ab-angle->ABCF A

Percentage Accurate: 79.8% → 79.8%

Time: 16.0s

Alternatives: 12

Speedup: 1.0×

• 36.5% 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: 79.8% 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: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \sin \left(\frac{\pi \cdot 0.005555555555555556}{\frac{1}{angle}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (sin (/ (* PI 0.005555555555555556) (/ 1.0 angle)))) 2.0)
  (pow (* b (cos (* PI (/ angle 180.0)))) 2.0)))

C

double code(double a, double b, double angle) {
	return pow((a * sin(((((double) M_PI) * 0.005555555555555556) / (1.0 / angle)))), 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.PI * 0.005555555555555556) / (1.0 / angle)))), 2.0) + Math.pow((b * Math.cos((Math.PI * (angle / 180.0)))), 2.0);
}

Python

def code(a, b, angle):
	return math.pow((a * math.sin(((math.pi * 0.005555555555555556) / (1.0 / angle)))), 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(Float64(Float64(pi * 0.005555555555555556) / Float64(1.0 / angle)))) ^ 2.0) + (Float64(b * cos(Float64(pi * Float64(angle / 180.0)))) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a * sin(((pi * 0.005555555555555556) / (1.0 / angle)))) ^ 2.0) + ((b * cos((pi * (angle / 180.0)))) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(N[(Pi * 0.005555555555555556), $MachinePrecision] / N[(1.0 / angle), $MachinePrecision]), $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 80.8%

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

   1. add-cube-cbrt 80.4%

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

   2. pow3 80.4%

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

   3. associate-*l/ 80.5%

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

   4. associate-*r/ 80.5%

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

   5. div-inv 80.5%

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

   6. metadata-eval 80.5%

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

4. Applied egg-rr 80.5%

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

   1. rem-cube-cbrt 80.8%

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

   2. *-commutative 80.8%

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

   3. metadata-eval 80.8%

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

   4. div-inv 80.8%

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

   5. associate-/r/ 80.9%

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

   6. div-inv 80.9%

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

   7. associate-/r* 80.9%

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

   8. div-inv 80.9%

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

   9. metadata-eval 80.9%

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

6. Applied egg-rr 80.9%

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

7. Final simplification 80.9%

   \[\leadsto {\left(a \cdot \sin \left(\frac{\pi \cdot 0.005555555555555556}{\frac{1}{angle}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
8. Add Preprocessing

Alternative 2: 79.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.8%

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

   1. *-commutative 80.8%

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

   2. clear-num 80.8%

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

   3. un-div-inv 80.9%

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

4. Applied egg-rr 80.9%

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

5. Final simplification 80.9%

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

Alternative 3: 79.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * angle), $MachinePrecision]}, N[Power[N[Sqrt[N[(a * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(b * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Initial program 80.8%

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

   1. unpow2 80.8%

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

   2. associate-*l/ 80.8%

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

   3. associate-/l* 80.8%

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

   4. unpow2 80.8%

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

3. Simplified 80.9%

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

6. Applied egg-rr 80.9%

   \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(a \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right), b \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}^{2}} \]

7. Final simplification 80.9%

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

Alternative 4: 63.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.06 \cdot 10^{+127}:\\ \;\;\;\;{b}^{2} \cdot {\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 2.06e+127)
   (* (pow b 2.0) (pow (cos (* PI (* 0.005555555555555556 angle))) 2.0))
   (pow (* a (sin (* angle (* PI -0.005555555555555556)))) 2.0)))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.06e+127) {
		tmp = pow(b, 2.0) * pow(cos((((double) M_PI) * (0.005555555555555556 * angle))), 2.0);
	} else {
		tmp = pow((a * sin((angle * (((double) M_PI) * -0.005555555555555556)))), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.06e+127) {
		tmp = Math.pow(b, 2.0) * Math.pow(Math.cos((Math.PI * (0.005555555555555556 * angle))), 2.0);
	} else {
		tmp = Math.pow((a * Math.sin((angle * (Math.PI * -0.005555555555555556)))), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 2.06e+127:
		tmp = math.pow(b, 2.0) * math.pow(math.cos((math.pi * (0.005555555555555556 * angle))), 2.0)
	else:
		tmp = math.pow((a * math.sin((angle * (math.pi * -0.005555555555555556)))), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 2.06e+127)
		tmp = Float64((b ^ 2.0) * (cos(Float64(pi * Float64(0.005555555555555556 * angle))) ^ 2.0));
	else
		tmp = Float64(a * sin(Float64(angle * Float64(pi * -0.005555555555555556)))) ^ 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 2.06e+127)
		tmp = (b ^ 2.0) * (cos((pi * (0.005555555555555556 * angle))) ^ 2.0);
	else
		tmp = (a * sin((angle * (pi * -0.005555555555555556)))) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 2.06e+127], N[(N[Power[b, 2.0], $MachinePrecision] * N[Power[N[Cos[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[Power[N[(a * N[Sin[N[(angle * N[(Pi * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.06000000000000001e127

