ab-angle->ABCF B

Percentage Accurate: 53.9% → 68.0%

Time: 37.7s

Alternatives: 18

Speedup: 27.9×

• 32.2% 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: 53.9% 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: 68.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := angle\_m \cdot \left(\pi \cdot 0.005555555555555556\right)\\ t_1 := \sin \left(\frac{angle\_m \cdot \pi}{180}\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 2 \cdot 10^{+218}:\\ \;\;\;\;2 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\cos t\_0 \cdot t\_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(t\_1 \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(t\_0\right)\right)\right)\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* angle_m (* PI 0.005555555555555556)))
        (t_1 (sin (/ (* angle_m PI) 180.0))))
   (*
    angle_s
    (if (<= (pow a 2.0) 2e+218)
      (* 2.0 (* (+ a b) (* (- b a) (* (cos t_0) t_1))))
      (* 2.0 (* (+ a b) (* (- b a) (* t_1 (cos (expm1 (log1p t_0)))))))))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = angle_m * (((double) M_PI) * 0.005555555555555556);
	double t_1 = sin(((angle_m * ((double) M_PI)) / 180.0));
	double tmp;
	if (pow(a, 2.0) <= 2e+218) {
		tmp = 2.0 * ((a + b) * ((b - a) * (cos(t_0) * t_1)));
	} else {
		tmp = 2.0 * ((a + b) * ((b - a) * (t_1 * cos(expm1(log1p(t_0))))));
	}
	return angle_s * tmp;
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = angle_m * (Math.PI * 0.005555555555555556);
	double t_1 = Math.sin(((angle_m * Math.PI) / 180.0));
	double tmp;
	if (Math.pow(a, 2.0) <= 2e+218) {
		tmp = 2.0 * ((a + b) * ((b - a) * (Math.cos(t_0) * t_1)));
	} else {
		tmp = 2.0 * ((a + b) * ((b - a) * (t_1 * Math.cos(Math.expm1(Math.log1p(t_0))))));
	}
	return angle_s * tmp;
}

Python

angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = angle_m * (math.pi * 0.005555555555555556)
	t_1 = math.sin(((angle_m * math.pi) / 180.0))
	tmp = 0
	if math.pow(a, 2.0) <= 2e+218:
		tmp = 2.0 * ((a + b) * ((b - a) * (math.cos(t_0) * t_1)))
	else:
		tmp = 2.0 * ((a + b) * ((b - a) * (t_1 * math.cos(math.expm1(math.log1p(t_0))))))
	return angle_s * tmp

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(angle_m * Float64(pi * 0.005555555555555556))
	t_1 = sin(Float64(Float64(angle_m * pi) / 180.0))
	tmp = 0.0
	if ((a ^ 2.0) <= 2e+218)
		tmp = Float64(2.0 * Float64(Float64(a + b) * Float64(Float64(b - a) * Float64(cos(t_0) * t_1))));
	else
		tmp = Float64(2.0 * Float64(Float64(a + b) * Float64(Float64(b - a) * Float64(t_1 * cos(expm1(log1p(t_0)))))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[Power[a, 2.0], $MachinePrecision], 2e+218], N[(2.0 * N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(N[Cos[t$95$0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(t$95$1 * N[Cos[N[(Exp[N[Log[1 + t$95$0], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 a 2) < 2.00000000000000017e218

   1. Initial program 57.3%

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

   3. Simplified 56.8%

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

      1. unpow2 56.8%

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

      2. unpow2 56.8%

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

      3. difference-of-squares 56.8%

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

   6. Applied egg-rr 56.8%

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

      1. add-log-exp 28.8%

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

      2. associate-*l* 28.8%

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

      3. exp-prod 27.3%

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

      4. *-commutative 27.3%

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

   8. Applied egg-rr 27.8%

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

      1. log-pow 27.9%

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

      2. +-commutative 27.9%

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

      3. rem-log-exp 62.5%

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

      4. *-commutative 62.5%

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

      5. associate-*r* 61.4%

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

      6. *-commutative 61.4%

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

      7. associate-*r* 62.7%

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

      8. associate-*r* 61.9%

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

      9. *-commutative 61.9%

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

      10. associate-*r* 62.8%

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

   10. Simplified 62.8%

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

       1. associate-*r* 61.9%

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

       2. metadata-eval 61.9%

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

       3. div-inv 63.0%

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

   12. Applied egg-rr 63.0%

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

   if 2.00000000000000017e218 < (pow.f64 a 2)

