ab-angle->ABCF B

Percentage Accurate: 54.2% → 67.5%

Time: 44.2s

Alternatives: 17

Speedup: 5.5×

• 33.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 67.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \begin{array}{l} t_0 := \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(a + b\right)\\ t_1 := \sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\\ \mathbf{if}\;{a}^{2} \leq 2 \cdot 10^{+294}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\left(t_1 \cdot \left(t_1 \cdot t_1\right)\right) \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(t_0 \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \left(\sqrt{\pi} \cdot \sqrt{\pi}\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (sin (* 0.005555555555555556 (* PI angle))) (+ a b)))
        (t_1 (cbrt (cos (* PI (* 0.005555555555555556 angle))))))
   (if (<= (pow a 2.0) 2e+294)
     (* 2.0 (* (- b a) (* (* t_1 (* t_1 t_1)) t_0)))
     (*
      2.0
      (*
       (- b a)
       (*
        t_0
        (cos (* 0.005555555555555556 (* angle (* (sqrt PI) (sqrt PI)))))))))))

C

b = abs(b);
double code(double a, double b, double angle) {
	double t_0 = sin((0.005555555555555556 * (((double) M_PI) * angle))) * (a + b);
	double t_1 = cbrt(cos((((double) M_PI) * (0.005555555555555556 * angle))));
	double tmp;
	if (pow(a, 2.0) <= 2e+294) {
		tmp = 2.0 * ((b - a) * ((t_1 * (t_1 * t_1)) * t_0));
	} else {
		tmp = 2.0 * ((b - a) * (t_0 * cos((0.005555555555555556 * (angle * (sqrt(((double) M_PI)) * sqrt(((double) M_PI))))))));
	}
	return tmp;
}

Java

b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double t_0 = Math.sin((0.005555555555555556 * (Math.PI * angle))) * (a + b);
	double t_1 = Math.cbrt(Math.cos((Math.PI * (0.005555555555555556 * angle))));
	double tmp;
	if (Math.pow(a, 2.0) <= 2e+294) {
		tmp = 2.0 * ((b - a) * ((t_1 * (t_1 * t_1)) * t_0));
	} else {
		tmp = 2.0 * ((b - a) * (t_0 * Math.cos((0.005555555555555556 * (angle * (Math.sqrt(Math.PI) * Math.sqrt(Math.PI)))))));
	}
	return tmp;
}

Julia

b = abs(b)
function code(a, b, angle)
	t_0 = Float64(sin(Float64(0.005555555555555556 * Float64(pi * angle))) * Float64(a + b))
	t_1 = cbrt(cos(Float64(pi * Float64(0.005555555555555556 * angle))))
	tmp = 0.0
	if ((a ^ 2.0) <= 2e+294)
		tmp = Float64(2.0 * Float64(Float64(b - a) * Float64(Float64(t_1 * Float64(t_1 * t_1)) * t_0)));
	else
		tmp = Float64(2.0 * Float64(Float64(b - a) * Float64(t_0 * cos(Float64(0.005555555555555556 * Float64(angle * Float64(sqrt(pi) * sqrt(pi))))))));
	end
	return tmp
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Sin[N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Cos[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[Power[a, 2.0], $MachinePrecision], 2e+294], N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(N[(t$95$1 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(t$95$0 * N[Cos[N[(0.005555555555555556 * N[(angle * N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 a 2) < 2.00000000000000013e294

   1. Initial program 59.7%

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

      1. associate-*l* 59.7%

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

      2. associate-*l* 59.7%

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

      3. unpow2 59.7%

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

      4. unpow2 59.7%

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

      5. difference-of-squares 59.7%

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

   3. Simplified 59.7%

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

   4. Taylor expanded in angle around inf 68.7%

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

      1. *-commutative 68.7%

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

      2. *-commutative 68.7%

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

      3. associate-*r* 69.1%

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

      4. add-cube-cbrt 66.7%

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

      5. unpow3 67.2%

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

      6. add-cube-cbrt 67.2%

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

   6. Applied egg-rr 69.1%

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

   if 2.00000000000000013e294 < (pow.f64 a 2)

   1. Initial program 30.2%

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

      1. associate-*l* 30.2%

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

      2. associate-*l* 30.2%

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

      3. unpow2 30.2%

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

      4. unpow2 30.2%

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

      5. difference-of-squares 51.8%

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

   3. Simplified 51.8%

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

   4. Taylor expanded in angle around inf 76.6%

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

      1. add-sqr-sqrt 86.3%

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

   6. Applied egg-rr 86.3%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 2 \cdot 10^{+294}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\left(\sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)} \cdot \left(\sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)} \cdot \sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(a + b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(a + b\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \left(\sqrt{\pi} \cdot \sqrt{\pi}\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 2: 67.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \begin{array}{l} t_0 := \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(a + b\right)\\ \mathbf{if}\;{a}^{2} \leq 2 \cdot 10^{+294}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(t_0 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(t_0 \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \left(\sqrt{\pi} \cdot \sqrt{\pi}\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (sin (* 0.005555555555555556 (* PI angle))) (+ a b))))
   (if (<= (pow a 2.0) 2e+294)
     (*
      2.0
      (*
       (- b a)
       (* t_0 (log1p (expm1 (cos (* PI (* 0.005555555555555556 angle))))))))
     (*
      2.0
      (*
       (- b a)
       (*
        t_0
        (cos (* 0.005555555555555556 (* angle (* (sqrt PI) (sqrt PI)))))))))))

C

b = abs(b);
double code(double a, double b, double angle) {
	double t_0 = sin((0.005555555555555556 * (((double) M_PI) * angle))) * (a + b);
	double tmp;
	if (pow(a, 2.0) <= 2e+294) {
		tmp = 2.0 * ((b - a) * (t_0 * log1p(expm1(cos((((double) M_PI) * (0.005555555555555556 * angle)))))));
	} else {
		tmp = 2.0 * ((b - a) * (t_0 * cos((0.005555555555555556 * (angle * (sqrt(((double) M_PI)) * sqrt(((double) M_PI))))))));
	}
	return tmp;
}

Java

b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double t_0 = Math.sin((0.005555555555555556 * (Math.PI * angle))) * (a + b);
	double tmp;
	if (Math.pow(a, 2.0) <= 2e+294) {
		tmp = 2.0 * ((b - a) * (t_0 * Math.log1p(Math.expm1(Math.cos((Math.PI * (0.005555555555555556 * angle)))))));
	} else {
		tmp = 2.0 * ((b - a) * (t_0 * Math.cos((0.005555555555555556 * (angle * (Math.sqrt(Math.PI) * Math.sqrt(Math.PI)))))));
	}
	return tmp;
}

Python

b = abs(b)
def code(a, b, angle):
	t_0 = math.sin((0.005555555555555556 * (math.pi * angle))) * (a + b)
	tmp = 0
	if math.pow(a, 2.0) <= 2e+294:
		tmp = 2.0 * ((b - a) * (t_0 * math.log1p(math.expm1(math.cos((math.pi * (0.005555555555555556 * angle)))))))
	else:
		tmp = 2.0 * ((b - a) * (t_0 * math.cos((0.005555555555555556 * (angle * (math.sqrt(math.pi) * math.sqrt(math.pi)))))))
	return tmp

Julia

b = abs(b)
function code(a, b, angle)
	t_0 = Float64(sin(Float64(0.005555555555555556 * Float64(pi * angle))) * Float64(a + b))
	tmp = 0.0
	if ((a ^ 2.0) <= 2e+294)
		tmp = Float64(2.0 * Float64(Float64(b - a) * Float64(t_0 * log1p(expm1(cos(Float64(pi * Float64(0.005555555555555556 * angle))))))));
	else
		tmp = Float64(2.0 * Float64(Float64(b - a) * Float64(t_0 * cos(Float64(0.005555555555555556 * Float64(angle * Float64(sqrt(pi) * sqrt(pi))))))));
	end
	return tmp
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Sin[N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[a, 2.0], $MachinePrecision], 2e+294], N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(t$95$0 * N[Log[1 + N[(Exp[N[Cos[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(t$95$0 * N[Cos[N[(0.005555555555555556 * N[(angle * N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 a 2) < 2.00000000000000013e294

   1. Initial program 59.7%

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

      1. associate-*l* 59.7%

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

      2. associate-*l* 59.7%

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

      3. unpow2 59.7%

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

      4. unpow2 59.7%

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

      5. difference-of-squares 59.7%

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

   3. Simplified 59.7%

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

   4. Taylor expanded in angle around inf 68.7%

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

      1. *-commutative 68.7%

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

      2. *-commutative 68.7%

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

      3. associate-*r* 69.1%

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

      4. add-cube-cbrt 66.7%

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

      5. unpow3 67.2%

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

      6. log1p-expm1-u 67.2%

         \[\leadsto 2 \cdot \left(\left(b - a\right) \cdot \left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left({\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{3}\right)\right)\right)} \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right)\right)\right) \]

      7. unpow3 66.7%

         \[\leadsto 2 \cdot \left(\left(b - a\right) \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \color{blue}{\left(\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)} \cdot \sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right) \cdot \sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right)\right)\right) \]

      8. add-cube-cbrt 69.1%

         \[\leadsto 2 \cdot \left(\left(b - a\right) \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \color{blue}{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right)\right)\right) \]

      9. *-commutative 69.1%

         \[\leadsto 2 \cdot \left(\left(b - a\right) \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\pi \cdot \color{blue}{\left(0.005555555555555556 \cdot angle\right)}\right)\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right)\right)\right) \]