   1. Initial program 77.4%

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

      1. unpow2 77.4%

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

      2. associate-*l/ 77.5%

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

      3. associate-/l* 77.4%

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

      4. unpow2 77.4%

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

   3. Simplified 77.5%

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

   5. Taylor expanded in a around 0 58.3%

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

      1. associate-*r* 58.3%

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

   7. Simplified 58.3%

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

   if 2.06000000000000001e127 < a

   1. Initial program 96.2%

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

      1. *-commutative 96.2%

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

      2. clear-num 96.2%

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

      3. un-div-inv 96.3%

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

   4. Applied egg-rr 96.3%

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

   6. Applied egg-rr 96.3%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(b \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right), a \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}^{2}} \]

   7. Taylor expanded in a around -inf 81.6%

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

      1. mul-1-neg 81.6%

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

      2. distribute-rgt-neg-in 81.6%

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

      3. *-commutative 81.6%

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

      4. associate-*r* 81.7%

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

      5. *-commutative 81.7%

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

      6. *-commutative 81.7%

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

      7. associate-*r* 81.6%

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

      8. metadata-eval 81.6%

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

      9. metadata-eval 81.6%

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

      10. rem-cube-cbrt 81.6%

          \[\leadsto {\left(a \cdot \left(-\sin \left(\left(angle \cdot \pi\right) \cdot \left(-\color{blue}{{\left(\sqrt[3]{-1}\right)}^{3}} \cdot 0.005555555555555556\right)\right)\right)\right)}^{2} \]

      11. rem-cube-cbrt 80.6%

          \[\leadsto {\left(a \cdot \left(-\sin \left(\left(angle \cdot \pi\right) \cdot \left(-{\left(\sqrt[3]{-1}\right)}^{3} \cdot \color{blue}{{\left(\sqrt[3]{0.005555555555555556}\right)}^{3}}\right)\right)\right)\right)}^{2} \]

      12. distribute-rgt-neg-in 80.6%

          \[\leadsto {\left(a \cdot \left(-\sin \color{blue}{\left(-\left(angle \cdot \pi\right) \cdot \left({\left(\sqrt[3]{-1}\right)}^{3} \cdot {\left(\sqrt[3]{0.005555555555555556}\right)}^{3}\right)\right)}\right)\right)}^{2} \]

      13. associate-*r* 80.7%

          \[\leadsto {\left(a \cdot \left(-\sin \left(-\color{blue}{angle \cdot \left(\pi \cdot \left({\left(\sqrt[3]{-1}\right)}^{3} \cdot {\left(\sqrt[3]{0.005555555555555556}\right)}^{3}\right)\right)}\right)\right)\right)}^{2} \]

      14. sin-neg 80.7%

          \[\leadsto {\left(a \cdot \left(-\color{blue}{\left(-\sin \left(angle \cdot \left(\pi \cdot \left({\left(\sqrt[3]{-1}\right)}^{3} \cdot {\left(\sqrt[3]{0.005555555555555556}\right)}^{3}\right)\right)\right)\right)}\right)\right)}^{2} \]

      15. remove-double-neg 80.7%

          \[\leadsto {\left(a \cdot \color{blue}{\sin \left(angle \cdot \left(\pi \cdot \left({\left(\sqrt[3]{-1}\right)}^{3} \cdot {\left(\sqrt[3]{0.005555555555555556}\right)}^{3}\right)\right)\right)}\right)}^{2} \]

