ab-angle->ABCF C

Percentage Accurate: 79.4% → 79.5%

Time: 10.4s

Alternatives: 9

Speedup: 1.4×

• 36.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle / 180.0);
	return pow((a * cos(t_0)), 2.0) + pow((b * sin(t_0)), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle / 180.0);
	return Math.pow((a * Math.cos(t_0)), 2.0) + Math.pow((b * Math.sin(t_0)), 2.0);
}

Python

def code(a, b, angle):
	t_0 = math.pi * (angle / 180.0)
	return math.pow((a * math.cos(t_0)), 2.0) + math.pow((b * math.sin(t_0)), 2.0)

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle / 180.0))
	return Float64((Float64(a * cos(t_0)) ^ 2.0) + (Float64(b * sin(t_0)) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	t_0 = pi * (angle / 180.0);
	tmp = ((a * cos(t_0)) ^ 2.0) + ((b * sin(t_0)) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, N[(N[Power[N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Initial Program: 79.4% accurate, 1.0× speedup

Math

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

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))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle / 180.0);
	return pow((a * cos(t_0)), 2.0) + pow((b * sin(t_0)), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle / 180.0);
	return Math.pow((a * Math.cos(t_0)), 2.0) + Math.pow((b * Math.sin(t_0)), 2.0);
}

Python

def code(a, b, angle):
	t_0 = math.pi * (angle / 180.0)
	return math.pow((a * math.cos(t_0)), 2.0) + math.pow((b * math.sin(t_0)), 2.0)

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle / 180.0))
	return Float64((Float64(a * cos(t_0)) ^ 2.0) + (Float64(b * sin(t_0)) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	t_0 = pi * (angle / 180.0);
	tmp = ((a * cos(t_0)) ^ 2.0) + ((b * sin(t_0)) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, N[(N[Power[N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Alternative 1: 79.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

double code(double a, double b, double angle) {
	return pow(a, 2.0) + pow((sin((angle * (((double) M_PI) * 0.005555555555555556))) * b), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow(a, 2.0) + Math.pow((Math.sin((angle * (Math.PI * 0.005555555555555556))) * b), 2.0);
}

Python

def code(a, b, angle):
	return math.pow(a, 2.0) + math.pow((math.sin((angle * (math.pi * 0.005555555555555556))) * b), 2.0)

Julia

function code(a, b, angle)
	return Float64((a ^ 2.0) + (Float64(sin(Float64(angle * Float64(pi * 0.005555555555555556))) * b) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (a ^ 2.0) + ((sin((angle * (pi * 0.005555555555555556))) * b) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[a, 2.0], $MachinePrecision] + N[Power[N[(N[Sin[N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.0%

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

3. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-sin.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-PI.f64 81.1

       \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(\sin \left(angle \cdot \left(\color{blue}{\pi} \cdot 0.005555555555555556\right)\right) \cdot b\right)}^{2} \]

5. Simplified 81.1%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\color{blue}{\left(\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot b\right)}}^{2} \]

6. Taylor expanded in angle around 0

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

8. Simplified 81.5%

   \[\leadsto {\left(a \cdot \color{blue}{1}\right)}^{2} + {\left(\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot b\right)}^{2} \]

9. Final simplification 81.5%

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

Alternative 2: 79.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

double code(double a, double b, double angle) {
	return pow(a, 2.0) + pow((b * sin((0.005555555555555556 * (angle * ((double) M_PI))))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow(a, 2.0) + Math.pow((b * Math.sin((0.005555555555555556 * (angle * Math.PI)))), 2.0);
}

Python

def code(a, b, angle):
	return math.pow(a, 2.0) + math.pow((b * math.sin((0.005555555555555556 * (angle * math.pi)))), 2.0)

Julia

function code(a, b, angle)
	return Float64((a ^ 2.0) + (Float64(b * sin(Float64(0.005555555555555556 * Float64(angle * pi)))) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (a ^ 2.0) + ((b * sin((0.005555555555555556 * (angle * pi)))) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[a, 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.0%

