ab-angle->ABCF C

Percentage Accurate: 80.2% → 80.1%

Time: 19.6s

Alternatives: 8

Speedup: N/A×

• 36.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

3. Simplified 79.3%

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

5. Taylor expanded in angle around 0 79.7%

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

6. Taylor expanded in angle around inf 79.7%

   \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \color{blue}{\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}^{2} \]
7. Step-by-step derivation

   1. *-commutative 79.7%

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

   2. associate-*r* 79.7%

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

   3. *-commutative 79.7%

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

   4. associate-*l* 79.7%

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

8. Simplified 79.7%

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

9. Final simplification 79.7%

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

Alternative 2: 80.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

3. Simplified 79.3%

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

5. Taylor expanded in angle around 0 79.7%

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

6. Taylor expanded in angle around 0 79.7%

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

7. Final simplification 79.7%

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

Alternative 3: 80.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_] := N[(N[Power[a, 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(angle * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.1%

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

3. Simplified 79.3%

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

5. Taylor expanded in angle around 0 79.7%

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

6. Final simplification 79.7%

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

Alternative 4: 67.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{-90}:\\ \;\;\;\;{a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + {\left(angle \cdot \left(b \cdot \pi\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 2.8e-90)
   (pow a 2.0)
   (+ (pow a 2.0) (* (pow (* angle (* b PI)) 2.0) 3.08641975308642e-5))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2.8e-90) {
		tmp = pow(a, 2.0);
	} else {
		tmp = pow(a, 2.0) + (pow((angle * (b * ((double) M_PI))), 2.0) * 3.08641975308642e-5);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2.8e-90) {
		tmp = Math.pow(a, 2.0);
	} else {
		tmp = Math.pow(a, 2.0) + (Math.pow((angle * (b * Math.PI)), 2.0) * 3.08641975308642e-5);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 2.8e-90:
		tmp = math.pow(a, 2.0)
	else:
		tmp = math.pow(a, 2.0) + (math.pow((angle * (b * math.pi)), 2.0) * 3.08641975308642e-5)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 2.8e-90)
		tmp = a ^ 2.0;
	else
		tmp = Float64((a ^ 2.0) + Float64((Float64(angle * Float64(b * pi)) ^ 2.0) * 3.08641975308642e-5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 2.8e-90)
		tmp = a ^ 2.0;
	else
		tmp = (a ^ 2.0) + (((angle * (b * pi)) ^ 2.0) * 3.08641975308642e-5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 2.8e-90], N[Power[a, 2.0], $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[(N[Power[N[(angle * N[(b * Pi), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.7999999999999999e-90

   1. Initial program 76.9%

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

   3. Simplified 77.1%

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

   5. Taylor expanded in angle around 0 77.7%

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

   6. Taylor expanded in angle around 0 71.0%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\color{blue}{\left(-0.005555555555555556 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)}}^{2} \]
   7. Step-by-step derivation

      1. unpow2 71.0%

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

      2. associate-*r* 71.0%

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

      3. associate-*l* 69.7%

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

      4. *-commutative 69.7%

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

      5. *-commutative 69.7%

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

      6. add-sqr-sqrt 40.2%

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

      7. sqrt-prod 59.1%

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

      8. unpow2 59.1%

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

      9. unpow-prod-down 59.1%

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

      10. metadata-eval 59.1%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{\color{blue}{3.08641975308642 \cdot 10^{-5}} \cdot {\left(angle \cdot \left(b \cdot \pi\right)\right)}^{2}}\right) \]

      11. unpow-prod-down 51.1%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left({angle}^{2} \cdot {\left(b \cdot \pi\right)}^{2}\right)}}\right) \]

      12. *-commutative 51.1%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot {\color{blue}{\left(\pi \cdot b\right)}}^{2}\right)}\right) \]

      13. pow-prod-down 51.0%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \color{blue}{\left({\pi}^{2} \cdot {b}^{2}\right)}\right)}\right) \]

      14. associate-*l* 51.0%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{\color{blue}{\left(3.08641975308642 \cdot 10^{-5} \cdot {angle}^{2}\right) \cdot \left({\pi}^{2} \cdot {b}^{2}\right)}}\right) \]

      15. *-commutative 51.0%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{\color{blue}{\left({angle}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\right)} \cdot \left({\pi}^{2} \cdot {b}^{2}\right)}\right) \]

      16. *-commutative 51.0%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{\color{blue}{\left({\pi}^{2} \cdot {b}^{2}\right) \cdot \left({angle}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\right)}}\right) \]