   9. Simplified 81.7%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.06 \cdot 10^{+127}:\\ \;\;\;\;{b}^{2} \cdot {\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \sin \left(\frac{\pi \cdot 0.005555555555555556}{\frac{1}{angle}}\right)\right)}^{2} + {b}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (sin (/ (* PI 0.005555555555555556) (/ 1.0 angle)))) 2.0)
  (pow b 2.0)))

C

double code(double a, double b, double angle) {
	return pow((a * sin(((((double) M_PI) * 0.005555555555555556) / (1.0 / angle)))), 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 * 0.005555555555555556) / (1.0 / angle)))), 2.0) + Math.pow(b, 2.0);
}

Python

def code(a, b, angle):
	return math.pow((a * math.sin(((math.pi * 0.005555555555555556) / (1.0 / angle)))), 2.0) + math.pow(b, 2.0)

Julia

function code(a, b, angle)
	return Float64((Float64(a * sin(Float64(Float64(pi * 0.005555555555555556) / Float64(1.0 / angle)))) ^ 2.0) + (b ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a * sin(((pi * 0.005555555555555556) / (1.0 / angle)))) ^ 2.0) + (b ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(N[(Pi * 0.005555555555555556), $MachinePrecision] / N[(1.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.8%

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

   1. add-cube-cbrt 80.4%

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

   2. pow3 80.4%

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

   3. associate-*l/ 80.5%

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

   4. associate-*r/ 80.5%

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

   5. div-inv 80.5%

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

   6. metadata-eval 80.5%

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

4. Applied egg-rr 80.5%

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

   1. rem-cube-cbrt 80.8%

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

   2. *-commutative 80.8%

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

   3. metadata-eval 80.8%

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

   4. div-inv 80.8%

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

   5. associate-/r/ 80.9%

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

   6. div-inv 80.9%

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

   7. associate-/r* 80.9%

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

   8. div-inv 80.9%

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

   9. metadata-eval 80.9%

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

6. Applied egg-rr 80.9%

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

   1. add-cube-cbrt 80.9%

      \[\leadsto {\left(a \cdot \sin \left(\frac{\pi \cdot 0.005555555555555556}{\frac{1}{angle}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)}\right)\right)}^{2} \]

   2. pow2 80.9%

      \[\leadsto {\left(a \cdot \sin \left(\frac{\pi \cdot 0.005555555555555556}{\frac{1}{angle}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2}} \cdot \sqrt[3]{\pi}\right)\right)\right)}^{2} \]

8. Applied egg-rr 80.9%

   \[\leadsto {\left(a \cdot \sin \left(\frac{\pi \cdot 0.005555555555555556}{\frac{1}{angle}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \color{blue}{\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right)}\right)\right)}^{2} \]

9. Taylor expanded in angle around 0 80.7%

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

Alternative 6: 79.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.8%

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

   1. associate-*l/ 80.8%

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

   2. clear-num 80.8%

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

4. Applied egg-rr 80.8%

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

5. Taylor expanded in angle around 0 80.6%

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

6. Final simplification 80.6%

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

Alternative 7: 79.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.8%

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

   1. *-commutative 80.8%

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

   2. clear-num 80.8%

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

   3. un-div-inv 80.9%

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

4. Applied egg-rr 80.9%

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

6. Applied egg-rr 80.9%

   \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(b \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right), a \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}^{2}} \]

