ab-angle->ABCF C

Percentage Accurate: 79.5% → 77.8%

Time: 6.8s

Alternatives: 10

Speedup: 2.0×

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

Specification

Math

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

FPCore

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

Initial Program: 79.5% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 77.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;angle\_m \leq 7.6 \cdot 10^{-27}:\\ \;\;\;\;a \cdot a + {\left(b \cdot \left(\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\_m\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\frac{angle\_m}{180} \cdot \mathsf{PI}\left(\right)\right)\right), b \cdot b, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= angle_m 7.6e-27)
   (+ (* a a) (pow (* b (* (* 0.005555555555555556 (PI)) angle_m)) 2.0))
   (fma
    (- 0.5 (* 0.5 (cos (* 2.0 (* (/ angle_m 180.0) (PI))))))
    (* b b)
    (* a a))))

Derivation

1. Split input into 2 regimes

2. if angle < 7.60000000000000001e-27

   1. Initial program 85.9%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 86.6%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 84.6%

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

   if 7.60000000000000001e-27 < angle

   1. Initial program 56.0%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 56.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow-prod-down N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lower-*.f64 56.6

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

   7. Applied rewrites 56.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. sqr-sin-a N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. *-commutative N/A

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

      21. lift-*.f64 56.6

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

   9. Applied rewrites 56.6%

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

Alternative 2: 79.5% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 77.5%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 78.1%

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

Alternative 3: 79.6% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 77.5%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 78.1%

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

6. Taylor expanded in angle around 0

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

8. Applied rewrites 78.1%

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

Alternative 4: 67.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{-56}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + {\left(b \cdot \left(\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\_m\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 5e-56)
   (* a a)
   (+ (* a a) (pow (* b (* (* 0.005555555555555556 (PI)) angle_m)) 2.0))))

Derivation

1. Split input into 2 regimes

2. if b < 4.99999999999999997e-56

   1. Initial program 74.2%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 59.7%

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

   if 4.99999999999999997e-56 < b

   1. Initial program 85.0%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 81.9%

      \[\leadsto a \cdot a + {\left(b \cdot \color{blue}{\left(\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}\right)}^{2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 71.7% accurate, 10.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{+128}:\\ \;\;\;\;a \cdot a + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(\left(b \cdot b\right) \cdot angle\_m\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot angle\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + \left(\left(\left(\left(angle\_m \cdot angle\_m\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot b\right)\right) \cdot b\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 5e+128)
   (+
    (* a a)
    (* (* (PI) (PI)) (* (* (* b b) angle_m) (* 3.08641975308642e-5 angle_m))))
   (+
    (* a a)
    (*
     (* (* (* (* angle_m angle_m) 3.08641975308642e-5) (PI)) (* (PI) b))
     b))))

Derivation

1. Split input into 2 regimes

2. if b < 5e128

   1. Initial program 73.8%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 62.9%

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

   10. Applied rewrites 63.4%

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

   12. Applied rewrites 69.3%

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

   if 5e128 < b

   1. Initial program 96.3%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 96.3%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 63.3%

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

   10. Applied rewrites 91.1%

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

Alternative 6: 71.7% accurate, 10.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;b \leq 5 \cdot 10^{+128}:\\ \;\;\;\;a \cdot a + t\_0 \cdot \left(\left(\left(b \cdot b\right) \cdot angle\_m\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot angle\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + t\_0 \cdot \left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle\_m \cdot angle\_m\right)\right) \cdot b\right) \cdot b\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (PI) (PI))))
   (if (<= b 5e+128)
     (+
      (* a a)
      (* t_0 (* (* (* b b) angle_m) (* 3.08641975308642e-5 angle_m))))
     (+
      (* a a)
      (* t_0 (* (* (* 3.08641975308642e-5 (* angle_m angle_m)) b) b))))))

Derivation

1. Split input into 2 regimes

2. if b < 5e128

   1. Initial program 73.8%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 62.9%

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

   10. Applied rewrites 63.4%

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

   12. Applied rewrites 69.3%

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

   if 5e128 < b

   1. Initial program 96.3%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 96.3%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 63.3%

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

   10. Applied rewrites 63.3%

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

   12. Applied rewrites 91.0%

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

Alternative 7: 71.7% accurate, 10.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;b \leq 5 \cdot 10^{+128}:\\ \;\;\;\;a \cdot a + t\_0 \cdot \left(angle\_m \cdot \left(\left(3.08641975308642 \cdot 10^{-5} \cdot angle\_m\right) \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + t\_0 \cdot \left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle\_m \cdot angle\_m\right)\right) \cdot b\right) \cdot b\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (PI) (PI))))
   (if (<= b 5e+128)
     (+
      (* a a)
      (* t_0 (* angle_m (* (* 3.08641975308642e-5 angle_m) (* b b)))))
     (+
      (* a a)
      (* t_0 (* (* (* 3.08641975308642e-5 (* angle_m angle_m)) b) b))))))

Derivation

1. Split input into 2 regimes

2. if b < 5e128

   1. Initial program 73.8%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 62.9%

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

   10. Applied rewrites 63.4%

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

   12. Applied rewrites 69.3%

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

   if 5e128 < b

   1. Initial program 96.3%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 96.3%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 63.3%

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

   10. Applied rewrites 63.3%

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

   12. Applied rewrites 91.0%

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

Alternative 8: 67.4% accurate, 10.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;angle\_m \leq 3.7 \cdot 10^{-157}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(angle\_m \cdot angle\_m\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), b \cdot b, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= angle_m 3.7e-157)
   (* a a)
   (fma
    (* (* (* angle_m angle_m) 3.08641975308642e-5) (* (PI) (PI)))
    (* b b)
    (* a a))))

Derivation

1. Split input into 2 regimes

2. if angle < 3.6999999999999998e-157

   1. Initial program 84.4%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 61.8%

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

   if 3.6999999999999998e-157 < angle

   1. Initial program 64.7%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 65.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow-prod-down N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lower-*.f64 63.2

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

   7. Applied rewrites 63.2%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 54.8%

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

Alternative 9: 70.1% accurate, 11.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ a \cdot a + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(angle\_m \cdot \left(\left(3.08641975308642 \cdot 10^{-5} \cdot angle\_m\right) \cdot \left(b \cdot b\right)\right)\right) \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (* a a)
  (* (* (PI) (PI)) (* angle_m (* (* 3.08641975308642e-5 angle_m) (* b b))))))

Derivation

1. Initial program 77.5%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 78.1%

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

6. Taylor expanded in angle around 0

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

8. Applied rewrites 63.0%

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

10. Applied rewrites 63.4%

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

12. Applied rewrites 69.6%

    \[\leadsto a \cdot a + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(angle \cdot \color{blue}{\left(\left(3.08641975308642 \cdot 10^{-5} \cdot angle\right) \cdot \left(b \cdot b\right)\right)}\right) \]
13. Add Preprocessing

Alternative 10: 57.1% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.5%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 54.4%

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

Reproduce

herbie shell --seed 2025019 
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* (PI) (/ angle 180.0)))) 2.0) (pow (* b (sin (* (PI) (/ angle 180.0)))) 2.0)))