   8. Applied egg-rr 48.4%

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

   9. Taylor expanded in a around inf 60.4%

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

   if 2.7999999999999999e-90 < b

   1. Initial program 83.0%

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

   3. Simplified 83.3%

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

   5. Taylor expanded in angle around 0 83.4%

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

   6. Taylor expanded in angle around 0 81.1%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\color{blue}{\left(-0.005555555555555556 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)}}^{2} \]
   7. Step-by-step derivation

      1. *-commutative 81.1%

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

      2. unpow-prod-down 81.1%

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

      3. *-commutative 81.1%

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

      4. metadata-eval 81.1%

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

   8. Applied egg-rr 81.1%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{-90}:\\ \;\;\;\;{a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + {\left(angle \cdot \left(b \cdot \pi\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{-89}:\\ \;\;\;\;{a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + {\left(-0.005555555555555556 \cdot \left(b \cdot \left(\pi \cdot angle\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.15e-89)
   (pow a 2.0)
   (+ (pow a 2.0) (pow (* -0.005555555555555556 (* b (* PI angle))) 2.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.15e-89) {
		tmp = pow(a, 2.0);
	} else {
		tmp = pow(a, 2.0) + pow((-0.005555555555555556 * (b * (((double) M_PI) * angle))), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.15e-89) {
		tmp = Math.pow(a, 2.0);
	} else {
		tmp = Math.pow(a, 2.0) + Math.pow((-0.005555555555555556 * (b * (Math.PI * angle))), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 1.15e-89:
		tmp = math.pow(a, 2.0)
	else:
		tmp = math.pow(a, 2.0) + math.pow((-0.005555555555555556 * (b * (math.pi * angle))), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.15e-89)
		tmp = a ^ 2.0;
	else
		tmp = Float64((a ^ 2.0) + (Float64(-0.005555555555555556 * Float64(b * Float64(pi * angle))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 1.15e-89)
		tmp = a ^ 2.0;
	else
		tmp = (a ^ 2.0) + ((-0.005555555555555556 * (b * (pi * angle))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.15e-89], N[Power[a, 2.0], $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[Power[N[(-0.005555555555555556 * N[(b * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.15e-89

   1. Initial program 76.9%

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

   3. Simplified 77.1%

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

   5. Taylor expanded in angle around 0 77.7%

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

   6. Taylor expanded in angle around 0 71.0%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\color{blue}{\left(-0.005555555555555556 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)}}^{2} \]
   7. Step-by-step derivation

      1. unpow2 71.0%

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

      2. associate-*r* 71.0%

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

      3. associate-*l* 69.7%

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

      4. *-commutative 69.7%

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

      5. *-commutative 69.7%

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

      6. add-sqr-sqrt 40.2%

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

      7. sqrt-prod 59.1%

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

      8. unpow2 59.1%

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

      9. unpow-prod-down 59.1%

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

      10. metadata-eval 59.1%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{\color{blue}{3.08641975308642 \cdot 10^{-5}} \cdot {\left(angle \cdot \left(b \cdot \pi\right)\right)}^{2}}\right) \]

      11. unpow-prod-down 51.1%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left({angle}^{2} \cdot {\left(b \cdot \pi\right)}^{2}\right)}}\right) \]

      12. *-commutative 51.1%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot {\color{blue}{\left(\pi \cdot b\right)}}^{2}\right)}\right) \]

      13. pow-prod-down 51.0%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \color{blue}{\left({\pi}^{2} \cdot {b}^{2}\right)}\right)}\right) \]

      14. associate-*l* 51.0%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{\color{blue}{\left(3.08641975308642 \cdot 10^{-5} \cdot {angle}^{2}\right) \cdot \left({\pi}^{2} \cdot {b}^{2}\right)}}\right) \]

      15. *-commutative 51.0%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{\color{blue}{\left({angle}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\right)} \cdot \left({\pi}^{2} \cdot {b}^{2}\right)}\right) \]

      16. *-commutative 51.0%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{\color{blue}{\left({\pi}^{2} \cdot {b}^{2}\right) \cdot \left({angle}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\right)}}\right) \]

   8. Applied egg-rr 48.4%

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

   9. Taylor expanded in a around inf 60.4%

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

   if 1.15e-89 < b

   1. Initial program 83.0%

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

   3. Simplified 83.3%

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

   5. Taylor expanded in angle around 0 83.4%

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

   6. Taylor expanded in angle around 0 81.1%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\color{blue}{\left(-0.005555555555555556 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)}}^{2} \]
   7. Step-by-step derivation