7. Taylor expanded in angle around 0 80.6%

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

8. Final simplification 80.6%

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

Alternative 8: 63.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 8.2 \cdot 10^{+123}:\\ \;\;\;\;{\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 8.2e+123)
   (pow (* b (cos (* 0.005555555555555556 (* PI angle)))) 2.0)
   (pow (* a (sin (* angle (* PI -0.005555555555555556)))) 2.0)))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 8.2e+123) {
		tmp = pow((b * cos((0.005555555555555556 * (((double) M_PI) * angle)))), 2.0);
	} else {
		tmp = pow((a * sin((angle * (((double) M_PI) * -0.005555555555555556)))), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 8.2e+123) {
		tmp = Math.pow((b * Math.cos((0.005555555555555556 * (Math.PI * angle)))), 2.0);
	} else {
		tmp = Math.pow((a * Math.sin((angle * (Math.PI * -0.005555555555555556)))), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 8.2e+123:
		tmp = math.pow((b * math.cos((0.005555555555555556 * (math.pi * angle)))), 2.0)
	else:
		tmp = math.pow((a * math.sin((angle * (math.pi * -0.005555555555555556)))), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 8.2e+123)
		tmp = Float64(b * cos(Float64(0.005555555555555556 * Float64(pi * angle)))) ^ 2.0;
	else
		tmp = Float64(a * sin(Float64(angle * Float64(pi * -0.005555555555555556)))) ^ 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 8.2e+123)
		tmp = (b * cos((0.005555555555555556 * (pi * angle)))) ^ 2.0;
	else
		tmp = (a * sin((angle * (pi * -0.005555555555555556)))) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 8.2e+123], N[Power[N[(b * N[Cos[N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[Power[N[(a * N[Sin[N[(angle * N[(Pi * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 8.19999999999999979e123

   1. Initial program 77.4%

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

      1. unpow2 77.4%

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

      2. associate-*l/ 77.5%

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

      3. associate-/l* 77.4%

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

      4. unpow2 77.4%

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

   3. Simplified 77.5%

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

   5. Taylor expanded in a around 0 58.3%

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

      1. *-commutative 58.3%

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

      2. associate-*r* 58.3%

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

      3. unpow2 58.3%

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

      4. unpow2 58.3%

         \[\leadsto \color{blue}{\left(b \cdot b\right)} \cdot \left(\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right) \]

      5. swap-sqr 58.3%

         \[\leadsto \color{blue}{\left(b \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)} \]

      6. unpow2 58.3%

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

      7. associate-*r* 58.3%

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

      8. *-commutative 58.3%

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

   7. Simplified 58.3%

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

   if 8.19999999999999979e123 < a

   1. Initial program 96.2%

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

      1. *-commutative 96.2%

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

      2. clear-num 96.2%

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

      3. un-div-inv 96.3%

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

   4. Applied egg-rr 96.3%

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

   6. Applied egg-rr 96.3%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(b \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right), a \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}^{2}} \]

   7. Taylor expanded in a around -inf 81.6%

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

      1. mul-1-neg 81.6%

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

      2. distribute-rgt-neg-in 81.6%

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

      3. *-commutative 81.6%

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

      4. associate-*r* 81.7%

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

      5. *-commutative 81.7%

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

      6. *-commutative 81.7%

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

      7. associate-*r* 81.6%

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

      8. metadata-eval 81.6%

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

      9. metadata-eval 81.6%

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

      10. rem-cube-cbrt 81.6%

          \[\leadsto {\left(a \cdot \left(-\sin \left(\left(angle \cdot \pi\right) \cdot \left(-\color{blue}{{\left(\sqrt[3]{-1}\right)}^{3}} \cdot 0.005555555555555556\right)\right)\right)\right)}^{2} \]

      11. rem-cube-cbrt 80.6%

          \[\leadsto {\left(a \cdot \left(-\sin \left(\left(angle \cdot \pi\right) \cdot \left(-{\left(\sqrt[3]{-1}\right)}^{3} \cdot \color{blue}{{\left(\sqrt[3]{0.005555555555555556}\right)}^{3}}\right)\right)\right)\right)}^{2} \]

      12. distribute-rgt-neg-in 80.6%

          \[\leadsto {\left(a \cdot \left(-\sin \color{blue}{\left(-\left(angle \cdot \pi\right) \cdot \left({\left(\sqrt[3]{-1}\right)}^{3} \cdot {\left(\sqrt[3]{0.005555555555555556}\right)}^{3}\right)\right)}\right)\right)}^{2} \]

      13. associate-*r* 80.7%

          \[\leadsto {\left(a \cdot \left(-\sin \left(-\color{blue}{angle \cdot \left(\pi \cdot \left({\left(\sqrt[3]{-1}\right)}^{3} \cdot {\left(\sqrt[3]{0.005555555555555556}\right)}^{3}\right)\right)}\right)\right)\right)}^{2} \]

      14. sin-neg 80.7%

          \[\leadsto {\left(a \cdot \left(-\color{blue}{\left(-\sin \left(angle \cdot \left(\pi \cdot \left({\left(\sqrt[3]{-1}\right)}^{3} \cdot {\left(\sqrt[3]{0.005555555555555556}\right)}^{3}\right)\right)\right)\right)}\right)\right)}^{2} \]

