ab-angle->ABCF C

Percentage Accurate: 79.8% → 79.8%

Time: 4.9s

Alternatives: 8

Speedup: 2.0×

• 36.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

Initial Program: 79.8% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(\sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle\_m, 0.005555555555555556, 0.5 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot a\right)}^{2} + {\left(\sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow
   (* (sin (fma (* (PI) angle_m) 0.005555555555555556 (* 0.5 (PI)))) a)
   2.0)
  (pow (* (sin (* (* 0.005555555555555556 angle_m) (PI))) b) 2.0)))

Derivation

1. Initial program 79.5%

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

   1. lift-cos.f64 N/A

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

   2. sin-+PI/2-rev N/A

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

   3. lower-sin.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-PI.f64 79.7

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

4. Applied rewrites 79.7%

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

5. Taylor expanded in angle around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{angle}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} \]

   4. lower-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{angle}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} \]

   5. lift-PI.f64 79.7

      \[\leadsto {\left(a \cdot \sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{angle}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} \]

7. Applied rewrites 79.7%

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

8. Taylor expanded in a around 0

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

   1. lower-+.f64 N/A

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

10. Applied rewrites 79.8%

    \[\leadsto \color{blue}{{\left(\sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, 0.005555555555555556, 0.5 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot a\right)}^{2} + {\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2}} \]

11. Final simplification 79.8%

    \[\leadsto {\left(\sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, 0.005555555555555556, 0.5 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot a\right)}^{2} + {\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2} \]
12. Add Preprocessing

Alternative 2: 64.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{-50}:\\ \;\;\;\;{\left(\sin \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \left(\mathsf{PI}\left(\right) \cdot angle\_m\right) \cdot 0.005555555555555556\right)\right) \cdot a\right)}^{2}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot b\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}, angle\_m \cdot angle\_m, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(angle\_m \cdot b\right)}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 1.15e-50)
   (pow
    (* (sin (fma 0.5 (PI) (* (* (PI) angle_m) 0.005555555555555556))) a)
    2.0)
   (if (<= b 4.4e+153)
     (fma
      (* (pow (* (PI) b) 2.0) 3.08641975308642e-5)
      (* angle_m angle_m)
      (* a a))
     (* (* (pow (* angle_m b) 2.0) (* (PI) (PI))) 3.08641975308642e-5))))

Derivation

1. Split input into 3 regimes

2. if b < 1.1500000000000001e-50

   1. Initial program 80.4%

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

      1. lift-cos.f64 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. lower-sin.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-PI.f64 80.6

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

   4. Applied rewrites 80.6%

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

   5. Taylor expanded in a around inf

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

      1. pow-prod-down N/A

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

      2. lower-pow.f64 N/A

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

   7. Applied rewrites 62.6%

      \[\leadsto \color{blue}{{\left(\sin \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot a\right)}^{2}} \]

   if 1.1500000000000001e-50 < b < 4.3999999999999999e153

   1. Initial program 67.5%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2} + {\color{blue}{a}}^{2} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \color{blue}{{angle}^{2}}, {a}^{2}\right) \]

   5. Applied rewrites 23.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot b\right)}^{2}, 3.08641975308642 \cdot 10^{-5}, {\left(\mathsf{PI}\left(\right) \cdot a\right)}^{2} \cdot -3.08641975308642 \cdot 10^{-5}\right), angle \cdot angle, a \cdot a\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \color{blue}{angle} \cdot angle, a \cdot a\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}, angle \cdot angle, a \cdot a\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}, angle \cdot angle, a \cdot a\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left({\mathsf{PI}\left(\right)}^{2} \cdot {b}^{2}\right) \cdot \frac{1}{32400}, angle \cdot angle, a \cdot a\right) \]

      4. unpow-prod-down N/A

         \[\leadsto \mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot b\right)}^{2} \cdot \frac{1}{32400}, angle \cdot angle, a \cdot a\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot b\right)}^{2} \cdot \frac{1}{32400}, angle \cdot angle, a \cdot a\right) \]

      6. lift-PI.f64 N/A

         \[\leadsto \mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot b\right)}^{2} \cdot \frac{1}{32400}, angle \cdot angle, a \cdot a\right) \]

      7. lift-pow.f64 62.5

         \[\leadsto \mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot b\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}, angle \cdot angle, a \cdot a\right) \]