   6. Applied egg-rr 69.1%

      \[\leadsto 2 \cdot \left(\left(b - a\right) \cdot \left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)} \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right)\right)\right) \]

   if 2.00000000000000013e294 < (pow.f64 a 2)

   1. Initial program 30.2%

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

      1. associate-*l* 30.2%

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

      2. associate-*l* 30.2%

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

      3. unpow2 30.2%

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

      4. unpow2 30.2%

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

      5. difference-of-squares 51.8%

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

   3. Simplified 51.8%

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

   4. Taylor expanded in angle around inf 76.6%

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

      1. add-sqr-sqrt 86.3%

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

   6. Applied egg-rr 86.3%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 2 \cdot 10^{+294}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(a + b\right)\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(a + b\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \left(\sqrt{\pi} \cdot \sqrt{\pi}\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 3: 67.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\\ \mathbf{if}\;{b}^{2} - {a}^{2} \leq -\infty:\\ \;\;\;\;\left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\left(\sin t_0 \cdot \left(a + b\right)\right) \cdot \cos t_0\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* PI angle))))
   (if (<= (- (pow b 2.0) (pow a 2.0)) (- INFINITY))
     (* (* (* (- b a) angle) (* PI (+ a b))) 0.011111111111111112)
     (* 2.0 (* (- b a) (* (* (sin t_0) (+ a b)) (cos t_0)))))))

C

b = abs(b);
double code(double a, double b, double angle) {
	double t_0 = 0.005555555555555556 * (((double) M_PI) * angle);
	double tmp;
	if ((pow(b, 2.0) - pow(a, 2.0)) <= -((double) INFINITY)) {
		tmp = (((b - a) * angle) * (((double) M_PI) * (a + b))) * 0.011111111111111112;
	} else {
		tmp = 2.0 * ((b - a) * ((sin(t_0) * (a + b)) * cos(t_0)));
	}
	return tmp;
}

Java

b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double t_0 = 0.005555555555555556 * (Math.PI * angle);
	double tmp;
	if ((Math.pow(b, 2.0) - Math.pow(a, 2.0)) <= -Double.POSITIVE_INFINITY) {
		tmp = (((b - a) * angle) * (Math.PI * (a + b))) * 0.011111111111111112;
	} else {
		tmp = 2.0 * ((b - a) * ((Math.sin(t_0) * (a + b)) * Math.cos(t_0)));
	}
	return tmp;
}

Python

b = abs(b)
def code(a, b, angle):
	t_0 = 0.005555555555555556 * (math.pi * angle)
	tmp = 0
	if (math.pow(b, 2.0) - math.pow(a, 2.0)) <= -math.inf:
		tmp = (((b - a) * angle) * (math.pi * (a + b))) * 0.011111111111111112
	else:
		tmp = 2.0 * ((b - a) * ((math.sin(t_0) * (a + b)) * math.cos(t_0)))
	return tmp

Julia

b = abs(b)
function code(a, b, angle)
	t_0 = Float64(0.005555555555555556 * Float64(pi * angle))
	tmp = 0.0
	if (Float64((b ^ 2.0) - (a ^ 2.0)) <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(b - a) * angle) * Float64(pi * Float64(a + b))) * 0.011111111111111112);
	else
		tmp = Float64(2.0 * Float64(Float64(b - a) * Float64(Float64(sin(t_0) * Float64(a + b)) * cos(t_0))));
	end
	return tmp
end

MATLAB

b = abs(b)
function tmp_2 = code(a, b, angle)
	t_0 = 0.005555555555555556 * (pi * angle);
	tmp = 0.0;
	if (((b ^ 2.0) - (a ^ 2.0)) <= -Inf)
		tmp = (((b - a) * angle) * (pi * (a + b))) * 0.011111111111111112;
	else
		tmp = 2.0 * ((b - a) * ((sin(t_0) * (a + b)) * cos(t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(N[(N[(N[(b - a), $MachinePrecision] * angle), $MachinePrecision] * N[(Pi * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(N[(N[Sin[t$95$0], $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b 2) (pow.f64 a 2)) < -inf.0

   1. Initial program 45.1%

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

      1. associate-*l* 45.1%

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

      2. unpow2 45.1%

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

      3. unpow2 45.1%

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

      4. difference-of-squares 45.1%

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

   3. Simplified 45.1%

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

   4. Taylor expanded in angle around 0 54.9%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative 54.9%

         \[\leadsto \color{blue}{\left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right) \cdot 0.011111111111111112} \]

      2. associate-*r* 80.3%

         \[\leadsto \color{blue}{\left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)} \cdot 0.011111111111111112 \]

   6. Simplified 80.3%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112} \]

   if -inf.0 < (-.f64 (pow.f64 b 2) (pow.f64 a 2))

   1. Initial program 51.7%

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

      1. associate-*l* 51.7%

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

      2. associate-*l* 51.7%

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

      3. unpow2 51.7%

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

      4. unpow2 51.7%

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

      5. difference-of-squares 60.2%

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

   3. Simplified 60.2%

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

   4. Taylor expanded in angle around inf 71.4%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -\infty:\\ \;\;\;\;\left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(a + b\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\\ \end{array} \]

Alternative 4: 67.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ 2 \cdot \left(\left(b - a\right) \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(a + b\right)\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (*
  2.0
  (*
   (- b a)
   (*
    (* (sin (* 0.005555555555555556 (* PI angle))) (+ a b))
    (log1p (expm1 (cos (* PI (* 0.005555555555555556 angle)))))))))

C

b = abs(b);
double code(double a, double b, double angle) {
	return 2.0 * ((b - a) * ((sin((0.005555555555555556 * (((double) M_PI) * angle))) * (a + b)) * log1p(expm1(cos((((double) M_PI) * (0.005555555555555556 * angle)))))));
}

Java

b = Math.abs(b);
public static double code(double a, double b, double angle) {
	return 2.0 * ((b - a) * ((Math.sin((0.005555555555555556 * (Math.PI * angle))) * (a + b)) * Math.log1p(Math.expm1(Math.cos((Math.PI * (0.005555555555555556 * angle)))))));
}

Python

b = abs(b)
def code(a, b, angle):
	return 2.0 * ((b - a) * ((math.sin((0.005555555555555556 * (math.pi * angle))) * (a + b)) * math.log1p(math.expm1(math.cos((math.pi * (0.005555555555555556 * angle)))))))

Julia

b = abs(b)
function code(a, b, angle)
	return Float64(2.0 * Float64(Float64(b - a) * Float64(Float64(sin(Float64(0.005555555555555556 * Float64(pi * angle))) * Float64(a + b)) * log1p(expm1(cos(Float64(pi * Float64(0.005555555555555556 * angle))))))))
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_] := N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(N[(N[Sin[N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[Log[1 + N[(Exp[N[Cos[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.4%

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

   1. associate-*l* 50.4%

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

   2. associate-*l* 50.4%

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

   3. unpow2 50.4%

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

   4. unpow2 50.4%

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

   5. difference-of-squares 57.2%

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

3. Simplified 57.2%

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

4. Taylor expanded in angle around inf 71.2%

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

   1. *-commutative 71.2%

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

   2. *-commutative 71.2%

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

   3. associate-*r* 71.8%

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

   4. add-cube-cbrt 69.4%

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

   5. unpow3 70.9%

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

   6. log1p-expm1-u 70.9%

      \[\leadsto 2 \cdot \left(\left(b - a\right) \cdot \left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left({\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{3}\right)\right)\right)} \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right)\right)\right) \]

   7. unpow3 69.4%

      \[\leadsto 2 \cdot \left(\left(b - a\right) \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \color{blue}{\left(\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)} \cdot \sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right) \cdot \sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right)\right)\right) \]

   8. add-cube-cbrt 71.8%

      \[\leadsto 2 \cdot \left(\left(b - a\right) \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \color{blue}{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right)\right)\right) \]

   9. *-commutative 71.8%

      \[\leadsto 2 \cdot \left(\left(b - a\right) \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\pi \cdot \color{blue}{\left(0.005555555555555556 \cdot angle\right)}\right)\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right)\right)\right) \]

6. Applied egg-rr 71.8%

   \[\leadsto 2 \cdot \left(\left(b - a\right) \cdot \left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)} \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right)\right)\right) \]

7. Final simplification 71.8%

   \[\leadsto 2 \cdot \left(\left(b - a\right) \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(a + b\right)\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)\right) \]

Alternative 5: 65.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+116} \lor \neg \left(\frac{angle}{180} \leq 2 \cdot 10^{+55}\right):\\ \;\;\;\;\left(\sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot 0.5\right) \cdot \left(2 \cdot {\left(a + b\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(a + b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (or (<= (/ angle 180.0) -5e+116) (not (<= (/ angle 180.0) 2e+55)))
   (*
    (* (sin (* 2.0 (* PI (* 0.005555555555555556 angle)))) 0.5)
    (* 2.0 (pow (+ a b) 2.0)))
   (*
    2.0
    (* (- b a) (* (sin (* 0.005555555555555556 (* PI angle))) (+ a b))))))

C

b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (((angle / 180.0) <= -5e+116) || !((angle / 180.0) <= 2e+55)) {
		tmp = (sin((2.0 * (((double) M_PI) * (0.005555555555555556 * angle)))) * 0.5) * (2.0 * pow((a + b), 2.0));
	} else {
		tmp = 2.0 * ((b - a) * (sin((0.005555555555555556 * (((double) M_PI) * angle))) * (a + b)));
	}
	return tmp;
}

