ab-angle->ABCF A

Percentage Accurate: 79.7% → 79.7%

Time: 2.3s

Alternatives: 7

Speedup: 1.0×

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

Specification

\[\mathsf{TRUE}\left(\right)\]

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: 79.7% 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))))

Derivation

1. Initial program 81.3%

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

Alternative 2: 66.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \mathsf{PI}\left(\right)\\ t_1 := {\left(b \cdot \cos t\_0\right)}^{2}\\ t_2 := \sin t\_0\\ \mathbf{if}\;b \leq -3.2 \cdot 10^{-28}:\\ \;\;\;\;{t\_2}^{2} + t\_1\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{+24}:\\ \;\;\;\;{\left(a \cdot t\_2\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;t\_2 + t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) (PI)))
        (t_1 (pow (* b (cos t_0)) 2.0))
        (t_2 (sin t_0)))
   (if (<= b -3.2e-28)
     (+ (pow t_2 2.0) t_1)
     (if (<= b 7.2e+24) (pow (* a t_2) 2.0) (+ t_2 t_1)))))

Derivation

1. Split input into 3 regimes

2. if b < -3.19999999999999982e-28

   1. Initial program 82.7%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 73.5%

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

   if -3.19999999999999982e-28 < b < 7.19999999999999966e24

   1. Initial program 79.5%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 65.1%

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

   if 7.19999999999999966e24 < b

   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 a around 0

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

   5. Applied rewrites 76.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 76.2%

      \[\leadsto \color{blue}{\sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 66.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \mathsf{PI}\left(\right)\\ t_1 := \sin t\_0\\ t_2 := t\_1 + {\left(b \cdot \cos t\_0\right)}^{2}\\ \mathbf{if}\;b \leq -3.25 \cdot 10^{-28}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{+24}:\\ \;\;\;\;{\left(a \cdot t\_1\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) (PI)))
        (t_1 (sin t_0))
        (t_2 (+ t_1 (pow (* b (cos t_0)) 2.0))))
   (if (<= b -3.25e-28) t_2 (if (<= b 7.2e+24) (pow (* a t_1) 2.0) t_2))))

Derivation

1. Split input into 2 regimes

2. if b < -3.25000000000000021e-28 or 7.19999999999999966e24 < b

   1. Initial program 83.1%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 74.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 74.7%

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

   if -3.25000000000000021e-28 < b < 7.19999999999999966e24

   1. Initial program 79.5%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 65.1%

      \[\leadsto \color{blue}{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 64.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \mathsf{PI}\left(\right)\\ t_1 := t\_0 + {\left(b \cdot \cos t\_0\right)}^{2}\\ \mathbf{if}\;b \leq -3.3 \cdot 10^{-28}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 10^{+106}:\\ \;\;\;\;{\left(a \cdot \sin t\_0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) (PI))) (t_1 (+ t_0 (pow (* b (cos t_0)) 2.0))))
   (if (<= b -3.3e-28) t_1 (if (<= b 1e+106) (pow (* a (sin t_0)) 2.0) t_1))))

Derivation

1. Split input into 2 regimes

2. if b < -3.3000000000000002e-28 or 1.00000000000000009e106 < b

   1. Initial program 86.8%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 80.7%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 80.6%

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

   9. Taylor expanded in angle around 0

      \[\leadsto angle \cdot \color{blue}{\left(\frac{-1}{34992000} \cdot \left(a \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + \frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   11. Applied rewrites 80.0%

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

   if -3.3000000000000002e-28 < b < 1.00000000000000009e106

   1. Initial program 76.9%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 61.4%

      \[\leadsto \color{blue}{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 40.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \mathsf{PI}\left(\right)\\ t_1 := {t\_0}^{2}\\ \mathbf{if}\;b \leq -2.3 \cdot 10^{+130}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{+192}:\\ \;\;\;\;{\left(a \cdot \sin t\_0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) (PI))) (t_1 (pow t_0 2.0)))
   (if (<= b -2.3e+130)
     t_1
     (if (<= b 1.7e+192) (pow (* a (sin t_0)) 2.0) t_1))))

Derivation

1. Split input into 2 regimes

2. if b < -2.30000000000000021e130 or 1.69999999999999998e192 < b

   1. Initial program 95.9%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 95.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 2.5%

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

   9. Taylor expanded in angle around 0

      \[\leadsto {\left(angle \cdot \left(\frac{-1}{34992000} \cdot \left(a \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + \frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2} \]

   11. Applied rewrites 31.8%

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

   if -2.30000000000000021e130 < b < 1.69999999999999998e192

   1. Initial program 76.8%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 50.0%

      \[\leadsto \color{blue}{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 17.9% accurate, 3.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 81.3%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 48.8%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 7.0%

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

9. Taylor expanded in angle around 0

   \[\leadsto {\left(angle \cdot \left(\frac{-1}{34992000} \cdot \left(a \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + \frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2} \]

11. Applied rewrites 18.2%

    \[\leadsto {\left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}^{2} \]
12. Add Preprocessing

Alternative 7: 4.4% accurate, 448.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{PI}\left(\right) \end{array} \]

FPCore

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

Derivation

1. Initial program 81.3%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 48.8%

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

6. Taylor expanded in angle around 0

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

8. Applied rewrites 4.2%

   \[\leadsto \color{blue}{\mathsf{PI}\left(\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024321 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  :pre (TRUE)
  (+ (pow (* a (sin (* (/ angle 180.0) (PI)))) 2.0) (pow (* b (cos (* (/ angle 180.0) (PI)))) 2.0)))