ab-angle->ABCF C

Percentage Accurate: 79.6% → 79.6%

Time: 4.9s

Alternatives: 11

Speedup: 2.0×

• 35.8% 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: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 79.6% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 80.1%

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

   1. unpow2 N/A

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

   2. lower-*.f64 80.2

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

5. Applied rewrites 80.2%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*r/ N/A

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

   5. *-commutative N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-PI.f64 80.3

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

7. Applied rewrites 80.3%

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

Alternative 2: 79.6% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 80.1%

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

   1. unpow2 N/A

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

   2. lower-*.f64 80.2

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

5. Applied rewrites 80.2%

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

Alternative 3: 79.6% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 80.1%

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

   1. unpow2 N/A

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

   2. lower-*.f64 80.2

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

5. Applied rewrites 80.2%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*r/ N/A

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

   5. *-commutative N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-PI.f64 80.3

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

7. Applied rewrites 80.3%

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

   1. lift-/.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lift-PI.f64 80.2

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

9. Applied rewrites 80.2%

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

Alternative 4: 79.6% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 80.1%

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

   1. unpow2 N/A

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

   2. lower-*.f64 80.2

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

5. Applied rewrites 80.2%

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

6. Taylor expanded in angle around 0

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

   1. lower-*.f64 80.2

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

8. Applied rewrites 80.2%

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

Alternative 5: 77.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot 0.005555555555555556, \left(\left(angle \cdot b\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.005555555555555556, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 - \cos \left(\left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right) \cdot 0.5, b \cdot b, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 0.005)
   (fma
    (* (* (* b (PI)) angle) 0.005555555555555556)
    (* (* (* angle b) (PI)) 0.005555555555555556)
    (* a a))
   (fma
    (- 0.5 (* (cos (* (* (/ angle 180.0) (PI)) 2.0)) 0.5))
    (* b b)
    (* a a))))

Derivation

1. Split input into 2 regimes

2. if angle < 0.0050000000000000001

   1. Initial program 87.7%

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

      1. unpow2 N/A

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

      2. lower-*.f64 87.9

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

   5. Applied rewrites 87.9%

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

   6. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-PI.f64 85.1

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

   8. Applied rewrites 85.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 85.1

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

   10. Applied rewrites 85.1%

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

       1. lift-*.f64 N/A

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

       2. lift-PI.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lift-PI.f64 85.2

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

   12. Applied rewrites 85.2%

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

   if 0.0050000000000000001 < angle

   1. Initial program 51.8%

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

      1. unpow2 N/A

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

      2. lower-*.f64 51.4

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

   5. Applied rewrites 51.4%

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

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. 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. Applied rewrites 51.4%

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

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. sqr-sin-a-rev N/A

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

      14. lift-PI.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 N/A

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

   9. Applied rewrites 51.4%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot 0.005555555555555556, \left(\left(angle \cdot b\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.005555555555555556, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 - \cos \left(\left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right) \cdot 0.5, b \cdot b, a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{-121}:\\ \;\;\;\;\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(0.011111111111111112, angle \cdot \mathsf{PI}\left(\right), 0.5 \cdot \mathsf{PI}\left(\right)\right)\right), 0.5, 0.5\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot 0.005555555555555556, \left(\left(angle \cdot b\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.005555555555555556, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 3.8e-121)
   (*
    (fma (sin (fma 0.011111111111111112 (* angle (PI)) (* 0.5 (PI)))) 0.5 0.5)
    (* a a))
   (fma
    (* (* (* b (PI)) angle) 0.005555555555555556)
    (* (* (* angle b) (PI)) 0.005555555555555556)
    (* a a))))

Derivation

1. Split input into 2 regimes

2. if b < 3.8000000000000001e-121

   1. Initial program 77.4%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow-prod-down N/A

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

      10. lift-pow.f64 N/A

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

   4. Applied rewrites 77.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. sqr-cos-a N/A

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

      14. lower-+.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. lower-*.f64 77.4

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

   6. Applied rewrites 77.4%

      \[\leadsto \mathsf{fma}\left(\color{blue}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}, a \cdot a, {\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot b\right)}^{2}\right) \]
   7. Step-by-step derivation

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-PI.f64 77.1

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

   8. Applied rewrites 77.1%

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

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 58.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(0.011111111111111112, angle \cdot \mathsf{PI}\left(\right), 0.5 \cdot \mathsf{PI}\left(\right)\right)\right), 0.5, 0.5\right) \cdot \left(a \cdot a\right)} \]

   if 3.8000000000000001e-121 < b

   1. Initial program 84.3%

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

      1. unpow2 N/A

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

      2. lower-*.f64 84.5

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-PI.f64 80.9

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

   8. Applied rewrites 80.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 80.9

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

   10. Applied rewrites 80.9%

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

       1. lift-*.f64 N/A

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

       2. lift-PI.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lift-PI.f64 81.0

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

   12. Applied rewrites 81.0%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{-121}:\\ \;\;\;\;\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(0.011111111111111112, angle \cdot \mathsf{PI}\left(\right), 0.5 \cdot \mathsf{PI}\left(\right)\right)\right), 0.5, 0.5\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot 0.005555555555555556, \left(\left(angle \cdot b\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.005555555555555556, a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{-121}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), 0.5, 0.5\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot 0.005555555555555556, \left(\left(angle \cdot b\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.005555555555555556, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 3.8e-121)
   (* (fma (cos (* 0.011111111111111112 (* angle (PI)))) 0.5 0.5) (* a a))
   (fma
    (* (* (* b (PI)) angle) 0.005555555555555556)
    (* (* (* angle b) (PI)) 0.005555555555555556)
    (* a a))))