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

3. Taylor expanded in angle around 0

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

5. Simplified 81.4%

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

6. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-sin.f64 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-PI.f64 81.0

       \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right) \cdot b\right)}^{2} \]

8. Simplified 81.0%

   \[\leadsto {\left(a \cdot 1\right)}^{2} + {\color{blue}{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b\right)}}^{2} \]

9. Final simplification 81.0%

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

Alternative 3: 76.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 0.001:\\ \;\;\;\;{a}^{2} + {\left(b \cdot \left(angle \cdot \left(\pi \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 0.005555555555555556\right)\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) 0.001)
   (+
    (pow a 2.0)
    (pow
     (*
      b
      (*
       angle
       (*
        PI
        (fma
         (* -2.8577960676726107e-8 (* angle angle))
         (* PI PI)
         0.005555555555555556))))
     2.0))
   (+
    (pow a 2.0)
    (*
     (- 0.5 (* 0.5 (cos (* 2.0 (* angle (* PI 0.005555555555555556))))))
     (* b b)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= 0.001) {
		tmp = pow(a, 2.0) + pow((b * (angle * (((double) M_PI) * fma((-2.8577960676726107e-8 * (angle * angle)), (((double) M_PI) * ((double) M_PI)), 0.005555555555555556)))), 2.0);
	} else {
		tmp = pow(a, 2.0) + ((0.5 - (0.5 * cos((2.0 * (angle * (((double) M_PI) * 0.005555555555555556)))))) * (b * b));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 0.001)
		tmp = Float64((a ^ 2.0) + (Float64(b * Float64(angle * Float64(pi * fma(Float64(-2.8577960676726107e-8 * Float64(angle * angle)), Float64(pi * pi), 0.005555555555555556)))) ^ 2.0));
	else
		tmp = Float64((a ^ 2.0) + Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(angle * Float64(pi * 0.005555555555555556)))))) * Float64(b * b)));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 0.001], N[(N[Power[a, 2.0], $MachinePrecision] + N[Power[N[(b * N[(angle * N[(Pi * N[(N[(-2.8577960676726107e-8 * N[(angle * angle), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 1e-3

   1. Initial program 88.8%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 89.1%

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

   6. Taylor expanded in angle around 0

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. unpow3 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

      12. +-commutative N/A

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

      13. lower-*.f64 N/A

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

   8. Simplified 83.7%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 0.005555555555555556\right)\right)\right)}\right)}^{2} \]

   if 1e-3 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 60.8%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 60.8

          \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(\sin \left(angle \cdot \left(\color{blue}{\pi} \cdot 0.005555555555555556\right)\right) \cdot b\right)}^{2} \]

   5. Simplified 60.8%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\color{blue}{\left(\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot b\right)}}^{2} \]

   6. Taylor expanded in angle around 0

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

   8. Simplified 61.5%

      \[\leadsto {\left(a \cdot \color{blue}{1}\right)}^{2} + {\left(\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot b\right)}^{2} \]

