ab-angle->ABCF C

Percentage Accurate: 79.4% → 79.4%

Time: 10.5s

Alternatives: 9

Speedup: 2.0×

• Could not converge on a ground truth

  1. a = 1.162737645138401e+156
  2. b = 1.3123285653020884e-58
  3. angle = 6.124715124841019e+23

• 36.1% 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: 79.4% 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: 79.4% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 77.5%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 77.9

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

5. Applied rewrites 77.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 78.0

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

7. Applied rewrites 78.0%

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

8. Final simplification 78.0%

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

Alternative 2: 66.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.2 \cdot 10^{-66}:\\ \;\;\;\;{\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2} + a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 7.2e-66)
   (* (pow (cos (* (* -0.005555555555555556 angle) (PI))) 2.0) (* a a))
   (+ (pow (* (* (* 0.005555555555555556 angle) (PI)) b) 2.0) (* a a))))

Derivation

1. Split input into 2 regimes

2. if b < 7.20000000000000025e-66

   1. Initial program 72.9%

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

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

      2. lower-*.f64 73.0

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

   5. Applied rewrites 73.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. rem-square-sqrt N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. div-inv N/A

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

      9. unpow-1 N/A

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

      10. lift-pow.f64 N/A

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

      11. *-commutative N/A

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

      12. frac-times N/A

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

      13. lift-pow.f64 N/A

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

      14. sqr-pow N/A

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

      15. associate-/r* N/A

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

      16. frac-times N/A

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

      17. lower-/.f64 N/A

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

   7. Applied rewrites 31.3%

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

   8. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 56.5%

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

   if 7.20000000000000025e-66 < b

   1. Initial program 86.2%

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

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

      2. lower-*.f64 87.2

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

   5. Applied rewrites 87.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. rem-square-sqrt N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. div-inv N/A

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

      9. unpow-1 N/A

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

      10. lift-pow.f64 N/A

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

      11. *-commutative N/A

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

      12. frac-times N/A

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

      13. lift-pow.f64 N/A

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

      14. sqr-pow N/A

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

      15. associate-/r* N/A

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

      16. frac-times N/A

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

      17. lower-/.f64 N/A

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

   7. Applied rewrites 48.3%

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

   8. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 83.9

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

   10. Applied rewrites 83.9%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.2 \cdot 10^{-66}:\\ \;\;\;\;{\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2} + a \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.2 \cdot 10^{-66}:\\ \;\;\;\;{\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot -0.005555555555555556\right)}^{2} \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2} + a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 7.2e-66)
   (* (pow (cos (* (* (PI) angle) -0.005555555555555556)) 2.0) (* a a))
   (+ (pow (* (* (* 0.005555555555555556 angle) (PI)) b) 2.0) (* a a))))

Derivation

1. Split input into 2 regimes

2. if b < 7.20000000000000025e-66

   1. Initial program 72.9%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. unpow-prod-down N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 73.1%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 56.6

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

   7. Applied rewrites 56.6%

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

   if 7.20000000000000025e-66 < b

   1. Initial program 86.2%

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

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

      2. lower-*.f64 87.2

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

   5. Applied rewrites 87.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. rem-square-sqrt N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. div-inv N/A

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

      9. unpow-1 N/A

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

      10. lift-pow.f64 N/A

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

      11. *-commutative N/A

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

      12. frac-times N/A

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

      13. lift-pow.f64 N/A

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

      14. sqr-pow N/A

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

      15. associate-/r* N/A

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

      16. frac-times N/A

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

      17. lower-/.f64 N/A

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

   7. Applied rewrites 48.3%

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

   8. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 83.9

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

   10. Applied rewrites 83.9%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.2 \cdot 10^{-66}:\\ \;\;\;\;{\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot -0.005555555555555556\right)}^{2} \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2} + a \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.4% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 77.5%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 77.9

