ab-angle->ABCF B

Percentage Accurate: 54.9% → 67.9%

Time: 12.2s

Alternatives: 16

Speedup: 16.8×

• Could not converge on a ground truth

  1. a = 1.2159102849775724e+269
  2. b = 8.152698048794684e+123
  3. angle = 3.230783783069709e+300

• 32.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

Initial Program: 54.9% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 67.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 1.4 \cdot 10^{+252}:\\ \;\;\;\;\cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{\left(\left(a - b\_m\right) \cdot 2\right) \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right)\right)}{\frac{-1}{a + b\_m}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \left(b\_m - a\right)\right) \cdot \left(a + b\_m\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= b_m 1.4e+252)
   (*
    (cos (* (/ angle 180.0) (PI)))
    (/
     (* (* (- a b_m) 2.0) (sin (* angle (* 0.005555555555555556 (PI)))))
     (/ -1.0 (+ a b_m))))
   (* (* (* (* 0.011111111111111112 (PI)) angle) (- b_m a)) (+ a b_m))))

Derivation

1. Split input into 2 regimes

2. if b < 1.40000000000000002e252

   1. Initial program 52.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 66.7

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

   4. Applied rewrites 67.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. rem-square-sqrt N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lower-fma.f64 31.2

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

   6. Applied rewrites 31.2%

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

   8. Applied rewrites 67.7%

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

   if 1.40000000000000002e252 < b

   1. Initial program 20.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 90.9

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

   5. Applied rewrites 90.9%

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

   7. Applied rewrites 100.0%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{+252}:\\ \;\;\;\;\cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{\left(\left(a - b\right) \cdot 2\right) \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right)\right)}{\frac{-1}{a + b}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \left(b - a\right)\right) \cdot \left(a + b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 68.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 4.8 \cdot 10^{+238}:\\ \;\;\;\;\left(\left(\left(\sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right) \cdot \left(b\_m - a\right)\right) \cdot \left(a + b\_m\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \left(b\_m - a\right)\right) \cdot \left(a + b\_m\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= b_m 4.8e+238)
   (*
    (*
     (* (* (sin (* (* angle 0.005555555555555556) (PI))) 2.0) (- b_m a))
     (+ a b_m))
    (cos (* (/ angle 180.0) (PI))))
   (* (* (* (* 0.011111111111111112 (PI)) angle) (- b_m a)) (+ a b_m))))

Derivation

1. Split input into 2 regimes

2. if b < 4.8e238

   1. Initial program 53.8%

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

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 67.0

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

   4. Applied rewrites 67.3%

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

   if 4.8e238 < b

   1. Initial program 18.7%

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

   3. Taylor expanded in angle around 0

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

      1. *-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 83.4

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

   5. Applied rewrites 83.4%

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

   7. Applied rewrites 94.1%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.8 \cdot 10^{+238}:\\ \;\;\;\;\left(\left(\left(\sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right) \cdot \left(b - a\right)\right) \cdot \left(a + b\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \left(b - a\right)\right) \cdot \left(a + b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 10^{+244}:\\ \;\;\;\;\cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot 0.005555555555555556\right) \cdot \left(\left(\left(\sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right) \cdot \left(b\_m - a\right)\right) \cdot \left(a + b\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \left(b\_m - a\right)\right) \cdot \left(a + b\_m\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= b_m 1e+244)
   (*
    (cos (* (* angle (PI)) 0.005555555555555556))
    (*
     (* (* (sin (* (* angle 0.005555555555555556) (PI))) 2.0) (- b_m a))
     (+ a b_m)))
   (* (* (* (* 0.011111111111111112 (PI)) angle) (- b_m a)) (+ a b_m))))