   10. Applied egg-rr 61.5%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right) \cdot \left(b \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 0.001:\\ \;\;\;\;{a}^{2} + {\left(b \cdot \left(angle \cdot \left(\pi \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 0.005555555555555556\right)\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right) \cdot \left(b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;{a}^{2} + \left(angle \cdot b\right) \cdot \left(\left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(angle \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + b \cdot \left(b \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) 2e-5)
   (+
    (pow a 2.0)
    (* (* angle b) (* (* PI (* PI 3.08641975308642e-5)) (* angle b))))
   (+
    (pow a 2.0)
    (*
     b
     (*
      b
      (- 0.5 (* 0.5 (cos (* 2.0 (* angle (* PI 0.005555555555555556)))))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= 2e-5) {
		tmp = pow(a, 2.0) + ((angle * b) * ((((double) M_PI) * (((double) M_PI) * 3.08641975308642e-5)) * (angle * b)));
	} else {
		tmp = pow(a, 2.0) + (b * (b * (0.5 - (0.5 * cos((2.0 * (angle * (((double) M_PI) * 0.005555555555555556))))))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= 2e-5) {
		tmp = Math.pow(a, 2.0) + ((angle * b) * ((Math.PI * (Math.PI * 3.08641975308642e-5)) * (angle * b)));
	} else {
		tmp = Math.pow(a, 2.0) + (b * (b * (0.5 - (0.5 * Math.cos((2.0 * (angle * (Math.PI * 0.005555555555555556))))))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if (angle / 180.0) <= 2e-5:
		tmp = math.pow(a, 2.0) + ((angle * b) * ((math.pi * (math.pi * 3.08641975308642e-5)) * (angle * b)))
	else:
		tmp = math.pow(a, 2.0) + (b * (b * (0.5 - (0.5 * math.cos((2.0 * (angle * (math.pi * 0.005555555555555556))))))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 2e-5)
		tmp = Float64((a ^ 2.0) + Float64(Float64(angle * b) * Float64(Float64(pi * Float64(pi * 3.08641975308642e-5)) * Float64(angle * b))));
	else
		tmp = Float64((a ^ 2.0) + Float64(b * Float64(b * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(angle * Float64(pi * 0.005555555555555556)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if ((angle / 180.0) <= 2e-5)
		tmp = (a ^ 2.0) + ((angle * b) * ((pi * (pi * 3.08641975308642e-5)) * (angle * b)));
	else
		tmp = (a ^ 2.0) + (b * (b * (0.5 - (0.5 * cos((2.0 * (angle * (pi * 0.005555555555555556))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 2e-5], N[(N[Power[a, 2.0], $MachinePrecision] + N[(N[(angle * b), $MachinePrecision] * N[(N[(Pi * N[(Pi * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision] * N[(angle * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[(b * N[(b * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2.00000000000000016e-5

   1. Initial program 88.7%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 89.1%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. lower-PI.f64 73.4

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right) \cdot \left(\left(\pi \cdot \color{blue}{\pi}\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \]

   8. Simplified 73.4%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(angle \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. associate-*l* N/A

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

      7. rem-square-sqrt N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. associate-*l* N/A

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

      11. associate-*l* N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. rem-square-sqrt N/A

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

      15. associate-*l* N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

   10. Applied egg-rr 84.8%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(\left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(angle \cdot b\right)\right) \cdot \left(angle \cdot b\right)} \]

   if 2.00000000000000016e-5 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 61.3%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 61.3

          \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(\sin \left(angle \cdot \left(\color{blue}{\pi} \cdot 0.005555555555555556\right)\right) \cdot b\right)}^{2} \]

   5. Simplified 61.3%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\color{blue}{\left(\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot b\right)}}^{2} \]

   6. Taylor expanded in angle around 0

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

   8. Simplified 62.0%

      \[\leadsto {\left(a \cdot \color{blue}{1}\right)}^{2} + {\left(\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot b\right)}^{2} \]