      15. remove-double-neg 80.7%

          \[\leadsto {\left(a \cdot \color{blue}{\sin \left(angle \cdot \left(\pi \cdot \left({\left(\sqrt[3]{-1}\right)}^{3} \cdot {\left(\sqrt[3]{0.005555555555555556}\right)}^{3}\right)\right)\right)}\right)}^{2} \]

   9. Simplified 81.7%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 8.2 \cdot 10^{+123}:\\ \;\;\;\;{\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.12 \cdot 10^{+126}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.12e+126)
   (* b b)
   (pow (* a (sin (* angle (* PI -0.005555555555555556)))) 2.0)))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.12e+126) {
		tmp = b * b;
	} else {
		tmp = pow((a * sin((angle * (((double) M_PI) * -0.005555555555555556)))), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.12e+126) {
		tmp = b * b;
	} else {
		tmp = Math.pow((a * Math.sin((angle * (Math.PI * -0.005555555555555556)))), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 1.12e+126:
		tmp = b * b
	else:
		tmp = math.pow((a * math.sin((angle * (math.pi * -0.005555555555555556)))), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.12e+126)
		tmp = Float64(b * b);
	else
		tmp = Float64(a * sin(Float64(angle * Float64(pi * -0.005555555555555556)))) ^ 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 1.12e+126)
		tmp = b * b;
	else
		tmp = (a * sin((angle * (pi * -0.005555555555555556)))) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.12e+126], N[(b * b), $MachinePrecision], N[Power[N[(a * N[Sin[N[(angle * N[(Pi * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.12e126

   1. Initial program 77.4%

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

      1. unpow2 77.4%

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

      2. associate-*l/ 77.5%

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

      3. associate-/l* 77.4%

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

      4. unpow2 77.4%

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

   3. Simplified 77.5%

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

   5. Taylor expanded in angle around 0 58.0%

      \[\leadsto \color{blue}{{b}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 58.0%

         \[\leadsto \color{blue}{b \cdot b} \]

   7. Applied egg-rr 58.0%

      \[\leadsto \color{blue}{b \cdot b} \]

   if 1.12e126 < a

   1. Initial program 96.2%

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

      1. *-commutative 96.2%

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

      2. clear-num 96.2%

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

      3. un-div-inv 96.3%

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

   4. Applied egg-rr 96.3%

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

   6. Applied egg-rr 96.3%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(b \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right), a \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}^{2}} \]

   7. Taylor expanded in a around -inf 81.6%

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

      1. mul-1-neg 81.6%

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

      2. distribute-rgt-neg-in 81.6%

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

      3. *-commutative 81.6%

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

      4. associate-*r* 81.7%

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

      5. *-commutative 81.7%

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

      6. *-commutative 81.7%

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

      7. associate-*r* 81.6%

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

      8. metadata-eval 81.6%

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

      9. metadata-eval 81.6%

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

      10. rem-cube-cbrt 81.6%

          \[\leadsto {\left(a \cdot \left(-\sin \left(\left(angle \cdot \pi\right) \cdot \left(-\color{blue}{{\left(\sqrt[3]{-1}\right)}^{3}} \cdot 0.005555555555555556\right)\right)\right)\right)}^{2} \]

      11. rem-cube-cbrt 80.6%

          \[\leadsto {\left(a \cdot \left(-\sin \left(\left(angle \cdot \pi\right) \cdot \left(-{\left(\sqrt[3]{-1}\right)}^{3} \cdot \color{blue}{{\left(\sqrt[3]{0.005555555555555556}\right)}^{3}}\right)\right)\right)\right)}^{2} \]

      12. distribute-rgt-neg-in 80.6%

          \[\leadsto {\left(a \cdot \left(-\sin \color{blue}{\left(-\left(angle \cdot \pi\right) \cdot \left({\left(\sqrt[3]{-1}\right)}^{3} \cdot {\left(\sqrt[3]{0.005555555555555556}\right)}^{3}\right)\right)}\right)\right)}^{2} \]