   1. Initial program 40.9%

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

   3. Simplified 44.4%

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

      1. unpow2 44.4%

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

      2. unpow2 44.4%

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

      3. difference-of-squares 52.8%

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

   6. Applied egg-rr 52.8%

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

      1. add-log-exp 42.7%

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

      2. associate-*l* 42.7%

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

      3. exp-prod 40.9%

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

      4. *-commutative 40.9%

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

   8. Applied egg-rr 38.5%

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

      1. log-pow 38.5%

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

      2. +-commutative 38.5%

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

      3. rem-log-exp 67.8%

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

      4. *-commutative 67.8%

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

      5. associate-*r* 66.7%

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

      6. *-commutative 66.7%

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

      7. associate-*r* 68.2%

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

      8. associate-*r* 70.1%

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

      9. *-commutative 70.1%

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

      10. associate-*r* 67.9%

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

   10. Simplified 67.9%

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

       1. associate-*r* 70.1%

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

       2. metadata-eval 70.1%

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

       3. div-inv 71.6%

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

   12. Applied egg-rr 71.6%

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

       1. expm1-log1p-u 67.7%

          \[\leadsto 2 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)} \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)\right)\right) \]

       2. expm1-undefine 68.9%

          \[\leadsto 2 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\cos \color{blue}{\left(e^{\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)} - 1\right)} \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)\right)\right) \]

   14. Applied egg-rr 68.9%

       \[\leadsto 2 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\cos \color{blue}{\left(e^{\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)} - 1\right)} \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)\right)\right) \]
   15. Step-by-step derivation

       1. expm1-define 67.7%

          \[\leadsto 2 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)} \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)\right)\right) \]

   16. Simplified 67.7%

       \[\leadsto 2 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)} \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 2 \cdot 10^{+218}:\\ \;\;\;\;2 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{angle \cdot \pi}{180}\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 67.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := angle\_m \cdot \left(\pi \cdot 0.005555555555555556\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 2 \cdot 10^{+218}:\\ \;\;\;\;2 \cdot \left(\left(a + b\right) \cdot \left(\cos t\_0 \cdot \left(\left(b - a\right) \cdot \sin t\_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{angle\_m \cdot \pi}{180}\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* angle_m (* PI 0.005555555555555556))))
   (*
    angle_s
    (if (<= (pow a 2.0) 2e+218)
      (* 2.0 (* (+ a b) (* (cos t_0) (* (- b a) (sin t_0)))))
      (* 2.0 (* (+ a b) (* (- b a) (sin (/ (* angle_m PI) 180.0)))))))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = angle_m * (((double) M_PI) * 0.005555555555555556);
	double tmp;
	if (pow(a, 2.0) <= 2e+218) {
		tmp = 2.0 * ((a + b) * (cos(t_0) * ((b - a) * sin(t_0))));
	} else {
		tmp = 2.0 * ((a + b) * ((b - a) * sin(((angle_m * ((double) M_PI)) / 180.0))));
	}
	return angle_s * tmp;
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = angle_m * (Math.PI * 0.005555555555555556);
	double tmp;
	if (Math.pow(a, 2.0) <= 2e+218) {
		tmp = 2.0 * ((a + b) * (Math.cos(t_0) * ((b - a) * Math.sin(t_0))));
	} else {
		tmp = 2.0 * ((a + b) * ((b - a) * Math.sin(((angle_m * Math.PI) / 180.0))));
	}
	return angle_s * tmp;
}

Python

angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = angle_m * (math.pi * 0.005555555555555556)
	tmp = 0
	if math.pow(a, 2.0) <= 2e+218:
		tmp = 2.0 * ((a + b) * (math.cos(t_0) * ((b - a) * math.sin(t_0))))
	else:
		tmp = 2.0 * ((a + b) * ((b - a) * math.sin(((angle_m * math.pi) / 180.0))))
	return angle_s * tmp

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(angle_m * Float64(pi * 0.005555555555555556))
	tmp = 0.0
	if ((a ^ 2.0) <= 2e+218)
		tmp = Float64(2.0 * Float64(Float64(a + b) * Float64(cos(t_0) * Float64(Float64(b - a) * sin(t_0)))));
	else
		tmp = Float64(2.0 * Float64(Float64(a + b) * Float64(Float64(b - a) * sin(Float64(Float64(angle_m * pi) / 180.0)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = angle_m * (pi * 0.005555555555555556);
	tmp = 0.0;
	if ((a ^ 2.0) <= 2e+218)
		tmp = 2.0 * ((a + b) * (cos(t_0) * ((b - a) * sin(t_0))));
	else
		tmp = 2.0 * ((a + b) * ((b - a) * sin(((angle_m * pi) / 180.0))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[Power[a, 2.0], $MachinePrecision], 2e+218], N[(2.0 * N[(N[(a + b), $MachinePrecision] * N[(N[Cos[t$95$0], $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 a 2) < 2.00000000000000017e218