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

5. Applied rewrites 77.9%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 78.0

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 78.0

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

7. Applied rewrites 78.0%

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

8. Final simplification 78.0%

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

Alternative 5: 66.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.8 \cdot 10^{-66}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2} + a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 7.8e-66)
   (* a a)
   (+ (pow (* (* (* 0.005555555555555556 angle) (PI)) b) 2.0) (* a a))))

Derivation

1. Split input into 2 regimes

2. if b < 7.79999999999999965e-66

   1. Initial program 72.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.1

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

   5. Applied rewrites 57.1%

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

   if 7.79999999999999965e-66 < b

   1. Initial program 86.2%

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

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

      2. lower-*.f64 87.2

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

   5. Applied rewrites 87.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. rem-square-sqrt N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. div-inv N/A

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

      9. unpow-1 N/A

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

      10. lift-pow.f64 N/A

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

      11. *-commutative N/A

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

      12. frac-times N/A

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

      13. lift-pow.f64 N/A

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

      14. sqr-pow N/A

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

      15. associate-/r* N/A

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

      16. frac-times N/A

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

      17. lower-/.f64 N/A

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

   7. Applied rewrites 48.3%

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

   8. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 83.9

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

   10. Applied rewrites 83.9%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.8 \cdot 10^{-66}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)}^{2} + a \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.8 \cdot 10^{-66}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot b\right)}^{2} + a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 7.8e-66)
   (* a a)
   (+ (pow (* (* (* 0.005555555555555556 (PI)) angle) b) 2.0) (* a a))))

Derivation

1. Split input into 2 regimes

2. if b < 7.79999999999999965e-66

   1. Initial program 72.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.1

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

   5. Applied rewrites 57.1%

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

   if 7.79999999999999965e-66 < b

   1. Initial program 86.2%

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

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

      2. lower-*.f64 87.2

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

   5. Applied rewrites 87.2%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 83.9

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

   8. Applied rewrites 83.9%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.8 \cdot 10^{-66}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot b\right)}^{2} + a \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.8% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{-166}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right) \cdot b\right) \cdot \left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) + a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) 2e-166)
   (* a a)
   (+
    (* (* (* (* (PI) (PI)) b) b) (* (* angle angle) 3.08641975308642e-5))
    (* a a))))

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2.00000000000000008e-166

   1. Initial program 83.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 60.0

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

   5. Applied rewrites 60.0%

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

   if 2.00000000000000008e-166 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 68.9%

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

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

      2. lower-*.f64 68.9

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

   5. Applied rewrites 68.9%

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

   6. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-PI.f64 N/A

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

      14. lower-PI.f64 58.8

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

   8. Applied rewrites 58.8%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{-166}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right) \cdot b\right) \cdot \left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) + a \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.4% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.8 \cdot 10^{-66}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right) \cdot \left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot angle\right) \cdot b\right) \cdot angle\right) + a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 7.8e-66)
   (* a a)
   (+
    (* (* (* (PI) (PI)) b) (* (* (* 3.08641975308642e-5 angle) b) angle))
    (* a a))))

Derivation

1. Split input into 2 regimes

2. if b < 7.79999999999999965e-66

   1. Initial program 72.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.1

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

   5. Applied rewrites 57.1%

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

   if 7.79999999999999965e-66 < b

   1. Initial program 86.2%

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

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

      2. lower-*.f64 87.2

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

   5. Applied rewrites 87.2%

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

   6. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-PI.f64 N/A

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

      14. lower-PI.f64 66.4

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

   8. Applied rewrites 66.4%

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

   10. Applied rewrites 81.8%

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

   12. Applied rewrites 84.0%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.8 \cdot 10^{-66}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right) \cdot \left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot angle\right) \cdot b\right) \cdot angle\right) + a \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.0% 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

real(8) function code(a, b, angle)
    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 77.5%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 54.9

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

5. Applied rewrites 54.9%

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

Reproduce

herbie shell --seed 2024276 
(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)))