ab-angle->ABCF C

Percentage Accurate: 79.7% → 79.7%

Time: 2.7s

Alternatives: 11

Speedup: 1.0×

• 36.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

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

Math

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

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: 79.7% 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: 79.7% 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))))

Derivation

1. Initial program 81.5%

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

Alternative 2: 68.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \frac{angle}{180}\\ t_1 := {\left(b \cdot \sin t\_0\right)}^{2}\\ \mathbf{if}\;b \leq -1.6 \cdot 10^{+144}:\\ \;\;\;\;t\_0 + t\_1\\ \mathbf{elif}\;b \leq 10^{+108}:\\ \;\;\;\;{\left(a \cdot \cos t\_0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{t\_0}^{2} + t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (PI) (/ angle 180.0))) (t_1 (pow (* b (sin t_0)) 2.0)))
   (if (<= b -1.6e+144)
     (+ t_0 t_1)
     (if (<= b 1e+108) (pow (* a (cos t_0)) 2.0) (+ (pow t_0 2.0) t_1)))))

Derivation

1. Split input into 3 regimes

2. if b < -1.6e144

   1. Initial program 94.4%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 78.3%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 78.8%

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

   9. Taylor expanded in angle around 0

      \[\leadsto \left(a + \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{64800} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{2}\right) + {angle}^{2} \cdot \left(\frac{-1}{24488801280000000} \cdot \left(a \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{6}\right)\right) + \frac{1}{25194240000} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)}\right) + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   11. Applied rewrites 78.9%

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

   if -1.6e144 < b < 1e108

   1. Initial program 77.2%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 72.5%

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

   if 1e108 < b

   1. Initial program 89.9%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 72.9%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 75.2%

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

Alternative 3: 68.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \frac{angle}{180}\\ t_1 := t\_0 + {\left(b \cdot \sin t\_0\right)}^{2}\\ \mathbf{if}\;b \leq -1.6 \cdot 10^{+144}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{+108}:\\ \;\;\;\;{\left(a \cdot \cos t\_0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (PI) (/ angle 180.0))) (t_1 (+ t_0 (pow (* b (sin t_0)) 2.0))))
   (if (<= b -1.6e+144)
     t_1
     (if (<= b 2.6e+108) (pow (* a (cos t_0)) 2.0) t_1))))

Derivation

1. Split input into 2 regimes

2. if b < -1.6e144 or 2.6000000000000002e108 < b

   1. Initial program 91.7%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 76.7%

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

   9. Taylor expanded in angle around 0

      \[\leadsto \left(a + \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{64800} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{2}\right) + {angle}^{2} \cdot \left(\frac{-1}{24488801280000000} \cdot \left(a \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{6}\right)\right) + \frac{1}{25194240000} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)}\right) + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   11. Applied rewrites 76.7%

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

   if -1.6e144 < b < 2.6000000000000002e108

   1. Initial program 77.2%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 72.5%

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

Alternative 4: 66.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\\ t_1 := \mathsf{PI}\left(\right) \cdot \frac{angle}{180}\\ \mathbf{if}\;b \leq -2.3 \cdot 10^{+167}:\\ \;\;\;\;t\_1 + {\left(b \cdot \sin t\_0\right)}^{2}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{+155}:\\ \;\;\;\;{\left(a \cdot \cos t\_1\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + {\left(b \cdot \sin t\_1\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (PI) (PI))) (t_1 (* (PI) (/ angle 180.0))))
   (if (<= b -2.3e+167)
     (+ t_1 (pow (* b (sin t_0)) 2.0))
     (if (<= b 1.25e+155)
       (pow (* a (cos t_1)) 2.0)
       (+ t_0 (pow (* b (sin t_1)) 2.0))))))

Derivation

1. Split input into 3 regimes

2. if b < -2.29999999999999988e167

   1. Initial program 96.4%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 84.7%

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

   9. Taylor expanded in angle around 0

      \[\leadsto \left(a + \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{64800} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{2}\right) + {angle}^{2} \cdot \left(\frac{-1}{24488801280000000} \cdot \left(a \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{6}\right)\right) + \frac{1}{25194240000} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)}\right) + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   11. Applied rewrites 84.7%

