ab-angle->ABCF A

Percentage Accurate: 79.6% → 78.3%

Time: 4.3s

Alternatives: 7

Speedup: N/A×

• 36.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 79.6% 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: 78.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot angle\_m\\ \mathbf{if}\;angle\_m \leq 6.5 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.005555555555555556 \cdot a\right) \cdot t\_0, 0.005555555555555556 \cdot \left(t\_0 \cdot a\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{angle\_m}{180}\right)\right)\right) \cdot a, a, b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (PI) angle_m)))
   (if (<= angle_m 6.5e-25)
     (fma
      (* (* 0.005555555555555556 a) t_0)
      (* 0.005555555555555556 (* t_0 a))
      (* b b))
     (fma
      (* (- 0.5 (* 0.5 (cos (* 2.0 (* (PI) (/ angle_m 180.0)))))) a)
      a
      (* b b)))))

Derivation

1. Split input into 2 regimes

2. if angle < 6.5e-25

   1. Initial program 85.6%

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

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 80.1

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

   8. Applied rewrites 80.1%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 80.1

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

   10. Applied rewrites 80.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lift-PI.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. lift-*.f64 N/A

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

       10. lift-PI.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 80.2

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

   12. Applied rewrites 80.2%

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

   if 6.5e-25 < angle

   1. Initial program 58.6%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
   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 62.3

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

   5. Applied rewrites 62.3%

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

      1. lift-+.f64 N/A

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

   7. Applied rewrites 62.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-sin-a N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-PI.f64 62.3

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

   9. Applied rewrites 62.3%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 6.5 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.005555555555555556 \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right), 0.005555555555555556 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right) \cdot a, a, b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.3% accurate, 1.9× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+ (pow (* a (sin (/ (* (PI) angle_m) 180.0))) 2.0) (* b b)))

Derivation

1. Initial program 77.9%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
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 79.2

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

5. Applied rewrites 79.2%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + \color{blue}{b \cdot b} \]
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. lower-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-PI.f64 79.3

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

7. Applied rewrites 79.3%

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

8. Final simplification 79.3%

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

Alternative 3: 79.4% accurate, 1.9× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+ (pow (* a (sin (* (/ angle_m 180.0) (PI)))) 2.0) (* b b)))

Derivation

1. Initial program 77.9%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
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 79.2

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

5. Applied rewrites 79.2%

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

6. Final simplification 79.2%

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

Alternative 4: 66.6% accurate, 9.3× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot angle\_m\\ \mathbf{if}\;a \leq 1.6 \cdot 10^{-48}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.005555555555555556 \cdot a\right) \cdot t\_0, 0.005555555555555556 \cdot \left(t\_0 \cdot a\right), b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (PI) angle_m)))
   (if (<= a 1.6e-48)
     (* b b)
     (fma
      (* (* 0.005555555555555556 a) t_0)
      (* 0.005555555555555556 (* t_0 a))
      (* b b)))))

Derivation

1. Split input into 2 regimes

2. if a < 1.5999999999999999e-48

   1. Initial program 76.0%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
   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 58.9

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

   5. Applied rewrites 58.9%

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

   if 1.5999999999999999e-48 < a

   1. Initial program 82.4%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
   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 83.6

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

   5. Applied rewrites 83.6%

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

   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. *-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 81.4

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

   8. Applied rewrites 81.4%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 81.4

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

   10. Applied rewrites 81.4%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lift-PI.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. lift-*.f64 N/A

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

       10. lift-PI.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 81.4

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

   12. Applied rewrites 81.4%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.6 \cdot 10^{-48}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.005555555555555556 \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right), 0.005555555555555556 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right), b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.6% accurate, 9.3× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 1.6 \cdot 10^{-48}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.005555555555555556 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\_m\right) \cdot a\right), 0.005555555555555556 \cdot \left(\left(a \cdot angle\_m\right) \cdot \mathsf{PI}\left(\right)\right), b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 1.6e-48)
   (* b b)
   (fma
    (* 0.005555555555555556 (* (* (PI) angle_m) a))
    (* 0.005555555555555556 (* (* a angle_m) (PI)))
    (* b b))))

Derivation

1. Split input into 2 regimes

2. if a < 1.5999999999999999e-48

   1. Initial program 76.0%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
   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 58.9

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

   5. Applied rewrites 58.9%

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

   if 1.5999999999999999e-48 < a

   1. Initial program 82.4%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
   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 83.6

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

   5. Applied rewrites 83.6%

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

   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. *-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 81.4

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

   8. Applied rewrites 81.4%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 81.4

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

   10. Applied rewrites 81.4%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-PI.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lift-PI.f64 81.4

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

   12. Applied rewrites 81.4%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.6 \cdot 10^{-48}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.005555555555555556 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right), 0.005555555555555556 \cdot \left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right), b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.6% accurate, 9.3× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \left(\left(\mathsf{PI}\left(\right) \cdot angle\_m\right) \cdot 0.005555555555555556\right) \cdot a\\ \mathbf{if}\;a \leq 1.6 \cdot 10^{-48}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_0, b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (* (* (PI) angle_m) 0.005555555555555556) a)))
   (if (<= a 1.6e-48) (* b b) (fma t_0 t_0 (* b b)))))

Derivation

1. Split input into 2 regimes

2. if a < 1.5999999999999999e-48

   1. Initial program 76.0%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
   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 58.9

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

   5. Applied rewrites 58.9%

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

   if 1.5999999999999999e-48 < a

   1. Initial program 82.4%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
   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 83.6

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

   5. Applied rewrites 83.6%

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

   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. *-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 81.4

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

   8. Applied rewrites 81.4%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 81.4

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

   10. Applied rewrites 81.4%

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

   12. Applied rewrites 81.4%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.6 \cdot 10^{-48}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot a, \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot a, b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.9% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

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 = b * b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
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 57.5

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

5. Applied rewrites 57.5%

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

Reproduce

herbie shell --seed 2025038 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) (PI)))) 2.0) (pow (* b (cos (* (/ angle 180.0) (PI)))) 2.0)))