   1. Initial program 57.3%

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

   3. Simplified 56.8%

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

      1. unpow2 56.8%

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

      2. unpow2 56.8%

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

      3. difference-of-squares 56.8%

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

   6. Applied egg-rr 56.8%

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

      1. add-log-exp 28.8%

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

      2. associate-*l* 28.8%

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

      3. exp-prod 27.3%

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

      4. *-commutative 27.3%

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

   8. Applied egg-rr 27.8%

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

      1. log-pow 27.9%

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

      2. +-commutative 27.9%

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

      3. rem-log-exp 62.5%

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

      4. *-commutative 62.5%

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

      5. associate-*r* 61.4%

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

      6. *-commutative 61.4%

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

      7. associate-*r* 62.7%

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

      8. associate-*r* 61.9%

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

      9. *-commutative 61.9%

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

      10. associate-*r* 62.8%

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

   10. Simplified 62.8%

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

   11. Taylor expanded in b around 0 62.1%

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

       1. +-commutative 62.1%

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

       2. *-commutative 62.1%

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

       3. *-commutative 62.1%

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

       4. *-commutative 62.1%

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

       5. associate-*r* 61.1%

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

       6. *-commutative 61.1%

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

       7. associate-*r* 62.7%

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

       8. mul-1-neg 62.7%

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

   13. Simplified 62.8%

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

   if 2.00000000000000017e218 < (pow.f64 a 2)

   1. Initial program 40.9%

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

   3. Simplified 44.4%

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

      1. unpow2 44.4%

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

      2. unpow2 44.4%

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

      3. difference-of-squares 52.8%

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

   6. Applied egg-rr 52.8%

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

      1. add-log-exp 42.7%

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

      2. associate-*l* 42.7%

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

      3. exp-prod 40.9%

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

      4. *-commutative 40.9%

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

   8. Applied egg-rr 38.5%

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

      1. log-pow 38.5%

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

      2. +-commutative 38.5%

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

      3. rem-log-exp 67.8%

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

      4. *-commutative 67.8%

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

      5. associate-*r* 66.7%

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

      6. *-commutative 66.7%

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

      7. associate-*r* 68.2%

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

      8. associate-*r* 70.1%

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

      9. *-commutative 70.1%

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

      10. associate-*r* 67.9%

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

   10. Simplified 67.9%

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

       1. associate-*r* 70.1%

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

       2. metadata-eval 70.1%

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

       3. div-inv 71.6%

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

   12. Applied egg-rr 71.6%

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

   13. Taylor expanded in angle around 0 75.6%

       \[\leadsto 2 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\color{blue}{1} \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 2 \cdot 10^{+218}:\\ \;\;\;\;2 \cdot \left(\left(a + b\right) \cdot \left(\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := angle\_m \cdot \left(\pi \cdot 0.005555555555555556\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 2 \cdot 10^{+218}:\\ \;\;\;\;2 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\cos t\_0 \cdot \sin t\_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{angle\_m \cdot \pi}{180}\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* angle_m (* PI 0.005555555555555556))))
   (*
    angle_s
    (if (<= (pow a 2.0) 2e+218)
      (* 2.0 (* (+ a b) (* (- b a) (* (cos t_0) (sin t_0)))))
      (* 2.0 (* (+ a b) (* (- b a) (sin (/ (* angle_m PI) 180.0)))))))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = angle_m * (((double) M_PI) * 0.005555555555555556);
	double tmp;
	if (pow(a, 2.0) <= 2e+218) {
		tmp = 2.0 * ((a + b) * ((b - a) * (cos(t_0) * sin(t_0))));
	} else {
		tmp = 2.0 * ((a + b) * ((b - a) * sin(((angle_m * ((double) M_PI)) / 180.0))));
	}
	return angle_s * tmp;
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = angle_m * (Math.PI * 0.005555555555555556);
	double tmp;
	if (Math.pow(a, 2.0) <= 2e+218) {
		tmp = 2.0 * ((a + b) * ((b - a) * (Math.cos(t_0) * Math.sin(t_0))));
	} else {
		tmp = 2.0 * ((a + b) * ((b - a) * Math.sin(((angle_m * Math.PI) / 180.0))));
	}
	return angle_s * tmp;
}

