (pow b 2.0) (pow a 2.0)) 5e-230)
      (* (* (- a) t_0) a)
      (* (* t_0 b) b)))))

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < 5.00000000000000035e-230

   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-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 64.5%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lift-*.f64 N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. clear-num N/A

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

      7. lift-*.f64 N/A

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

      8. flip-+ N/A

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

      9. lift-+.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 64.6

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 64.6

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

   6. Applied rewrites 64.6%

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

      1. lift-PI.f64 N/A

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

      2. add-sqr-sqrt N/A

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

      3. lower-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lower-sqrt.f64 62.4

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

   8. Applied rewrites 62.4%

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

   9. Taylor expanded in b around 0

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. *-commutative N/A

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

       7. *-commutative N/A

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

       8. distribute-rgt1-in N/A

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

       9. metadata-eval N/A

          \[\leadsto a \cdot \left(-1 \cdot \left(a \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + \left(\left(\color{blue}{0} \cdot a\right) \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot b \]

       10. mul0-lft N/A

           \[\leadsto a \cdot \left(-1 \cdot \left(a \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + \left(\color{blue}{0} \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot b \]

       11. associate-*l* N/A

           \[\leadsto a \cdot \left(-1 \cdot \left(a \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + \color{blue}{0 \cdot \left(\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b\right)} \]

       12. *-commutative N/A

           \[\leadsto a \cdot \left(-1 \cdot \left(a \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + 0 \cdot \color{blue}{\left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

       13. associate-*l* N/A

           \[\leadsto a \cdot \left(-1 \cdot \left(a \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + \color{blue}{\left(0 \cdot b\right) \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   11. Applied rewrites 64.6%

       \[\leadsto \color{blue}{\left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(0 + \left(-a\right)\right)\right) \cdot a} \]

   if 5.00000000000000035e-230 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 49.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 69.1%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lift-*.f64 N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. clear-num N/A

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

      7. lift-*.f64 N/A

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

      8. flip-+ N/A

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

      9. lift-+.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 69.1

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 69.1

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

   6. Applied rewrites 69.1%

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

      1. lift-PI.f64 N/A

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

      2. add-sqr-sqrt N/A

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

      3. lower-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lower-sqrt.f64 62.8

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

   8. Applied rewrites 62.8%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. distribute-rgt1-in N/A

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

       6. metadata-eval N/A

          \[\leadsto b \cdot \left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \left(\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{0} \cdot b\right)\right) \cdot a \]

       7. mul0-lft N/A

          \[\leadsto b \cdot \left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \left(\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{0}\right) \cdot a \]

       8. associate-*r* N/A

          \[\leadsto b \cdot \left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \color{blue}{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(0 \cdot a\right)} \]

       9. mul0-lft N/A

          \[\leadsto b \cdot \left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{0} \]

       10. mul0-lft N/A

           \[\leadsto b \cdot \left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(0 \cdot b\right)} \]

       11. associate-*r* N/A

           \[\leadsto b \cdot \left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \color{blue}{\left(\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 0\right) \cdot b} \]

       12. mul0-lft N/A

           \[\leadsto b \cdot \left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \left(\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(0 \cdot a\right)}\right) \cdot b \]

       13. metadata-eval N/A

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

       14. distribute-rgt1-in N/A

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

       15. *-commutative N/A

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

   11. Applied rewrites 65.0%

       \[\leadsto \color{blue}{\left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(0 + b\right)\right) \cdot b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq 5 \cdot 10^{-230}:\\ \;\;\;\;\left(\left(-a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot b\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(\sin \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\_m\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{{\left(a + b\right)}^{-1}}\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (*
   (* (sin (* 0.011111111111111112 (* (PI) angle_m))) (- b a))
   (/ 1.0 (pow (+ a b) -1.0)))))

Derivation

1. Initial program 51.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lift--.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. lift-pow.f64 N/A

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

   11. unpow2 N/A

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

   12. difference-of-squares N/A

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

   13. associate-*l* N/A

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

   14. lower-*.f64 N/A

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

4. Applied rewrites 66.7%

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

   1. lift-+.f64 N/A

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

   2. flip-+ N/A

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

   3. lift-*.f64 N/A

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

   4. clear-num N/A

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

   5. lower-/.f64 N/A

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

   6. clear-num N/A

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

   7. lift-*.f64 N/A

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

   8. flip-+ N/A

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

   9. lift-+.f64 N/A

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

   10. inv-pow N/A

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

   11. lower-pow.f64 66.7

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-+.f64 66.7

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

6. Applied rewrites 66.7%

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

7. Final simplification 66.7%

   \[\leadsto \left(\sin \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{{\left(a + b\right)}^{-1}} \]
8. Add Preprocessing

