ab-angle->ABCF A

Percentage Accurate: 79.4% → 79.3%

Time: 37.9s

Alternatives: 7

Speedup: N/A×

• 37.1% 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: 79.4% 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: 79.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.5%

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

   1. add-sqr-sqrt 40.3%

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

   2. pow2 40.3%

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

   3. associate-*l/ 40.3%

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

   4. associate-*r/ 40.4%

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

   5. div-inv 40.4%

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

   6. metadata-eval 40.4%

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

4. Applied egg-rr 40.4%

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

5. Taylor expanded in angle around 0 85.1%

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

6. Final simplification 85.1%

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

Alternative 2: 79.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.5%

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

   1. unpow2 84.5%

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

   2. associate-*l/ 84.4%

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

   3. associate-/l* 84.5%

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

   4. unpow2 84.5%

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

3. Simplified 84.5%

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

5. Taylor expanded in angle around 0 85.1%

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

6. Taylor expanded in angle around inf 85.0%

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

7. Final simplification 85.0%

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

Alternative 3: 79.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.5%

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

   1. unpow2 84.5%

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

   2. associate-*l/ 84.4%

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

   3. associate-/l* 84.5%

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

   4. unpow2 84.5%

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

3. Simplified 84.5%

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

5. Taylor expanded in angle around 0 85.1%

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

6. Final simplification 85.1%

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

Alternative 4: 73.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 10^{+147}:\\ \;\;\;\;{b}^{2} + \frac{\left(angle \cdot 0.005555555555555556\right) \cdot {\left(a \cdot \pi\right)}^{2}}{\frac{180}{angle}}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(0.005555555555555556 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \left(a \cdot \pi\right)\right)\right) \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1e+147)
   (+
    (pow b 2.0)
    (/ (* (* angle 0.005555555555555556) (pow (* a PI) 2.0)) (/ 180.0 angle)))
   (+
    (pow b 2.0)
    (*
     (* 0.005555555555555556 (* (* angle 0.005555555555555556) (* a PI)))
     (* a (* angle PI))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1e+147) {
		tmp = pow(b, 2.0) + (((angle * 0.005555555555555556) * pow((a * ((double) M_PI)), 2.0)) / (180.0 / angle));
	} else {
		tmp = pow(b, 2.0) + ((0.005555555555555556 * ((angle * 0.005555555555555556) * (a * ((double) M_PI)))) * (a * (angle * ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1e+147) {
		tmp = Math.pow(b, 2.0) + (((angle * 0.005555555555555556) * Math.pow((a * Math.PI), 2.0)) / (180.0 / angle));
	} else {
		tmp = Math.pow(b, 2.0) + ((0.005555555555555556 * ((angle * 0.005555555555555556) * (a * Math.PI))) * (a * (angle * Math.PI)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 1e+147:
		tmp = math.pow(b, 2.0) + (((angle * 0.005555555555555556) * math.pow((a * math.pi), 2.0)) / (180.0 / angle))
	else:
		tmp = math.pow(b, 2.0) + ((0.005555555555555556 * ((angle * 0.005555555555555556) * (a * math.pi))) * (a * (angle * math.pi)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 1e+147)
		tmp = Float64((b ^ 2.0) + Float64(Float64(Float64(angle * 0.005555555555555556) * (Float64(a * pi) ^ 2.0)) / Float64(180.0 / angle)));
	else
		tmp = Float64((b ^ 2.0) + Float64(Float64(0.005555555555555556 * Float64(Float64(angle * 0.005555555555555556) * Float64(a * pi))) * Float64(a * Float64(angle * pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 1e+147)
		tmp = (b ^ 2.0) + (((angle * 0.005555555555555556) * ((a * pi) ^ 2.0)) / (180.0 / angle));
	else
		tmp = (b ^ 2.0) + ((0.005555555555555556 * ((angle * 0.005555555555555556) * (a * pi))) * (a * (angle * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1e+147], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(N[(angle * 0.005555555555555556), $MachinePrecision] * N[Power[N[(a * Pi), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(0.005555555555555556 * N[(N[(angle * 0.005555555555555556), $MachinePrecision] * N[(a * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 9.9999999999999998e146