   8. Applied rewrites 62.5%

      \[\leadsto \mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot b\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}, \color{blue}{angle} \cdot angle, a \cdot a\right) \]

   if 4.3999999999999999e153 < b

   1. Initial program 90.0%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2} + {\color{blue}{a}}^{2} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \color{blue}{{angle}^{2}}, {a}^{2}\right) \]

   5. Applied rewrites 52.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot b\right)}^{2}, 3.08641975308642 \cdot 10^{-5}, {\left(\mathsf{PI}\left(\right) \cdot a\right)}^{2} \cdot -3.08641975308642 \cdot 10^{-5}\right), angle \cdot angle, a \cdot a\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto \frac{1}{32400} \cdot \color{blue}{\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400} \]

      2. lower-*.f64 N/A

         \[\leadsto \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400} \]

      3. *-commutative N/A

         \[\leadsto \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {b}^{2}\right)\right) \cdot \frac{1}{32400} \]

      4. unpow-prod-down N/A

         \[\leadsto \left({angle}^{2} \cdot {\left(\mathsf{PI}\left(\right) \cdot b\right)}^{2}\right) \cdot \frac{1}{32400} \]

      5. pow-prod-down N/A

         \[\leadsto {\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot b\right)\right)}^{2} \cdot \frac{1}{32400} \]

      6. *-commutative N/A

         \[\leadsto {\left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \frac{1}{32400} \]

      7. lower-pow.f64 N/A

         \[\leadsto {\left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \frac{1}{32400} \]

      8. *-commutative N/A

         \[\leadsto {\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      9. *-commutative N/A

         \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      10. lower-*.f64 N/A

          \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      11. lift-*.f64 N/A

          \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      12. lift-PI.f64 80.3

          \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5} \]

   8. Applied rewrites 80.3%

      \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \color{blue}{3.08641975308642 \cdot 10^{-5}} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      2. lift-*.f64 N/A

         \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      3. lift-PI.f64 N/A

         \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      4. lift-*.f64 N/A

         \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      5. unpow-prod-down N/A

         \[\leadsto \left({\left(\mathsf{PI}\left(\right) \cdot b\right)}^{2} \cdot {angle}^{2}\right) \cdot \frac{1}{32400} \]

      6. unpow-prod-down N/A

         \[\leadsto \left(\left({\mathsf{PI}\left(\right)}^{2} \cdot {b}^{2}\right) \cdot {angle}^{2}\right) \cdot \frac{1}{32400} \]

      7. *-commutative N/A

         \[\leadsto \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) \cdot \frac{1}{32400} \]

      8. *-commutative N/A

         \[\leadsto \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400} \]

      9. associate-*r* N/A

         \[\leadsto \left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400} \]

      11. pow-prod-down N/A

          \[\leadsto \left({\left(angle \cdot b\right)}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400} \]

      12. lower-pow.f64 N/A

          \[\leadsto \left({\left(angle \cdot b\right)}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400} \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(angle \cdot b\right)}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400} \]

      14. unpow2 N/A

          \[\leadsto \left({\left(angle \cdot b\right)}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{32400} \]

      15. lower-*.f64 N/A

          \[\leadsto \left({\left(angle \cdot b\right)}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{32400} \]

      16. lift-PI.f64 N/A

          \[\leadsto \left({\left(angle \cdot b\right)}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{32400} \]

      17. lift-PI.f64 80.4

          \[\leadsto \left({\left(angle \cdot b\right)}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} \]

   10. Applied rewrites 80.4%

       \[\leadsto \left({\left(angle \cdot b\right)}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{-50}:\\ \;\;\;\;{\left(\sin \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot a\right)}^{2}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot b\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}, angle \cdot angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(angle \cdot b\right)}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ a \cdot a + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+ (* a a) (pow (* b (sin (* (PI) (* 0.005555555555555556 angle_m)))) 2.0)))

Derivation

1. Initial program 79.5%

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

3. Taylor expanded in angle around 0

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

   1. lower-*.f64 79.6

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

5. Applied rewrites 79.6%

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

6. Taylor expanded in angle around 0

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

   1. sin-+PI/2-rev N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 79.2

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

8. Applied rewrites 79.2%

   \[\leadsto \color{blue}{a \cdot a} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}^{2} \]
9. Add Preprocessing