Alternative 7: 60.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq 10^{-158}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot a\right) \cdot angle\_m\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0, \mathsf{PI}\left(\right), \mathsf{PI}\left(\right) \cdot b\right) \cdot \left(0.011111111111111112 \cdot angle\_m\right)\right) \cdot b\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (- (pow b 2.0) (pow a 2.0)) 1e-158)
    (* (* (* -0.011111111111111112 a) angle_m) (* (PI) a))
    (* (* (fma 0.0 (PI) (* (PI) b)) (* 0.011111111111111112 angle_m)) b))))

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < 1.00000000000000006e-158

   1. Initial program 54.1%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 53.5

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

   5. Applied rewrites 53.5%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 53.5%

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

   10. Applied rewrites 58.2%

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

   12. Applied rewrites 58.3%

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

   if 1.00000000000000006e-158 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 49.4%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 45.2

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

   5. Applied rewrites 45.2%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 13.1%

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

   10. Applied rewrites 13.8%

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

   11. Taylor expanded in a around 0

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

   13. Applied rewrites 53.8%

       \[\leadsto \left(\left(0.011111111111111112 \cdot angle\right) \cdot \mathsf{fma}\left(0, \mathsf{PI}\left(\right), \mathsf{PI}\left(\right) \cdot b\right)\right) \cdot \color{blue}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq 10^{-158}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot a\right) \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0, \mathsf{PI}\left(\right), \mathsf{PI}\left(\right) \cdot b\right) \cdot \left(0.011111111111111112 \cdot angle\right)\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -1 \cdot 10^{-266}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot a\right) \cdot angle\_m\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot b\right) \cdot angle\_m\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (- (pow b 2.0) (pow a 2.0)) -1e-266)
    (* (* (* -0.011111111111111112 a) angle_m) (* (PI) a))
    (* (* (* (* (PI) b) b) angle_m) 0.011111111111111112))))

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -9.9999999999999998e-267

   1. Initial program 51.4%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 50.9

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

   5. Applied rewrites 50.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 50.9%

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

   10. Applied rewrites 59.6%

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

   12. Applied rewrites 59.6%

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

   if -9.9999999999999998e-267 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 52.3%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 48.6

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

   5. Applied rewrites 48.6%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 21.8%

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

   10. Applied rewrites 20.0%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 48.6%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -1 \cdot 10^{-266}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot a\right) \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot b\right) \cdot angle\right) \cdot 0.011111111111111112\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -1 \cdot 10^{-266}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot a\right) \cdot angle\_m\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(b \cdot b\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\_m\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (- (pow b 2.0) (pow a 2.0)) -1e-266)
    (* (* (* -0.011111111111111112 a) angle_m) (* (PI) a))
    (* (* (* (* b b) (PI)) angle_m) 0.011111111111111112))))

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -9.9999999999999998e-267

   1. Initial program 51.4%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 50.9

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

   5. Applied rewrites 50.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 50.9%

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

   10. Applied rewrites 59.6%

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

   12. Applied rewrites 59.6%

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

   if -9.9999999999999998e-267 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 52.3%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 48.6

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

   5. Applied rewrites 48.6%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 48.6%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -1 \cdot 10^{-266}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot a\right) \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(b \cdot b\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot 0.011111111111111112\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 5 \cdot 10^{-242}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \sin \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\_m\right) \cdot \left(b - a\right)\right) \cdot \left(a + b\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (pow a 2.0) 5e-242)
    (* (* b b) (sin (* 0.011111111111111112 (* (PI) angle_m))))
    (* (* (* (* 0.011111111111111112 (PI)) angle_m) (- b a)) (+ a b)))))

Derivation

1. Split input into 2 regimes

2. if (pow.f64 a #s(literal 2 binary64)) < 4.9999999999999998e-242

   1. Initial program 62.4%

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

   4. Applied rewrites 13.9%

      \[\leadsto \color{blue}{\frac{\left(\sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot 0.011111111111111112\right) + 0\right) \cdot \left(\left({b}^{4} - {a}^{4}\right) \cdot 2\right)}{2 \cdot \mathsf{fma}\left(a, a, b \cdot b\right)}} \]

   5. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 61.3

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

   7. Applied rewrites 61.3%

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

   if 4.9999999999999998e-242 < (pow.f64 a #s(literal 2 binary64))