Java

b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (((angle / 180.0) <= -5e+116) || !((angle / 180.0) <= 2e+55)) {
		tmp = (Math.sin((2.0 * (Math.PI * (0.005555555555555556 * angle)))) * 0.5) * (2.0 * Math.pow((a + b), 2.0));
	} else {
		tmp = 2.0 * ((b - a) * (Math.sin((0.005555555555555556 * (Math.PI * angle))) * (a + b)));
	}
	return tmp;
}

Python

b = abs(b)
def code(a, b, angle):
	tmp = 0
	if ((angle / 180.0) <= -5e+116) or not ((angle / 180.0) <= 2e+55):
		tmp = (math.sin((2.0 * (math.pi * (0.005555555555555556 * angle)))) * 0.5) * (2.0 * math.pow((a + b), 2.0))
	else:
		tmp = 2.0 * ((b - a) * (math.sin((0.005555555555555556 * (math.pi * angle))) * (a + b)))
	return tmp

Julia

b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if ((Float64(angle / 180.0) <= -5e+116) || !(Float64(angle / 180.0) <= 2e+55))
		tmp = Float64(Float64(sin(Float64(2.0 * Float64(pi * Float64(0.005555555555555556 * angle)))) * 0.5) * Float64(2.0 * (Float64(a + b) ^ 2.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(b - a) * Float64(sin(Float64(0.005555555555555556 * Float64(pi * angle))) * Float64(a + b))));
	end
	return tmp
end

MATLAB

b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (((angle / 180.0) <= -5e+116) || ~(((angle / 180.0) <= 2e+55)))
		tmp = (sin((2.0 * (pi * (0.005555555555555556 * angle)))) * 0.5) * (2.0 * ((a + b) ^ 2.0));
	else
		tmp = 2.0 * ((b - a) * (sin((0.005555555555555556 * (pi * angle))) * (a + b)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[Or[LessEqual[N[(angle / 180.0), $MachinePrecision], -5e+116], N[Not[LessEqual[N[(angle / 180.0), $MachinePrecision], 2e+55]], $MachinePrecision]], N[(N[(N[Sin[N[(2.0 * N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 * N[Power[N[(a + b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(N[Sin[N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < -5.00000000000000025e116 or 2.00000000000000002e55 < (/.f64 angle 180)

   1. Initial program 26.7%

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

      1. *-commutative 26.7%

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

      2. associate-*l* 26.7%

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

      3. unpow2 26.7%

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

      4. fma-neg 27.8%

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

      5. unpow2 27.8%

         \[\leadsto \sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \mathsf{fma}\left(b, b, -\color{blue}{a \cdot a}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. Simplified 27.8%

      \[\leadsto \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \mathsf{fma}\left(b, b, -a \cdot a\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   5. Applied egg-rr 23.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(\sin 0 + \sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right) \cdot 0.5\right) \cdot \left(2 \cdot {\left(b + a\right)}^{2}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 24.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(\sin 0 + \sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right) \cdot 0.5\right) \cdot \left(2 \cdot {\left(b + a\right)}^{2}\right)\right)\right)} \]

      2. expm1-log1p 42.6%

         \[\leadsto \color{blue}{\left(\left(\sin 0 + \sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right) \cdot 0.5\right) \cdot \left(2 \cdot {\left(b + a\right)}^{2}\right)} \]

      3. sin-0 42.6%

         \[\leadsto \left(\left(\color{blue}{0} + \sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right) \cdot 0.5\right) \cdot \left(2 \cdot {\left(b + a\right)}^{2}\right) \]

      4. +-lft-identity 42.6%

         \[\leadsto \left(\color{blue}{\sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)} \cdot 0.5\right) \cdot \left(2 \cdot {\left(b + a\right)}^{2}\right) \]

      5. *-commutative 42.6%

         \[\leadsto \left(\sin \color{blue}{\left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \cdot 0.5\right) \cdot \left(2 \cdot {\left(b + a\right)}^{2}\right) \]

      6. +-commutative 42.6%

         \[\leadsto \left(\sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot 0.5\right) \cdot \left(2 \cdot {\color{blue}{\left(a + b\right)}}^{2}\right) \]

   7. Simplified 42.6%

      \[\leadsto \color{blue}{\left(\sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot 0.5\right) \cdot \left(2 \cdot {\left(a + b\right)}^{2}\right)} \]

   if -5.00000000000000025e116 < (/.f64 angle 180) < 2.00000000000000002e55

   1. Initial program 63.2%

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

      1. associate-*l* 63.2%

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

      2. unpow2 63.2%

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

      3. unpow2 63.2%

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

      4. difference-of-squares 72.0%

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

   3. Simplified 72.0%

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

      1. add-cube-cbrt 70.2%

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

      2. pow3 70.7%

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

      3. div-inv 71.3%

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

      4. metadata-eval 71.3%

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

   5. Applied egg-rr 71.3%

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

   6. Taylor expanded in angle around inf 89.6%

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

      1. *-commutative 89.6%

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

      2. +-commutative 89.6%

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

   8. Simplified 89.6%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+116} \lor \neg \left(\frac{angle}{180} \leq 2 \cdot 10^{+55}\right):\\ \;\;\;\;\left(\sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot 0.5\right) \cdot \left(2 \cdot {\left(a + b\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(a + b\right)\right)\right)\\ \end{array} \]

Alternative 6: 65.9% accurate, 2.9× speedup

Math

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

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (* 2.0 (* (+ a b) (* (- b a) (sin (* PI (* 0.005555555555555556 angle)))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_] := N[(2.0 * N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.4%

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

   1. associate-*l* 50.4%

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

   2. unpow2 50.4%

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

   3. unpow2 50.4%

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

   4. difference-of-squares 57.2%

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

3. Simplified 57.2%

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

   1. add-cube-cbrt 57.4%

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

   2. pow3 58.0%

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

   3. div-inv 58.9%

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

   4. metadata-eval 58.9%

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

5. Applied egg-rr 58.9%

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

6. Taylor expanded in angle around inf 67.5%

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

   1. associate-*r* 67.5%

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

   2. associate-*r* 68.2%

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

   3. *-commutative 68.2%

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

   4. +-commutative 68.2%

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

8. Simplified 68.2%

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

9. Final simplification 68.2%

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

Alternative 7: 49.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 2.15 \cdot 10^{-166}:\\ \;\;\;\;\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= a 2.15e-166)
   (* (sin (* PI (/ angle 180.0))) (* 2.0 (* b b)))
   (* (* (* (- b a) angle) (* PI (+ a b))) 0.011111111111111112)))

C

b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.15e-166) {
		tmp = sin((((double) M_PI) * (angle / 180.0))) * (2.0 * (b * b));
	} else {
		tmp = (((b - a) * angle) * (((double) M_PI) * (a + b))) * 0.011111111111111112;
	}
	return tmp;
}

Java

b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.15e-166) {
		tmp = Math.sin((Math.PI * (angle / 180.0))) * (2.0 * (b * b));
	} else {
		tmp = (((b - a) * angle) * (Math.PI * (a + b))) * 0.011111111111111112;
	}
	return tmp;
}

Python

b = abs(b)
def code(a, b, angle):
	tmp = 0
	if a <= 2.15e-166:
		tmp = math.sin((math.pi * (angle / 180.0))) * (2.0 * (b * b))
	else:
		tmp = (((b - a) * angle) * (math.pi * (a + b))) * 0.011111111111111112
	return tmp

Julia

b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (a <= 2.15e-166)
		tmp = Float64(sin(Float64(pi * Float64(angle / 180.0))) * Float64(2.0 * Float64(b * b)));
	else
		tmp = Float64(Float64(Float64(Float64(b - a) * angle) * Float64(pi * Float64(a + b))) * 0.011111111111111112);
	end
	return tmp
end

MATLAB

b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 2.15e-166)
		tmp = sin((pi * (angle / 180.0))) * (2.0 * (b * b));
	else
		tmp = (((b - a) * angle) * (pi * (a + b))) * 0.011111111111111112;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[a, 2.15e-166], N[(N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(b - a), $MachinePrecision] * angle), $MachinePrecision] * N[(Pi * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.15e-166

   1. Initial program 54.7%

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

      1. *-commutative 54.7%

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

      2. associate-*l* 54.7%

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

      3. unpow2 54.7%

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

      4. fma-neg 56.7%

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

      5. unpow2 56.7%

         \[\leadsto \sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \mathsf{fma}\left(b, b, -\color{blue}{a \cdot a}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. Simplified 56.7%

      \[\leadsto \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \mathsf{fma}\left(b, b, -a \cdot a\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in b around inf 42.4%

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

      1. unpow2 42.4%

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

   6. Simplified 42.4%

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

   7. Taylor expanded in angle around 0 40.1%

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

   if 2.15e-166 < a

   1. Initial program 44.2%

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

      1. associate-*l* 44.2%

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

      2. unpow2 44.2%

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

      3. unpow2 44.2%

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

      4. difference-of-squares 53.1%

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

   3. Simplified 53.1%

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

   4. Taylor expanded in angle around 0 53.0%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative 53.0%

         \[\leadsto \color{blue}{\left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right) \cdot 0.011111111111111112} \]

      2. associate-*r* 65.4%

         \[\leadsto \color{blue}{\left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)} \cdot 0.011111111111111112 \]

   6. Simplified 65.4%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.15 \cdot 10^{-166}:\\ \;\;\;\;\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112\\ \end{array} \]