      1. add-log-exp 62.3%

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

      2. *-commutative 62.3%

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

      3. exp-prod 62.3%

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

      4. *-commutative 62.3%

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

      5. exp-prod 55.0%

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

      6. *-commutative 55.0%

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

      7. exp-prod 55.0%

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

   8. Applied egg-rr 55.0%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\color{blue}{\log \left({\left({\left({\left(e^{\pi}\right)}^{b}\right)}^{angle}\right)}^{-0.005555555555555556}\right)}}^{2} \]
   9. Step-by-step derivation

      1. log-pow 55.0%

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

      2. log-pow 55.0%

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

      3. log-pow 81.1%

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

      4. rem-log-exp 81.1%

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

      5. *-commutative 81.1%

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

      6. associate-*r* 81.1%

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

   10. Simplified 81.1%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{-89}:\\ \;\;\;\;{a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + {\left(-0.005555555555555556 \cdot \left(b \cdot \left(\pi \cdot angle\right)\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{-90}:\\ \;\;\;\;{a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(a, b \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 2e-90)
   (pow a 2.0)
   (pow (hypot a (* b (* angle (* PI 0.005555555555555556)))) 2.0)))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2e-90) {
		tmp = pow(a, 2.0);
	} else {
		tmp = pow(hypot(a, (b * (angle * (((double) M_PI) * 0.005555555555555556)))), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2e-90) {
		tmp = Math.pow(a, 2.0);
	} else {
		tmp = Math.pow(Math.hypot(a, (b * (angle * (Math.PI * 0.005555555555555556)))), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 2e-90:
		tmp = math.pow(a, 2.0)
	else:
		tmp = math.pow(math.hypot(a, (b * (angle * (math.pi * 0.005555555555555556)))), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 2e-90)
		tmp = a ^ 2.0;
	else
		tmp = hypot(a, Float64(b * Float64(angle * Float64(pi * 0.005555555555555556)))) ^ 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 2e-90)
		tmp = a ^ 2.0;
	else
		tmp = hypot(a, (b * (angle * (pi * 0.005555555555555556)))) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 2e-90], N[Power[a, 2.0], $MachinePrecision], N[Power[N[Sqrt[a ^ 2 + N[(b * N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.99999999999999999e-90

   1. Initial program 76.9%

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

   3. Simplified 77.1%

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

   5. Taylor expanded in angle around 0 77.7%

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

   6. Taylor expanded in angle around 0 71.0%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\color{blue}{\left(-0.005555555555555556 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)}}^{2} \]
   7. Step-by-step derivation

      1. unpow2 71.0%

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

      2. associate-*r* 71.0%

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

      3. associate-*l* 69.7%

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

      4. *-commutative 69.7%

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

      5. *-commutative 69.7%

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

      6. add-sqr-sqrt 40.2%

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

      7. sqrt-prod 59.1%

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

      8. unpow2 59.1%

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

      9. unpow-prod-down 59.1%

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

      10. metadata-eval 59.1%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{\color{blue}{3.08641975308642 \cdot 10^{-5}} \cdot {\left(angle \cdot \left(b \cdot \pi\right)\right)}^{2}}\right) \]

      11. unpow-prod-down 51.1%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left({angle}^{2} \cdot {\left(b \cdot \pi\right)}^{2}\right)}}\right) \]

      12. *-commutative 51.1%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot {\color{blue}{\left(\pi \cdot b\right)}}^{2}\right)}\right) \]

      13. pow-prod-down 51.0%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \color{blue}{\left({\pi}^{2} \cdot {b}^{2}\right)}\right)}\right) \]

      14. associate-*l* 51.0%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{\color{blue}{\left(3.08641975308642 \cdot 10^{-5} \cdot {angle}^{2}\right) \cdot \left({\pi}^{2} \cdot {b}^{2}\right)}}\right) \]

      15. *-commutative 51.0%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{\color{blue}{\left({angle}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\right)} \cdot \left({\pi}^{2} \cdot {b}^{2}\right)}\right) \]

      16. *-commutative 51.0%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{\color{blue}{\left({\pi}^{2} \cdot {b}^{2}\right) \cdot \left({angle}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\right)}}\right) \]

   8. Applied egg-rr 48.4%

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

   9. Taylor expanded in a around inf 60.4%

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

   if 1.99999999999999999e-90 < b

   1. Initial program 83.0%

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

   3. Simplified 83.3%

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

   5. Taylor expanded in angle around 0 83.4%

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

   6. Taylor expanded in angle around 0 81.1%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\color{blue}{\left(-0.005555555555555556 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)}}^{2} \]
   7. Step-by-step derivation