   1. Initial program 47.5%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 50.5

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

   5. Applied rewrites 50.5%

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

   7. Applied rewrites 60.7%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 5 \cdot 10^{-242}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \sin \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \left(b - a\right)\right) \cdot \left(a + b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 10^{-47}:\\ \;\;\;\;\left(\left(\left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\_m\right) \cdot \left(b - a\right)\right) \cdot \left(a + b\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\_m\right)\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 1e-47)
    (* (* (* (* 0.011111111111111112 (PI)) angle_m) (- b a)) (+ a b))
    (* (sin (* 0.011111111111111112 (* (PI) angle_m))) (* (- b a) (+ a b))))))

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 9.9999999999999997e-48

   1. Initial program 56.7%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 57.6

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

   5. Applied rewrites 57.6%

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

   7. Applied rewrites 68.3%

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

   if 9.9999999999999997e-48 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 41.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.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--.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

   4. Applied rewrites 48.6%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{-47}:\\ \;\;\;\;\left(\left(\left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \left(b - a\right)\right) \cdot \left(a + b\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(\sin \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\_m\right)\right) \cdot \left(b - a\right)\right) \cdot \left(a + b\right)\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (* (* (sin (* 0.011111111111111112 (* (PI) angle_m))) (- b a)) (+ a b))))

Derivation

1. Initial program 51.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lift--.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. lift-pow.f64 N/A

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

   11. unpow2 N/A

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

   12. difference-of-squares N/A

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

   13. associate-*l* N/A

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

   14. lower-*.f64 N/A

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

4. Applied rewrites 66.7%

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

5. Final simplification 66.7%

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

Alternative 13: 67.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(\sin \left(\left(0.011111111111111112 \cdot angle\_m\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(b - a\right)\right) \cdot \left(a + b\right)\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (* (* (sin (* (* 0.011111111111111112 angle_m) (PI))) (- b a)) (+ a b))))

Derivation

1. Initial program 51.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lift--.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. lift-pow.f64 N/A

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

   11. unpow2 N/A

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

   12. difference-of-squares N/A

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

   13. associate-*l* N/A

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

   14. lower-*.f64 N/A

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

4. Applied rewrites 66.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 65.8

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

6. Applied rewrites 65.8%

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

7. Final simplification 65.8%

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

Alternative 14: 61.9% accurate, 10.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\_m\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 23000000000000:\\ \;\;\;\;\left(t\_0 \cdot \left(b - a\right)\right) \cdot \left(a + b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-a\right) \cdot \left(a + b\right)\right) \cdot t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* (* 0.011111111111111112 (PI)) angle_m)))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 23000000000000.0)
      (* (* t_0 (- b a)) (+ a b))
      (* (* (- a) (+ a b)) t_0)))))

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2.3e13

   1. Initial program 58.7%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 58.3

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

   5. Applied rewrites 58.3%

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

   7. Applied rewrites 68.4%

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

   if 2.3e13 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 34.4%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 27.1

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

   5. Applied rewrites 27.1%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 25.3%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 23000000000000:\\ \;\;\;\;\left(\left(\left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \left(b - a\right)\right) \cdot \left(a + b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-a\right) \cdot \left(a + b\right)\right) \cdot \left(\left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 37.4% accurate, 21.6× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(\left(-0.011111111111111112 \cdot a\right) \cdot angle\_m\right) \cdot \left(\mathsf{PI}\left(\right) \cdot a\right)\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (* angle_s (* (* (* -0.011111111111111112 a) angle_m) (* (PI) a))))

Derivation

1. Initial program 51.9%

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

3. Taylor expanded in angle around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-PI.f64 N/A

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

   10. unpow2 N/A

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

   11. unpow2 N/A

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

   12. difference-of-squares N/A

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

   13. lower-*.f64 N/A

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

   14. lower-+.f64 N/A

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

   15. lower--.f64 49.7

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

5. Applied rewrites 49.7%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 34.9%

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

10. Applied rewrites 37.8%

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

12. Applied rewrites 37.8%

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

13. Final simplification 37.8%

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

Alternative 16: 37.5% accurate, 21.6× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\_m\right) \cdot a\right) \cdot \left(-0.011111111111111112 \cdot a\right)\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (* angle_s (* (* (* (PI) angle_m) a) (* -0.011111111111111112 a))))

Derivation

1. Initial program 51.9%

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

3. Taylor expanded in angle around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-PI.f64 N/A

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

   10. unpow2 N/A

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

   11. unpow2 N/A

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

   12. difference-of-squares N/A

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

   13. lower-*.f64 N/A

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

   14. lower-+.f64 N/A

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

   15. lower--.f64 49.7

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

5. Applied rewrites 49.7%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 34.9%

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

10. Applied rewrites 37.8%

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

11. Final simplification 37.8%

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

Reproduce

herbie shell --seed 2024276 
(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)))))