Alternative 8: 41.2% accurate, 5.4× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \begin{array}{l} t_0 := 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{if}\;a \leq 1.16 \cdot 10^{-27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{-11}:\\ \;\;\;\;\pi \cdot \left(\left(a \cdot a\right) \cdot \left(angle \cdot -0.011111111111111112\right)\right)\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{+28}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\pi \cdot \left(\left(a \cdot a\right) \cdot -0.011111111111111112\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* 0.011111111111111112 (* angle (* PI (* b b))))))
   (if (<= a 1.16e-27)
     t_0
     (if (<= a 9.5e-11)
       (* PI (* (* a a) (* angle -0.011111111111111112)))
       (if (<= a 4.5e+28)
         t_0
         (* angle (* PI (* (* a a) -0.011111111111111112))))))))

C

b = abs(b);
double code(double a, double b, double angle) {
	double t_0 = 0.011111111111111112 * (angle * (((double) M_PI) * (b * b)));
	double tmp;
	if (a <= 1.16e-27) {
		tmp = t_0;
	} else if (a <= 9.5e-11) {
		tmp = ((double) M_PI) * ((a * a) * (angle * -0.011111111111111112));
	} else if (a <= 4.5e+28) {
		tmp = t_0;
	} else {
		tmp = angle * (((double) M_PI) * ((a * a) * -0.011111111111111112));
	}
	return tmp;
}

Java

b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double t_0 = 0.011111111111111112 * (angle * (Math.PI * (b * b)));
	double tmp;
	if (a <= 1.16e-27) {
		tmp = t_0;
	} else if (a <= 9.5e-11) {
		tmp = Math.PI * ((a * a) * (angle * -0.011111111111111112));
	} else if (a <= 4.5e+28) {
		tmp = t_0;
	} else {
		tmp = angle * (Math.PI * ((a * a) * -0.011111111111111112));
	}
	return tmp;
}

Python

b = abs(b)
def code(a, b, angle):
	t_0 = 0.011111111111111112 * (angle * (math.pi * (b * b)))
	tmp = 0
	if a <= 1.16e-27:
		tmp = t_0
	elif a <= 9.5e-11:
		tmp = math.pi * ((a * a) * (angle * -0.011111111111111112))
	elif a <= 4.5e+28:
		tmp = t_0
	else:
		tmp = angle * (math.pi * ((a * a) * -0.011111111111111112))
	return tmp

Julia

b = abs(b)
function code(a, b, angle)
	t_0 = Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(b * b))))
	tmp = 0.0
	if (a <= 1.16e-27)
		tmp = t_0;
	elseif (a <= 9.5e-11)
		tmp = Float64(pi * Float64(Float64(a * a) * Float64(angle * -0.011111111111111112)));
	elseif (a <= 4.5e+28)
		tmp = t_0;
	else
		tmp = Float64(angle * Float64(pi * Float64(Float64(a * a) * -0.011111111111111112)));
	end
	return tmp
end

MATLAB

b = abs(b)
function tmp_2 = code(a, b, angle)
	t_0 = 0.011111111111111112 * (angle * (pi * (b * b)));
	tmp = 0.0;
	if (a <= 1.16e-27)
		tmp = t_0;
	elseif (a <= 9.5e-11)
		tmp = pi * ((a * a) * (angle * -0.011111111111111112));
	elseif (a <= 4.5e+28)
		tmp = t_0;
	else
		tmp = angle * (pi * ((a * a) * -0.011111111111111112));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_] := Block[{t$95$0 = N[(0.011111111111111112 * N[(angle * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 1.16e-27], t$95$0, If[LessEqual[a, 9.5e-11], N[(Pi * N[(N[(a * a), $MachinePrecision] * N[(angle * -0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 4.5e+28], t$95$0, N[(angle * N[(Pi * N[(N[(a * a), $MachinePrecision] * -0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 1.16000000000000005e-27 or 9.49999999999999951e-11 < a < 4.4999999999999997e28

   1. Initial program 55.9%

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

      1. associate-*l* 55.9%

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

      2. unpow2 55.9%

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

      3. unpow2 55.9%

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

      4. difference-of-squares 60.3%

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

   3. Simplified 60.3%

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

   4. Taylor expanded in angle around 0 56.4%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]

   5. Taylor expanded in b around inf 39.6%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 39.6%

         \[\leadsto \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \pi\right)\right) \cdot 0.011111111111111112} \]

      2. *-commutative 39.6%

         \[\leadsto \left(angle \cdot \color{blue}{\left(\pi \cdot {b}^{2}\right)}\right) \cdot 0.011111111111111112 \]

      3. unpow2 39.6%

         \[\leadsto \left(angle \cdot \left(\pi \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \cdot 0.011111111111111112 \]

   7. Simplified 39.6%

      \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right) \cdot 0.011111111111111112} \]

   if 1.16000000000000005e-27 < a < 9.49999999999999951e-11

   1. Initial program 67.1%

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

      1. associate-*l* 67.1%

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

      2. unpow2 67.1%

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

      3. unpow2 67.1%

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

      4. difference-of-squares 67.1%

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

   3. Simplified 67.1%

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

   4. Taylor expanded in angle around 0 47.4%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]

   5. Taylor expanded in b around 0 47.4%

      \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 47.4%

         \[\leadsto \color{blue}{\left(angle \cdot \left({a}^{2} \cdot \pi\right)\right) \cdot -0.011111111111111112} \]

      2. *-commutative 47.4%

         \[\leadsto \left(angle \cdot \color{blue}{\left(\pi \cdot {a}^{2}\right)}\right) \cdot -0.011111111111111112 \]

      3. unpow2 47.4%

         \[\leadsto \left(angle \cdot \left(\pi \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot -0.011111111111111112 \]

   7. Simplified 47.4%

      \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right) \cdot -0.011111111111111112} \]

   8. Taylor expanded in angle around 0 47.4%

      \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 47.4%

         \[\leadsto \color{blue}{\left(angle \cdot \left({a}^{2} \cdot \pi\right)\right) \cdot -0.011111111111111112} \]

      2. *-commutative 47.4%

         \[\leadsto \left(angle \cdot \color{blue}{\left(\pi \cdot {a}^{2}\right)}\right) \cdot -0.011111111111111112 \]

      3. associate-*r* 47.7%

         \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot {a}^{2}\right)} \cdot -0.011111111111111112 \]

      4. associate-*l* 48.0%

         \[\leadsto \color{blue}{\left(angle \cdot \pi\right) \cdot \left({a}^{2} \cdot -0.011111111111111112\right)} \]

      5. associate-*r* 48.0%

         \[\leadsto \color{blue}{angle \cdot \left(\pi \cdot \left({a}^{2} \cdot -0.011111111111111112\right)\right)} \]

      6. unpow2 48.0%

         \[\leadsto angle \cdot \left(\pi \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot -0.011111111111111112\right)\right) \]

   10. Simplified 48.0%

       \[\leadsto \color{blue}{angle \cdot \left(\pi \cdot \left(\left(a \cdot a\right) \cdot -0.011111111111111112\right)\right)} \]

   11. Taylor expanded in angle around 0 47.4%

       \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*r* 48.0%

          \[\leadsto \color{blue}{\left(-0.011111111111111112 \cdot angle\right) \cdot \left({a}^{2} \cdot \pi\right)} \]

       2. associate-*r* 48.0%

          \[\leadsto \color{blue}{\left(\left(-0.011111111111111112 \cdot angle\right) \cdot {a}^{2}\right) \cdot \pi} \]

       3. *-commutative 48.0%

          \[\leadsto \color{blue}{\left({a}^{2} \cdot \left(-0.011111111111111112 \cdot angle\right)\right)} \cdot \pi \]

       4. *-commutative 48.0%

          \[\leadsto \color{blue}{\pi \cdot \left({a}^{2} \cdot \left(-0.011111111111111112 \cdot angle\right)\right)} \]

       5. unpow2 48.0%

          \[\leadsto \pi \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \left(-0.011111111111111112 \cdot angle\right)\right) \]

       6. *-commutative 48.0%

          \[\leadsto \pi \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(angle \cdot -0.011111111111111112\right)}\right) \]

   13. Simplified 48.0%

       \[\leadsto \color{blue}{\pi \cdot \left(\left(a \cdot a\right) \cdot \left(angle \cdot -0.011111111111111112\right)\right)} \]

   if 4.4999999999999997e28 < a

   1. Initial program 32.6%

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

      1. associate-*l* 32.6%

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

      2. unpow2 32.6%

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

      3. unpow2 32.6%

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

      4. difference-of-squares 47.4%

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

   3. Simplified 47.4%

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

   4. Taylor expanded in angle around 0 49.0%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]

   5. Taylor expanded in b around 0 45.4%

      \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 45.4%

         \[\leadsto \color{blue}{\left(angle \cdot \left({a}^{2} \cdot \pi\right)\right) \cdot -0.011111111111111112} \]

      2. *-commutative 45.4%

         \[\leadsto \left(angle \cdot \color{blue}{\left(\pi \cdot {a}^{2}\right)}\right) \cdot -0.011111111111111112 \]

      3. unpow2 45.4%

         \[\leadsto \left(angle \cdot \left(\pi \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot -0.011111111111111112 \]

   7. Simplified 45.4%

      \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right) \cdot -0.011111111111111112} \]