   1. Initial program 81.8%

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

      1. unpow2 81.8%

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

      2. associate-*l/ 81.6%

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

      3. associate-/l* 81.8%

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

      4. unpow2 81.8%

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

   3. Simplified 81.8%

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

   5. Taylor expanded in angle around 0 82.5%

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

   6. Taylor expanded in angle around 0 75.7%

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

      1. *-commutative 75.7%

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

      2. associate-*l* 75.8%

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

   8. Simplified 75.8%

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

      1. unpow2 75.8%

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

      2. associate-*r* 75.8%

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

      3. associate-*l* 75.4%

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

      4. *-commutative 75.4%

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

      5. *-commutative 75.4%

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

      6. *-commutative 75.4%

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

      7. *-commutative 75.4%

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

      8. associate-*l* 75.4%

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

      9. *-commutative 75.4%

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

   10. Applied egg-rr 75.4%

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

       1. *-commutative 75.4%

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

       2. metadata-eval 75.4%

          \[\leadsto \left(\left(a \cdot \pi\right) \cdot \left(\left(a \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right) + {\left(b \cdot 1\right)}^{2} \]

       3. div-inv 75.4%

          \[\leadsto \left(\left(a \cdot \pi\right) \cdot \left(\left(a \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \color{blue}{\frac{angle}{180}} + {\left(b \cdot 1\right)}^{2} \]

       4. clear-num 75.4%

          \[\leadsto \left(\left(a \cdot \pi\right) \cdot \left(\left(a \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \color{blue}{\frac{1}{\frac{180}{angle}}} + {\left(b \cdot 1\right)}^{2} \]

       5. un-div-inv 75.4%

          \[\leadsto \color{blue}{\frac{\left(a \cdot \pi\right) \cdot \left(\left(a \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)}{\frac{180}{angle}}} + {\left(b \cdot 1\right)}^{2} \]

       6. associate-*r* 76.0%

          \[\leadsto \frac{\color{blue}{\left(\left(a \cdot \pi\right) \cdot \left(a \cdot \pi\right)\right) \cdot \left(angle \cdot 0.005555555555555556\right)}}{\frac{180}{angle}} + {\left(b \cdot 1\right)}^{2} \]

       7. *-commutative 76.0%

          \[\leadsto \frac{\color{blue}{\left(angle \cdot 0.005555555555555556\right) \cdot \left(\left(a \cdot \pi\right) \cdot \left(a \cdot \pi\right)\right)}}{\frac{180}{angle}} + {\left(b \cdot 1\right)}^{2} \]

       8. pow2 76.0%

          \[\leadsto \frac{\left(angle \cdot 0.005555555555555556\right) \cdot \color{blue}{{\left(a \cdot \pi\right)}^{2}}}{\frac{180}{angle}} + {\left(b \cdot 1\right)}^{2} \]

   12. Applied egg-rr 76.0%

       \[\leadsto \color{blue}{\frac{\left(angle \cdot 0.005555555555555556\right) \cdot {\left(a \cdot \pi\right)}^{2}}{\frac{180}{angle}}} + {\left(b \cdot 1\right)}^{2} \]

   if 9.9999999999999998e146 < a

   1. Initial program 97.9%

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

      1. unpow2 97.9%

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

      2. associate-*l/ 98.0%

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

      3. associate-/l* 98.0%

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

      4. unpow2 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in angle around 0 98.0%

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

   6. Taylor expanded in angle around 0 97.9%

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

      1. *-commutative 97.9%

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

      2. associate-*l* 97.9%

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

   8. Simplified 97.9%

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

      1. unpow2 97.9%

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

      2. associate-*r* 97.9%

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

      3. *-commutative 97.9%

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

      4. *-commutative 97.9%

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

      5. associate-*l* 97.9%

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

      6. *-commutative 97.9%

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

      7. associate-*r* 98.0%

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

      8. *-commutative 98.0%

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

   10. Applied egg-rr 98.0%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 10^{+147}:\\ \;\;\;\;{b}^{2} + \frac{\left(angle \cdot 0.005555555555555556\right) \cdot {\left(a \cdot \pi\right)}^{2}}{\frac{180}{angle}}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(0.005555555555555556 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \left(a \cdot \pi\right)\right)\right) \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ {b}^{2} + \left(angle \cdot 0.005555555555555556\right) \cdot \left(\left(a \cdot \pi\right) \cdot \left(a \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow b 2.0)
  (*
   (* angle 0.005555555555555556)
   (* (* a PI) (* a (* angle (/ PI 180.0)))))))