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

   12. Taylor expanded in angle around 0

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

   14. Applied rewrites 85.3%

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

   if -2.29999999999999988e167 < b < 1.25e155

   1. Initial program 76.5%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 69.1%

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

   if 1.25e155 < b

   1. Initial program 99.6%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 85.2%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 88.2%

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

   9. Taylor expanded in angle around 0

      \[\leadsto \left(a + \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{64800} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{2}\right) + {angle}^{2} \cdot \left(\frac{-1}{24488801280000000} \cdot \left(a \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{6}\right)\right) + \frac{1}{25194240000} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)}\right) + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   11. Applied rewrites 88.2%

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

   12. Taylor expanded in angle around 0

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

   14. Applied rewrites 77.6%

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

Alternative 5: 67.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \frac{angle}{180}\\ t_1 := \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right) + {\left(b \cdot \sin t\_0\right)}^{2}\\ \mathbf{if}\;b \leq -1.95 \cdot 10^{+154}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{+155}:\\ \;\;\;\;{\left(a \cdot \cos t\_0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (PI) (/ angle 180.0)))
        (t_1 (+ (* (PI) (PI)) (pow (* b (sin t_0)) 2.0))))
   (if (<= b -1.95e+154)
     t_1
     (if (<= b 1.25e+155) (pow (* a (cos t_0)) 2.0) t_1))))

Derivation

1. Split input into 2 regimes

2. if b < -1.9500000000000001e154 or 1.25e155 < b

   1. Initial program 98.3%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 84.2%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 85.8%

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

   9. Taylor expanded in angle around 0

      \[\leadsto \left(a + \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{64800} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{2}\right) + {angle}^{2} \cdot \left(\frac{-1}{24488801280000000} \cdot \left(a \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{6}\right)\right) + \frac{1}{25194240000} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)}\right) + {\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} \]

   11. Applied rewrites 85.8%

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

   12. Taylor expanded in angle around 0

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

   14. Applied rewrites 78.6%

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

   if -1.9500000000000001e154 < b < 1.25e155

   1. Initial program 76.1%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 69.6%

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

Alternative 6: 57.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \frac{angle}{180}\\ \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{+131}:\\ \;\;\;\;{\left({t\_0}^{2}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot \cos t\_0\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (PI) (/ angle 180.0))))
   (if (<= (/ angle 180.0) -2e+131)
     (pow (pow t_0 2.0) 2.0)
     (pow (* a (cos t_0)) 2.0))))

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < -1.9999999999999998e131

   1. Initial program 62.9%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 33.8%

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

   6. Taylor expanded in angle around 0

      \[\leadsto {a}^{2} \]

   8. Applied rewrites 49.9%

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

   if -1.9999999999999998e131 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 85.1%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 64.8%

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

Alternative 7: 23.6% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 81.5%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 59.7%

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

6. Taylor expanded in angle around 0

   \[\leadsto {a}^{2} \]

8. Applied rewrites 24.3%

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

Alternative 8: 17.8% accurate, 3.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 81.5%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 27.8%

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

6. Taylor expanded in angle around 0

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

8. Applied rewrites 34.3%

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

9. Taylor expanded in a around 0

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

11. Applied rewrites 17.9%

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

Alternative 9: 4.4% accurate, 4.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 81.5%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 27.8%

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

6. Taylor expanded in angle around 0

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

8. Applied rewrites 34.3%

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

9. Taylor expanded in a around 0

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

11. Applied rewrites 17.9%

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

12. Taylor expanded in angle around 0

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

14. Applied rewrites 4.3%

    \[\leadsto {\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}^{2} \]
15. Add Preprocessing

Alternative 10: 4.4% accurate, 74.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 81.5%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 59.7%

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

6. Taylor expanded in b around inf

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

8. Applied rewrites 3.1%

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

9. Taylor expanded in angle around 0

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

11. Applied rewrites 4.3%

    \[\leadsto \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right) \]
12. Add Preprocessing

Alternative 11: 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 81.5%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 59.7%

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

6. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{{a}^{2}} \]

8. Applied rewrites 4.3%

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

Reproduce

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