Derivation

1. Split input into 2 regimes

2. if b < 1.00000000000000007e244

   1. Initial program 53.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 67.1

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

   4. Applied rewrites 67.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 66.8

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 66.8

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

   6. Applied rewrites 66.8%

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

   if 1.00000000000000007e244 < b

   1. Initial program 16.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 93.0

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

   5. Applied rewrites 93.0%

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

   7. Applied rewrites 100.0%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10^{+244}:\\ \;\;\;\;\cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot 0.005555555555555556\right) \cdot \left(\left(\left(\sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right) \cdot \left(b - a\right)\right) \cdot \left(a + b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \left(b - a\right)\right) \cdot \left(a + b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(angle \cdot 0.005555555555555556\right) \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;b\_m \leq 4.8 \cdot 10^{+238}:\\ \;\;\;\;\left(\cos t\_0 \cdot \left(a + b\_m\right)\right) \cdot \left(\left(\sin t\_0 \cdot \left(b\_m - a\right)\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \left(b\_m - a\right)\right) \cdot \left(a + b\_m\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (* (* angle 0.005555555555555556) (PI))))
   (if (<= b_m 4.8e+238)
     (* (* (cos t_0) (+ a b_m)) (* (* (sin t_0) (- b_m a)) 2.0))
     (* (* (* (* 0.011111111111111112 (PI)) angle) (- b_m a)) (+ a b_m)))))

Derivation

1. Split input into 2 regimes

2. if b < 4.8e238

   1. Initial program 53.8%

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

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 67.0

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

   4. Applied rewrites 67.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. rem-square-sqrt N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lower-fma.f64 30.0

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

   6. Applied rewrites 30.0%

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

   8. Applied rewrites 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r/ N/A

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

      5. associate-*l* N/A

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

      6. lift-neg.f64 N/A

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

      7. metadata-eval N/A

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

      8. frac-2neg N/A

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

      9. /-rgt-identity N/A

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

   10. Applied rewrites 66.3%

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

   if 4.8e238 < b

   1. Initial program 18.7%

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

   3. Taylor expanded in angle around 0

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

      1. *-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 83.4

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

   5. Applied rewrites 83.4%

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

   7. Applied rewrites 94.1%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.8 \cdot 10^{+238}:\\ \;\;\;\;\left(\cos \left(\left(angle \cdot 0.005555555555555556\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a + b\right)\right) \cdot \left(\left(\sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(b - a\right)\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \left(b - a\right)\right) \cdot \left(a + b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;{b\_m}^{2} - {a}^{2} \leq -2 \cdot 10^{+119}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot a\right) \cdot angle\right) \cdot \left(-0.011111111111111112 \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a + b\_m\right) \cdot \left(b\_m - a\right)\right) \cdot 0.011111111111111112\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= (- (pow b_m 2.0) (pow a 2.0)) -2e+119)
   (* (* (* (PI) a) angle) (* -0.011111111111111112 a))
   (* (* (* (+ a b_m) (- b_m a)) 0.011111111111111112) (* angle (PI)))))

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -1.99999999999999989e119

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 39.4

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

   5. Applied rewrites 39.4%

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

   6. Taylor expanded in 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 39.4%

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

   10. Applied rewrites 54.2%

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

   12. Applied rewrites 54.3%

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

   if -1.99999999999999989e119 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 50.8%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 55.2

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

   5. Applied rewrites 55.2%

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

   7. Applied rewrites 55.2%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -2 \cdot 10^{+119}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot a\right) \cdot angle\right) \cdot \left(-0.011111111111111112 \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;{b\_m}^{2} - {a}^{2} \leq -2 \cdot 10^{+119}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot a\right) \cdot angle\right) \cdot \left(-0.011111111111111112 \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a + b\_m\right) \cdot \left(b\_m - a\right)\right) \cdot \left(\left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= (- (pow b_m 2.0) (pow a 2.0)) -2e+119)
   (* (* (* (PI) a) angle) (* -0.011111111111111112 a))
   (* (* (+ a b_m) (- b_m a)) (* (* 0.011111111111111112 (PI)) angle))))

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -1.99999999999999989e119

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 39.4

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

   5. Applied rewrites 39.4%

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

   6. Taylor expanded in 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 39.4%

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

   10. Applied rewrites 54.2%

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

   12. Applied rewrites 54.3%

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

   if -1.99999999999999989e119 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 50.8%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 55.2

