ab-angle->ABCF A

Percentage Accurate: 79.9% → 79.4%

Time: 12.3s

Alternatives: 7

Speedup: N/A×

• 36.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;{b}^{2} + \left(a \cdot angle\_m\right) \cdot \left(angle\_m \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)\right) \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= (/ angle_m 180.0) 2e-13)
   (+
    (pow b 2.0)
    (* (* a angle_m) (* angle_m (* 3.08641975308642e-5 (* a (* PI PI))))))
   (+
    (pow b 2.0)
    (*
     (- 0.5 (* 0.5 (cos (* 2.0 (* PI (* angle_m 0.005555555555555556))))))
     (* a a)))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 2e-13) {
		tmp = pow(b, 2.0) + ((a * angle_m) * (angle_m * (3.08641975308642e-5 * (a * (((double) M_PI) * ((double) M_PI))))));
	} else {
		tmp = pow(b, 2.0) + ((0.5 - (0.5 * cos((2.0 * (((double) M_PI) * (angle_m * 0.005555555555555556)))))) * (a * a));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 2e-13) {
		tmp = Math.pow(b, 2.0) + ((a * angle_m) * (angle_m * (3.08641975308642e-5 * (a * (Math.PI * Math.PI)))));
	} else {
		tmp = Math.pow(b, 2.0) + ((0.5 - (0.5 * Math.cos((2.0 * (Math.PI * (angle_m * 0.005555555555555556)))))) * (a * a));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 2e-13:
		tmp = math.pow(b, 2.0) + ((a * angle_m) * (angle_m * (3.08641975308642e-5 * (a * (math.pi * math.pi)))))
	else:
		tmp = math.pow(b, 2.0) + ((0.5 - (0.5 * math.cos((2.0 * (math.pi * (angle_m * 0.005555555555555556)))))) * (a * a))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e-13)
		tmp = Float64((b ^ 2.0) + Float64(Float64(a * angle_m) * Float64(angle_m * Float64(3.08641975308642e-5 * Float64(a * Float64(pi * pi))))));
	else
		tmp = Float64((b ^ 2.0) + Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(pi * Float64(angle_m * 0.005555555555555556)))))) * Float64(a * a)));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if ((angle_m / 180.0) <= 2e-13)
		tmp = (b ^ 2.0) + ((a * angle_m) * (angle_m * (3.08641975308642e-5 * (a * (pi * pi)))));
	else
		tmp = (b ^ 2.0) + ((0.5 - (0.5 * cos((2.0 * (pi * (angle_m * 0.005555555555555556)))))) * (a * a));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e-13], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(a * angle$95$m), $MachinePrecision] * N[(angle$95$m * N[(3.08641975308642e-5 * N[(a * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2.0000000000000001e-13

   1. Initial program 87.0%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 87.0%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-PI.f64 N/A

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

      18. lower-PI.f64 69.8

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

   8. Simplified 69.8%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

   10. Applied egg-rr 83.8%

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

   11. Taylor expanded in a around 0

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-PI.f64 N/A

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

       6. lower-PI.f64 83.9

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

   13. Simplified 83.9%

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

   if 2.0000000000000001e-13 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 59.1%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 59.1%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-sin.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 59.2

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

   8. Simplified 59.2%

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

   10. Applied egg-rr 59.3%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;{b}^{2} + \left(a \cdot angle\right) \cdot \left(angle \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \cdot \left(a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(a \cdot \sin \left(\frac{1}{\frac{180}{angle\_m \cdot \pi}}\right)\right)}^{2} + {b}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+ (pow (* a (sin (/ 1.0 (/ 180.0 (* angle_m PI))))) 2.0) (pow b 2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((a * sin((1.0 / (180.0 / (angle_m * ((double) M_PI)))))), 2.0) + pow(b, 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return Math.pow((a * Math.sin((1.0 / (180.0 / (angle_m * Math.PI))))), 2.0) + Math.pow(b, 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return math.pow((a * math.sin((1.0 / (180.0 / (angle_m * math.pi))))), 2.0) + math.pow(b, 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(a * sin(Float64(1.0 / Float64(180.0 / Float64(angle_m * pi))))) ^ 2.0) + (b ^ 2.0))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((a * sin((1.0 / (180.0 / (angle_m * pi))))) ^ 2.0) + (b ^ 2.0);
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(a * N[Sin[N[(1.0 / N[(180.0 / N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.9%