Python

angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = angle_m * (math.pi * 0.005555555555555556)
	tmp = 0
	if math.pow(a, 2.0) <= 2e+218:
		tmp = 2.0 * ((a + b) * ((b - a) * (math.cos(t_0) * math.sin(t_0))))
	else:
		tmp = 2.0 * ((a + b) * ((b - a) * math.sin(((angle_m * math.pi) / 180.0))))
	return angle_s * tmp

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(angle_m * Float64(pi * 0.005555555555555556))
	tmp = 0.0
	if ((a ^ 2.0) <= 2e+218)
		tmp = Float64(2.0 * Float64(Float64(a + b) * Float64(Float64(b - a) * Float64(cos(t_0) * sin(t_0)))));
	else
		tmp = Float64(2.0 * Float64(Float64(a + b) * Float64(Float64(b - a) * sin(Float64(Float64(angle_m * pi) / 180.0)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = angle_m * (pi * 0.005555555555555556);
	tmp = 0.0;
	if ((a ^ 2.0) <= 2e+218)
		tmp = 2.0 * ((a + b) * ((b - a) * (cos(t_0) * sin(t_0))));
	else
		tmp = 2.0 * ((a + b) * ((b - a) * sin(((angle_m * pi) / 180.0))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[Power[a, 2.0], $MachinePrecision], 2e+218], N[(2.0 * N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(N[Cos[t$95$0], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 a 2) < 2.00000000000000017e218

   1. Initial program 57.3%

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

   3. Simplified 56.8%

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

      1. unpow2 56.8%

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

      2. unpow2 56.8%

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

      3. difference-of-squares 56.8%

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

   6. Applied egg-rr 56.8%

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

      1. add-log-exp 28.8%

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

      2. associate-*l* 28.8%

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

      3. exp-prod 27.3%

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

      4. *-commutative 27.3%

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

   8. Applied egg-rr 27.8%

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

      1. log-pow 27.9%

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

      2. +-commutative 27.9%

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

      3. rem-log-exp 62.5%

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

      4. *-commutative 62.5%

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

      5. associate-*r* 61.4%

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

      6. *-commutative 61.4%

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

      7. associate-*r* 62.7%

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

      8. associate-*r* 61.9%

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

      9. *-commutative 61.9%

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

      10. associate-*r* 62.8%

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

   10. Simplified 62.8%

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

   if 2.00000000000000017e218 < (pow.f64 a 2)

   1. Initial program 40.9%

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

   3. Simplified 44.4%

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

      1. unpow2 44.4%

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

      2. unpow2 44.4%

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

      3. difference-of-squares 52.8%

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

   6. Applied egg-rr 52.8%

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

      1. add-log-exp 42.7%

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

      2. associate-*l* 42.7%

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

      3. exp-prod 40.9%

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

      4. *-commutative 40.9%

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

   8. Applied egg-rr 38.5%

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

      1. log-pow 38.5%

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

      2. +-commutative 38.5%

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

      3. rem-log-exp 67.8%

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

      4. *-commutative 67.8%

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

      5. associate-*r* 66.7%

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

      6. *-commutative 66.7%

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

      7. associate-*r* 68.2%

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

      8. associate-*r* 70.1%

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

      9. *-commutative 70.1%

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

      10. associate-*r* 67.9%

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

   10. Simplified 67.9%

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

       1. associate-*r* 70.1%

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

       2. metadata-eval 70.1%

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

       3. div-inv 71.6%

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

   12. Applied egg-rr 71.6%

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

   13. Taylor expanded in angle around 0 75.6%

       \[\leadsto 2 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\color{blue}{1} \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 2 \cdot 10^{+218}:\\ \;\;\;\;2 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 5 \cdot 10^{-114}:\\ \;\;\;\;2 \cdot \left(\left(a + b\right) \cdot \left(b \cdot \left(\sin t\_0 \cdot \cos t\_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{angle\_m \cdot \pi}{180}\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle_m PI))))
   (*
    angle_s
    (if (<= (pow a 2.0) 5e-114)
      (* 2.0 (* (+ a b) (* b (* (sin t_0) (cos t_0)))))
      (* 2.0 (* (+ a b) (* (- b a) (sin (/ (* angle_m PI) 180.0)))))))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = 0.005555555555555556 * (angle_m * ((double) M_PI));
	double tmp;
	if (pow(a, 2.0) <= 5e-114) {
		tmp = 2.0 * ((a + b) * (b * (sin(t_0) * cos(t_0))));
	} else {
		tmp = 2.0 * ((a + b) * ((b - a) * sin(((angle_m * ((double) M_PI)) / 180.0))));
	}
	return angle_s * tmp;
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = 0.005555555555555556 * (angle_m * Math.PI);
	double tmp;
	if (Math.pow(a, 2.0) <= 5e-114) {
		tmp = 2.0 * ((a + b) * (b * (Math.sin(t_0) * Math.cos(t_0))));
	} else {
		tmp = 2.0 * ((a + b) * ((b - a) * Math.sin(((angle_m * Math.PI) / 180.0))));
	}
	return angle_s * tmp;
}