      13. associate-*r* 80.7%

          \[\leadsto {\left(a \cdot \left(-\sin \left(-\color{blue}{angle \cdot \left(\pi \cdot \left({\left(\sqrt[3]{-1}\right)}^{3} \cdot {\left(\sqrt[3]{0.005555555555555556}\right)}^{3}\right)\right)}\right)\right)\right)}^{2} \]

      14. sin-neg 80.7%

          \[\leadsto {\left(a \cdot \left(-\color{blue}{\left(-\sin \left(angle \cdot \left(\pi \cdot \left({\left(\sqrt[3]{-1}\right)}^{3} \cdot {\left(\sqrt[3]{0.005555555555555556}\right)}^{3}\right)\right)\right)\right)}\right)\right)}^{2} \]

      15. remove-double-neg 80.7%

          \[\leadsto {\left(a \cdot \color{blue}{\sin \left(angle \cdot \left(\pi \cdot \left({\left(\sqrt[3]{-1}\right)}^{3} \cdot {\left(\sqrt[3]{0.005555555555555556}\right)}^{3}\right)\right)\right)}\right)}^{2} \]

   9. Simplified 81.7%

      \[\leadsto {\color{blue}{\left(a \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)}}^{2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 63.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 7.1 \cdot 10^{+129}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 7.1e+129)
   (* b b)
   (pow (* a (sin (* 0.005555555555555556 (* PI angle)))) 2.0)))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 7.1e+129) {
		tmp = b * b;
	} else {
		tmp = pow((a * sin((0.005555555555555556 * (((double) M_PI) * angle)))), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 7.1e+129) {
		tmp = b * b;
	} else {
		tmp = Math.pow((a * Math.sin((0.005555555555555556 * (Math.PI * angle)))), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 7.1e+129:
		tmp = b * b
	else:
		tmp = math.pow((a * math.sin((0.005555555555555556 * (math.pi * angle)))), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 7.1e+129)
		tmp = Float64(b * b);
	else
		tmp = Float64(a * sin(Float64(0.005555555555555556 * Float64(pi * angle)))) ^ 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 7.1e+129)
		tmp = b * b;
	else
		tmp = (a * sin((0.005555555555555556 * (pi * angle)))) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 7.1e+129], N[(b * b), $MachinePrecision], N[Power[N[(a * N[Sin[N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 7.09999999999999996e129

   1. Initial program 77.4%

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

      1. unpow2 77.4%

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

      2. associate-*l/ 77.5%

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

      3. associate-/l* 77.4%

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

      4. unpow2 77.4%

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

   3. Simplified 77.5%

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

   5. Taylor expanded in angle around 0 58.0%

      \[\leadsto \color{blue}{{b}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 58.0%

         \[\leadsto \color{blue}{b \cdot b} \]

   7. Applied egg-rr 58.0%

      \[\leadsto \color{blue}{b \cdot b} \]

   if 7.09999999999999996e129 < a

   1. Initial program 96.2%

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

      1. unpow2 96.2%

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

      2. associate-*l/ 96.3%

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

      3. associate-/l* 96.3%

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

      4. unpow2 96.3%

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

   3. Simplified 96.3%

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

   5. Taylor expanded in a around inf 65.4%

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

      1. unpow2 65.4%

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

      2. *-commutative 65.4%

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

      3. associate-*r* 65.4%

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

      4. unpow2 65.4%

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)} \]

      5. swap-sqr 81.7%

         \[\leadsto \color{blue}{\left(a \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right) \cdot \left(a \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)} \]

      6. unpow2 81.7%

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

      7. associate-*r* 81.6%

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

      8. *-commutative 81.6%

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

   7. Simplified 81.6%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 7.1 \cdot 10^{+129}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.95 \cdot 10^{+157}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{b}^{6}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.95e+157) (* b b) (cbrt (pow b 6.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.95e+157) {
		tmp = b * b;
	} else {
		tmp = cbrt(pow(b, 6.0));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.95e+157) {
		tmp = b * b;
	} else {
		tmp = Math.cbrt(Math.pow(b, 6.0));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.95e+157)
		tmp = Float64(b * b);
	else
		tmp = cbrt((b ^ 6.0));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.95e+157], N[(b * b), $MachinePrecision], N[Power[N[Power[b, 6.0], $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.94999999999999985e157

