ab-angle->ABCF C

Percentage Accurate: 80.4% → 80.0%

Time: 9.4s

Alternatives: 7

Speedup: N/A×

• 37.3% 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: 80.4% 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: 80.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ \left(\left(1 \cdot a\right) \cdot 1\right) \cdot a + {\left(b \cdot \sin \left(\frac{\left(angle \cdot t\_0\right) \cdot t\_0}{180}\right)\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (sqrt (PI))))
   (+
    (* (* (* 1.0 a) 1.0) a)
    (pow (* b (sin (/ (* (* angle t_0) t_0) 180.0))) 2.0))))

Derivation

1. Initial program 82.7%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 82.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 82.9

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

7. Applied rewrites 82.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f64 N/A

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

   4. add-sqr-sqrt N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lower-sqrt.f64 83.0

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

9. Applied rewrites 83.0%

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

    1. lift-pow.f64 N/A

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

    2. unpow2 N/A

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

    3. lift-*.f64 N/A

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

    4. *-commutative N/A

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

    5. associate-*r* N/A

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

    6. lower-*.f64 N/A

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

    7. lower-*.f64 83.0

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

    8. lift-*.f64 N/A

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

    9. *-commutative N/A

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

    10. lower-*.f64 83.0

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

11. Applied rewrites 83.0%

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

Alternative 2: 80.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(1 \cdot a\right) \cdot 1, a, {\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot b\right)}^{2}\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (fma (* (* 1.0 a) 1.0) a (pow (* (sin (/ (* (PI) angle) 180.0)) b) 2.0)))

Derivation

1. Initial program 82.7%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 82.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 82.9

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

7. Applied rewrites 82.9%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r/ N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

9. Applied rewrites 82.9%

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

Alternative 3: 80.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(1 \cdot a\right) \cdot 1, a, {\left(\sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2}\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (fma (* (* 1.0 a) 1.0) a (pow (* (sin (* (/ angle 180.0) (PI))) b) 2.0)))

Derivation

1. Initial program 82.7%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 82.9%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

7. Applied rewrites 82.9%

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

Alternative 4: 67.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.45 \cdot 10^{-49}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(-0.011111111111111112 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), 0.5, 0.5\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot 1\right)}^{2} + {\left(b \cdot \left(\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.45e-49)
   (* (fma (cos (* -0.011111111111111112 (* angle (PI)))) 0.5 0.5) (* a a))
   (+
    (pow (* a 1.0) 2.0)
    (pow (* b (* (* 0.005555555555555556 (PI)) angle)) 2.0))))

Derivation

1. Split input into 2 regimes

2. if b < 1.45e-49

   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. 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. unpow2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 84.4%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. metadata-eval N/A

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

      4. unpow1 N/A

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

      5. lift-cos.f64 N/A

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

      6. cos-neg-rev N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. cos-neg-rev N/A

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

      11. metadata-eval N/A

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

      12. unpow1 N/A

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

      13. lift-cos.f64 N/A

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

      14. cos-neg-rev N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. cos-neg-rev N/A

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

   6. Applied rewrites 84.4%

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

   7. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. cos-neg-rev N/A

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

      7. lower-cos.f64 N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower-PI.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 65.9

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

   9. Applied rewrites 65.9%

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

   if 1.45e-49 < b

   1. Initial program 79.0%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 79.0%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 75.8

         \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \left(\left(0.005555555555555556 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot angle\right)\right)}^{2} \]

   8. Applied rewrites 75.8%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \color{blue}{\left(\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}\right)}^{2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 51.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 5.5 \cdot 10^{-152}:\\ \;\;\;\;{\sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{2} \cdot \left(b \cdot b\right)\\ \mathbf{elif}\;a \leq 1.04 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(\left(-3.08641975308642 \cdot 10^{-5} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(a \cdot a - b \cdot b\right), angle \cdot angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 5.5e-152)
   (* (pow (sin (* (* (PI) 0.005555555555555556) angle)) 2.0) (* b b))
   (if (<= a 1.04e+154)
     (fma
      (* (* -3.08641975308642e-5 (* (PI) (PI))) (- (* a a) (* b b)))
      (* angle angle)
      (* a a))
     (* a a))))

Derivation

1. Split input into 3 regimes

2. if a < 5.4999999999999998e-152

   1. Initial program 83.0%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-PI.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 46.1

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

   5. Applied rewrites 46.1%

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

   if 5.4999999999999998e-152 < a < 1.04e154

   1. Initial program 70.3%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 52.8%

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

   if 1.04e154 < a

   1. Initial program 100.0%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{a \cdot a} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 53.4% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.04 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(\left(-3.08641975308642 \cdot 10^{-5} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(a \cdot a - b \cdot b\right), angle \cdot angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.04e+154)
   (fma
    (* (* -3.08641975308642e-5 (* (PI) (PI))) (- (* a a) (* b b)))
    (* angle angle)
    (* a a))
   (* a a)))

Derivation

1. Split input into 2 regimes

2. if a < 1.04e154

   1. Initial program 79.4%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 47.5%

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

   if 1.04e154 < a

   1. Initial program 100.0%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{a \cdot a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 58.0% 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 82.7%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 58.8

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

5. Applied rewrites 58.8%

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

Reproduce

herbie shell --seed 2024354 
(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)))