ab-angle->ABCF A

Percentage Accurate: 79.9% → 79.8%

Time: 4.3s

Alternatives: 7

Speedup: N/A×

• 36.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[\mathsf{TRUE}\left(\right)\]

Math

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

FPCore

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

Initial Program: 79.9% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 79.8% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 81.0%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 81.2

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

5. Applied rewrites 81.2%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. associate-*r/ N/A

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

   6. lift-*.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-/.f64 81.3

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

7. Applied rewrites 81.3%

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

8. Final simplification 81.3%

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

Alternative 2: 79.8% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+ (pow (* a (sin (* (* 0.005555555555555556 angle) (PI)))) 2.0) (* b b)))

Derivation

1. Initial program 81.0%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 81.2

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

5. Applied rewrites 81.2%

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

6. Taylor expanded in angle around 0

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

   1. lower-*.f64 81.3

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

8. Applied rewrites 81.3%

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

9. Final simplification 81.3%

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

Alternative 3: 77.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot angle\\ \mathbf{if}\;angle \leq 0.0045:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right) \cdot a, \left(0.005555555555555556 \cdot t\_0\right) \cdot a, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \frac{t\_0}{180}\right), a \cdot a, b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (PI) angle)))
   (if (<= angle 0.0045)
     (fma
      (* (* (* (PI) 0.005555555555555556) angle) a)
      (* (* 0.005555555555555556 t_0) a)
      (* b b))
     (fma (- 0.5 (* 0.5 (cos (* 2.0 (/ t_0 180.0))))) (* a a) (* b b)))))

Derivation

1. Split input into 2 regimes

2. if angle < 0.00449999999999999966

   1. Initial program 85.7%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 86.0

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

   5. Applied rewrites 86.0%

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

   6. Taylor expanded in angle around 0

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

      1. associate-*l/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 83.6

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

   8. Applied rewrites 83.6%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 83.6

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

   10. Applied rewrites 83.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot a, \left(0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot a, b \cdot b\right)} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-PI.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*r* N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. lift-PI.f64 83.6

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right) \cdot a, \left(0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot a, b \cdot b\right) \]

   12. Applied rewrites 83.6%

       \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right) \cdot a, \left(0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot a, b \cdot b\right) \]

   if 0.00449999999999999966 < angle

   1. Initial program 67.4%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 67.3

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

   5. Applied rewrites 67.3%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow-prod-down N/A

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

      10. lower-fma.f64 N/A

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

   7. Applied rewrites 67.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-sin-a N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lift-*.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. lift-/.f64 67.1

          \[\leadsto \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \color{blue}{\frac{\mathsf{PI}\left(\right) \cdot angle}{180}}\right), a \cdot a, b \cdot b\right) \]

   9. Applied rewrites 67.1%

      \[\leadsto \mathsf{fma}\left(\color{blue}{0.5 - 0.5 \cdot \cos \left(2 \cdot \frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)}, a \cdot a, b \cdot b\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 0.0045:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right) \cdot a, \left(0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot a, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right), a \cdot a, b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 0.0045:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right) \cdot a, \left(0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot a, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 0.0045)
   (fma
    (* (* (* (PI) 0.005555555555555556) angle) a)
    (* (* 0.005555555555555556 (* (PI) angle)) a)
    (* b b))
   (fma
    (- 0.5 (* 0.5 (cos (* 2.0 (* (PI) (* 0.005555555555555556 angle))))))
    (* a a)
    (* b b))))

Derivation

1. Split input into 2 regimes

2. if angle < 0.00449999999999999966

   1. Initial program 85.7%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 86.0

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

   5. Applied rewrites 86.0%

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

   6. Taylor expanded in angle around 0

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

      1. associate-*l/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 83.6

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

   8. Applied rewrites 83.6%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 83.6

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

   10. Applied rewrites 83.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot a, \left(0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot a, b \cdot b\right)} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-PI.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*r* N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. lift-PI.f64 83.6