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

   5. Applied rewrites 55.2%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -2 \cdot 10^{+119}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot a\right) \cdot angle\right) \cdot \left(-0.011111111111111112 \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(\left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;{b\_m}^{2} - {a}^{2} \leq -5 \cdot 10^{-208}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot a\right) \cdot angle\right) \cdot \left(-0.011111111111111112 \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.011111111111111112, angle \cdot \mathsf{PI}\left(\right), 0\right) \cdot \left(b\_m \cdot b\_m\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= (- (pow b_m 2.0) (pow a 2.0)) -5e-208)
   (* (* (* (PI) a) angle) (* -0.011111111111111112 a))
   (* (fma 0.011111111111111112 (* angle (PI)) 0.0) (* b_m b_m))))

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -4.99999999999999963e-208

   1. Initial program 54.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 43.7

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

   5. Applied rewrites 43.7%

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

   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 43.3%

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

   10. Applied rewrites 52.7%

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

   12. Applied rewrites 52.8%

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

   if -4.99999999999999963e-208 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 49.6%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 56.1

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

   5. Applied rewrites 56.1%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 21.7%

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

   10. Applied rewrites 21.5%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 52.9%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -5 \cdot 10^{-208}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot a\right) \cdot angle\right) \cdot \left(-0.011111111111111112 \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.011111111111111112, angle \cdot \mathsf{PI}\left(\right), 0\right) \cdot \left(b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;{b\_m}^{2} - {a}^{2} \leq -5 \cdot 10^{-208}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot a\right) \cdot angle\right) \cdot \left(-0.011111111111111112 \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(b\_m \cdot b\_m\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= (- (pow b_m 2.0) (pow a 2.0)) -5e-208)
   (* (* (* (PI) a) angle) (* -0.011111111111111112 a))
   (* (* (* (* b_m b_m) (PI)) angle) 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))) < -4.99999999999999963e-208

   1. Initial program 54.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 43.7

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

   5. Applied rewrites 43.7%

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

   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 43.3%

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

   10. Applied rewrites 52.7%

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

   12. Applied rewrites 52.8%

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

   if -4.99999999999999963e-208 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 49.6%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 56.1

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

   5. Applied rewrites 56.1%

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

   6. Taylor expanded in b around 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 52.8%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -5 \cdot 10^{-208}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot a\right) \cdot angle\right) \cdot \left(-0.011111111111111112 \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(b \cdot b\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot 0.011111111111111112\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 5 \cdot 10^{-301}:\\ \;\;\;\;\left(b\_m \cdot b\_m\right) \cdot \sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \left(b\_m - a\right)\right) \cdot \left(a + b\_m\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= (pow a 2.0) 5e-301)
   (* (* b_m b_m) (sin (* (* angle (PI)) 0.011111111111111112)))
   (* (* (* (* 0.011111111111111112 (PI)) angle) (- b_m a)) (+ a b_m))))

Derivation

1. Split input into 2 regimes

2. if (pow.f64 a #s(literal 2 binary64)) < 5.00000000000000013e-301

   1. Initial program 67.1%

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

   4. Applied rewrites 10.7%

      \[\leadsto \color{blue}{\left({b}^{6} - {a}^{6}\right) \cdot \left({\left(\mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), a \cdot a, {b}^{4}\right)\right)}^{-1} \cdot \sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot 0.011111111111111112\right)\right)} \]

   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. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 66.3

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

   7. Applied rewrites 66.3%

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

   if 5.00000000000000013e-301 < (pow.f64 a #s(literal 2 binary64))

   1. Initial program 45.8%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 47.9

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

   5. Applied rewrites 47.9%

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

   7. Applied rewrites 59.8%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 5 \cdot 10^{-301}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \left(b - a\right)\right) \cdot \left(a + b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-37}:\\ \;\;\;\;\left(\left(\left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \left(b\_m - a\right)\right) \cdot \left(a + b\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a + b\_m\right) \cdot \left(b\_m - a\right)\right) \cdot \sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot 0.011111111111111112\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= (/ angle 180.0) 5e-37)
   (* (* (* (* 0.011111111111111112 (PI)) angle) (- b_m a)) (+ a b_m))
   (* (* (+ a b_m) (- b_m a)) (sin (* (* angle (PI)) 0.011111111111111112)))))