Alternative 4: 64.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{-50}:\\ \;\;\;\;{\left(\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\_m\right) \cdot 0.005555555555555556\right) \cdot a\right)}^{2}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot b\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}, angle\_m \cdot angle\_m, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(angle\_m \cdot b\right)}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 1.15e-50)
   (pow (* (cos (* (* (PI) angle_m) 0.005555555555555556)) a) 2.0)
   (if (<= b 4.4e+153)
     (fma
      (* (pow (* (PI) b) 2.0) 3.08641975308642e-5)
      (* angle_m angle_m)
      (* a a))
     (* (* (pow (* angle_m b) 2.0) (* (PI) (PI))) 3.08641975308642e-5))))

Derivation

1. Split input into 3 regimes

2. if b < 1.1500000000000001e-50

   1. Initial program 80.4%

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

   3. Taylor expanded in a around inf

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

      1. pow-prod-down N/A

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

      2. lower-pow.f64 N/A

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

      3. *-commutative N/A

         \[\leadsto {\left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot a\right)}^{2} \]

      4. lower-*.f64 N/A

         \[\leadsto {\left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot a\right)}^{2} \]

      5. lower-cos.f64 N/A

         \[\leadsto {\left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot a\right)}^{2} \]

      6. *-commutative N/A

         \[\leadsto {\left(\cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right) \cdot a\right)}^{2} \]

      7. lower-*.f64 N/A

         \[\leadsto {\left(\cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right) \cdot a\right)}^{2} \]

      8. *-commutative N/A

         \[\leadsto {\left(\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right) \cdot a\right)}^{2} \]

      9. lower-*.f64 N/A

         \[\leadsto {\left(\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right) \cdot a\right)}^{2} \]

      10. lift-PI.f64 62.4

          \[\leadsto {\left(\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot a\right)}^{2} \]

   5. Applied rewrites 62.4%

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

   if 1.1500000000000001e-50 < b < 4.3999999999999999e153

   1. Initial program 67.5%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2} + {\color{blue}{a}}^{2} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \color{blue}{{angle}^{2}}, {a}^{2}\right) \]

   5. Applied rewrites 23.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot b\right)}^{2}, 3.08641975308642 \cdot 10^{-5}, {\left(\mathsf{PI}\left(\right) \cdot a\right)}^{2} \cdot -3.08641975308642 \cdot 10^{-5}\right), angle \cdot angle, a \cdot a\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \color{blue}{angle} \cdot angle, a \cdot a\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}, angle \cdot angle, a \cdot a\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}, angle \cdot angle, a \cdot a\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left({\mathsf{PI}\left(\right)}^{2} \cdot {b}^{2}\right) \cdot \frac{1}{32400}, angle \cdot angle, a \cdot a\right) \]

      4. unpow-prod-down N/A

         \[\leadsto \mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot b\right)}^{2} \cdot \frac{1}{32400}, angle \cdot angle, a \cdot a\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot b\right)}^{2} \cdot \frac{1}{32400}, angle \cdot angle, a \cdot a\right) \]

      6. lift-PI.f64 N/A

         \[\leadsto \mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot b\right)}^{2} \cdot \frac{1}{32400}, angle \cdot angle, a \cdot a\right) \]

      7. lift-pow.f64 62.5

         \[\leadsto \mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot b\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}, angle \cdot angle, a \cdot a\right) \]

   8. Applied rewrites 62.5%

      \[\leadsto \mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot b\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}, \color{blue}{angle} \cdot angle, a \cdot a\right) \]

   if 4.3999999999999999e153 < b

   1. Initial program 90.0%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2} + {\color{blue}{a}}^{2} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \color{blue}{{angle}^{2}}, {a}^{2}\right) \]

   5. Applied rewrites 52.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot b\right)}^{2}, 3.08641975308642 \cdot 10^{-5}, {\left(\mathsf{PI}\left(\right) \cdot a\right)}^{2} \cdot -3.08641975308642 \cdot 10^{-5}\right), angle \cdot angle, a \cdot a\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto \frac{1}{32400} \cdot \color{blue}{\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400} \]

      2. lower-*.f64 N/A

         \[\leadsto \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400} \]

      3. *-commutative N/A

         \[\leadsto \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {b}^{2}\right)\right) \cdot \frac{1}{32400} \]

      4. unpow-prod-down N/A

         \[\leadsto \left({angle}^{2} \cdot {\left(\mathsf{PI}\left(\right) \cdot b\right)}^{2}\right) \cdot \frac{1}{32400} \]

      5. pow-prod-down N/A

         \[\leadsto {\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot b\right)\right)}^{2} \cdot \frac{1}{32400} \]