   8. Taylor expanded in angle around 0 45.4%

      \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 45.4%

         \[\leadsto \color{blue}{\left(angle \cdot \left({a}^{2} \cdot \pi\right)\right) \cdot -0.011111111111111112} \]

      2. *-commutative 45.4%

         \[\leadsto \left(angle \cdot \color{blue}{\left(\pi \cdot {a}^{2}\right)}\right) \cdot -0.011111111111111112 \]

      3. associate-*r* 45.4%

         \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot {a}^{2}\right)} \cdot -0.011111111111111112 \]

      4. associate-*l* 45.4%

         \[\leadsto \color{blue}{\left(angle \cdot \pi\right) \cdot \left({a}^{2} \cdot -0.011111111111111112\right)} \]

      5. associate-*r* 45.4%

         \[\leadsto \color{blue}{angle \cdot \left(\pi \cdot \left({a}^{2} \cdot -0.011111111111111112\right)\right)} \]

      6. unpow2 45.4%

         \[\leadsto angle \cdot \left(\pi \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot -0.011111111111111112\right)\right) \]

   10. Simplified 45.4%

       \[\leadsto \color{blue}{angle \cdot \left(\pi \cdot \left(\left(a \cdot a\right) \cdot -0.011111111111111112\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.16 \cdot 10^{-27}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{-11}:\\ \;\;\;\;\pi \cdot \left(\left(a \cdot a\right) \cdot \left(angle \cdot -0.011111111111111112\right)\right)\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{+28}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\pi \cdot \left(\left(a \cdot a\right) \cdot -0.011111111111111112\right)\right)\\ \end{array} \]

Alternative 9: 41.2% accurate, 5.4× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \begin{array}{l} t_0 := \pi \cdot \left(b \cdot b\right)\\ \mathbf{if}\;a \leq 7.2 \cdot 10^{-27}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot t_0\right)\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{-11}:\\ \;\;\;\;\pi \cdot \left(\left(a \cdot a\right) \cdot \left(angle \cdot -0.011111111111111112\right)\right)\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{+28}:\\ \;\;\;\;\left(angle \cdot 0.011111111111111112\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\pi \cdot \left(\left(a \cdot a\right) \cdot -0.011111111111111112\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (* b b))))
   (if (<= a 7.2e-27)
     (* 0.011111111111111112 (* angle t_0))
     (if (<= a 9.5e-11)
       (* PI (* (* a a) (* angle -0.011111111111111112)))
       (if (<= a 5.2e+28)
         (* (* angle 0.011111111111111112) t_0)
         (* angle (* PI (* (* a a) -0.011111111111111112))))))))

C

b = abs(b);
double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (b * b);
	double tmp;
	if (a <= 7.2e-27) {
		tmp = 0.011111111111111112 * (angle * t_0);
	} else if (a <= 9.5e-11) {
		tmp = ((double) M_PI) * ((a * a) * (angle * -0.011111111111111112));
	} else if (a <= 5.2e+28) {
		tmp = (angle * 0.011111111111111112) * t_0;
	} else {
		tmp = angle * (((double) M_PI) * ((a * a) * -0.011111111111111112));
	}
	return tmp;
}

Java

b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (b * b);
	double tmp;
	if (a <= 7.2e-27) {
		tmp = 0.011111111111111112 * (angle * t_0);
	} else if (a <= 9.5e-11) {
		tmp = Math.PI * ((a * a) * (angle * -0.011111111111111112));
	} else if (a <= 5.2e+28) {
		tmp = (angle * 0.011111111111111112) * t_0;
	} else {
		tmp = angle * (Math.PI * ((a * a) * -0.011111111111111112));
	}
	return tmp;
}

Python

b = abs(b)
def code(a, b, angle):
	t_0 = math.pi * (b * b)
	tmp = 0
	if a <= 7.2e-27:
		tmp = 0.011111111111111112 * (angle * t_0)
	elif a <= 9.5e-11:
		tmp = math.pi * ((a * a) * (angle * -0.011111111111111112))
	elif a <= 5.2e+28:
		tmp = (angle * 0.011111111111111112) * t_0
	else:
		tmp = angle * (math.pi * ((a * a) * -0.011111111111111112))
	return tmp

Julia

b = abs(b)
function code(a, b, angle)
	t_0 = Float64(pi * Float64(b * b))
	tmp = 0.0
	if (a <= 7.2e-27)
		tmp = Float64(0.011111111111111112 * Float64(angle * t_0));
	elseif (a <= 9.5e-11)
		tmp = Float64(pi * Float64(Float64(a * a) * Float64(angle * -0.011111111111111112)));
	elseif (a <= 5.2e+28)
		tmp = Float64(Float64(angle * 0.011111111111111112) * t_0);
	else
		tmp = Float64(angle * Float64(pi * Float64(Float64(a * a) * -0.011111111111111112)));
	end
	return tmp
end

MATLAB

b = abs(b)
function tmp_2 = code(a, b, angle)
	t_0 = pi * (b * b);
	tmp = 0.0;
	if (a <= 7.2e-27)
		tmp = 0.011111111111111112 * (angle * t_0);
	elseif (a <= 9.5e-11)
		tmp = pi * ((a * a) * (angle * -0.011111111111111112));
	elseif (a <= 5.2e+28)
		tmp = (angle * 0.011111111111111112) * t_0;
	else
		tmp = angle * (pi * ((a * a) * -0.011111111111111112));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 7.2e-27], N[(0.011111111111111112 * N[(angle * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 9.5e-11], N[(Pi * N[(N[(a * a), $MachinePrecision] * N[(angle * -0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 5.2e+28], N[(N[(angle * 0.011111111111111112), $MachinePrecision] * t$95$0), $MachinePrecision], N[(angle * N[(Pi * N[(N[(a * a), $MachinePrecision] * -0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if a < 7.1999999999999997e-27

   1. Initial program 56.3%

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

      1. associate-*l* 56.3%

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

      2. unpow2 56.3%

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

      3. unpow2 56.3%

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

      4. difference-of-squares 60.9%

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

   3. Simplified 60.9%

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

   4. Taylor expanded in angle around 0 56.7%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]

   5. Taylor expanded in b around inf 40.0%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 40.0%

         \[\leadsto \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \pi\right)\right) \cdot 0.011111111111111112} \]

      2. *-commutative 40.0%

         \[\leadsto \left(angle \cdot \color{blue}{\left(\pi \cdot {b}^{2}\right)}\right) \cdot 0.011111111111111112 \]

      3. unpow2 40.0%

         \[\leadsto \left(angle \cdot \left(\pi \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \cdot 0.011111111111111112 \]

   7. Simplified 40.0%

      \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right) \cdot 0.011111111111111112} \]

   if 7.1999999999999997e-27 < a < 9.49999999999999951e-11

   1. Initial program 67.1%

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

      1. associate-*l* 67.1%

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

      2. unpow2 67.1%

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

      3. unpow2 67.1%

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

      4. difference-of-squares 67.1%

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

   3. Simplified 67.1%

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

   4. Taylor expanded in angle around 0 47.4%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]

   5. Taylor expanded in b around 0 47.4%

      \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 47.4%

         \[\leadsto \color{blue}{\left(angle \cdot \left({a}^{2} \cdot \pi\right)\right) \cdot -0.011111111111111112} \]

      2. *-commutative 47.4%

         \[\leadsto \left(angle \cdot \color{blue}{\left(\pi \cdot {a}^{2}\right)}\right) \cdot -0.011111111111111112 \]

      3. unpow2 47.4%

         \[\leadsto \left(angle \cdot \left(\pi \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot -0.011111111111111112 \]

   7. Simplified 47.4%

      \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right) \cdot -0.011111111111111112} \]

   8. Taylor expanded in angle around 0 47.4%

      \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 47.4%

         \[\leadsto \color{blue}{\left(angle \cdot \left({a}^{2} \cdot \pi\right)\right) \cdot -0.011111111111111112} \]

      2. *-commutative 47.4%

         \[\leadsto \left(angle \cdot \color{blue}{\left(\pi \cdot {a}^{2}\right)}\right) \cdot -0.011111111111111112 \]

      3. associate-*r* 47.7%

         \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot {a}^{2}\right)} \cdot -0.011111111111111112 \]

      4. associate-*l* 48.0%

         \[\leadsto \color{blue}{\left(angle \cdot \pi\right) \cdot \left({a}^{2} \cdot -0.011111111111111112\right)} \]

      5. associate-*r* 48.0%

         \[\leadsto \color{blue}{angle \cdot \left(\pi \cdot \left({a}^{2} \cdot -0.011111111111111112\right)\right)} \]

      6. unpow2 48.0%

         \[\leadsto angle \cdot \left(\pi \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot -0.011111111111111112\right)\right) \]

   10. Simplified 48.0%

       \[\leadsto \color{blue}{angle \cdot \left(\pi \cdot \left(\left(a \cdot a\right) \cdot -0.011111111111111112\right)\right)} \]

   11. Taylor expanded in angle around 0 47.4%

       \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*r* 48.0%

          \[\leadsto \color{blue}{\left(-0.011111111111111112 \cdot angle\right) \cdot \left({a}^{2} \cdot \pi\right)} \]

       2. associate-*r* 48.0%

          \[\leadsto \color{blue}{\left(\left(-0.011111111111111112 \cdot angle\right) \cdot {a}^{2}\right) \cdot \pi} \]

       3. *-commutative 48.0%

          \[\leadsto \color{blue}{\left({a}^{2} \cdot \left(-0.011111111111111112 \cdot angle\right)\right)} \cdot \pi \]