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.9999999999999997e-37

   1. Initial program 58.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 59.5

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

   5. Applied rewrites 59.5%

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

   7. Applied rewrites 71.7%

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

   if 4.9999999999999997e-37 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 37.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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. 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 42.9%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-37}:\\ \;\;\;\;\left(\left(\left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \left(b - a\right)\right) \cdot \left(a + b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot 0.011111111111111112\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 5 \cdot 10^{+238}:\\ \;\;\;\;\left(\sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot 0.011111111111111112\right) \cdot \left(b\_m - a\right)\right) \cdot \left(a + b\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \left(b\_m - a\right)\right) \cdot \left(a + b\_m\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= b_m 5e+238)
   (* (* (sin (* (* angle (PI)) 0.011111111111111112)) (- b_m a)) (+ a b_m))
   (* (* (* (* 0.011111111111111112 (PI)) angle) (- b_m a)) (+ a b_m))))

Derivation

1. Split input into 2 regimes

2. if b < 4.99999999999999995e238

   1. Initial program 53.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   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 66.2%

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

   if 4.99999999999999995e238 < b

   1. Initial program 18.7%

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

   3. Taylor expanded in angle around 0

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

      1. *-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 83.4

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

   5. Applied rewrites 83.4%

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

   7. Applied rewrites 94.1%

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

4. Final simplification 68.0%

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

Alternative 12: 63.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 1.65 \cdot 10^{-127}:\\ \;\;\;\;\left(\left(-a\right) \cdot a\right) \cdot \sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \left(b\_m - a\right)\right) \cdot \left(a + b\_m\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= b_m 1.65e-127)
   (* (* (- a) a) (sin (* (* angle (PI)) 0.011111111111111112)))
   (* (* (* (* 0.011111111111111112 (PI)) angle) (- b_m a)) (+ a b_m))))

Derivation

1. Split input into 2 regimes

2. if b < 1.6499999999999999e-127

   1. Initial program 56.5%

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

   4. Applied rewrites 8.8%

      \[\leadsto \color{blue}{\left({b}^{6} - {a}^{6}\right) \cdot \left({\left(\mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), a \cdot a, {b}^{4}\right)\right)}^{-1} \cdot \sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot 0.011111111111111112\right)\right)} \]

   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. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. mul-1-neg N/A

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

      12. unpow2 N/A

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

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

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

      14. mul-1-neg N/A

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

      15. lower-*.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 42.2

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

   7. Applied rewrites 42.2%

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

   if 1.6499999999999999e-127 < b

   1. Initial program 41.2%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 49.1

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

   5. Applied rewrites 49.1%

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

   7. Applied rewrites 57.9%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.65 \cdot 10^{-127}:\\ \;\;\;\;\left(\left(-a\right) \cdot a\right) \cdot \sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(0.011111111111111112 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \left(b - a\right)\right) \cdot \left(a + b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.4% accurate, 16.8× speedup

Math

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

FPCore

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

Derivation

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 51.3

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

5. Applied rewrites 51.3%

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

7. Applied rewrites 59.4%

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

8. Final simplification 59.4%

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

Alternative 14: 38.3% accurate, 21.6× speedup

Math

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

FPCore

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

Derivation

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 51.3

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

5. Applied rewrites 51.3%

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

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 30.1%

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

10. Applied rewrites 33.6%

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

12. Applied rewrites 33.6%

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

13. Final simplification 33.6%

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

Alternative 15: 38.3% accurate, 21.6× speedup

Math

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

FPCore

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

Derivation

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 51.3

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

5. Applied rewrites 51.3%

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

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 30.1%

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

10. Applied rewrites 33.6%

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

12. Applied rewrites 33.6%

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

13. Final simplification 33.6%

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

Alternative 16: 38.3% accurate, 21.6× speedup

Math

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

FPCore

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

Derivation

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 51.3

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

5. Applied rewrites 51.3%

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

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 30.1%

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

10. Applied rewrites 33.6%

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

11. Final simplification 33.6%

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

Reproduce

herbie shell --seed 2024273 
(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)))))