ab-angle->ABCF C

Percentage Accurate: 79.9% → 80.0%

Time: 4.7s

Alternatives: 8

Speedup: 2.0×

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

Specification

Math

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

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: 79.9% 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, 1.9× 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 78.9%

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

3. Taylor expanded in angle around 0

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

   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 79.3

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

5. Applied rewrites 79.3%

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

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-PI.f64 79.5

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

7. Applied rewrites 79.5%

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

Alternative 2: 80.0% 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 78.9%

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

3. Taylor expanded in angle around 0

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

   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 79.3

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

5. Applied rewrites 79.3%

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

6. Taylor expanded in angle around 0

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

   1. lower-*.f64 79.4

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

8. Applied rewrites 79.4%

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

Alternative 3: 58.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.95 \cdot 10^{-147}:\\ \;\;\;\;{\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot 0.005555555555555556\right) \cdot angle, \left(b \cdot \left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot angle, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.95e-147)
   (pow (* (sin (* (* 0.005555555555555556 angle) (PI))) b) 2.0)
   (fma
    (* (* (* b (PI)) 0.005555555555555556) angle)
    (* (* b (* 0.005555555555555556 (PI))) angle)
    (* a a))))

Derivation

1. Split input into 2 regimes

2. if a < 1.9499999999999999e-147

   1. Initial program 77.9%

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

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

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

      2. lower-sin.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-PI.f64 77.6

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

   4. Applied rewrites 77.6%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

   7. Applied rewrites 45.2%

      \[\leadsto \color{blue}{\left(\frac{{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b\right)}^{2}}{a \cdot a} + {\sin \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), 0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}\right) \cdot \left(a \cdot a\right)} \]

   8. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. unpow-prod-down N/A

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

      3. lower-pow.f64 N/A

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

   10. Applied rewrites 45.9%

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

   if 1.9499999999999999e-147 < a

   1. Initial program 81.2%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{a}^{2}} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 81.4

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

   5. Applied rewrites 81.4%

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

   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. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 79.1

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

   8. Applied rewrites 79.1%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

   10. Applied rewrites 79.1%

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

       1. associate-*l* N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-PI.f64 79.2

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

   12. Applied rewrites 79.2%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.95 \cdot 10^{-147}:\\ \;\;\;\;{\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot 0.005555555555555556\right) \cdot angle, \left(b \cdot \left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot angle, a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.95 \cdot 10^{-147}:\\ \;\;\;\;{\left(\sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot 0.005555555555555556\right) \cdot b\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot 0.005555555555555556\right) \cdot angle, \left(b \cdot \left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot angle, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.95e-147)
   (pow (* (sin (* (* angle (PI)) 0.005555555555555556)) b) 2.0)
   (fma
    (* (* (* b (PI)) 0.005555555555555556) angle)
    (* (* b (* 0.005555555555555556 (PI))) angle)
    (* a a))))

Derivation

1. Split input into 2 regimes

2. if a < 1.9499999999999999e-147

   1. Initial program 77.9%

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

   3. Taylor expanded in 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. pow-prod-down N/A

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

      2. lower-pow.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 46.0

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

   5. Applied rewrites 46.0%

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

   if 1.9499999999999999e-147 < a

   1. Initial program 81.2%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{a}^{2}} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 81.4

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

   5. Applied rewrites 81.4%

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

   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. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 79.1

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

   8. Applied rewrites 79.1%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

   10. Applied rewrites 79.1%

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

       1. associate-*l* N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-PI.f64 79.2

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

   12. Applied rewrites 79.2%

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

Alternative 5: 67.0% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.05 \cdot 10^{-86}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot 0.005555555555555556\right) \cdot angle, \left(b \cdot \left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot angle, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 3.05e-86)
   (* a a)
   (fma
    (* (* (* b (PI)) 0.005555555555555556) angle)
    (* (* b (* 0.005555555555555556 (PI))) angle)
    (* a a))))

Derivation

1. Split input into 2 regimes

2. if b < 3.05000000000000016e-86

   1. Initial program 79.1%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 59.8

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

   5. Applied rewrites 59.8%

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

   if 3.05000000000000016e-86 < b

   1. Initial program 78.6%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 78.5

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

   5. Applied rewrites 78.5%

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

   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. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 75.2

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

   8. Applied rewrites 75.2%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

   10. Applied rewrites 75.3%

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

       1. associate-*l* N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-PI.f64 75.3

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

   12. Applied rewrites 75.3%

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

Alternative 6: 67.0% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;b \leq 3.05 \cdot 10^{-86}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(t\_0 \cdot 0.005555555555555556\right) \cdot angle, t\_0 \cdot \left(0.005555555555555556 \cdot angle\right), a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* b (PI))))
   (if (<= b 3.05e-86)
     (* a a)
     (fma
      (* (* t_0 0.005555555555555556) angle)
      (* t_0 (* 0.005555555555555556 angle))
      (* a a)))))

Derivation

1. Split input into 2 regimes

2. if b < 3.05000000000000016e-86

   1. Initial program 79.1%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 59.8

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

   5. Applied rewrites 59.8%

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

   if 3.05000000000000016e-86 < b

   1. Initial program 78.6%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 78.5

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

   5. Applied rewrites 78.5%

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

   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. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 75.2

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

   8. Applied rewrites 75.2%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

   10. Applied rewrites 75.3%

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

       1. associate-*l* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-PI.f64 N/A

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

       5. lower-*.f64 75.3

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

   12. Applied rewrites 75.3%

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

Alternative 7: 67.0% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;b \leq 3.05 \cdot 10^{-86}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(t\_0 \cdot 0.005555555555555556\right) \cdot angle, \left(t\_0 \cdot angle\right) \cdot 0.005555555555555556, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* b (PI))))
   (if (<= b 3.05e-86)
     (* a a)
     (fma
      (* (* t_0 0.005555555555555556) angle)
      (* (* t_0 angle) 0.005555555555555556)
      (* a a)))))

Derivation

1. Split input into 2 regimes

2. if b < 3.05000000000000016e-86

   1. Initial program 79.1%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 59.8

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

   5. Applied rewrites 59.8%

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

   if 3.05000000000000016e-86 < b

   1. Initial program 78.6%

      \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 78.5

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

   5. Applied rewrites 78.5%

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

   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. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 75.2

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

   8. Applied rewrites 75.2%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

   10. Applied rewrites 75.3%

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

   11. Taylor expanded in b around 0

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

       1. *-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-PI.f64 75.3

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

   13. Applied rewrites 75.3%

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

Alternative 8: 56.8% 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 78.9%

   \[{\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]
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 53.1

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

5. Applied rewrites 53.1%

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

Reproduce

herbie shell --seed 2025044 
(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)))