      6. *-commutative N/A

         \[\leadsto {\left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \frac{1}{32400} \]

      7. lower-pow.f64 N/A

         \[\leadsto {\left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \frac{1}{32400} \]

      8. *-commutative N/A

         \[\leadsto {\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      9. *-commutative N/A

         \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      10. lower-*.f64 N/A

          \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      11. lift-*.f64 N/A

          \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      12. lift-PI.f64 80.3

          \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5} \]

   8. Applied rewrites 80.3%

      \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \color{blue}{3.08641975308642 \cdot 10^{-5}} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      2. lift-*.f64 N/A

         \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      3. lift-PI.f64 N/A

         \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      4. lift-*.f64 N/A

         \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      5. unpow-prod-down N/A

         \[\leadsto \left({\left(\mathsf{PI}\left(\right) \cdot b\right)}^{2} \cdot {angle}^{2}\right) \cdot \frac{1}{32400} \]

      6. unpow-prod-down N/A

         \[\leadsto \left(\left({\mathsf{PI}\left(\right)}^{2} \cdot {b}^{2}\right) \cdot {angle}^{2}\right) \cdot \frac{1}{32400} \]

      7. *-commutative N/A

         \[\leadsto \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) \cdot \frac{1}{32400} \]

      8. *-commutative N/A

         \[\leadsto \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400} \]

      9. associate-*r* N/A

         \[\leadsto \left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400} \]

      11. pow-prod-down N/A

          \[\leadsto \left({\left(angle \cdot b\right)}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400} \]

      12. lower-pow.f64 N/A

          \[\leadsto \left({\left(angle \cdot b\right)}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400} \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(angle \cdot b\right)}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400} \]

      14. unpow2 N/A

          \[\leadsto \left({\left(angle \cdot b\right)}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{32400} \]

      15. lower-*.f64 N/A

          \[\leadsto \left({\left(angle \cdot b\right)}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{32400} \]

      16. lift-PI.f64 N/A

          \[\leadsto \left({\left(angle \cdot b\right)}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{32400} \]

      17. lift-PI.f64 80.4

          \[\leadsto \left({\left(angle \cdot b\right)}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} \]

   10. Applied rewrites 80.4%

       \[\leadsto \left({\left(angle \cdot b\right)}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 64.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.65 \cdot 10^{-50}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot b\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}, angle\_m \cdot angle\_m, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(angle\_m \cdot b\right)}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 1.65e-50)
   (* a a)
   (if (<= b 4.4e+153)
     (fma
      (* (pow (* (PI) b) 2.0) 3.08641975308642e-5)
      (* angle_m angle_m)
      (* a a))
     (* (* (pow (* angle_m b) 2.0) (* (PI) (PI))) 3.08641975308642e-5))))

Derivation

1. Split input into 3 regimes

2. if b < 1.6499999999999999e-50

   1. Initial program 80.4%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{a}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto a \cdot \color{blue}{a} \]

      2. lower-*.f64 62.1

         \[\leadsto a \cdot \color{blue}{a} \]

   5. Applied rewrites 62.1%

      \[\leadsto \color{blue}{a \cdot a} \]

   if 1.6499999999999999e-50 < b < 4.3999999999999999e153

   1. Initial program 67.5%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2} + {\color{blue}{a}}^{2} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \color{blue}{{angle}^{2}}, {a}^{2}\right) \]

   5. Applied rewrites 23.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot b\right)}^{2}, 3.08641975308642 \cdot 10^{-5}, {\left(\mathsf{PI}\left(\right) \cdot a\right)}^{2} \cdot -3.08641975308642 \cdot 10^{-5}\right), angle \cdot angle, a \cdot a\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \color{blue}{angle} \cdot angle, a \cdot a\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}, angle \cdot angle, a \cdot a\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}, angle \cdot angle, a \cdot a\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left({\mathsf{PI}\left(\right)}^{2} \cdot {b}^{2}\right) \cdot \frac{1}{32400}, angle \cdot angle, a \cdot a\right) \]

      4. unpow-prod-down N/A

         \[\leadsto \mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot b\right)}^{2} \cdot \frac{1}{32400}, angle \cdot angle, a \cdot a\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot b\right)}^{2} \cdot \frac{1}{32400}, angle \cdot angle, a \cdot a\right) \]