       4. *-commutative 48.0%

          \[\leadsto \color{blue}{\pi \cdot \left({a}^{2} \cdot \left(-0.011111111111111112 \cdot angle\right)\right)} \]

       5. unpow2 48.0%

          \[\leadsto \pi \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \left(-0.011111111111111112 \cdot angle\right)\right) \]

       6. *-commutative 48.0%

          \[\leadsto \pi \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(angle \cdot -0.011111111111111112\right)}\right) \]

   13. Simplified 48.0%

       \[\leadsto \color{blue}{\pi \cdot \left(\left(a \cdot a\right) \cdot \left(angle \cdot -0.011111111111111112\right)\right)} \]

   if 9.49999999999999951e-11 < a < 5.2000000000000004e28

   1. Initial program 48.9%

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

      1. associate-*l* 48.9%

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

      2. unpow2 48.9%

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

      3. unpow2 48.9%

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

      4. difference-of-squares 48.9%

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

   3. Simplified 48.9%

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

   4. Taylor expanded in angle around 0 51.8%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]

   5. Taylor expanded in b around inf 32.9%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 32.9%

         \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left({b}^{2} \cdot \pi\right)} \]

      2. *-commutative 32.9%

         \[\leadsto \color{blue}{\left({b}^{2} \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot angle\right)} \]

      3. *-commutative 32.9%

         \[\leadsto \color{blue}{\left(\pi \cdot {b}^{2}\right)} \cdot \left(0.011111111111111112 \cdot angle\right) \]

      4. unpow2 32.9%

         \[\leadsto \left(\pi \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot \left(0.011111111111111112 \cdot angle\right) \]

      5. *-commutative 32.9%

         \[\leadsto \left(\pi \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(angle \cdot 0.011111111111111112\right)} \]

   7. Simplified 32.9%

      \[\leadsto \color{blue}{\left(\pi \cdot \left(b \cdot b\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)} \]

   if 5.2000000000000004e28 < a

   1. Initial program 32.6%

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

      1. associate-*l* 32.6%

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

      2. unpow2 32.6%

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

      3. unpow2 32.6%

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

      4. difference-of-squares 47.4%

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

   3. Simplified 47.4%

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

   4. Taylor expanded in angle around 0 49.0%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]

   5. Taylor expanded in b around 0 45.4%

      \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 45.4%

         \[\leadsto \color{blue}{\left(angle \cdot \left({a}^{2} \cdot \pi\right)\right) \cdot -0.011111111111111112} \]

      2. *-commutative 45.4%

         \[\leadsto \left(angle \cdot \color{blue}{\left(\pi \cdot {a}^{2}\right)}\right) \cdot -0.011111111111111112 \]

      3. unpow2 45.4%

         \[\leadsto \left(angle \cdot \left(\pi \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot -0.011111111111111112 \]

   7. Simplified 45.4%

      \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right) \cdot -0.011111111111111112} \]

   8. Taylor expanded in angle around 0 45.4%

      \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 45.4%

         \[\leadsto \color{blue}{\left(angle \cdot \left({a}^{2} \cdot \pi\right)\right) \cdot -0.011111111111111112} \]

      2. *-commutative 45.4%

         \[\leadsto \left(angle \cdot \color{blue}{\left(\pi \cdot {a}^{2}\right)}\right) \cdot -0.011111111111111112 \]

      3. associate-*r* 45.4%

         \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot {a}^{2}\right)} \cdot -0.011111111111111112 \]

      4. associate-*l* 45.4%

         \[\leadsto \color{blue}{\left(angle \cdot \pi\right) \cdot \left({a}^{2} \cdot -0.011111111111111112\right)} \]

      5. associate-*r* 45.4%

         \[\leadsto \color{blue}{angle \cdot \left(\pi \cdot \left({a}^{2} \cdot -0.011111111111111112\right)\right)} \]

      6. unpow2 45.4%

         \[\leadsto angle \cdot \left(\pi \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot -0.011111111111111112\right)\right) \]

   10. Simplified 45.4%

       \[\leadsto \color{blue}{angle \cdot \left(\pi \cdot \left(\left(a \cdot a\right) \cdot -0.011111111111111112\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 41.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 7.2 \cdot 10^{-27}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{-11}:\\ \;\;\;\;\pi \cdot \left(\left(a \cdot a\right) \cdot \left(angle \cdot -0.011111111111111112\right)\right)\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{+28}:\\ \;\;\;\;\left(angle \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\pi \cdot \left(\left(a \cdot a\right) \cdot -0.011111111111111112\right)\right)\\ \end{array} \]

Alternative 10: 42.3% accurate, 5.5× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 9 \cdot 10^{-34}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(a \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= a 9e-34)
   (* 0.011111111111111112 (* angle (* PI (* b b))))
   (* 0.011111111111111112 (* angle (* (- b a) (* a PI))))))

C

b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (a <= 9e-34) {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * (b * b)));
	} else {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * ((double) M_PI))));
	}
	return tmp;
}

Java

b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 9e-34) {
		tmp = 0.011111111111111112 * (angle * (Math.PI * (b * b)));
	} else {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * Math.PI)));
	}
	return tmp;
}

Python

b = abs(b)
def code(a, b, angle):
	tmp = 0
	if a <= 9e-34:
		tmp = 0.011111111111111112 * (angle * (math.pi * (b * b)))
	else:
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * math.pi)))
	return tmp

Julia

b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (a <= 9e-34)
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(b * b))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(Float64(b - a) * Float64(a * pi))));
	end
	return tmp
end

MATLAB

b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 9e-34)
		tmp = 0.011111111111111112 * (angle * (pi * (b * b)));
	else
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[a, 9e-34], N[(0.011111111111111112 * N[(angle * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle * N[(N[(b - a), $MachinePrecision] * N[(a * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 9.00000000000000085e-34

   1. Initial program 55.3%

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

      1. associate-*l* 55.3%

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

      2. unpow2 55.3%

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

      3. unpow2 55.3%

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

      4. difference-of-squares 60.0%

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

   3. Simplified 60.0%

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

   4. Taylor expanded in angle around 0 55.7%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]

   5. Taylor expanded in b around inf 39.7%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 39.7%

         \[\leadsto \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \pi\right)\right) \cdot 0.011111111111111112} \]

      2. *-commutative 39.7%

         \[\leadsto \left(angle \cdot \color{blue}{\left(\pi \cdot {b}^{2}\right)}\right) \cdot 0.011111111111111112 \]

      3. unpow2 39.7%

         \[\leadsto \left(angle \cdot \left(\pi \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \cdot 0.011111111111111112 \]

   7. Simplified 39.7%

      \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right) \cdot 0.011111111111111112} \]

   if 9.00000000000000085e-34 < a

   1. Initial program 40.0%

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

      1. associate-*l* 40.0%

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

      2. unpow2 40.0%

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

      3. unpow2 40.0%

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

      4. difference-of-squares 51.3%

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

   3. Simplified 51.3%

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

   4. Taylor expanded in angle around 0 51.7%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]

   5. Taylor expanded in a around inf 46.7%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(a \cdot \pi\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative 46.7%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(\pi \cdot a\right)}\right)\right) \]

   7. Simplified 46.7%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(\pi \cdot a\right)}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 9 \cdot 10^{-34}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(a \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 11: 48.6% accurate, 5.5× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{-52}:\\ \;\;\;\;\left(\pi \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot -0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(b \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= b 5e-52)
   (* (* PI (* a a)) (* angle -0.011111111111111112))
   (* 0.011111111111111112 (* angle (* (- b a) (* b PI))))))

C

b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (b <= 5e-52) {
		tmp = (((double) M_PI) * (a * a)) * (angle * -0.011111111111111112);
	} else {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (b * ((double) M_PI))));
	}
	return tmp;
}

Java

b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 5e-52) {
		tmp = (Math.PI * (a * a)) * (angle * -0.011111111111111112);
	} else {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (b * Math.PI)));
	}
	return tmp;
}

Python

b = abs(b)
def code(a, b, angle):
	tmp = 0
	if b <= 5e-52:
		tmp = (math.pi * (a * a)) * (angle * -0.011111111111111112)
	else:
		tmp = 0.011111111111111112 * (angle * ((b - a) * (b * math.pi)))
	return tmp

Julia

b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (b <= 5e-52)
		tmp = Float64(Float64(pi * Float64(a * a)) * Float64(angle * -0.011111111111111112));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(Float64(b - a) * Float64(b * pi))));
	end
	return tmp
end

MATLAB

b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 5e-52)
		tmp = (pi * (a * a)) * (angle * -0.011111111111111112);
	else
		tmp = 0.011111111111111112 * (angle * ((b - a) * (b * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[b, 5e-52], N[(N[(Pi * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(angle * -0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle * N[(N[(b - a), $MachinePrecision] * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5e-52

   1. Initial program 53.6%

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

      1. associate-*l* 53.6%

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

      2. unpow2 53.6%

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

      3. unpow2 53.6%

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

      4. difference-of-squares 58.2%

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

   3. Simplified 58.2%

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

   4. Taylor expanded in angle around 0 56.0%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]

   5. Taylor expanded in b around 0 42.1%

      \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 42.1%

         \[\leadsto \color{blue}{\left(-0.011111111111111112 \cdot angle\right) \cdot \left({a}^{2} \cdot \pi\right)} \]

      2. *-commutative 42.1%

         \[\leadsto \color{blue}{\left({a}^{2} \cdot \pi\right) \cdot \left(-0.011111111111111112 \cdot angle\right)} \]