   10. Applied egg-rr 61.7%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(b \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right) \cdot b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;{a}^{2} + \left(angle \cdot b\right) \cdot \left(\left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(angle \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + b \cdot \left(b \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{+112}:\\ \;\;\;\;{a}^{2} + \left(\pi \cdot \pi\right) \cdot \left(angle \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + b \cdot \left(\left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(angle \cdot \left(angle \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.5e+112)
   (+
    (pow a 2.0)
    (* (* PI PI) (* angle (* 3.08641975308642e-5 (* angle (* b b))))))
   (+
    (pow a 2.0)
    (* b (* (* PI (* PI 3.08641975308642e-5)) (* angle (* angle b)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.5e+112) {
		tmp = pow(a, 2.0) + ((((double) M_PI) * ((double) M_PI)) * (angle * (3.08641975308642e-5 * (angle * (b * b)))));
	} else {
		tmp = pow(a, 2.0) + (b * ((((double) M_PI) * (((double) M_PI) * 3.08641975308642e-5)) * (angle * (angle * b))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.5e+112) {
		tmp = Math.pow(a, 2.0) + ((Math.PI * Math.PI) * (angle * (3.08641975308642e-5 * (angle * (b * b)))));
	} else {
		tmp = Math.pow(a, 2.0) + (b * ((Math.PI * (Math.PI * 3.08641975308642e-5)) * (angle * (angle * b))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 1.5e+112:
		tmp = math.pow(a, 2.0) + ((math.pi * math.pi) * (angle * (3.08641975308642e-5 * (angle * (b * b)))))
	else:
		tmp = math.pow(a, 2.0) + (b * ((math.pi * (math.pi * 3.08641975308642e-5)) * (angle * (angle * b))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.5e+112)
		tmp = Float64((a ^ 2.0) + Float64(Float64(pi * pi) * Float64(angle * Float64(3.08641975308642e-5 * Float64(angle * Float64(b * b))))));
	else
		tmp = Float64((a ^ 2.0) + Float64(b * Float64(Float64(pi * Float64(pi * 3.08641975308642e-5)) * Float64(angle * Float64(angle * b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 1.5e+112)
		tmp = (a ^ 2.0) + ((pi * pi) * (angle * (3.08641975308642e-5 * (angle * (b * b)))));
	else
		tmp = (a ^ 2.0) + (b * ((pi * (pi * 3.08641975308642e-5)) * (angle * (angle * b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.5e+112], N[(N[Power[a, 2.0], $MachinePrecision] + N[(N[(Pi * Pi), $MachinePrecision] * N[(angle * N[(3.08641975308642e-5 * N[(angle * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[(b * N[(N[(Pi * N[(Pi * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision] * N[(angle * N[(angle * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.4999999999999999e112

   1. Initial program 77.9%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 78.4%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. lower-PI.f64 69.8

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right) \cdot \left(\left(\pi \cdot \color{blue}{\pi}\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \]

   8. Simplified 69.8%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(angle \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 69.8

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(\pi \cdot \pi\right) \cdot \left(angle \cdot \color{blue}{\left(\left(angle \cdot \left(b \cdot b\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)}\right) \]

   10. Applied egg-rr 69.8%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(\pi \cdot \pi\right) \cdot \left(angle \cdot \left(\left(angle \cdot \left(b \cdot b\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)} \]

   if 1.4999999999999999e112 < b

   1. Initial program 95.9%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 95.9%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. lower-PI.f64 65.5

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right) \cdot \left(\left(\pi \cdot \color{blue}{\pi}\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \]

   8. Simplified 65.5%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(angle \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. associate-*l* N/A

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

      7. rem-square-sqrt N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. associate-*l* N/A

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

      11. associate-*l* N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. rem-square-sqrt N/A