      6. lift-PI.f64 N/A

         \[\leadsto \mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot b\right)}^{2} \cdot \frac{1}{32400}, angle \cdot angle, a \cdot a\right) \]

      7. lift-pow.f64 62.5

         \[\leadsto \mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot b\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}, angle \cdot angle, a \cdot a\right) \]

   8. Applied rewrites 62.5%

      \[\leadsto \mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot b\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}, \color{blue}{angle} \cdot angle, a \cdot a\right) \]

   if 4.3999999999999999e153 < b

   1. Initial program 90.0%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2} + {\color{blue}{a}}^{2} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \color{blue}{{angle}^{2}}, {a}^{2}\right) \]

   5. Applied rewrites 52.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot b\right)}^{2}, 3.08641975308642 \cdot 10^{-5}, {\left(\mathsf{PI}\left(\right) \cdot a\right)}^{2} \cdot -3.08641975308642 \cdot 10^{-5}\right), angle \cdot angle, a \cdot a\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto \frac{1}{32400} \cdot \color{blue}{\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400} \]

      2. lower-*.f64 N/A

         \[\leadsto \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400} \]

      3. *-commutative N/A

         \[\leadsto \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {b}^{2}\right)\right) \cdot \frac{1}{32400} \]

      4. unpow-prod-down N/A

         \[\leadsto \left({angle}^{2} \cdot {\left(\mathsf{PI}\left(\right) \cdot b\right)}^{2}\right) \cdot \frac{1}{32400} \]

      5. pow-prod-down N/A

         \[\leadsto {\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot b\right)\right)}^{2} \cdot \frac{1}{32400} \]

      6. *-commutative N/A

         \[\leadsto {\left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \frac{1}{32400} \]

      7. lower-pow.f64 N/A

         \[\leadsto {\left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \frac{1}{32400} \]

      8. *-commutative N/A

         \[\leadsto {\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      9. *-commutative N/A

         \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      10. lower-*.f64 N/A

          \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      11. lift-*.f64 N/A

          \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      12. lift-PI.f64 80.3

          \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5} \]

   8. Applied rewrites 80.3%

      \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \color{blue}{3.08641975308642 \cdot 10^{-5}} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      2. lift-*.f64 N/A

         \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      3. lift-PI.f64 N/A

         \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      4. lift-*.f64 N/A

         \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      5. unpow-prod-down N/A

         \[\leadsto \left({\left(\mathsf{PI}\left(\right) \cdot b\right)}^{2} \cdot {angle}^{2}\right) \cdot \frac{1}{32400} \]

      6. unpow-prod-down N/A

         \[\leadsto \left(\left({\mathsf{PI}\left(\right)}^{2} \cdot {b}^{2}\right) \cdot {angle}^{2}\right) \cdot \frac{1}{32400} \]

      7. *-commutative N/A

         \[\leadsto \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) \cdot \frac{1}{32400} \]

      8. *-commutative N/A

         \[\leadsto \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400} \]

      9. associate-*r* N/A

         \[\leadsto \left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400} \]

      11. pow-prod-down N/A

          \[\leadsto \left({\left(angle \cdot b\right)}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400} \]

      12. lower-pow.f64 N/A

          \[\leadsto \left({\left(angle \cdot b\right)}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400} \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(angle \cdot b\right)}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400} \]

      14. unpow2 N/A

          \[\leadsto \left({\left(angle \cdot b\right)}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{32400} \]

      15. lower-*.f64 N/A

          \[\leadsto \left({\left(angle \cdot b\right)}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{32400} \]

      16. lift-PI.f64 N/A

          \[\leadsto \left({\left(angle \cdot b\right)}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{32400} \]

      17. lift-PI.f64 80.4

          \[\leadsto \left({\left(angle \cdot b\right)}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} \]

   10. Applied rewrites 80.4%

       \[\leadsto \left({\left(angle \cdot b\right)}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 62.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 2.85 \cdot 10^{+170}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left({\left(angle\_m \cdot b\right)}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 2.85e+170)
   (* a a)
   (* (* (pow (* angle_m b) 2.0) (* (PI) (PI))) 3.08641975308642e-5)))

Derivation

1. Split input into 2 regimes

2. if b < 2.84999999999999984e170

   1. Initial program 77.6%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{a}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto a \cdot \color{blue}{a} \]

      2. lower-*.f64 61.6

         \[\leadsto a \cdot \color{blue}{a} \]