      3. *-commutative 42.1%

         \[\leadsto \color{blue}{\left(\pi \cdot {a}^{2}\right)} \cdot \left(-0.011111111111111112 \cdot angle\right) \]

      4. unpow2 42.1%

         \[\leadsto \left(\pi \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \left(-0.011111111111111112 \cdot angle\right) \]

      5. *-commutative 42.1%

         \[\leadsto \left(\pi \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(angle \cdot -0.011111111111111112\right)} \]

   7. Simplified 42.1%

      \[\leadsto \color{blue}{\left(\pi \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot -0.011111111111111112\right)} \]

   if 5e-52 < b

   1. Initial program 42.4%

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

      1. associate-*l* 42.4%

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

      2. unpow2 42.4%

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

      3. unpow2 42.4%

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

      4. difference-of-squares 55.0%

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

   3. Simplified 55.0%

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

   4. Taylor expanded in angle around 0 50.5%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]

   5. Taylor expanded in a around 0 44.8%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(b \cdot \pi\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative 44.8%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(\pi \cdot b\right)}\right)\right) \]

   7. Simplified 44.8%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(\pi \cdot b\right)}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{-52}:\\ \;\;\;\;\left(\pi \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot -0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(b \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 12: 54.5% accurate, 5.5× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ 0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right) \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (* 0.011111111111111112 (* angle (* (- b a) (* PI (+ a b))))))

C

b = abs(b);
double code(double a, double b, double angle) {
	return 0.011111111111111112 * (angle * ((b - a) * (((double) M_PI) * (a + b))));
}

Java

b = Math.abs(b);
public static double code(double a, double b, double angle) {
	return 0.011111111111111112 * (angle * ((b - a) * (Math.PI * (a + b))));
}

Python

b = abs(b)
def code(a, b, angle):
	return 0.011111111111111112 * (angle * ((b - a) * (math.pi * (a + b))))

Julia

b = abs(b)
function code(a, b, angle)
	return Float64(0.011111111111111112 * Float64(angle * Float64(Float64(b - a) * Float64(pi * Float64(a + b)))))
end

MATLAB

b = abs(b)
function tmp = code(a, b, angle)
	tmp = 0.011111111111111112 * (angle * ((b - a) * (pi * (a + b))));
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_] := N[(0.011111111111111112 * N[(angle * N[(N[(b - a), $MachinePrecision] * N[(Pi * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.4%

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

   1. associate-*l* 50.4%

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

   2. unpow2 50.4%

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

   3. unpow2 50.4%

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

   4. difference-of-squares 57.2%

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

3. Simplified 57.2%

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

4. Taylor expanded in angle around 0 54.4%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]

5. Final simplification 54.4%

   \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right) \]

Alternative 13: 54.5% accurate, 5.5× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ angle \cdot \left(0.011111111111111112 \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right) \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (* angle (* 0.011111111111111112 (* (- b a) (* PI (+ a b))))))

C

b = abs(b);
double code(double a, double b, double angle) {
	return angle * (0.011111111111111112 * ((b - a) * (((double) M_PI) * (a + b))));
}

Java

b = Math.abs(b);
public static double code(double a, double b, double angle) {
	return angle * (0.011111111111111112 * ((b - a) * (Math.PI * (a + b))));
}

Python

b = abs(b)
def code(a, b, angle):
	return angle * (0.011111111111111112 * ((b - a) * (math.pi * (a + b))))

Julia

b = abs(b)
function code(a, b, angle)
	return Float64(angle * Float64(0.011111111111111112 * Float64(Float64(b - a) * Float64(pi * Float64(a + b)))))
end

MATLAB

b = abs(b)
function tmp = code(a, b, angle)
	tmp = angle * (0.011111111111111112 * ((b - a) * (pi * (a + b))));
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_] := N[(angle * N[(0.011111111111111112 * N[(N[(b - a), $MachinePrecision] * N[(Pi * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.4%

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

   1. associate-*l* 50.4%

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

   2. unpow2 50.4%

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

   3. unpow2 50.4%

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

   4. difference-of-squares 57.2%

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

3. Simplified 57.2%

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

4. Taylor expanded in angle around inf 58.0%

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

5. Taylor expanded in angle around 0 54.4%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative 54.4%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \pi\right)}\right)\right) \]

   2. *-commutative 54.4%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(\pi \cdot \left(a + b\right)\right)}\right)\right) \]

   3. +-commutative 54.4%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \color{blue}{\left(b + a\right)}\right)\right)\right) \]

   4. *-commutative 54.4%

      \[\leadsto \color{blue}{\left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right) \cdot 0.011111111111111112} \]

   5. associate-*l* 54.4%

      \[\leadsto \color{blue}{angle \cdot \left(\left(\left(b - a\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right) \cdot 0.011111111111111112\right)} \]

7. Simplified 54.4%

   \[\leadsto \color{blue}{angle \cdot \left(\left(\left(b - a\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right) \cdot 0.011111111111111112\right)} \]

8. Final simplification 54.4%

   \[\leadsto angle \cdot \left(0.011111111111111112 \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right) \]

Alternative 14: 54.5% accurate, 5.5× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \left(\pi \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right) \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (* (* PI (* (- b a) (+ a b))) (* angle 0.011111111111111112)))

C

b = abs(b);
double code(double a, double b, double angle) {
	return (((double) M_PI) * ((b - a) * (a + b))) * (angle * 0.011111111111111112);
}

Java

b = Math.abs(b);
public static double code(double a, double b, double angle) {
	return (Math.PI * ((b - a) * (a + b))) * (angle * 0.011111111111111112);
}

Python

b = abs(b)
def code(a, b, angle):
	return (math.pi * ((b - a) * (a + b))) * (angle * 0.011111111111111112)

Julia

b = abs(b)
function code(a, b, angle)
	return Float64(Float64(pi * Float64(Float64(b - a) * Float64(a + b))) * Float64(angle * 0.011111111111111112))
end

MATLAB

b = abs(b)
function tmp = code(a, b, angle)
	tmp = (pi * ((b - a) * (a + b))) * (angle * 0.011111111111111112);
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_] := N[(N[(Pi * N[(N[(b - a), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.4%

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

   1. *-commutative 50.4%

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

   2. associate-*l* 50.4%

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

   3. unpow2 50.4%

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

   4. fma-neg 53.9%

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

   5. unpow2 53.9%

      \[\leadsto \sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \mathsf{fma}\left(b, b, -\color{blue}{a \cdot a}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

3. Simplified 53.9%

   \[\leadsto \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \mathsf{fma}\left(b, b, -a \cdot a\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

4. Taylor expanded in angle around 0 48.3%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \pi\right)\right)} \]
5. Step-by-step derivation

   1. associate-*r* 48.4%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \pi\right)} \]

   2. *-commutative 48.4%

      \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot angle\right)} \]

   3. *-commutative 48.4%

      \[\leadsto \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \left(0.011111111111111112 \cdot angle\right) \]

   4. unpow2 48.4%

      \[\leadsto \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \left(0.011111111111111112 \cdot angle\right) \]

   5. unpow2 48.4%

      \[\leadsto \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \left(0.011111111111111112 \cdot angle\right) \]

   6. sub-neg 48.4%

      \[\leadsto \left(\pi \cdot \color{blue}{\left(b \cdot b + \left(-a \cdot a\right)\right)}\right) \cdot \left(0.011111111111111112 \cdot angle\right) \]

   7. sub-neg 48.4%

      \[\leadsto \left(\pi \cdot \color{blue}{\left(b \cdot b - a \cdot a\right)}\right) \cdot \left(0.011111111111111112 \cdot angle\right) \]

   8. difference-of-squares 54.4%

      \[\leadsto \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \left(0.011111111111111112 \cdot angle\right) \]

   9. *-commutative 54.4%

      \[\leadsto \left(\pi \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(b + a\right)\right)}\right) \cdot \left(0.011111111111111112 \cdot angle\right) \]

   10. +-commutative 54.4%

       \[\leadsto \left(\pi \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(a + b\right)}\right)\right) \cdot \left(0.011111111111111112 \cdot angle\right) \]

   11. *-commutative 54.4%

       \[\leadsto \left(\pi \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \color{blue}{\left(angle \cdot 0.011111111111111112\right)} \]

6. Simplified 54.4%

   \[\leadsto \color{blue}{\left(\pi \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)} \]

7. Final simplification 54.4%

   \[\leadsto \left(\pi \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right) \]

Alternative 15: 62.5% accurate, 5.5× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112 \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (* (* (* (- b a) angle) (* PI (+ a b))) 0.011111111111111112))

C

b = abs(b);
double code(double a, double b, double angle) {
	return (((b - a) * angle) * (((double) M_PI) * (a + b))) * 0.011111111111111112;
}

Java

b = Math.abs(b);
public static double code(double a, double b, double angle) {
	return (((b - a) * angle) * (Math.PI * (a + b))) * 0.011111111111111112;
}

Python

b = abs(b)
def code(a, b, angle):
	return (((b - a) * angle) * (math.pi * (a + b))) * 0.011111111111111112

Julia

b = abs(b)
function code(a, b, angle)
	return Float64(Float64(Float64(Float64(b - a) * angle) * Float64(pi * Float64(a + b))) * 0.011111111111111112)
end