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

      15. associate-*l* N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

   10. Applied egg-rr 95.7%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(\left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(angle \cdot \left(angle \cdot b\right)\right)\right) \cdot b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{+112}:\\ \;\;\;\;{a}^{2} + \left(\pi \cdot \pi\right) \cdot \left(angle \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + b \cdot \left(\left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(angle \cdot \left(angle \cdot b\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{+124}:\\ \;\;\;\;{a}^{2} + \left(\pi \cdot \pi\right) \cdot \left(angle \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + \left(\left(angle \cdot b\right) \cdot \left(angle \cdot b\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 2e+124)
   (+
    (pow a 2.0)
    (* (* PI PI) (* angle (* 3.08641975308642e-5 (* angle (* b b))))))
   (+
    (pow a 2.0)
    (* (* (* angle b) (* angle b)) (* (* PI PI) 3.08641975308642e-5)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2e+124) {
		tmp = pow(a, 2.0) + ((((double) M_PI) * ((double) M_PI)) * (angle * (3.08641975308642e-5 * (angle * (b * b)))));
	} else {
		tmp = pow(a, 2.0) + (((angle * b) * (angle * b)) * ((((double) M_PI) * ((double) M_PI)) * 3.08641975308642e-5));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2e+124) {
		tmp = Math.pow(a, 2.0) + ((Math.PI * Math.PI) * (angle * (3.08641975308642e-5 * (angle * (b * b)))));
	} else {
		tmp = Math.pow(a, 2.0) + (((angle * b) * (angle * b)) * ((Math.PI * Math.PI) * 3.08641975308642e-5));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 2e+124:
		tmp = math.pow(a, 2.0) + ((math.pi * math.pi) * (angle * (3.08641975308642e-5 * (angle * (b * b)))))
	else:
		tmp = math.pow(a, 2.0) + (((angle * b) * (angle * b)) * ((math.pi * math.pi) * 3.08641975308642e-5))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 2e+124)
		tmp = Float64((a ^ 2.0) + Float64(Float64(pi * pi) * Float64(angle * Float64(3.08641975308642e-5 * Float64(angle * Float64(b * b))))));
	else
		tmp = Float64((a ^ 2.0) + Float64(Float64(Float64(angle * b) * Float64(angle * b)) * Float64(Float64(pi * pi) * 3.08641975308642e-5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 2e+124)
		tmp = (a ^ 2.0) + ((pi * pi) * (angle * (3.08641975308642e-5 * (angle * (b * b)))));
	else
		tmp = (a ^ 2.0) + (((angle * b) * (angle * b)) * ((pi * pi) * 3.08641975308642e-5));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 2e+124], N[(N[Power[a, 2.0], $MachinePrecision] + N[(N[(Pi * Pi), $MachinePrecision] * N[(angle * N[(3.08641975308642e-5 * N[(angle * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[(N[(N[(angle * b), $MachinePrecision] * N[(angle * b), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.9999999999999999e124

   1. Initial program 78.1%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 78.6%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. lower-PI.f64 70.0

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right) \cdot \left(\left(\pi \cdot \color{blue}{\pi}\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \]

   8. Simplified 70.0%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(angle \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 70.1

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(\pi \cdot \pi\right) \cdot \left(angle \cdot \color{blue}{\left(\left(angle \cdot \left(b \cdot b\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)}\right) \]

   10. Applied egg-rr 70.1%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(\pi \cdot \pi\right) \cdot \left(angle \cdot \left(\left(angle \cdot \left(b \cdot b\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)} \]

   if 1.9999999999999999e124 < b

   1. Initial program 95.8%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 95.8%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. lower-PI.f64 63.9

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right) \cdot \left(\left(\pi \cdot \color{blue}{\pi}\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \]

   8. Simplified 63.9%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(angle \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 95.5

         \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(\left(angle \cdot b\right) \cdot \color{blue}{\left(angle \cdot b\right)}\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \]