   5. Applied rewrites 61.6%

      \[\leadsto \color{blue}{a \cdot a} \]

   if 2.84999999999999984e170 < b

   1. Initial program 95.9%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2} + {\color{blue}{a}}^{2} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \color{blue}{{angle}^{2}}, {a}^{2}\right) \]

   5. Applied rewrites 57.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot b\right)}^{2}, 3.08641975308642 \cdot 10^{-5}, {\left(\mathsf{PI}\left(\right) \cdot a\right)}^{2} \cdot -3.08641975308642 \cdot 10^{-5}\right), angle \cdot angle, a \cdot a\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto \frac{1}{32400} \cdot \color{blue}{\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400} \]

      2. lower-*.f64 N/A

         \[\leadsto \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400} \]

      3. *-commutative N/A

         \[\leadsto \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {b}^{2}\right)\right) \cdot \frac{1}{32400} \]

      4. unpow-prod-down N/A

         \[\leadsto \left({angle}^{2} \cdot {\left(\mathsf{PI}\left(\right) \cdot b\right)}^{2}\right) \cdot \frac{1}{32400} \]

      5. pow-prod-down N/A

         \[\leadsto {\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot b\right)\right)}^{2} \cdot \frac{1}{32400} \]

      6. *-commutative N/A

         \[\leadsto {\left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \frac{1}{32400} \]

      7. lower-pow.f64 N/A

         \[\leadsto {\left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \frac{1}{32400} \]

      8. *-commutative N/A

         \[\leadsto {\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      9. *-commutative N/A

         \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      10. lower-*.f64 N/A

          \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      11. lift-*.f64 N/A

          \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      12. lift-PI.f64 85.0

          \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5} \]

   8. Applied rewrites 85.0%

      \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \color{blue}{3.08641975308642 \cdot 10^{-5}} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      2. lift-*.f64 N/A

         \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      3. lift-PI.f64 N/A

         \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      4. lift-*.f64 N/A

         \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      5. unpow-prod-down N/A

         \[\leadsto \left({\left(\mathsf{PI}\left(\right) \cdot b\right)}^{2} \cdot {angle}^{2}\right) \cdot \frac{1}{32400} \]

      6. unpow-prod-down N/A

         \[\leadsto \left(\left({\mathsf{PI}\left(\right)}^{2} \cdot {b}^{2}\right) \cdot {angle}^{2}\right) \cdot \frac{1}{32400} \]

      7. *-commutative N/A

         \[\leadsto \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) \cdot \frac{1}{32400} \]

      8. *-commutative N/A

         \[\leadsto \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400} \]

      9. associate-*r* N/A

         \[\leadsto \left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400} \]

      11. pow-prod-down N/A

          \[\leadsto \left({\left(angle \cdot b\right)}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400} \]

      12. lower-pow.f64 N/A

          \[\leadsto \left({\left(angle \cdot b\right)}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400} \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(angle \cdot b\right)}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400} \]

      14. unpow2 N/A

          \[\leadsto \left({\left(angle \cdot b\right)}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{32400} \]

      15. lower-*.f64 N/A

          \[\leadsto \left({\left(angle \cdot b\right)}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{32400} \]

      16. lift-PI.f64 N/A

          \[\leadsto \left({\left(angle \cdot b\right)}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{32400} \]

      17. lift-PI.f64 85.0

          \[\leadsto \left({\left(angle \cdot b\right)}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} \]

   10. Applied rewrites 85.0%

       \[\leadsto \left({\left(angle \cdot b\right)}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 62.5% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\_m\\ \mathbf{if}\;b \leq 2.85 \cdot 10^{+170}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(t\_0 \cdot 3.08641975308642 \cdot 10^{-5}\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (* b (PI)) angle_m)))
   (if (<= b 2.85e+170) (* a a) (* t_0 (* t_0 3.08641975308642e-5)))))

Derivation

1. Split input into 2 regimes

2. if b < 2.84999999999999984e170

   1. Initial program 77.6%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{a}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto a \cdot \color{blue}{a} \]

      2. lower-*.f64 61.6

         \[\leadsto a \cdot \color{blue}{a} \]

   5. Applied rewrites 61.6%

      \[\leadsto \color{blue}{a \cdot a} \]

   if 2.84999999999999984e170 < b

   1. Initial program 95.9%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2} + {\color{blue}{a}}^{2} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \color{blue}{{angle}^{2}}, {a}^{2}\right) \]