Python

angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = 0.005555555555555556 * (angle_m * math.pi)
	tmp = 0
	if math.pow(a, 2.0) <= 5e-114:
		tmp = 2.0 * ((a + b) * (b * (math.sin(t_0) * math.cos(t_0))))
	else:
		tmp = 2.0 * ((a + b) * ((b - a) * math.sin(((angle_m * math.pi) / 180.0))))
	return angle_s * tmp

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(0.005555555555555556 * Float64(angle_m * pi))
	tmp = 0.0
	if ((a ^ 2.0) <= 5e-114)
		tmp = Float64(2.0 * Float64(Float64(a + b) * Float64(b * Float64(sin(t_0) * cos(t_0)))));
	else
		tmp = Float64(2.0 * Float64(Float64(a + b) * Float64(Float64(b - a) * sin(Float64(Float64(angle_m * pi) / 180.0)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = 0.005555555555555556 * (angle_m * pi);
	tmp = 0.0;
	if ((a ^ 2.0) <= 5e-114)
		tmp = 2.0 * ((a + b) * (b * (sin(t_0) * cos(t_0))));
	else
		tmp = 2.0 * ((a + b) * ((b - a) * sin(((angle_m * pi) / 180.0))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[Power[a, 2.0], $MachinePrecision], 5e-114], N[(2.0 * N[(N[(a + b), $MachinePrecision] * N[(b * N[(N[Sin[t$95$0], $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 a 2) < 4.99999999999999989e-114

   1. Initial program 57.1%

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

   3. Simplified 56.1%

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

      1. unpow2 56.1%

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

      2. unpow2 56.1%

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

      3. difference-of-squares 56.1%

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

   6. Applied egg-rr 56.1%

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

      1. add-log-exp 34.8%

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

      2. associate-*l* 34.8%

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

      3. exp-prod 34.0%

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

      4. *-commutative 34.0%

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

   8. Applied egg-rr 34.8%

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

      1. log-pow 35.1%

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

      2. +-commutative 35.1%

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

      3. rem-log-exp 63.9%

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

      4. *-commutative 63.9%

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

      5. associate-*r* 61.7%

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

      6. *-commutative 61.7%

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

      7. associate-*r* 63.7%

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

      8. associate-*r* 62.2%

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

      9. *-commutative 62.2%

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

      10. associate-*r* 63.7%

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

   10. Simplified 63.7%

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

   11. Taylor expanded in b around inf 61.5%

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

   if 4.99999999999999989e-114 < (pow.f64 a 2)