      1. unpow2 81.1%

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

      2. associate-*r* 81.1%

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

      3. associate-*l* 77.9%

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

      4. *-commutative 77.9%

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

      5. *-commutative 77.9%

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

      6. add-sqr-sqrt 37.4%

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

      7. sqrt-prod 50.1%

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

      8. unpow2 50.1%

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

      9. unpow-prod-down 50.1%

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

      10. metadata-eval 50.1%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{\color{blue}{3.08641975308642 \cdot 10^{-5}} \cdot {\left(angle \cdot \left(b \cdot \pi\right)\right)}^{2}}\right) \]

      11. unpow-prod-down 41.3%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left({angle}^{2} \cdot {\left(b \cdot \pi\right)}^{2}\right)}}\right) \]

      12. *-commutative 41.3%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot {\color{blue}{\left(\pi \cdot b\right)}}^{2}\right)}\right) \]

      13. pow-prod-down 41.3%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \color{blue}{\left({\pi}^{2} \cdot {b}^{2}\right)}\right)}\right) \]

      14. associate-*l* 41.3%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{\color{blue}{\left(3.08641975308642 \cdot 10^{-5} \cdot {angle}^{2}\right) \cdot \left({\pi}^{2} \cdot {b}^{2}\right)}}\right) \]

      15. *-commutative 41.3%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{\color{blue}{\left({angle}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\right)} \cdot \left({\pi}^{2} \cdot {b}^{2}\right)}\right) \]

      16. *-commutative 41.3%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{\color{blue}{\left({\pi}^{2} \cdot {b}^{2}\right) \cdot \left({angle}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\right)}}\right) \]

   8. Applied egg-rr 27.5%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \left(\left(\pi \cdot b\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 27.0%

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

      2. pow2 27.0%

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

   10. Applied egg-rr 81.1%

       \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(a, b \cdot \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right)\right)\right)}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{-90}:\\ \;\;\;\;{a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(a, b \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(b \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\\ \mathbf{if}\;b \leq 3.6 \cdot 10^{-91}:\\ \;\;\;\;{a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + t\_0 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* b PI) (* angle 0.005555555555555556))))
   (if (<= b 3.6e-91) (pow a 2.0) (+ (pow a 2.0) (* t_0 t_0)))))

C

double code(double a, double b, double angle) {
	double t_0 = (b * ((double) M_PI)) * (angle * 0.005555555555555556);
	double tmp;
	if (b <= 3.6e-91) {
		tmp = pow(a, 2.0);
	} else {
		tmp = pow(a, 2.0) + (t_0 * t_0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (b * Math.PI) * (angle * 0.005555555555555556);
	double tmp;
	if (b <= 3.6e-91) {
		tmp = Math.pow(a, 2.0);
	} else {
		tmp = Math.pow(a, 2.0) + (t_0 * t_0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = (b * math.pi) * (angle * 0.005555555555555556)
	tmp = 0
	if b <= 3.6e-91:
		tmp = math.pow(a, 2.0)
	else:
		tmp = math.pow(a, 2.0) + (t_0 * t_0)
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(b * pi) * Float64(angle * 0.005555555555555556))
	tmp = 0.0
	if (b <= 3.6e-91)
		tmp = a ^ 2.0;
	else
		tmp = Float64((a ^ 2.0) + Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = (b * pi) * (angle * 0.005555555555555556);
	tmp = 0.0;
	if (b <= 3.6e-91)
		tmp = a ^ 2.0;
	else
		tmp = (a ^ 2.0) + (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(b * Pi), $MachinePrecision] * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 3.6e-91], N[Power[a, 2.0], $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 3.6e-91

   1. Initial program 76.9%

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

   3. Simplified 77.1%

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

   5. Taylor expanded in angle around 0 77.7%

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

   6. Taylor expanded in angle around 0 71.0%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\color{blue}{\left(-0.005555555555555556 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)}}^{2} \]
   7. Step-by-step derivation

      1. unpow2 71.0%

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

      2. associate-*r* 71.0%

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

      3. associate-*l* 69.7%

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

      4. *-commutative 69.7%

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

      5. *-commutative 69.7%

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

      6. add-sqr-sqrt 40.2%

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

      7. sqrt-prod 59.1%

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

      8. unpow2 59.1%

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

      9. unpow-prod-down 59.1%

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

      10. metadata-eval 59.1%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{\color{blue}{3.08641975308642 \cdot 10^{-5}} \cdot {\left(angle \cdot \left(b \cdot \pi\right)\right)}^{2}}\right) \]

      11. unpow-prod-down 51.1%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left({angle}^{2} \cdot {\left(b \cdot \pi\right)}^{2}\right)}}\right) \]

      12. *-commutative 51.1%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot {\color{blue}{\left(\pi \cdot b\right)}}^{2}\right)}\right) \]