C

double code(double a, double b, double angle) {
	return pow(b, 2.0) + ((angle * 0.005555555555555556) * ((a * ((double) M_PI)) * (a * (angle * (((double) M_PI) / 180.0)))));
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow(b, 2.0) + ((angle * 0.005555555555555556) * ((a * Math.PI) * (a * (angle * (Math.PI / 180.0)))));
}

Python

def code(a, b, angle):
	return math.pow(b, 2.0) + ((angle * 0.005555555555555556) * ((a * math.pi) * (a * (angle * (math.pi / 180.0)))))

Julia

function code(a, b, angle)
	return Float64((b ^ 2.0) + Float64(Float64(angle * 0.005555555555555556) * Float64(Float64(a * pi) * Float64(a * Float64(angle * Float64(pi / 180.0))))))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (b ^ 2.0) + ((angle * 0.005555555555555556) * ((a * pi) * (a * (angle * (pi / 180.0)))));
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(angle * 0.005555555555555556), $MachinePrecision] * N[(N[(a * Pi), $MachinePrecision] * N[(a * N[(angle * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.5%

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

   1. unpow2 84.5%

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

   2. associate-*l/ 84.4%

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

   3. associate-/l* 84.5%

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

   4. unpow2 84.5%

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

3. Simplified 84.5%

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

5. Taylor expanded in angle around 0 85.1%

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

6. Taylor expanded in angle around 0 79.5%

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

   1. *-commutative 79.5%

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

   2. associate-*l* 79.5%

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

8. Simplified 79.5%

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

   1. unpow2 79.5%

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

   2. associate-*r* 79.5%

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

   3. associate-*l* 78.5%

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

   4. *-commutative 78.5%

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

   5. *-commutative 78.5%

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

   6. *-commutative 78.5%

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

   7. *-commutative 78.5%

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

   8. associate-*l* 78.4%

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

   9. *-commutative 78.4%

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

10. Applied egg-rr 78.4%

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

    1. metadata-eval 78.4%

       \[\leadsto \left(angle \cdot 0.005555555555555556\right) \cdot \left(\left(a \cdot \pi\right) \cdot \left(\left(a \cdot \pi\right) \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right) + {\left(b \cdot 1\right)}^{2} \]

    2. div-inv 78.5%

       \[\leadsto \left(angle \cdot 0.005555555555555556\right) \cdot \left(\left(a \cdot \pi\right) \cdot \left(\left(a \cdot \pi\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) + {\left(b \cdot 1\right)}^{2} \]

    3. clear-num 78.4%

       \[\leadsto \left(angle \cdot 0.005555555555555556\right) \cdot \left(\left(a \cdot \pi\right) \cdot \left(\left(a \cdot \pi\right) \cdot \color{blue}{\frac{1}{\frac{180}{angle}}}\right)\right) + {\left(b \cdot 1\right)}^{2} \]

    4. un-div-inv 78.5%

       \[\leadsto \left(angle \cdot 0.005555555555555556\right) \cdot \left(\left(a \cdot \pi\right) \cdot \color{blue}{\frac{a \cdot \pi}{\frac{180}{angle}}}\right) + {\left(b \cdot 1\right)}^{2} \]

12. Applied egg-rr 78.5%

    \[\leadsto \left(angle \cdot 0.005555555555555556\right) \cdot \left(\left(a \cdot \pi\right) \cdot \color{blue}{\frac{a \cdot \pi}{\frac{180}{angle}}}\right) + {\left(b \cdot 1\right)}^{2} \]
13. Step-by-step derivation

    1. associate-/l* 78.4%

       \[\leadsto \left(angle \cdot 0.005555555555555556\right) \cdot \left(\left(a \cdot \pi\right) \cdot \color{blue}{\left(a \cdot \frac{\pi}{\frac{180}{angle}}\right)}\right) + {\left(b \cdot 1\right)}^{2} \]