   5. Applied rewrites 57.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot b\right)}^{2}, 3.08641975308642 \cdot 10^{-5}, {\left(\mathsf{PI}\left(\right) \cdot a\right)}^{2} \cdot -3.08641975308642 \cdot 10^{-5}\right), angle \cdot angle, a \cdot a\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto \frac{1}{32400} \cdot \color{blue}{\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400} \]

      2. lower-*.f64 N/A

         \[\leadsto \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400} \]

      3. *-commutative N/A

         \[\leadsto \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {b}^{2}\right)\right) \cdot \frac{1}{32400} \]

      4. unpow-prod-down N/A

         \[\leadsto \left({angle}^{2} \cdot {\left(\mathsf{PI}\left(\right) \cdot b\right)}^{2}\right) \cdot \frac{1}{32400} \]

      5. pow-prod-down N/A

         \[\leadsto {\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot b\right)\right)}^{2} \cdot \frac{1}{32400} \]

      6. *-commutative N/A

         \[\leadsto {\left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \frac{1}{32400} \]

      7. lower-pow.f64 N/A

         \[\leadsto {\left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \frac{1}{32400} \]

      8. *-commutative N/A

         \[\leadsto {\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      9. *-commutative N/A

         \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      10. lower-*.f64 N/A

          \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      11. lift-*.f64 N/A

          \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      12. lift-PI.f64 85.0

          \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5} \]

   8. Applied rewrites 85.0%

      \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \color{blue}{3.08641975308642 \cdot 10^{-5}} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      2. lift-*.f64 N/A

         \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      3. lift-PI.f64 N/A

         \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      4. lift-*.f64 N/A

         \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      5. unpow2 N/A

         \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)\right) \cdot \frac{1}{32400} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)\right) \cdot \frac{1}{32400} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)\right) \cdot \frac{1}{32400} \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)\right) \cdot \frac{1}{32400} \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)\right) \cdot \frac{1}{32400} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)\right) \cdot \frac{1}{32400} \]

      11. lift-PI.f64 N/A

          \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)\right) \cdot \frac{1}{32400} \]

      12. lift-*.f64 85.0

          \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} \]

   10. Applied rewrites 85.0%

       \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)\right) \cdot \frac{1}{32400} \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)\right) \cdot \frac{1}{32400} \]

       3. associate-*l* N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right) \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right) \cdot \color{blue}{\frac{1}{32400}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right) \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right) \cdot \color{blue}{\frac{1}{32400}}\right) \]

       5. lift-PI.f64 N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right) \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right) \cdot \frac{1}{32400}\right) \]

       6. lift-*.f64 N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right) \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right) \cdot \frac{1}{32400}\right) \]

       7. *-commutative N/A

          \[\leadsto \left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right) \cdot \frac{1}{32400}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right) \cdot \frac{1}{32400}\right) \]

       9. lift-PI.f64 N/A

          \[\leadsto \left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right) \cdot \frac{1}{32400}\right) \]

       10. lower-*.f64 85.0

           \[\leadsto \left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \]

       11. lift-PI.f64 N/A

           \[\leadsto \left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right) \cdot \frac{1}{32400}\right) \]

       12. lift-*.f64 N/A

           \[\leadsto \left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right) \cdot \frac{1}{32400}\right) \]

       13. *-commutative N/A

           \[\leadsto \left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \left(\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \frac{1}{32400}\right) \]

       14. lower-*.f64 N/A

           \[\leadsto \left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \left(\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \frac{1}{32400}\right) \]

       15. lift-PI.f64 85.0

           \[\leadsto \left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \left(\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \]

   12. Applied rewrites 85.0%

       \[\leadsto \left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \left(\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \color{blue}{3.08641975308642 \cdot 10^{-5}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 57.0% accurate, 74.7× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ a \cdot a \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m) :precision binary64 (* a a))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return a * a;
}

Fortran

angle_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, angle_m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle_m
    code = a * a
end function

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return a * a;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return a * a

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64(a * a)
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = a * a;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(a * a), $MachinePrecision]

Derivation

1. Initial program 79.5%

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

3. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{{a}^{2}} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto a \cdot \color{blue}{a} \]

   2. lower-*.f64 58.0

      \[\leadsto a \cdot \color{blue}{a} \]

5. Applied rewrites 58.0%

   \[\leadsto \color{blue}{a \cdot a} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025073 
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* (PI) (/ angle 180.0)))) 2.0) (pow (* b (sin (* (PI) (/ angle 180.0)))) 2.0)))