      13. pow-prod-down 51.0%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \color{blue}{\left({\pi}^{2} \cdot {b}^{2}\right)}\right)}\right) \]

      14. associate-*l* 51.0%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{\color{blue}{\left(3.08641975308642 \cdot 10^{-5} \cdot {angle}^{2}\right) \cdot \left({\pi}^{2} \cdot {b}^{2}\right)}}\right) \]

      15. *-commutative 51.0%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{\color{blue}{\left({angle}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\right)} \cdot \left({\pi}^{2} \cdot {b}^{2}\right)}\right) \]

      16. *-commutative 51.0%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{\color{blue}{\left({\pi}^{2} \cdot {b}^{2}\right) \cdot \left({angle}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\right)}}\right) \]

   8. Applied egg-rr 48.4%

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

   9. Taylor expanded in a around inf 60.4%

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

   if 3.6e-91 < b

   1. Initial program 83.0%

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

   3. Simplified 83.3%

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

   5. Taylor expanded in angle around 0 83.4%

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

   6. Taylor expanded in angle around 0 81.1%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\color{blue}{\left(-0.005555555555555556 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)}}^{2} \]
   7. Step-by-step derivation

      1. unpow-prod-down 81.1%

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

      2. metadata-eval 81.1%

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

      3. *-commutative 81.1%

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

      4. *-commutative 81.1%

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

      5. pow-prod-down 66.0%

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

      6. *-commutative 66.0%

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

      7. pow-prod-down 66.0%

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

      8. associate-*r* 66.0%

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

      9. add-sqr-sqrt 66.0%

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

   8. Applied egg-rr 81.0%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.6 \cdot 10^{-91}:\\ \;\;\;\;{a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + \left(\left(b \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.0% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ {a}^{2} \end{array} \]

FPCore

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

C

double code(double a, double b, double angle) {
	return pow(a, 2.0);
}

Fortran

real(8) function code(a, b, angle)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    code = a ** 2.0d0
end function

Java

public static double code(double a, double b, double angle) {
	return Math.pow(a, 2.0);
}

Python

def code(a, b, angle):
	return math.pow(a, 2.0)

Julia

function code(a, b, angle)
	return a ^ 2.0
end

MATLAB

function tmp = code(a, b, angle)
	tmp = a ^ 2.0;
end

Wolfram

code[a_, b_, angle_] := N[Power[a, 2.0], $MachinePrecision]

Derivation

1. Initial program 79.1%

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

3. Simplified 79.3%

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

5. Taylor expanded in angle around 0 79.7%

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

6. Taylor expanded in angle around 0 74.6%

   \[\leadsto {\left(a \cdot 1\right)}^{2} + {\color{blue}{\left(-0.005555555555555556 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)}}^{2} \]
7. Step-by-step derivation

   1. unpow2 74.6%

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

   2. associate-*r* 74.6%

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

   3. associate-*l* 72.6%

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

   4. *-commutative 72.6%

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

   5. *-commutative 72.6%

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

   6. add-sqr-sqrt 39.2%

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

   7. sqrt-prod 55.9%

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

   8. unpow2 55.9%

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

   9. unpow-prod-down 55.9%

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

   10. metadata-eval 55.9%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{\color{blue}{3.08641975308642 \cdot 10^{-5}} \cdot {\left(angle \cdot \left(b \cdot \pi\right)\right)}^{2}}\right) \]

   11. unpow-prod-down 47.6%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left({angle}^{2} \cdot {\left(b \cdot \pi\right)}^{2}\right)}}\right) \]

   12. *-commutative 47.6%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot {\color{blue}{\left(\pi \cdot b\right)}}^{2}\right)}\right) \]

   13. pow-prod-down 47.6%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \color{blue}{\left({\pi}^{2} \cdot {b}^{2}\right)}\right)}\right) \]

   14. associate-*l* 47.6%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{\color{blue}{\left(3.08641975308642 \cdot 10^{-5} \cdot {angle}^{2}\right) \cdot \left({\pi}^{2} \cdot {b}^{2}\right)}}\right) \]

   15. *-commutative 47.6%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{\color{blue}{\left({angle}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\right)} \cdot \left({\pi}^{2} \cdot {b}^{2}\right)}\right) \]

   16. *-commutative 47.6%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot -0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \sqrt{\color{blue}{\left({\pi}^{2} \cdot {b}^{2}\right) \cdot \left({angle}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\right)}}\right) \]

8. Applied egg-rr 41.0%

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

9. Taylor expanded in a around inf 53.8%

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

10. Final simplification 53.8%

    \[\leadsto {a}^{2} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024075 
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* PI (/ angle 180.0)))) 2.0) (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))