   1. Initial program 77.5%

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

      1. unpow2 77.5%

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

      2. associate-*l/ 77.5%

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

      3. associate-/l* 77.4%

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

      4. unpow2 77.4%

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

   3. Simplified 77.5%

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

   5. Taylor expanded in angle around 0 57.2%

      \[\leadsto \color{blue}{{b}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 57.2%

         \[\leadsto \color{blue}{b \cdot b} \]

   7. Applied egg-rr 57.2%

      \[\leadsto \color{blue}{b \cdot b} \]

   if 1.94999999999999985e157 < a

   1. Initial program 99.5%

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

      1. unpow2 99.5%

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

      2. associate-*l/ 99.6%

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

      3. associate-/l* 99.6%

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

      4. unpow2 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in angle around 0 40.2%

      \[\leadsto \color{blue}{{b}^{2}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 40.2%

         \[\leadsto \color{blue}{\sqrt{{b}^{2}} \cdot \sqrt{{b}^{2}}} \]

      2. sqrt-unprod 42.6%

         \[\leadsto \color{blue}{\sqrt{{b}^{2} \cdot {b}^{2}}} \]

      3. pow-prod-up 42.6%

         \[\leadsto \sqrt{\color{blue}{{b}^{\left(2 + 2\right)}}} \]

      4. metadata-eval 42.6%

         \[\leadsto \sqrt{{b}^{\color{blue}{4}}} \]

   7. Applied egg-rr 42.6%

      \[\leadsto \color{blue}{\sqrt{{b}^{4}}} \]
   8. Step-by-step derivation

      1. add-cbrt-cube 42.5%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{{b}^{4}} \cdot \sqrt{{b}^{4}}\right) \cdot \sqrt{{b}^{4}}}} \]

      2. pow1/3 42.5%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{{b}^{4}} \cdot \sqrt{{b}^{4}}\right) \cdot \sqrt{{b}^{4}}\right)}^{0.3333333333333333}} \]

      3. add-sqr-sqrt 42.5%

         \[\leadsto {\left(\color{blue}{{b}^{4}} \cdot \sqrt{{b}^{4}}\right)}^{0.3333333333333333} \]

      4. sqrt-pow1 42.5%

         \[\leadsto {\left({b}^{4} \cdot \color{blue}{{b}^{\left(\frac{4}{2}\right)}}\right)}^{0.3333333333333333} \]

      5. metadata-eval 42.5%

         \[\leadsto {\left({b}^{4} \cdot {b}^{\color{blue}{2}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 42.5%

         \[\leadsto {\color{blue}{\left({b}^{\left(4 + 2\right)}\right)}}^{0.3333333333333333} \]

      7. metadata-eval 42.5%

         \[\leadsto {\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} \]

   9. Applied egg-rr 42.5%

      \[\leadsto \color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} \]
   10. Step-by-step derivation

       1. unpow1/3 42.5%

          \[\leadsto \color{blue}{\sqrt[3]{{b}^{6}}} \]

   11. Simplified 42.5%

       \[\leadsto \color{blue}{\sqrt[3]{{b}^{6}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 58.0% accurate, 139.0× speedup

Math

\[\begin{array}{l} \\ b \cdot b \end{array} \]

FPCore

(FPCore (a b angle) :precision binary64 (* b b))

C

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

Fortran

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

Java

public static double code(double a, double b, double angle) {
	return b * b;
}

Python

def code(a, b, angle):
	return b * b

Julia

function code(a, b, angle)
	return Float64(b * b)
end

MATLAB

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

Wolfram

code[a_, b_, angle_] := N[(b * b), $MachinePrecision]

Derivation

1. Initial program 80.8%

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

   1. unpow2 80.8%

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

   2. associate-*l/ 80.8%

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

   3. associate-/l* 80.8%

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

   4. unpow2 80.8%

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

3. Simplified 80.9%

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

5. Taylor expanded in angle around 0 54.6%

   \[\leadsto \color{blue}{{b}^{2}} \]
6. Step-by-step derivation

   1. unpow2 54.6%

      \[\leadsto \color{blue}{b \cdot b} \]

7. Applied egg-rr 54.6%

   \[\leadsto \color{blue}{b \cdot b} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024169 
(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)))