ab-angle->ABCF A

Percentage Accurate: 79.7% → 79.6%

Time: 4.8s

Alternatives: 6

Speedup: N/A×

• 37.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

Initial Program: 79.7% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 79.6% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 80.6%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 81.6

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

5. Applied rewrites 81.6%

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

6. Taylor expanded in angle around 0

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

   1. lower-*.f64 81.6

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

8. Applied rewrites 81.6%

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

Alternative 2: 66.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.15 \cdot 10^{-104}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left({\left(a \cdot angle\right)}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} + b \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 2.15e-104)
   (* b b)
   (+
    (* (* (pow (* a angle) 2.0) (* (PI) (PI))) 3.08641975308642e-5)
    (* b b))))

Derivation

1. Split input into 2 regimes

2. if a < 2.15000000000000005e-104

   1. Initial program 78.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 59.0

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

   5. Applied rewrites 59.0%

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

   if 2.15000000000000005e-104 < a

   1. Initial program 83.6%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 83.6

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

   5. Applied rewrites 83.6%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. pow-prod-down N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-*.f64 80.4

          \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5} + b \cdot b \]

   8. Applied rewrites 80.4%

      \[\leadsto \color{blue}{{\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}} + b \cdot b \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow-prod-down N/A

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

      9. pow-prod-down N/A

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

      10. lower-*.f64 N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-PI.f64 N/A

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

      17. lift-PI.f64 80.5

          \[\leadsto \left({\left(a \cdot angle\right)}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} + b \cdot b \]

   10. Applied rewrites 80.5%

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

Alternative 3: 66.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.15 \cdot 10^{-104}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5} + b \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 2.15e-104)
   (* b b)
   (+ (* (pow (* (* a angle) (PI)) 2.0) 3.08641975308642e-5) (* b b))))

Derivation

1. Split input into 2 regimes

2. if a < 2.15000000000000005e-104

   1. Initial program 78.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 59.0

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

   5. Applied rewrites 59.0%

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

   if 2.15000000000000005e-104 < a

   1. Initial program 83.6%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 83.6

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

   5. Applied rewrites 83.6%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. pow-prod-down N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-*.f64 80.4

          \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5} + b \cdot b \]

   8. Applied rewrites 80.4%

      \[\leadsto \color{blue}{{\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}} + b \cdot b \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 80.5

         \[\leadsto {\left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5} + b \cdot b \]

   10. Applied rewrites 80.5%

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

Alternative 4: 66.6% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\\ \mathbf{if}\;a \leq 2.15 \cdot 10^{-104}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot t\_0\right) \cdot 3.08641975308642 \cdot 10^{-5} + b \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* (PI) angle) a)))
   (if (<= a 2.15e-104)
     (* b b)
     (+ (* (* t_0 t_0) 3.08641975308642e-5) (* b b)))))

Derivation

1. Split input into 2 regimes

2. if a < 2.15000000000000005e-104

   1. Initial program 78.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 59.0

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

   5. Applied rewrites 59.0%

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

   if 2.15000000000000005e-104 < a

   1. Initial program 83.6%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 83.6

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

   5. Applied rewrites 83.6%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. pow-prod-down N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-*.f64 80.4

          \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5} + b \cdot b \]

   8. Applied rewrites 80.4%

      \[\leadsto \color{blue}{{\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}} + b \cdot b \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 80.4

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

   10. Applied rewrites 80.4%

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

Alternative 5: 57.2% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 9.2 \cdot 10^{+164}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{a} \cdot \frac{b}{a}\right) \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 9.2e+164) (* b b) (* (* (/ b a) (/ b a)) (* a a))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 9.2e+164) {
		tmp = b * b;
	} else {
		tmp = ((b / a) * (b / a)) * (a * a);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (a <= 9.2d+164) then
        tmp = b * b
    else
        tmp = ((b / a) * (b / a)) * (a * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 9.2e+164) {
		tmp = b * b;
	} else {
		tmp = ((b / a) * (b / a)) * (a * a);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 9.2e+164:
		tmp = b * b
	else:
		tmp = ((b / a) * (b / a)) * (a * a)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 9.2e+164)
		tmp = Float64(b * b);
	else
		tmp = Float64(Float64(Float64(b / a) * Float64(b / a)) * Float64(a * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 9.2e+164)
		tmp = b * b;
	else
		tmp = ((b / a) * (b / a)) * (a * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 9.2e+164], N[(b * b), $MachinePrecision], N[(N[(N[(b / a), $MachinePrecision] * N[(b / a), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 9.1999999999999998e164

   1. Initial program 77.2%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 58.4

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

   5. Applied rewrites 58.4%

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

   if 9.1999999999999998e164 < a

   1. Initial program 99.8%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 56.6%

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

   6. Taylor expanded in angle around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{{b}^{2}}{{a}^{2}} \cdot \left(a \cdot a\right) \]

      2. pow2 N/A

         \[\leadsto \frac{b \cdot b}{{a}^{2}} \cdot \left(a \cdot a\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{b \cdot b}{{a}^{2}} \cdot \left(a \cdot a\right) \]

      4. pow2 N/A

         \[\leadsto \frac{b \cdot b}{a \cdot a} \cdot \left(a \cdot a\right) \]

      5. lift-*.f64 0.0

         \[\leadsto \frac{b \cdot b}{a \cdot a} \cdot \left(a \cdot a\right) \]

   8. Applied rewrites 0.0%

      \[\leadsto \frac{b \cdot b}{a \cdot a} \cdot \left(\color{blue}{a} \cdot a\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{b \cdot b}{a \cdot a} \cdot \left(a \cdot a\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{b \cdot b}{a \cdot a} \cdot \left(a \cdot a\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{b \cdot b}{a \cdot a} \cdot \left(a \cdot a\right) \]

      4. times-frac N/A

         \[\leadsto \left(\frac{b}{a} \cdot \frac{b}{a}\right) \cdot \left(a \cdot a\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{b}{a} \cdot \frac{b}{a}\right) \cdot \left(a \cdot a\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\frac{b}{a} \cdot \frac{b}{a}\right) \cdot \left(a \cdot a\right) \]

      7. lower-/.f64 35.9

         \[\leadsto \left(\frac{b}{a} \cdot \frac{b}{a}\right) \cdot \left(a \cdot a\right) \]

   10. Applied rewrites 35.9%

       \[\leadsto \left(\frac{b}{a} \cdot \frac{b}{a}\right) \cdot \left(a \cdot a\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 57.2% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

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 = b * b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.6%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 54.9

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

5. Applied rewrites 54.9%

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

Reproduce

herbie shell --seed 2025073 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) (PI)))) 2.0) (pow (* b (cos (* (/ angle 180.0) (PI)))) 2.0)))