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

3. Taylor expanded in angle around 0

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

5. Simplified 78.9%

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

   1. lift-PI.f64 N/A

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

   2. associate-*l/ N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 79.0

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

7. Applied egg-rr 79.0%

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

8. Final simplification 79.0%

   \[\leadsto {\left(a \cdot \sin \left(\frac{1}{\frac{180}{angle \cdot \pi}}\right)\right)}^{2} + {b}^{2} \]
9. Add Preprocessing

Alternative 3: 79.9% accurate, 1.4× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+ (pow b 2.0) (pow (* a (sin (* PI (* angle_m 0.005555555555555556)))) 2.0)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[b, 2.0], $MachinePrecision] + N[Power[N[(a * N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.9%

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

3. Taylor expanded in angle around 0

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

5. Simplified 78.9%

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

   1. lift-/.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 78.9

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

   4. lift-/.f64 N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. lower-*.f64 79.0

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

7. Applied egg-rr 79.0%

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

8. Final simplification 79.0%

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

Alternative 4: 79.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{-20}:\\ \;\;\;\;{b}^{2} + \left(a \cdot angle\_m\right) \cdot \left(angle\_m \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + a \cdot \left(a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= (/ angle_m 180.0) 5e-20)
   (+
    (pow b 2.0)
    (* (* a angle_m) (* angle_m (* 3.08641975308642e-5 (* a (* PI PI))))))
   (+
    (pow b 2.0)
    (*
     a
     (*
      a
      (-
       0.5
       (* 0.5 (cos (* 2.0 (* PI (* angle_m 0.005555555555555556)))))))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 5e-20) {
		tmp = pow(b, 2.0) + ((a * angle_m) * (angle_m * (3.08641975308642e-5 * (a * (((double) M_PI) * ((double) M_PI))))));
	} else {
		tmp = pow(b, 2.0) + (a * (a * (0.5 - (0.5 * cos((2.0 * (((double) M_PI) * (angle_m * 0.005555555555555556))))))));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 5e-20) {
		tmp = Math.pow(b, 2.0) + ((a * angle_m) * (angle_m * (3.08641975308642e-5 * (a * (Math.PI * Math.PI)))));
	} else {
		tmp = Math.pow(b, 2.0) + (a * (a * (0.5 - (0.5 * Math.cos((2.0 * (Math.PI * (angle_m * 0.005555555555555556))))))));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 5e-20:
		tmp = math.pow(b, 2.0) + ((a * angle_m) * (angle_m * (3.08641975308642e-5 * (a * (math.pi * math.pi)))))
	else:
		tmp = math.pow(b, 2.0) + (a * (a * (0.5 - (0.5 * math.cos((2.0 * (math.pi * (angle_m * 0.005555555555555556))))))))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e-20)
		tmp = Float64((b ^ 2.0) + Float64(Float64(a * angle_m) * Float64(angle_m * Float64(3.08641975308642e-5 * Float64(a * Float64(pi * pi))))));
	else
		tmp = Float64((b ^ 2.0) + Float64(a * Float64(a * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(pi * Float64(angle_m * 0.005555555555555556)))))))));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if ((angle_m / 180.0) <= 5e-20)
		tmp = (b ^ 2.0) + ((a * angle_m) * (angle_m * (3.08641975308642e-5 * (a * (pi * pi)))));
	else
		tmp = (b ^ 2.0) + (a * (a * (0.5 - (0.5 * cos((2.0 * (pi * (angle_m * 0.005555555555555556))))))));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e-20], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(a * angle$95$m), $MachinePrecision] * N[(angle$95$m * N[(3.08641975308642e-5 * N[(a * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[(a * N[(a * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(Pi * N[(angle$95$m * 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)) < 4.9999999999999999e-20

   1. Initial program 86.9%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 87.0%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-PI.f64 N/A

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

      18. lower-PI.f64 69.6

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

   8. Simplified 69.6%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

   10. Applied egg-rr 83.7%

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

   11. Taylor expanded in a around 0

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-PI.f64 N/A

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

       6. lower-PI.f64 83.8

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

   13. Simplified 83.8%

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

   if 4.9999999999999999e-20 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 59.6%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 59.6%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-sin.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 59.8

