ab-angle->ABCF A

Percentage Accurate: 79.8% → 79.8%

Time: 14.1s

Alternatives: 5

Speedup: 2.0×

• 37.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.8% 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, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.0%

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

   1. unpow2 82.0%

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

   2. associate-*l/ 81.8%

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

   3. associate-/l* 82.0%

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

   4. unpow2 82.0%

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

3. Simplified 82.0%

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

5. Taylor expanded in angle around 0 82.4%

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

   1. associate-*r/ 82.1%

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

   2. associate-*l/ 82.4%

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

   3. add-sqr-sqrt 45.6%

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

   4. pow2 45.6%

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

   5. associate-*l/ 45.3%

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

   6. associate-*r/ 45.6%

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

   7. div-inv 45.6%

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

   8. metadata-eval 45.6%

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

7. Applied egg-rr 45.6%

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

8. Final simplification 45.6%

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

Alternative 2: 79.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.0%

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

   1. unpow2 82.0%

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

   2. associate-*l/ 81.8%

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

   3. associate-/l* 82.0%

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

   4. unpow2 82.0%

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

3. Simplified 82.0%

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

5. Taylor expanded in angle around 0 82.4%

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

   1. *-commutative 82.4%

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

   2. associate-/r/ 82.4%

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

7. Applied egg-rr 82.4%

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

8. Final simplification 82.4%

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

Alternative 3: 79.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.0%

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

   1. unpow2 82.0%

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

   2. associate-*l/ 81.8%

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

   3. associate-/l* 82.0%

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

   4. unpow2 82.0%

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

3. Simplified 82.0%

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

5. Taylor expanded in angle around 0 82.4%

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

   1. *-rgt-identity 82.4%

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

   2. unpow2 82.4%

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

7. Applied egg-rr 82.4%

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

Alternative 4: 62.5% accurate, 3.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 8.2 \cdot 10^{+182}:\\ \;\;\;\;{b}^{2}\\ \mathbf{else}:\\ \;\;\;\;3.08641975308642 \cdot 10^{-5} \cdot \left(\left(a \cdot angle\_m\right) \cdot \left(\pi \cdot \left(a \cdot \left(angle\_m \cdot \pi\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 8.2e+182)
   (pow b 2.0)
   (* 3.08641975308642e-5 (* (* a angle_m) (* PI (* a (* angle_m PI)))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 8.2e+182) {
		tmp = pow(b, 2.0);
	} else {
		tmp = 3.08641975308642e-5 * ((a * angle_m) * (((double) M_PI) * (a * (angle_m * ((double) M_PI)))));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 8.2e+182) {
		tmp = Math.pow(b, 2.0);
	} else {
		tmp = 3.08641975308642e-5 * ((a * angle_m) * (Math.PI * (a * (angle_m * Math.PI))));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 8.2e+182:
		tmp = math.pow(b, 2.0)
	else:
		tmp = 3.08641975308642e-5 * ((a * angle_m) * (math.pi * (a * (angle_m * math.pi))))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 8.2e+182)
		tmp = b ^ 2.0;
	else
		tmp = Float64(3.08641975308642e-5 * Float64(Float64(a * angle_m) * Float64(pi * Float64(a * Float64(angle_m * pi)))));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (a <= 8.2e+182)
		tmp = b ^ 2.0;
	else
		tmp = 3.08641975308642e-5 * ((a * angle_m) * (pi * (a * (angle_m * pi))));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 8.2e+182], N[Power[b, 2.0], $MachinePrecision], N[(3.08641975308642e-5 * N[(N[(a * angle$95$m), $MachinePrecision] * N[(Pi * N[(a * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 8.20000000000000006e182

   1. Initial program 80.0%

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

      1. unpow2 80.0%

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

      2. associate-*l/ 79.7%

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

      3. associate-/l* 80.0%

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

      4. unpow2 80.0%

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

   3. Simplified 80.0%

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

   5. Taylor expanded in angle around 0 66.5%

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

   if 8.20000000000000006e182 < a

   1. Initial program 99.9%

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

      1. unpow2 99.9%

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

      2. associate-*l/ 99.9%

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

      3. associate-/l* 99.9%

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

      4. unpow2 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in a around inf 77.7%

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

      1. unpow2 77.7%

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

      2. *-commutative 77.7%

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

      3. associate-*r* 77.7%

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

      4. unpow2 77.7%

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

      5. swap-sqr 81.4%

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

      6. unpow2 81.4%

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

      7. associate-*r* 81.4%

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

      8. *-commutative 81.4%

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

   7. Simplified 81.4%

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

   8. Taylor expanded in angle around 0 77.7%

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

      1. unpow2 77.7%

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

      2. unpow2 77.7%

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

      3. unpow2 77.7%

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

      4. swap-sqr 77.7%

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

      5. swap-sqr 81.3%

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

      6. unpow2 81.3%

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

   10. Simplified 81.3%

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

       1. unpow2 81.3%

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

       2. associate-*r* 81.4%

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

       3. associate-*l* 81.3%

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

       4. *-commutative 81.3%

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

   12. Applied egg-rr 81.3%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 8.2 \cdot 10^{+182}:\\ \;\;\;\;{b}^{2}\\ \mathbf{else}:\\ \;\;\;\;3.08641975308642 \cdot 10^{-5} \cdot \left(\left(a \cdot angle\right) \cdot \left(\pi \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 37.7% accurate, 32.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.0%

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

   1. unpow2 82.0%

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

   2. associate-*l/ 81.8%

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

   3. associate-/l* 82.0%

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

   4. unpow2 82.0%

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

3. Simplified 82.0%

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

5. Taylor expanded in a around inf 34.5%

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

   1. unpow2 34.5%

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

   2. *-commutative 34.5%

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

   3. associate-*r* 34.8%

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

   4. unpow2 34.8%

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

   5. swap-sqr 36.8%

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

   6. unpow2 36.8%

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

   7. associate-*r* 36.5%

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

   8. *-commutative 36.5%

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

7. Simplified 36.5%

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

8. Taylor expanded in angle around 0 33.4%

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

   1. unpow2 33.4%

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

   2. unpow2 33.4%

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

   3. unpow2 33.4%

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

   4. swap-sqr 33.4%

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

   5. swap-sqr 34.7%

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

   6. unpow2 34.7%

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

10. Simplified 34.7%

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

    1. unpow2 34.7%

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

    2. associate-*r* 34.7%

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

    3. associate-*l* 34.7%

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

    4. *-commutative 34.7%

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

12. Applied egg-rr 34.7%

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

13. Final simplification 34.7%

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

Reproduce

herbie shell --seed 2024173 
(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)))