   1. Initial program 48.0%

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

   3. Simplified 50.2%

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

      1. unpow2 50.2%

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

      2. unpow2 50.2%

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

      3. difference-of-squares 55.0%

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

   6. Applied egg-rr 55.0%

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

      1. add-log-exp 32.4%

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

      2. associate-*l* 32.4%

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

      3. exp-prod 30.3%

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

      4. *-commutative 30.3%

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

   8. Applied egg-rr 28.8%

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

      1. log-pow 28.8%

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

      2. +-commutative 28.8%

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

      3. rem-log-exp 64.5%

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

      4. *-commutative 64.5%

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

      5. associate-*r* 64.2%

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

      6. *-commutative 64.2%

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

      7. associate-*r* 65.1%

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

      8. associate-*r* 66.4%

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

      9. *-commutative 66.4%

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

      10. associate-*r* 65.2%

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

   10. Simplified 65.2%

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

       1. associate-*r* 66.4%

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

       2. metadata-eval 66.4%

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

       3. div-inv 67.3%

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

   12. Applied egg-rr 67.3%

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

   13. Taylor expanded in angle around 0 66.5%

       \[\leadsto 2 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\color{blue}{1} \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 5 \cdot 10^{-114}:\\ \;\;\;\;2 \cdot \left(\left(a + b\right) \cdot \left(b \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(a + b\right) \cdot \left(b - a\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;2 \cdot \left(\left(a + b\right) \cdot \left(\left(angle\_m \cdot 0.005555555555555556\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+95}:\\ \;\;\;\;2 \cdot \left(\cos \left(\left(angle\_m \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \left(t\_0 \cdot \sin \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t\_0 \cdot \sin \left(\pi \cdot \frac{angle\_m}{180}\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* (+ a b) (- b a))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 2e-13)
      (* 2.0 (* (+ a b) (* (* angle_m 0.005555555555555556) (* (- b a) PI))))
      (if (<= (/ angle_m 180.0) 5e+95)
        (*
         2.0
         (*
          (cos (* (* angle_m PI) -0.005555555555555556))
          (* t_0 (sin (* 0.005555555555555556 (* angle_m PI))))))
        (* 2.0 (* t_0 (sin (* PI (/ angle_m 180.0))))))))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (a + b) * (b - a);
	double tmp;
	if ((angle_m / 180.0) <= 2e-13) {
		tmp = 2.0 * ((a + b) * ((angle_m * 0.005555555555555556) * ((b - a) * ((double) M_PI))));
	} else if ((angle_m / 180.0) <= 5e+95) {
		tmp = 2.0 * (cos(((angle_m * ((double) M_PI)) * -0.005555555555555556)) * (t_0 * sin((0.005555555555555556 * (angle_m * ((double) M_PI))))));
	} else {
		tmp = 2.0 * (t_0 * sin((((double) M_PI) * (angle_m / 180.0))));
	}
	return angle_s * tmp;
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (a + b) * (b - a);
	double tmp;
	if ((angle_m / 180.0) <= 2e-13) {
		tmp = 2.0 * ((a + b) * ((angle_m * 0.005555555555555556) * ((b - a) * Math.PI)));
	} else if ((angle_m / 180.0) <= 5e+95) {
		tmp = 2.0 * (Math.cos(((angle_m * Math.PI) * -0.005555555555555556)) * (t_0 * Math.sin((0.005555555555555556 * (angle_m * Math.PI)))));
	} else {
		tmp = 2.0 * (t_0 * Math.sin((Math.PI * (angle_m / 180.0))));
	}
	return angle_s * tmp;
}

Python

angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = (a + b) * (b - a)
	tmp = 0
	if (angle_m / 180.0) <= 2e-13:
		tmp = 2.0 * ((a + b) * ((angle_m * 0.005555555555555556) * ((b - a) * math.pi)))
	elif (angle_m / 180.0) <= 5e+95:
		tmp = 2.0 * (math.cos(((angle_m * math.pi) * -0.005555555555555556)) * (t_0 * math.sin((0.005555555555555556 * (angle_m * math.pi)))))
	else:
		tmp = 2.0 * (t_0 * math.sin((math.pi * (angle_m / 180.0))))
	return angle_s * tmp

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(Float64(a + b) * Float64(b - a))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e-13)
		tmp = Float64(2.0 * Float64(Float64(a + b) * Float64(Float64(angle_m * 0.005555555555555556) * Float64(Float64(b - a) * pi))));
	elseif (Float64(angle_m / 180.0) <= 5e+95)
		tmp = Float64(2.0 * Float64(cos(Float64(Float64(angle_m * pi) * -0.005555555555555556)) * Float64(t_0 * sin(Float64(0.005555555555555556 * Float64(angle_m * pi))))));
	else
		tmp = Float64(2.0 * Float64(t_0 * sin(Float64(pi * Float64(angle_m / 180.0)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = (a + b) * (b - a);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 2e-13)
		tmp = 2.0 * ((a + b) * ((angle_m * 0.005555555555555556) * ((b - a) * pi)));
	elseif ((angle_m / 180.0) <= 5e+95)
		tmp = 2.0 * (cos(((angle_m * pi) * -0.005555555555555556)) * (t_0 * sin((0.005555555555555556 * (angle_m * pi)))));
	else
		tmp = 2.0 * (t_0 * sin((pi * (angle_m / 180.0))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e-13], N[(2.0 * N[(N[(a + b), $MachinePrecision] * N[(N[(angle$95$m * 0.005555555555555556), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+95], N[(2.0 * N[(N[Cos[N[(N[(angle$95$m * Pi), $MachinePrecision] * -0.005555555555555556), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$0 * N[Sin[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < 2.0000000000000001e-13

