ab-angle->ABCF A

Percentage Accurate: 79.6% → 79.4%

Time: 1.1min

Alternatives: 10

Speedup: 1.0×

• 36.7% 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: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \sin \left({\left(\sqrt[3]{angle \cdot \left(\sqrt[3]{{\pi}^{3}} \cdot 0.005555555555555556\right)}\right)}^{3}\right)\right)}^{2} + {\left(b \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow
   (*
    a
    (sin
     (pow (cbrt (* angle (* (cbrt (pow PI 3.0)) 0.005555555555555556))) 3.0)))
   2.0)
  (pow (* b (cos (* PI (/ angle 180.0)))) 2.0)))

C

double code(double a, double b, double angle) {
	return pow((a * sin(pow(cbrt((angle * (cbrt(pow(((double) M_PI), 3.0)) * 0.005555555555555556))), 3.0))), 2.0) + pow((b * cos((((double) M_PI) * (angle / 180.0)))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.sin(Math.pow(Math.cbrt((angle * (Math.cbrt(Math.pow(Math.PI, 3.0)) * 0.005555555555555556))), 3.0))), 2.0) + Math.pow((b * Math.cos((Math.PI * (angle / 180.0)))), 2.0);
}

Julia

function code(a, b, angle)
	return Float64((Float64(a * sin((cbrt(Float64(angle * Float64(cbrt((pi ^ 3.0)) * 0.005555555555555556))) ^ 3.0))) ^ 2.0) + (Float64(b * cos(Float64(pi * Float64(angle / 180.0)))) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[Power[N[Power[N[(angle * N[(N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision] * 0.005555555555555556), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.2%

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

   1. associate-*l/ 80.1%

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

   2. associate-*r/ 80.1%

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

   3. add-cube-cbrt 80.1%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\sqrt[3]{angle \cdot \frac{\pi}{180}} \cdot \sqrt[3]{angle \cdot \frac{\pi}{180}}\right) \cdot \sqrt[3]{angle \cdot \frac{\pi}{180}}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   4. pow3 80.2%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left({\left(\sqrt[3]{angle \cdot \frac{\pi}{180}}\right)}^{3}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   5. div-inv 80.2%

      \[\leadsto {\left(a \cdot \sin \left({\left(\sqrt[3]{angle \cdot \color{blue}{\left(\pi \cdot \frac{1}{180}\right)}}\right)}^{3}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   6. metadata-eval 80.2%

      \[\leadsto {\left(a \cdot \sin \left({\left(\sqrt[3]{angle \cdot \left(\pi \cdot \color{blue}{0.005555555555555556}\right)}\right)}^{3}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

3. Applied egg-rr 80.2%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left({\left(\sqrt[3]{angle \cdot \left(\pi \cdot 0.005555555555555556\right)}\right)}^{3}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
4. Step-by-step derivation

   1. add-cbrt-cube 80.3%

      \[\leadsto {\left(a \cdot \sin \left({\left(\sqrt[3]{angle \cdot \left(\color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}} \cdot 0.005555555555555556\right)}\right)}^{3}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   2. pow3 80.3%

      \[\leadsto {\left(a \cdot \sin \left({\left(\sqrt[3]{angle \cdot \left(\sqrt[3]{\color{blue}{{\pi}^{3}}} \cdot 0.005555555555555556\right)}\right)}^{3}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

5. Applied egg-rr 80.3%

   \[\leadsto {\left(a \cdot \sin \left({\left(\sqrt[3]{angle \cdot \left(\color{blue}{\sqrt[3]{{\pi}^{3}}} \cdot 0.005555555555555556\right)}\right)}^{3}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

6. Final simplification 80.3%

   \[\leadsto {\left(a \cdot \sin \left({\left(\sqrt[3]{angle \cdot \left(\sqrt[3]{{\pi}^{3}} \cdot 0.005555555555555556\right)}\right)}^{3}\right)\right)}^{2} + {\left(b \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

Alternative 2: 38.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

   1. associate-*l/ 80.1%

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

   2. associate-*r/ 80.3%

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

   3. add-sqr-sqrt 38.5%

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

   4. pow2 38.5%

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

   5. div-inv 38.5%

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

   6. metadata-eval 38.5%

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

3. Applied egg-rr 38.5%

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

4. Final simplification 38.5%

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

Alternative 3: 79.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (sin (* PI (/ angle 180.0)))) 2.0)
  (pow (* b (cos (/ 1.0 (/ (/ 180.0 angle) PI)))) 2.0)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[N[(1.0 / N[(N[(180.0 / angle), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.2%