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

   8. Simplified 59.8%

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

   10. Applied egg-rr 59.9%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-20}:\\ \;\;\;\;{b}^{2} + \left(a \cdot angle\right) \cdot \left(angle \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + a \cdot \left(a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 2 \cdot 10^{+120}:\\ \;\;\;\;{b}^{2} + \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle\_m \cdot \left(a \cdot a\right)\right)\right) \cdot \left(angle\_m \cdot \left(\pi \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(a \cdot angle\_m\right) \cdot \left(angle\_m \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 2e+120)
   (+
    (pow b 2.0)
    (* (* 3.08641975308642e-5 (* angle_m (* a a))) (* angle_m (* PI PI))))
   (+
    (pow b 2.0)
    (* (* a angle_m) (* angle_m (* 3.08641975308642e-5 (* a (* PI PI))))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 2e+120) {
		tmp = pow(b, 2.0) + ((3.08641975308642e-5 * (angle_m * (a * a))) * (angle_m * (((double) M_PI) * ((double) M_PI))));
	} else {
		tmp = pow(b, 2.0) + ((a * angle_m) * (angle_m * (3.08641975308642e-5 * (a * (((double) M_PI) * ((double) M_PI))))));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 2e+120) {
		tmp = Math.pow(b, 2.0) + ((3.08641975308642e-5 * (angle_m * (a * a))) * (angle_m * (Math.PI * Math.PI)));
	} else {
		tmp = Math.pow(b, 2.0) + ((a * angle_m) * (angle_m * (3.08641975308642e-5 * (a * (Math.PI * Math.PI)))));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 2e+120:
		tmp = math.pow(b, 2.0) + ((3.08641975308642e-5 * (angle_m * (a * a))) * (angle_m * (math.pi * math.pi)))
	else:
		tmp = math.pow(b, 2.0) + ((a * angle_m) * (angle_m * (3.08641975308642e-5 * (a * (math.pi * math.pi)))))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 2e+120)
		tmp = Float64((b ^ 2.0) + Float64(Float64(3.08641975308642e-5 * Float64(angle_m * Float64(a * a))) * Float64(angle_m * Float64(pi * pi))));
	else
		tmp = Float64((b ^ 2.0) + Float64(Float64(a * angle_m) * Float64(angle_m * Float64(3.08641975308642e-5 * Float64(a * Float64(pi * pi))))));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (a <= 2e+120)
		tmp = (b ^ 2.0) + ((3.08641975308642e-5 * (angle_m * (a * a))) * (angle_m * (pi * pi)));
	else
		tmp = (b ^ 2.0) + ((a * angle_m) * (angle_m * (3.08641975308642e-5 * (a * (pi * pi)))));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 2e+120], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(3.08641975308642e-5 * N[(angle$95$m * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(angle$95$m * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(a * angle$95$m), $MachinePrecision] * N[(angle$95$m * N[(3.08641975308642e-5 * N[(a * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2e120

   1. Initial program 75.3%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 75.3%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-PI.f64 N/A

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

      18. lower-PI.f64 62.0

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

   8. Simplified 62.0%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. associate-*r* N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   10. Applied egg-rr 68.3%

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

   if 2e120 < a

   1. Initial program 96.1%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 96.1%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-PI.f64 N/A