   1. Initial program 60.4%

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

   3. Simplified 61.8%

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

      1. unpow2 61.8%

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

      2. unpow2 61.8%

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

      3. difference-of-squares 65.7%

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

   6. Applied egg-rr 65.7%

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

      1. add-log-exp 38.5%

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

      2. associate-*l* 38.5%

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

      3. exp-prod 36.3%

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

      4. *-commutative 36.3%

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

   8. Applied egg-rr 35.7%

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

      1. log-pow 35.9%

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

      2. +-commutative 35.9%

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

      3. rem-log-exp 78.5%

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

      4. *-commutative 78.5%

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

      5. associate-*r* 77.7%

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

      6. *-commutative 77.7%

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

      7. associate-*r* 78.1%

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

      8. associate-*r* 78.8%

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

      9. *-commutative 78.8%

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

      10. associate-*r* 78.6%

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

   10. Simplified 78.6%

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

       1. associate-*r* 78.8%

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

       2. metadata-eval 78.8%

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

       3. div-inv 78.7%

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

   12. Applied egg-rr 78.7%

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

   13. Taylor expanded in angle around 0 69.3%

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

       1. associate-*r* 69.3%

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

   15. Simplified 69.3%

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

   if 2.0000000000000001e-13 < (/.f64 angle 180) < 5.00000000000000025e95

   1. Initial program 44.1%

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

   3. Simplified 40.7%

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

      1. unpow2 40.7%

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

      2. unpow2 40.7%

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

      3. difference-of-squares 40.7%

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

   6. Applied egg-rr 40.7%

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

   7. Taylor expanded in angle around inf 48.5%

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

   if 5.00000000000000025e95 < (/.f64 angle 180)

   1. Initial program 23.3%

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

   3. Simplified 24.3%

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

      1. unpow2 24.3%

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

      2. unpow2 24.3%

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

      3. difference-of-squares 24.3%

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

   6. Applied egg-rr 24.3%

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

   7. Taylor expanded in angle around 0 32.2%

      \[\leadsto 2 \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{1}\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;2 \cdot \left(\left(a + b\right) \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+95}:\\ \;\;\;\;2 \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := angle\_m \cdot \frac{\pi}{-180}\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 10^{-50}:\\ \;\;\;\;2 \cdot \left(\left(a + b\right) \cdot \left(\left(angle\_m \cdot 0.005555555555555556\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{+96}:\\ \;\;\;\;2 \cdot \left(\cos t\_0 \cdot \left(\sin t\_0 \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\pi \cdot \frac{angle\_m}{180}\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* angle_m (/ PI -180.0))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 1e-50)
      (* 2.0 (* (+ a b) (* (* angle_m 0.005555555555555556) (* (- b a) PI))))
      (if (<= (/ angle_m 180.0) 4e+96)
        (* 2.0 (* (cos t_0) (* (sin t_0) (* (+ a b) (- a b)))))
        (* 2.0 (* (* (+ a b) (- b a)) (sin (* PI (/ angle_m 180.0))))))))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = angle_m * (((double) M_PI) / -180.0);
	double tmp;
	if ((angle_m / 180.0) <= 1e-50) {
		tmp = 2.0 * ((a + b) * ((angle_m * 0.005555555555555556) * ((b - a) * ((double) M_PI))));
	} else if ((angle_m / 180.0) <= 4e+96) {
		tmp = 2.0 * (cos(t_0) * (sin(t_0) * ((a + b) * (a - b))));
	} else {
		tmp = 2.0 * (((a + b) * (b - a)) * sin((((double) M_PI) * (angle_m / 180.0))));
	}
	return angle_s * tmp;
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = angle_m * (Math.PI / -180.0);
	double tmp;
	if ((angle_m / 180.0) <= 1e-50) {
		tmp = 2.0 * ((a + b) * ((angle_m * 0.005555555555555556) * ((b - a) * Math.PI)));
	} else if ((angle_m / 180.0) <= 4e+96) {
		tmp = 2.0 * (Math.cos(t_0) * (Math.sin(t_0) * ((a + b) * (a - b))));
	} else {
		tmp = 2.0 * (((a + b) * (b - a)) * Math.sin((Math.PI * (angle_m / 180.0))));
	}
	return angle_s * tmp;
}