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

   1. associate-*l/ 80.1%

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

   2. clear-num 80.2%

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

   3. associate-/r* 80.3%

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

3. Applied egg-rr 80.3%

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

4. Final simplification 80.3%

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

Alternative 4: 66.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.5 \cdot 10^{-28}:\\ \;\;\;\;{\left(b \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(a \cdot 0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(\pi \cdot \left(a \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot \left(a \cdot angle\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 3.5e-28)
   (+ (pow (* b (cos (* PI (/ angle 180.0)))) 2.0) (pow (* a 0.0) 2.0))
   (+
    (pow b 2.0)
    (*
     (* PI (* a (* angle 0.005555555555555556)))
     (* 0.005555555555555556 (* PI (* a angle)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 3.5e-28) {
		tmp = pow((b * cos((((double) M_PI) * (angle / 180.0)))), 2.0) + pow((a * 0.0), 2.0);
	} else {
		tmp = pow(b, 2.0) + ((((double) M_PI) * (a * (angle * 0.005555555555555556))) * (0.005555555555555556 * (((double) M_PI) * (a * angle))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 3.5e-28) {
		tmp = Math.pow((b * Math.cos((Math.PI * (angle / 180.0)))), 2.0) + Math.pow((a * 0.0), 2.0);
	} else {
		tmp = Math.pow(b, 2.0) + ((Math.PI * (a * (angle * 0.005555555555555556))) * (0.005555555555555556 * (Math.PI * (a * angle))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 3.5e-28:
		tmp = math.pow((b * math.cos((math.pi * (angle / 180.0)))), 2.0) + math.pow((a * 0.0), 2.0)
	else:
		tmp = math.pow(b, 2.0) + ((math.pi * (a * (angle * 0.005555555555555556))) * (0.005555555555555556 * (math.pi * (a * angle))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 3.5e-28)
		tmp = Float64((Float64(b * cos(Float64(pi * Float64(angle / 180.0)))) ^ 2.0) + (Float64(a * 0.0) ^ 2.0));
	else
		tmp = Float64((b ^ 2.0) + Float64(Float64(pi * Float64(a * Float64(angle * 0.005555555555555556))) * Float64(0.005555555555555556 * Float64(pi * Float64(a * angle)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 3.5e-28)
		tmp = ((b * cos((pi * (angle / 180.0)))) ^ 2.0) + ((a * 0.0) ^ 2.0);
	else
		tmp = (b ^ 2.0) + ((pi * (a * (angle * 0.005555555555555556))) * (0.005555555555555556 * (pi * (a * angle))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 3.5e-28], N[(N[Power[N[(b * N[Cos[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * 0.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(Pi * N[(a * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.005555555555555556 * N[(Pi * N[(a * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.5e-28

   1. Initial program 76.9%

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

      1. associate-*l/ 76.8%

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

      2. associate-*r/ 76.8%

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

      3. add-cube-cbrt 76.9%

         \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\sqrt[3]{angle \cdot \frac{\pi}{180}} \cdot \sqrt[3]{angle \cdot \frac{\pi}{180}}\right) \cdot \sqrt[3]{angle \cdot \frac{\pi}{180}}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      4. pow3 77.0%

         \[\leadsto {\left(a \cdot \sin \color{blue}{\left({\left(\sqrt[3]{angle \cdot \frac{\pi}{180}}\right)}^{3}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      5. div-inv 77.0%

         \[\leadsto {\left(a \cdot \sin \left({\left(\sqrt[3]{angle \cdot \color{blue}{\left(\pi \cdot \frac{1}{180}\right)}}\right)}^{3}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      6. metadata-eval 77.0%

         \[\leadsto {\left(a \cdot \sin \left({\left(\sqrt[3]{angle \cdot \left(\pi \cdot \color{blue}{0.005555555555555556}\right)}\right)}^{3}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   3. Applied egg-rr 77.0%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left({\left(\sqrt[3]{angle \cdot \left(\pi \cdot 0.005555555555555556\right)}\right)}^{3}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   4. Taylor expanded in angle around 0 61.2%

      \[\leadsto {\left(a \cdot \color{blue}{0}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   if 3.5e-28 < a

   1. Initial program 90.5%

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

      1. associate-*l/ 90.3%

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

      2. associate-*r/ 90.4%

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

      3. associate-*l/ 90.4%

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

      4. associate-*r/ 90.4%

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

   3. Simplified 90.4%

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

   4. Taylor expanded in angle around 0 90.4%

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

   5. Taylor expanded in angle around 0 89.2%

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

      1. *-commutative 89.2%

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

      2. associate-*l* 89.2%

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

   7. Simplified 89.2%

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

      1. unpow2 89.2%

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

      2. associate-*l* 89.2%

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

      3. associate-*r* 89.3%

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

      4. *-commutative 89.3%

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

      5. associate-*r* 89.3%

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

      6. *-commutative 89.3%

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

   9. Applied egg-rr 89.3%

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

       1. associate-*r* 89.3%

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

       2. *-commutative 89.3%

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

       3. associate-*l* 89.2%

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

       4. *-commutative 89.2%

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

       5. associate-*r* 89.3%

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

       6. *-commutative 89.3%

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

       7. associate-*r* 89.3%

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

   11. Simplified 89.3%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.5 \cdot 10^{-28}:\\ \;\;\;\;{\left(b \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(a \cdot 0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(\pi \cdot \left(a \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot \left(a \cdot angle\right)\right)\right)\\ \end{array} \]