MATLAB

b = abs(b)
function tmp = code(a, b, angle)
	tmp = (((b - a) * angle) * (pi * (a + b))) * 0.011111111111111112;
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_] := N[(N[(N[(N[(b - a), $MachinePrecision] * angle), $MachinePrecision] * N[(Pi * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]

Derivation

1. Initial program 50.4%

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

   1. associate-*l* 50.4%

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

   2. unpow2 50.4%

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

   3. unpow2 50.4%

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

   4. difference-of-squares 57.2%

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

3. Simplified 57.2%

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

4. Taylor expanded in angle around 0 54.4%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]
5. Step-by-step derivation

   1. *-commutative 54.4%

      \[\leadsto \color{blue}{\left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right) \cdot 0.011111111111111112} \]

   2. associate-*r* 64.4%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)} \cdot 0.011111111111111112 \]

6. Simplified 64.4%

   \[\leadsto \color{blue}{\left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112} \]

7. Final simplification 64.4%

   \[\leadsto \left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112 \]

Alternative 16: 34.4% accurate, 5.7× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ angle \cdot \left(\pi \cdot \left(\left(a \cdot a\right) \cdot -0.011111111111111112\right)\right) \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (* angle (* PI (* (* a a) -0.011111111111111112))))

C

b = abs(b);
double code(double a, double b, double angle) {
	return angle * (((double) M_PI) * ((a * a) * -0.011111111111111112));
}

Java

b = Math.abs(b);
public static double code(double a, double b, double angle) {
	return angle * (Math.PI * ((a * a) * -0.011111111111111112));
}

Python

b = abs(b)
def code(a, b, angle):
	return angle * (math.pi * ((a * a) * -0.011111111111111112))

Julia

b = abs(b)
function code(a, b, angle)
	return Float64(angle * Float64(pi * Float64(Float64(a * a) * -0.011111111111111112)))
end

MATLAB

b = abs(b)
function tmp = code(a, b, angle)
	tmp = angle * (pi * ((a * a) * -0.011111111111111112));
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_] := N[(angle * N[(Pi * N[(N[(a * a), $MachinePrecision] * -0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.4%

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

   1. associate-*l* 50.4%

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

   2. unpow2 50.4%

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

   3. unpow2 50.4%

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

   4. difference-of-squares 57.2%

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

3. Simplified 57.2%

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

4. Taylor expanded in angle around 0 54.4%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]

5. Taylor expanded in b around 0 36.2%

   \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative 36.2%

      \[\leadsto \color{blue}{\left(angle \cdot \left({a}^{2} \cdot \pi\right)\right) \cdot -0.011111111111111112} \]

   2. *-commutative 36.2%

      \[\leadsto \left(angle \cdot \color{blue}{\left(\pi \cdot {a}^{2}\right)}\right) \cdot -0.011111111111111112 \]

   3. unpow2 36.2%

      \[\leadsto \left(angle \cdot \left(\pi \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot -0.011111111111111112 \]

7. Simplified 36.2%

   \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right) \cdot -0.011111111111111112} \]

8. Taylor expanded in angle around 0 36.2%

   \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)} \]
9. Step-by-step derivation

   1. *-commutative 36.2%

      \[\leadsto \color{blue}{\left(angle \cdot \left({a}^{2} \cdot \pi\right)\right) \cdot -0.011111111111111112} \]

   2. *-commutative 36.2%

      \[\leadsto \left(angle \cdot \color{blue}{\left(\pi \cdot {a}^{2}\right)}\right) \cdot -0.011111111111111112 \]

   3. associate-*r* 36.2%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot {a}^{2}\right)} \cdot -0.011111111111111112 \]

   4. associate-*l* 36.2%

      \[\leadsto \color{blue}{\left(angle \cdot \pi\right) \cdot \left({a}^{2} \cdot -0.011111111111111112\right)} \]

   5. associate-*r* 36.2%

      \[\leadsto \color{blue}{angle \cdot \left(\pi \cdot \left({a}^{2} \cdot -0.011111111111111112\right)\right)} \]

   6. unpow2 36.2%

      \[\leadsto angle \cdot \left(\pi \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot -0.011111111111111112\right)\right) \]

10. Simplified 36.2%

    \[\leadsto \color{blue}{angle \cdot \left(\pi \cdot \left(\left(a \cdot a\right) \cdot -0.011111111111111112\right)\right)} \]

11. Final simplification 36.2%

    \[\leadsto angle \cdot \left(\pi \cdot \left(\left(a \cdot a\right) \cdot -0.011111111111111112\right)\right) \]

Alternative 17: 34.4% accurate, 5.7× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \pi \cdot \left(\left(a \cdot a\right) \cdot \left(angle \cdot -0.011111111111111112\right)\right) \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (* PI (* (* a a) (* angle -0.011111111111111112))))

C

b = abs(b);
double code(double a, double b, double angle) {
	return ((double) M_PI) * ((a * a) * (angle * -0.011111111111111112));
}

Java

b = Math.abs(b);
public static double code(double a, double b, double angle) {
	return Math.PI * ((a * a) * (angle * -0.011111111111111112));
}

Python

b = abs(b)
def code(a, b, angle):
	return math.pi * ((a * a) * (angle * -0.011111111111111112))

Julia

b = abs(b)
function code(a, b, angle)
	return Float64(pi * Float64(Float64(a * a) * Float64(angle * -0.011111111111111112)))
end

MATLAB

b = abs(b)
function tmp = code(a, b, angle)
	tmp = pi * ((a * a) * (angle * -0.011111111111111112));
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_] := N[(Pi * N[(N[(a * a), $MachinePrecision] * N[(angle * -0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.4%

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

   1. associate-*l* 50.4%

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

   2. unpow2 50.4%

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

   3. unpow2 50.4%

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

   4. difference-of-squares 57.2%

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

3. Simplified 57.2%

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

4. Taylor expanded in angle around 0 54.4%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]

5. Taylor expanded in b around 0 36.2%

   \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative 36.2%

      \[\leadsto \color{blue}{\left(angle \cdot \left({a}^{2} \cdot \pi\right)\right) \cdot -0.011111111111111112} \]

   2. *-commutative 36.2%

      \[\leadsto \left(angle \cdot \color{blue}{\left(\pi \cdot {a}^{2}\right)}\right) \cdot -0.011111111111111112 \]

   3. unpow2 36.2%

      \[\leadsto \left(angle \cdot \left(\pi \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot -0.011111111111111112 \]

7. Simplified 36.2%

   \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right) \cdot -0.011111111111111112} \]

8. Taylor expanded in angle around 0 36.2%

   \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)} \]
9. Step-by-step derivation

   1. *-commutative 36.2%

      \[\leadsto \color{blue}{\left(angle \cdot \left({a}^{2} \cdot \pi\right)\right) \cdot -0.011111111111111112} \]

   2. *-commutative 36.2%

      \[\leadsto \left(angle \cdot \color{blue}{\left(\pi \cdot {a}^{2}\right)}\right) \cdot -0.011111111111111112 \]

   3. associate-*r* 36.2%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot {a}^{2}\right)} \cdot -0.011111111111111112 \]

   4. associate-*l* 36.2%

      \[\leadsto \color{blue}{\left(angle \cdot \pi\right) \cdot \left({a}^{2} \cdot -0.011111111111111112\right)} \]

   5. associate-*r* 36.2%

      \[\leadsto \color{blue}{angle \cdot \left(\pi \cdot \left({a}^{2} \cdot -0.011111111111111112\right)\right)} \]

   6. unpow2 36.2%

      \[\leadsto angle \cdot \left(\pi \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot -0.011111111111111112\right)\right) \]

10. Simplified 36.2%

    \[\leadsto \color{blue}{angle \cdot \left(\pi \cdot \left(\left(a \cdot a\right) \cdot -0.011111111111111112\right)\right)} \]

11. Taylor expanded in angle around 0 36.2%

    \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)} \]
12. Step-by-step derivation

    1. associate-*r* 36.2%

       \[\leadsto \color{blue}{\left(-0.011111111111111112 \cdot angle\right) \cdot \left({a}^{2} \cdot \pi\right)} \]

    2. associate-*r* 36.2%

       \[\leadsto \color{blue}{\left(\left(-0.011111111111111112 \cdot angle\right) \cdot {a}^{2}\right) \cdot \pi} \]

    3. *-commutative 36.2%

       \[\leadsto \color{blue}{\left({a}^{2} \cdot \left(-0.011111111111111112 \cdot angle\right)\right)} \cdot \pi \]

    4. *-commutative 36.2%

       \[\leadsto \color{blue}{\pi \cdot \left({a}^{2} \cdot \left(-0.011111111111111112 \cdot angle\right)\right)} \]

    5. unpow2 36.2%

       \[\leadsto \pi \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \left(-0.011111111111111112 \cdot angle\right)\right) \]

    6. *-commutative 36.2%

       \[\leadsto \pi \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(angle \cdot -0.011111111111111112\right)}\right) \]

13. Simplified 36.2%

    \[\leadsto \color{blue}{\pi \cdot \left(\left(a \cdot a\right) \cdot \left(angle \cdot -0.011111111111111112\right)\right)} \]

14. Final simplification 36.2%

    \[\leadsto \pi \cdot \left(\left(a \cdot a\right) \cdot \left(angle \cdot -0.011111111111111112\right)\right) \]

Reproduce

herbie shell --seed 2023230 
(FPCore (a b angle)
  :name "ab-angle->ABCF B"
  :precision binary64
  (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* PI (/ angle 180.0)))) (cos (* PI (/ angle 180.0)))))