    2. associate-/r/ 78.5%

       \[\leadsto \left(angle \cdot 0.005555555555555556\right) \cdot \left(\left(a \cdot \pi\right) \cdot \left(a \cdot \color{blue}{\left(\frac{\pi}{180} \cdot angle\right)}\right)\right) + {\left(b \cdot 1\right)}^{2} \]

    3. *-commutative 78.5%

       \[\leadsto \left(angle \cdot 0.005555555555555556\right) \cdot \left(\left(a \cdot \pi\right) \cdot \left(a \cdot \color{blue}{\left(angle \cdot \frac{\pi}{180}\right)}\right)\right) + {\left(b \cdot 1\right)}^{2} \]

14. Simplified 78.5%

    \[\leadsto \left(angle \cdot 0.005555555555555556\right) \cdot \left(\left(a \cdot \pi\right) \cdot \color{blue}{\left(a \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)}\right) + {\left(b \cdot 1\right)}^{2} \]

15. Final simplification 78.5%

    \[\leadsto {b}^{2} + \left(angle \cdot 0.005555555555555556\right) \cdot \left(\left(a \cdot \pi\right) \cdot \left(a \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)\right) \]
16. Add Preprocessing

Alternative 6: 74.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(angle \cdot 0.005555555555555556\right) \cdot \left(a \cdot \pi\right)\\ {b}^{2} + t\_0 \cdot t\_0 \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* angle 0.005555555555555556) (* a PI))))
   (+ (pow b 2.0) (* t_0 t_0))))

C

double code(double a, double b, double angle) {
	double t_0 = (angle * 0.005555555555555556) * (a * ((double) M_PI));
	return pow(b, 2.0) + (t_0 * t_0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (angle * 0.005555555555555556) * (a * Math.PI);
	return Math.pow(b, 2.0) + (t_0 * t_0);
}

Python

def code(a, b, angle):
	t_0 = (angle * 0.005555555555555556) * (a * math.pi)
	return math.pow(b, 2.0) + (t_0 * t_0)

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(angle * 0.005555555555555556) * Float64(a * pi))
	return Float64((b ^ 2.0) + Float64(t_0 * t_0))
end

MATLAB

function tmp = code(a, b, angle)
	t_0 = (angle * 0.005555555555555556) * (a * pi);
	tmp = (b ^ 2.0) + (t_0 * t_0);
end

Wolfram

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

Derivation

1. Initial program 84.5%

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

   1. unpow2 84.5%

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

   2. associate-*l/ 84.4%

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

   3. associate-/l* 84.5%

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

   4. unpow2 84.5%

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

3. Simplified 84.5%

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

5. Taylor expanded in angle around 0 85.1%

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

6. Taylor expanded in angle around 0 79.5%

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

   1. *-commutative 79.5%

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

   2. associate-*l* 79.5%

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

8. Simplified 79.5%

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

   1. unpow2 79.5%

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

   2. *-commutative 79.5%

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

   3. *-commutative 79.5%

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

   4. *-commutative 79.5%

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

   5. associate-*l* 79.5%

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

   6. *-commutative 79.5%

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

   7. *-commutative 79.5%

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

   8. associate-*l* 79.5%

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

   9. *-commutative 79.5%

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

10. Applied egg-rr 79.5%

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

11. Final simplification 79.5%

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

Alternative 7: 74.3% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.5%

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

   1. unpow2 84.5%

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

   2. associate-*l/ 84.4%

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

   3. associate-/l* 84.5%

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

   4. unpow2 84.5%

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

3. Simplified 84.5%

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

5. Taylor expanded in angle around 0 85.1%

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

6. Taylor expanded in angle around 0 79.5%

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

   1. *-commutative 79.5%

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

   2. associate-*l* 79.5%

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

8. Simplified 79.5%

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

   1. unpow2 79.5%

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

   2. associate-*r* 79.5%

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

   3. *-commutative 79.5%

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

   4. *-commutative 79.5%

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

   5. associate-*l* 79.5%

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

   6. *-commutative 79.5%

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

   7. associate-*r* 79.5%

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

   8. *-commutative 79.5%

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

10. Applied egg-rr 79.5%

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

11. Final simplification 79.5%

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

Reproduce

herbie shell --seed 2024100 
(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)))