   10. Applied egg-rr 95.5%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(\left(angle \cdot b\right) \cdot \left(angle \cdot b\right)\right)} \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{+124}:\\ \;\;\;\;{a}^{2} + \left(\pi \cdot \pi\right) \cdot \left(angle \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + \left(\left(angle \cdot b\right) \cdot \left(angle \cdot b\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{+124}:\\ \;\;\;\;{a}^{2} + \left(\pi \cdot \pi\right) \cdot \left(angle \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + angle \cdot \left(b \cdot \left(b \cdot \left(angle \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 2e+124)
   (+
    (pow a 2.0)
    (* (* PI PI) (* angle (* 3.08641975308642e-5 (* angle (* b b))))))
   (+
    (pow a 2.0)
    (* angle (* b (* b (* angle (* PI (* PI 3.08641975308642e-5)))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2e+124) {
		tmp = pow(a, 2.0) + ((((double) M_PI) * ((double) M_PI)) * (angle * (3.08641975308642e-5 * (angle * (b * b)))));
	} else {
		tmp = pow(a, 2.0) + (angle * (b * (b * (angle * (((double) M_PI) * (((double) M_PI) * 3.08641975308642e-5))))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2e+124) {
		tmp = Math.pow(a, 2.0) + ((Math.PI * Math.PI) * (angle * (3.08641975308642e-5 * (angle * (b * b)))));
	} else {
		tmp = Math.pow(a, 2.0) + (angle * (b * (b * (angle * (Math.PI * (Math.PI * 3.08641975308642e-5))))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 2e+124:
		tmp = math.pow(a, 2.0) + ((math.pi * math.pi) * (angle * (3.08641975308642e-5 * (angle * (b * b)))))
	else:
		tmp = math.pow(a, 2.0) + (angle * (b * (b * (angle * (math.pi * (math.pi * 3.08641975308642e-5))))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 2e+124)
		tmp = Float64((a ^ 2.0) + Float64(Float64(pi * pi) * Float64(angle * Float64(3.08641975308642e-5 * Float64(angle * Float64(b * b))))));
	else
		tmp = Float64((a ^ 2.0) + Float64(angle * Float64(b * Float64(b * Float64(angle * Float64(pi * Float64(pi * 3.08641975308642e-5)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 2e+124)
		tmp = (a ^ 2.0) + ((pi * pi) * (angle * (3.08641975308642e-5 * (angle * (b * b)))));
	else
		tmp = (a ^ 2.0) + (angle * (b * (b * (angle * (pi * (pi * 3.08641975308642e-5))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 2e+124], N[(N[Power[a, 2.0], $MachinePrecision] + N[(N[(Pi * Pi), $MachinePrecision] * N[(angle * N[(3.08641975308642e-5 * N[(angle * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[(angle * N[(b * N[(b * N[(angle * N[(Pi * N[(Pi * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.9999999999999999e124

   1. Initial program 78.1%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 78.6%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. lower-PI.f64 70.0

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right) \cdot \left(\left(\pi \cdot \color{blue}{\pi}\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \]

   8. Simplified 70.0%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(angle \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 70.1

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(\pi \cdot \pi\right) \cdot \left(angle \cdot \color{blue}{\left(\left(angle \cdot \left(b \cdot b\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)}\right) \]

   10. Applied egg-rr 70.1%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(\pi \cdot \pi\right) \cdot \left(angle \cdot \left(\left(angle \cdot \left(b \cdot b\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)} \]

   if 1.9999999999999999e124 < b

   1. Initial program 95.8%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 95.8%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. lower-PI.f64 63.9

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right) \cdot \left(\left(\pi \cdot \color{blue}{\pi}\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \]