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right) \cdot a, \left(0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot a, b \cdot b\right) \]

   12. Applied rewrites 83.6%

       \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right) \cdot a, \left(0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot a, b \cdot b\right) \]

   if 0.00449999999999999966 < angle

   1. Initial program 67.4%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 67.3

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

   5. Applied rewrites 67.3%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow-prod-down N/A

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

      10. lower-fma.f64 N/A

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

   7. Applied rewrites 67.3%

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

   8. Taylor expanded in angle around 0

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

      1. lower-*.f64 67.3

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

   10. Applied rewrites 67.3%

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

       1. lift-pow.f64 N/A

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

       2. unpow2 N/A

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

       3. lift-sin.f64 N/A

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

       4. lift-sin.f64 N/A

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

       5. sqr-sin-a N/A

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

       6. lower--.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-cos.f64 N/A

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

       9. lower-*.f64 67.0

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

   12. Applied rewrites 67.0%

       \[\leadsto \mathsf{fma}\left(\color{blue}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}, a \cdot a, b \cdot b\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 0.0045:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right) \cdot a, \left(0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot a, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.8% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right) \cdot a, \left(0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot a, b \cdot b\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (fma
  (* (* (* (PI) 0.005555555555555556) angle) a)
  (* (* 0.005555555555555556 (* (PI) angle)) a)
  (* b b)))

Derivation

1. Initial program 81.0%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 81.2

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

5. Applied rewrites 81.2%

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

6. Taylor expanded in angle around 0

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

   1. associate-*l/ N/A

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

   2. *-commutative N/A

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

   3. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-*.f64 77.9

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

8. Applied rewrites 77.9%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-fma.f64 77.9

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

10. Applied rewrites 77.8%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\left(0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot a, \left(0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot a, b \cdot b\right)} \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. lift-PI.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. associate-*r* N/A

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

    5. lower-*.f64 N/A

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

    6. *-commutative N/A

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

    7. lower-*.f64 N/A

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

    8. lift-PI.f64 77.9

       \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right) \cdot a, \left(0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot a, b \cdot b\right) \]

12. Applied rewrites 77.9%

    \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right) \cdot a, \left(0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot a, b \cdot b\right) \]
13. Add Preprocessing

Alternative 6: 74.8% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right) \cdot 0.005555555555555556\\ \mathsf{fma}\left(t\_0, t\_0, b \cdot b\right) \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* (* (PI) angle) a) 0.005555555555555556)))
   (fma t_0 t_0 (* b b))))

Derivation

1. Initial program 81.0%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 81.2

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

5. Applied rewrites 81.2%

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

6. Taylor expanded in angle around 0

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

   1. associate-*l/ N/A

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

   2. *-commutative N/A

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

   3. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-*.f64 77.9

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

8. Applied rewrites 77.9%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-fma.f64 77.9

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

10. Applied rewrites 77.8%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\left(0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot a, \left(0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot a, b \cdot b\right)} \]
11. Step-by-step derivation

12. Applied rewrites 77.9%

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

13. Final simplification 77.9%

    \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right) \cdot 0.005555555555555556, \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right) \cdot 0.005555555555555556, b \cdot b\right) \]
14. Add Preprocessing

Alternative 7: 57.1% accurate, 74.7× speedup

Math

\[\begin{array}{l} \\ b \cdot b \end{array} \]

FPCore

(FPCore (a b angle) :precision binary64 (* b b))

C

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

Fortran

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

Java

public static double code(double a, double b, double angle) {
	return b * b;
}

Python

def code(a, b, angle):
	return b * b

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_] := N[(b * b), $MachinePrecision]

Derivation

1. Initial program 81.0%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 61.5

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

5. Applied rewrites 61.5%

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

Reproduce

herbie shell --seed 2025050 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  :pre (TRUE)
  (+ (pow (* a (sin (* (/ angle 180.0) (PI)))) 2.0) (pow (* b (cos (* (/ angle 180.0) (PI)))) 2.0)))