ab-angle->ABCF A

Percentage Accurate: 80.1% → 80.1%

Time: 5.3s

Alternatives: 7

Speedup: 2.0×

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

Specification

Math

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

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: 80.1% 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: 80.1% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 78.3%

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

   1. lift-cos.f64 N/A

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

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

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

   3. lower-sin.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-PI.f64 78.7

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

4. Applied rewrites 78.7%

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

5. Taylor expanded in angle around 0

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

   1. lower-*.f64 78.6

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

7. Applied rewrites 78.6%

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

8. Taylor expanded in angle around inf

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

   1. lower-sin.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 78.7

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

10. Applied rewrites 78.7%

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

Alternative 2: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \mathsf{fma}\left({\cos \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right)}^{2}, b \cdot b, {\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle\_m}{180}\right) \cdot a\right)}^{2}\right) \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (fma
  (pow (cos (* (PI) (* 0.005555555555555556 angle_m))) 2.0)
  (* b b)
  (pow (* (sin (* (PI) (/ angle_m 180.0))) a) 2.0)))

Derivation

1. Initial program 78.3%

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

   1. lift-cos.f64 N/A

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

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

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

   3. lower-sin.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-PI.f64 78.7

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

4. Applied rewrites 78.7%

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

5. Taylor expanded in angle around 0

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

   1. lower-*.f64 78.6

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

7. Applied rewrites 78.6%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. +-commutative N/A

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

9. Applied rewrites 78.6%

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

Alternative 3: 80.2% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 78.3%

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

   1. lift-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. pow-to-exp N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-*.f64 N/A

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

4. Applied rewrites 34.8%

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

5. Taylor expanded in angle around inf

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

   1. lower-sin.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 34.8

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

7. Applied rewrites 34.8%

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

8. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 78.6

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

10. Applied rewrites 78.6%

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

Alternative 4: 56.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.2 \cdot 10^{-156}:\\ \;\;\;\;{\left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\_m\right) \cdot 0.005555555555555556\right) \cdot a\right)}^{2}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{+72}:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot a\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}, angle\_m \cdot angle\_m, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle\_m \cdot angle\_m\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b, b\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 1.2e-156)
   (pow (* (sin (* (* (PI) angle_m) 0.005555555555555556)) a) 2.0)
   (if (<= b 1.8e+72)
     (fma
      (* (pow (* (PI) a) 2.0) 3.08641975308642e-5)
      (* angle_m angle_m)
      (* b b))
     (pow
      (fma (* -1.54320987654321e-5 (* angle_m angle_m)) (* (* (PI) (PI)) b) b)
      2.0))))

Derivation

1. Split input into 3 regimes

2. if b < 1.2e-156

   1. Initial program 75.5%

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

   3. Taylor expanded in a around inf

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

      1. pow-prod-down N/A

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

      2. lower-pow.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 38.2

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

   5. Applied rewrites 38.2%

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

   if 1.2e-156 < b < 1.80000000000000017e72

   1. Initial program 70.9%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 63.3%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow-prod-down N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-pow.f64 64.3

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

   8. Applied rewrites 64.3%

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

   if 1.80000000000000017e72 < b

   1. Initial program 91.6%

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

   3. Taylor expanded in a around 0

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

      1. pow-prod-down N/A

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

      2. lower-pow.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 84.9

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in angle around 0

      \[\leadsto {\left(b + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}^{2} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {\left(\frac{-1}{64800} \cdot \left({angle}^{2} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + b\right)}^{2} \]

      2. associate-*r* N/A

         \[\leadsto {\left(\left(\frac{-1}{64800} \cdot {angle}^{2}\right) \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right) + b\right)}^{2} \]

      3. lower-fma.f64 N/A

         \[\leadsto {\left(\mathsf{fma}\left(\frac{-1}{64800} \cdot {angle}^{2}, b \cdot {\mathsf{PI}\left(\right)}^{2}, b\right)\right)}^{2} \]

      4. lower-*.f64 N/A

         \[\leadsto {\left(\mathsf{fma}\left(\frac{-1}{64800} \cdot {angle}^{2}, b \cdot {\mathsf{PI}\left(\right)}^{2}, b\right)\right)}^{2} \]

      5. unpow2 N/A

         \[\leadsto {\left(\mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), b \cdot {\mathsf{PI}\left(\right)}^{2}, b\right)\right)}^{2} \]

      6. lower-*.f64 N/A

         \[\leadsto {\left(\mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), b \cdot {\mathsf{PI}\left(\right)}^{2}, b\right)\right)}^{2} \]

      7. *-commutative N/A

         \[\leadsto {\left(\mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), {\mathsf{PI}\left(\right)}^{2} \cdot b, b\right)\right)}^{2} \]

      8. lower-*.f64 N/A

         \[\leadsto {\left(\mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), {\mathsf{PI}\left(\right)}^{2} \cdot b, b\right)\right)}^{2} \]

      9. unpow2 N/A

         \[\leadsto {\left(\mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b, b\right)\right)}^{2} \]

      10. lower-*.f64 N/A

          \[\leadsto {\left(\mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b, b\right)\right)}^{2} \]

      11. lift-PI.f64 N/A

          \[\leadsto {\left(\mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b, b\right)\right)}^{2} \]

      12. lift-PI.f64 86.6

          \[\leadsto {\left(\mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b, b\right)\right)}^{2} \]

   8. Applied rewrites 86.6%

      \[\leadsto {\left(\mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b, b\right)\right)}^{2} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 65.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 4.4 \cdot 10^{-56}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;a \leq 4 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot a\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}, angle\_m \cdot angle\_m, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\mathsf{PI}\left(\right) \cdot angle\_m\right) \cdot a\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 4.4e-56)
   (* b b)
   (if (<= a 4e+153)
     (fma
      (* (pow (* (PI) a) 2.0) 3.08641975308642e-5)
      (* angle_m angle_m)
      (* b b))
     (* (pow (* (* (PI) angle_m) a) 2.0) 3.08641975308642e-5))))

Derivation

1. Split input into 3 regimes

2. if a < 4.40000000000000008e-56

   1. Initial program 75.1%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
   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 63.5

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

   5. Applied rewrites 63.5%

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

   if 4.40000000000000008e-56 < a < 4e153

   1. Initial program 74.2%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 31.4%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow-prod-down N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-pow.f64 66.1

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

   8. Applied rewrites 66.1%

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

   if 4e153 < a

   1. Initial program 99.7%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 43.7%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. pow-prod-down N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-pow.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 77.9

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

   8. Applied rewrites 77.9%

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

Alternative 6: 63.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 7.5 \cdot 10^{+141}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\mathsf{PI}\left(\right) \cdot angle\_m\right) \cdot a\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 7.5e+141)
   (* b b)
   (* (pow (* (* (PI) angle_m) a) 2.0) 3.08641975308642e-5)))

Derivation

1. Split input into 2 regimes

2. if a < 7.49999999999999937e141

   1. Initial program 74.8%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
   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 63.4

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

   5. Applied rewrites 63.4%

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

   if 7.49999999999999937e141 < a

   1. Initial program 97.6%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 39.7%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. pow-prod-down N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-pow.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 77.9

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

   8. Applied rewrites 77.9%

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

Alternative 7: 57.9% 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 78.3%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
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 59.1

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

5. Applied rewrites 59.1%

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

Reproduce

herbie shell --seed 2025056 
(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)))