Derivation

1. Split input into 2 regimes

2. if b < 3.8000000000000001e-121

   1. Initial program 77.4%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow-prod-down N/A

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

      10. lift-pow.f64 N/A

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

   4. Applied rewrites 77.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. sqr-cos-a N/A

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

      14. lower-+.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. lower-*.f64 77.4

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

   6. Applied rewrites 77.4%

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

   7. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

   9. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), 0.5, 0.5\right) \cdot \left(a \cdot a\right)} \]

   if 3.8000000000000001e-121 < b

   1. Initial program 84.3%

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

      1. unpow2 N/A

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

      2. lower-*.f64 84.5

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-PI.f64 80.9

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

   8. Applied rewrites 80.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 80.9

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

   10. Applied rewrites 80.9%

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

       1. lift-*.f64 N/A

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

       2. lift-PI.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lift-PI.f64 81.0

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

   12. Applied rewrites 81.0%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{-121}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), 0.5, 0.5\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot 0.005555555555555556, \left(\left(angle \cdot b\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.005555555555555556, a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.3% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{-113}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot 0.005555555555555556, \left(\left(angle \cdot b\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.005555555555555556, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.05e-113)
   (* a a)
   (fma
    (* (* (* b (PI)) angle) 0.005555555555555556)
    (* (* (* angle b) (PI)) 0.005555555555555556)
    (* a a))))

Derivation

1. Split input into 2 regimes

2. if b < 1.05e-113

   1. Initial program 76.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 59.1

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

   5. Applied rewrites 59.1%

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

   if 1.05e-113 < b

   1. Initial program 85.4%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 85.5

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

   5. Applied rewrites 85.5%

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

   6. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-PI.f64 82.0

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

   8. Applied rewrites 82.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 82.0

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

   10. Applied rewrites 82.1%

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

       1. lift-*.f64 N/A

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

       2. lift-PI.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lift-PI.f64 82.1

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

   12. Applied rewrites 82.1%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{-113}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot 0.005555555555555556, \left(\left(angle \cdot b\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.005555555555555556, a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.3% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{-113}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot 0.005555555555555556, \left(b \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 0.005555555555555556, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.05e-113)
   (* a a)
   (fma
    (* (* (* b (PI)) angle) 0.005555555555555556)
    (* (* b (* angle (PI))) 0.005555555555555556)
    (* a a))))

Derivation

1. Split input into 2 regimes

2. if b < 1.05e-113

   1. Initial program 76.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 59.1

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

   5. Applied rewrites 59.1%

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

   if 1.05e-113 < b

   1. Initial program 85.4%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 85.5

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

   5. Applied rewrites 85.5%

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

   6. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-PI.f64 82.0

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

   8. Applied rewrites 82.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 82.0

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

   10. Applied rewrites 82.1%

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

       1. lift-*.f64 N/A

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

       2. lift-PI.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-PI.f64 82.1

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

   12. Applied rewrites 82.1%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{-113}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot 0.005555555555555556, \left(b \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 0.005555555555555556, a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.3% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;b \leq 1.05 \cdot 10^{-113}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(t\_0 \cdot angle\right) \cdot 0.005555555555555556, t\_0 \cdot \left(0.005555555555555556 \cdot angle\right), a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* b (PI))))
   (if (<= b 1.05e-113)
     (* a a)
     (fma
      (* (* t_0 angle) 0.005555555555555556)
      (* t_0 (* 0.005555555555555556 angle))
      (* a a)))))

Derivation

1. Split input into 2 regimes

2. if b < 1.05e-113

   1. Initial program 76.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 59.1

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

   5. Applied rewrites 59.1%

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

   if 1.05e-113 < b

   1. Initial program 85.4%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 85.5

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

   5. Applied rewrites 85.5%

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

   6. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-PI.f64 82.0

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

   8. Applied rewrites 82.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 82.0

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

   10. Applied rewrites 82.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-PI.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-PI.f64 N/A

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

       10. lower-*.f64 82.1

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

   12. Applied rewrites 82.1%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{-113}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot 0.005555555555555556, \left(b \cdot \mathsf{PI}\left(\right)\right) \cdot \left(0.005555555555555556 \cdot angle\right), a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.5% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.1%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 52.8

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

5. Applied rewrites 52.8%

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

Reproduce

herbie shell --seed 2025051 
(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)))