   8. Simplified 63.9%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(angle \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   10. Applied egg-rr 88.8%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(b \cdot \left(b \cdot \left(angle \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right)\right) \cdot angle} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{+124}:\\ \;\;\;\;{a}^{2} + \left(\pi \cdot \pi\right) \cdot \left(angle \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + angle \cdot \left(b \cdot \left(b \cdot \left(angle \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{+15}:\\ \;\;\;\;{a}^{2} + \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(angle \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + angle \cdot \left(b \cdot \left(b \cdot \left(angle \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 5e+15)
   (+
    (pow a 2.0)
    (* (* (* PI PI) 3.08641975308642e-5) (* angle (* angle (* b b)))))
   (+
    (pow a 2.0)
    (* angle (* b (* b (* angle (* PI (* PI 3.08641975308642e-5)))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 5e+15) {
		tmp = pow(a, 2.0) + (((((double) M_PI) * ((double) M_PI)) * 3.08641975308642e-5) * (angle * (angle * (b * b))));
	} else {
		tmp = pow(a, 2.0) + (angle * (b * (b * (angle * (((double) M_PI) * (((double) M_PI) * 3.08641975308642e-5))))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 5e+15) {
		tmp = Math.pow(a, 2.0) + (((Math.PI * Math.PI) * 3.08641975308642e-5) * (angle * (angle * (b * b))));
	} else {
		tmp = Math.pow(a, 2.0) + (angle * (b * (b * (angle * (Math.PI * (Math.PI * 3.08641975308642e-5))))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 5e+15:
		tmp = math.pow(a, 2.0) + (((math.pi * math.pi) * 3.08641975308642e-5) * (angle * (angle * (b * b))))
	else:
		tmp = math.pow(a, 2.0) + (angle * (b * (b * (angle * (math.pi * (math.pi * 3.08641975308642e-5))))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 5e+15)
		tmp = Float64((a ^ 2.0) + Float64(Float64(Float64(pi * pi) * 3.08641975308642e-5) * Float64(angle * Float64(angle * Float64(b * b)))));
	else
		tmp = Float64((a ^ 2.0) + Float64(angle * Float64(b * Float64(b * Float64(angle * Float64(pi * Float64(pi * 3.08641975308642e-5)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 5e+15)
		tmp = (a ^ 2.0) + (((pi * pi) * 3.08641975308642e-5) * (angle * (angle * (b * b))));
	else
		tmp = (a ^ 2.0) + (angle * (b * (b * (angle * (pi * (pi * 3.08641975308642e-5))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 5e+15], N[(N[Power[a, 2.0], $MachinePrecision] + N[(N[(N[(Pi * Pi), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision] * N[(angle * N[(angle * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[(angle * N[(b * N[(b * N[(angle * N[(Pi * N[(Pi * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5e15

   1. Initial program 78.0%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 78.5%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. lower-PI.f64 69.1

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right) \cdot \left(\left(\pi \cdot \color{blue}{\pi}\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \]

   8. Simplified 69.1%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(angle \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)} \]

   if 5e15 < b

   1. Initial program 89.7%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 89.6%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. lower-PI.f64 68.8

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right) \cdot \left(\left(\pi \cdot \color{blue}{\pi}\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \]

   8. Simplified 68.8%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(angle \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   10. Applied egg-rr 84.4%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(b \cdot \left(b \cdot \left(angle \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right)\right) \cdot angle} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{+15}:\\ \;\;\;\;{a}^{2} + \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(angle \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + angle \cdot \left(b \cdot \left(b \cdot \left(angle \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ {a}^{2} + \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(angle \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow a 2.0)
  (* (* (* PI PI) 3.08641975308642e-5) (* angle (* angle (* b b))))))

C

double code(double a, double b, double angle) {
	return pow(a, 2.0) + (((((double) M_PI) * ((double) M_PI)) * 3.08641975308642e-5) * (angle * (angle * (b * b))));
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow(a, 2.0) + (((Math.PI * Math.PI) * 3.08641975308642e-5) * (angle * (angle * (b * b))));
}

Python

def code(a, b, angle):
	return math.pow(a, 2.0) + (((math.pi * math.pi) * 3.08641975308642e-5) * (angle * (angle * (b * b))))

Julia

function code(a, b, angle)
	return Float64((a ^ 2.0) + Float64(Float64(Float64(pi * pi) * 3.08641975308642e-5) * Float64(angle * Float64(angle * Float64(b * b)))))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (a ^ 2.0) + (((pi * pi) * 3.08641975308642e-5) * (angle * (angle * (b * b))));
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[a, 2.0], $MachinePrecision] + N[(N[(N[(Pi * Pi), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision] * N[(angle * N[(angle * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.0%

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

3. Taylor expanded in angle around 0

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

5. Simplified 81.4%

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

6. Taylor expanded in angle around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-PI.f64 N/A

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

   15. lower-PI.f64 69.0

       \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right) \cdot \left(\left(\pi \cdot \color{blue}{\pi}\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \]

8. Simplified 69.0%

   \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(angle \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)} \]

9. Final simplification 69.0%

   \[\leadsto {a}^{2} + \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(angle \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right) \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(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)))