Alternative 5: 79.5% accurate, 1.5× 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 80.2%

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

   1. associate-*l/ 80.1%

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

   2. associate-*r/ 80.1%

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

   3. associate-*l/ 80.1%

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

   4. associate-*r/ 80.2%

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

3. Simplified 80.2%

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

4. Taylor expanded in angle around 0 80.1%

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

5. Taylor expanded in angle around inf 80.0%

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

6. Final simplification 80.0%

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

Alternative 6: 79.6% accurate, 1.5× 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 80.2%

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

   1. associate-*l/ 80.1%

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

   2. associate-*r/ 80.1%

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

   3. associate-*l/ 80.1%

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

   4. associate-*r/ 80.2%

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

3. Simplified 80.2%

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

4. Taylor expanded in angle around 0 80.1%

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

5. Final simplification 80.1%

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

Alternative 7: 79.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

   1. associate-*l/ 80.1%

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

   2. associate-*r/ 80.3%

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

   3. add-sqr-sqrt 38.5%

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

   4. pow2 38.5%

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

   5. div-inv 38.5%

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

   6. metadata-eval 38.5%

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

3. Applied egg-rr 38.5%

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

4. Taylor expanded in angle around 0 80.2%

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

5. Final simplification 80.2%

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

Alternative 8: 74.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

   1. associate-*l/ 80.1%

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

   2. associate-*r/ 80.1%

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

   3. associate-*l/ 80.1%

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

   4. associate-*r/ 80.2%

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

3. Simplified 80.2%

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

4. Taylor expanded in angle around 0 80.1%

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

5. Taylor expanded in angle around 0 76.8%

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

   1. *-commutative 76.8%

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

   2. associate-*l* 76.4%

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

7. Simplified 76.4%

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

   1. unpow2 76.4%

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

   2. associate-*l* 76.4%

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

   3. associate-*r* 76.4%

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

   4. *-commutative 76.4%

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

   5. associate-*r* 76.4%

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

   6. *-commutative 76.4%

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

9. Applied egg-rr 76.4%

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

    1. associate-*r* 76.4%

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

    2. *-commutative 76.4%

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

    3. associate-*l* 76.4%

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

    4. *-commutative 76.4%

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

    5. associate-*r* 76.8%

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

    6. *-commutative 76.8%

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

    7. associate-*r* 76.8%

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

11. Simplified 76.8%

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

12. Final simplification 76.8%

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

Alternative 9: 74.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

   1. associate-*l/ 80.1%

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

   2. associate-*r/ 80.1%

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

   3. associate-*l/ 80.1%

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

   4. associate-*r/ 80.2%

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

3. Simplified 80.2%

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

4. Taylor expanded in angle around 0 80.1%

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

5. Taylor expanded in angle around 0 76.8%

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

   1. *-commutative 76.8%

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

   2. associate-*l* 76.4%

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

7. Simplified 76.4%

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

8. Taylor expanded in angle around 0 66.1%

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

   1. *-commutative 66.1%

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

   2. unpow2 66.1%

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

   3. unpow2 66.1%

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

   4. unswap-sqr 66.1%

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

   5. unpow2 66.1%

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

   6. swap-sqr 76.8%

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

   7. associate-*r* 76.4%

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

   8. associate-*r* 76.4%

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

   9. unpow2 76.4%

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

   10. associate-*r* 76.8%

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

   11. *-commutative 76.8%

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

10. Simplified 76.8%

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

    1. unpow2 76.8%

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

    2. associate-*r* 76.8%

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

    3. associate-*r* 76.8%

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

    4. swap-sqr 76.8%

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

    5. *-commutative 76.8%

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

    6. *-commutative 76.8%

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

    7. pow2 76.8%

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

12. Applied egg-rr 76.8%

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

13. Final simplification 76.8%

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

Alternative 10: 74.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

   1. associate-*l/ 80.1%

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

   2. associate-*r/ 80.1%

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

   3. associate-*l/ 80.1%

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

   4. associate-*r/ 80.2%

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

3. Simplified 80.2%

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

4. Taylor expanded in angle around 0 80.1%

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

5. Taylor expanded in angle around 0 76.8%

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

   1. *-commutative 76.8%

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

   2. associate-*l* 76.4%

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

7. Simplified 76.4%

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

8. Taylor expanded in angle around 0 66.1%

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

   1. *-commutative 66.1%

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

   2. unpow2 66.1%

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

   3. unpow2 66.1%

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

   4. unswap-sqr 66.1%

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

   5. unpow2 66.1%

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

   6. swap-sqr 76.8%

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

   7. associate-*r* 76.4%

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

   8. associate-*r* 76.4%

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

   9. unpow2 76.4%

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

   10. associate-*r* 76.8%

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

   11. *-commutative 76.8%

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

10. Simplified 76.8%

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

11. Final simplification 76.8%

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

Reproduce

herbie shell --seed 2023275 
(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)))