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

      18. lower-PI.f64 70.4

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

   8. Simplified 70.4%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

   10. Applied egg-rr 95.7%

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

   11. Taylor expanded in a around 0

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-PI.f64 N/A

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

       6. lower-PI.f64 95.8

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

   13. Simplified 95.8%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2 \cdot 10^{+120}:\\ \;\;\;\;{b}^{2} + \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right) \cdot \left(angle \cdot \left(\pi \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(a \cdot angle\right) \cdot \left(angle \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 1.85 \cdot 10^{-187}:\\ \;\;\;\;{b}^{2} + angle\_m \cdot \left(angle\_m \cdot \left(\pi \cdot \left(a \cdot \left(a \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(a \cdot angle\_m\right) \cdot \left(angle\_m \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 1.85e-187)
   (+
    (pow b 2.0)
    (* angle_m (* angle_m (* PI (* a (* a (* PI 3.08641975308642e-5)))))))
   (+
    (pow b 2.0)
    (* (* a angle_m) (* angle_m (* 3.08641975308642e-5 (* a (* PI PI))))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.85e-187) {
		tmp = pow(b, 2.0) + (angle_m * (angle_m * (((double) M_PI) * (a * (a * (((double) M_PI) * 3.08641975308642e-5))))));
	} else {
		tmp = pow(b, 2.0) + ((a * angle_m) * (angle_m * (3.08641975308642e-5 * (a * (((double) M_PI) * ((double) M_PI))))));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.85e-187) {
		tmp = Math.pow(b, 2.0) + (angle_m * (angle_m * (Math.PI * (a * (a * (Math.PI * 3.08641975308642e-5))))));
	} else {
		tmp = Math.pow(b, 2.0) + ((a * angle_m) * (angle_m * (3.08641975308642e-5 * (a * (Math.PI * Math.PI)))));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 1.85e-187:
		tmp = math.pow(b, 2.0) + (angle_m * (angle_m * (math.pi * (a * (a * (math.pi * 3.08641975308642e-5))))))
	else:
		tmp = math.pow(b, 2.0) + ((a * angle_m) * (angle_m * (3.08641975308642e-5 * (a * (math.pi * math.pi)))))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 1.85e-187)
		tmp = Float64((b ^ 2.0) + Float64(angle_m * Float64(angle_m * Float64(pi * Float64(a * Float64(a * Float64(pi * 3.08641975308642e-5)))))));
	else
		tmp = Float64((b ^ 2.0) + Float64(Float64(a * angle_m) * Float64(angle_m * Float64(3.08641975308642e-5 * Float64(a * Float64(pi * pi))))));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (a <= 1.85e-187)
		tmp = (b ^ 2.0) + (angle_m * (angle_m * (pi * (a * (a * (pi * 3.08641975308642e-5))))));
	else
		tmp = (b ^ 2.0) + ((a * angle_m) * (angle_m * (3.08641975308642e-5 * (a * (pi * pi)))));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 1.85e-187], N[(N[Power[b, 2.0], $MachinePrecision] + N[(angle$95$m * N[(angle$95$m * N[(Pi * N[(a * N[(a * N[(Pi * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(a * angle$95$m), $MachinePrecision] * N[(angle$95$m * N[(3.08641975308642e-5 * N[(a * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.85000000000000005e-187

   1. Initial program 78.6%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 78.7%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-PI.f64 N/A

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

      18. lower-PI.f64 63.4

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

   8. Simplified 63.4%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   10. Applied egg-rr 70.5%

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

   if 1.85000000000000005e-187 < a

   1. Initial program 79.4%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 79.3%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-PI.f64 N/A

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

      18. lower-PI.f64 63.6

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

   8. Simplified 63.6%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

   10. Applied egg-rr 76.3%

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

   11. Taylor expanded in a around 0

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-PI.f64 N/A

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

       6. lower-PI.f64 76.3

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

   13. Simplified 76.3%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.85 \cdot 10^{-187}:\\ \;\;\;\;{b}^{2} + angle \cdot \left(angle \cdot \left(\pi \cdot \left(a \cdot \left(a \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(a \cdot angle\right) \cdot \left(angle \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.6% accurate, 3.3× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow b 2.0)
  (* (* a angle_m) (* angle_m (* 3.08641975308642e-5 (* a (* PI PI)))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(a * angle$95$m), $MachinePrecision] * N[(angle$95$m * N[(3.08641975308642e-5 * N[(a * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.9%

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

3. Taylor expanded in angle around 0

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

5. Simplified 78.9%

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

6. Taylor expanded in angle around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-PI.f64 N/A

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

   18. lower-PI.f64 63.5

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

8. Simplified 63.5%

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*l* N/A

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

   12. associate-*r* N/A

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

   13. associate-*l* N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-*.f64 N/A

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

10. Applied egg-rr 74.4%

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

11. Taylor expanded in a around 0

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

    1. lower-*.f64 N/A

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

    2. lower-*.f64 N/A

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

    3. unpow2 N/A

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

    4. lower-*.f64 N/A

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

    5. lower-PI.f64 N/A

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

    6. lower-PI.f64 74.5

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

13. Simplified 74.5%

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

14. Final simplification 74.5%

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

Reproduce

herbie shell --seed 2024215 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))