Python

angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = angle_m * (math.pi / -180.0)
	tmp = 0
	if (angle_m / 180.0) <= 1e-50:
		tmp = 2.0 * ((a + b) * ((angle_m * 0.005555555555555556) * ((b - a) * math.pi)))
	elif (angle_m / 180.0) <= 4e+96:
		tmp = 2.0 * (math.cos(t_0) * (math.sin(t_0) * ((a + b) * (a - b))))
	else:
		tmp = 2.0 * (((a + b) * (b - a)) * math.sin((math.pi * (angle_m / 180.0))))
	return angle_s * tmp

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(angle_m * Float64(pi / -180.0))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e-50)
		tmp = Float64(2.0 * Float64(Float64(a + b) * Float64(Float64(angle_m * 0.005555555555555556) * Float64(Float64(b - a) * pi))));
	elseif (Float64(angle_m / 180.0) <= 4e+96)
		tmp = Float64(2.0 * Float64(cos(t_0) * Float64(sin(t_0) * Float64(Float64(a + b) * Float64(a - b)))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(a + b) * Float64(b - a)) * sin(Float64(pi * Float64(angle_m / 180.0)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = angle_m * (pi / -180.0);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 1e-50)
		tmp = 2.0 * ((a + b) * ((angle_m * 0.005555555555555556) * ((b - a) * pi)));
	elseif ((angle_m / 180.0) <= 4e+96)
		tmp = 2.0 * (cos(t_0) * (sin(t_0) * ((a + b) * (a - b))));
	else
		tmp = 2.0 * (((a + b) * (b - a)) * sin((pi * (angle_m / 180.0))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(angle$95$m * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e-50], N[(2.0 * N[(N[(a + b), $MachinePrecision] * N[(N[(angle$95$m * 0.005555555555555556), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+96], N[(2.0 * N[(N[Cos[t$95$0], $MachinePrecision] * N[(N[Sin[t$95$0], $MachinePrecision] * N[(N[(a + b), $MachinePrecision] * N[(a - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < 1.00000000000000001e-50

   1. Initial program 59.0%

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

   3. Simplified 60.4%

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

      1. unpow2 60.4%

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

      2. unpow2 60.4%

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

      3. difference-of-squares 63.9%

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

   6. Applied egg-rr 63.9%

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

      1. add-log-exp 37.6%

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

      2. associate-*l* 37.6%

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

      3. exp-prod 35.7%

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

      4. *-commutative 35.7%

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

   8. Applied egg-rr 35.1%

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

      1. log-pow 35.3%

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

      2. +-commutative 35.3%

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

      3. rem-log-exp 77.4%

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

      4. *-commutative 77.4%

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

      5. associate-*r* 76.6%

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

      6. *-commutative 76.6%

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

      7. associate-*r* 76.9%

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

      8. associate-*r* 77.7%

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

      9. *-commutative 77.7%

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

      10. associate-*r* 77.5%

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

   10. Simplified 77.5%

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

       1. associate-*r* 77.7%

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

       2. metadata-eval 77.7%

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

       3. div-inv 77.6%

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

   12. Applied egg-rr 77.6%

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

   13. Taylor expanded in angle around 0 67.7%

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

       1. associate-*r* 67.7%

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

   15. Simplified 67.7%

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

   if 1.00000000000000001e-50 < (/.f64 angle 180) < 4.0000000000000002e96

   1. Initial program 55.8%

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

   3. Simplified 56.5%

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

      1. unpow2 56.5%

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

      2. unpow2 56.5%

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

      3. difference-of-squares 59.4%

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

   6. Applied egg-rr 59.4%

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

   if 4.0000000000000002e96 < (/.f64 angle 180)

   1. Initial program 23.3%

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

   3. Simplified 24.3%

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

      1. unpow2 24.3%

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

      2. unpow2 24.3%

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

      3. difference-of-squares 24.3%

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

   6. Applied egg-rr 24.3%

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

   7. Taylor expanded in angle around 0 32.2%

      \[\leadsto 2 \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{1}\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{-50}:\\ \;\;\;\;2 \cdot \left(\left(a + b\right) \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 4 \cdot 10^{+96}:\\ \;\;\;\;2 \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\sin \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(a + b\right) \cdot \left(b - a\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 10^{-14}:\\ \;\;\;\;2 \cdot \left(\left(a + b\right) \cdot \left(\left(angle\_m \cdot 0.005555555555555556\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+95}:\\ \;\;\;\;2 \cdot \left(t\_0 \cdot \left(\sin \left(\frac{angle\_m \cdot \pi}{180}\right) \cdot \cos \left(\frac{angle\_m \cdot \pi}{-180}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t\_0 \cdot \sin \left(\pi \cdot \frac{angle\_m}{180}\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* (+ a b) (- b a))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 1e-14)
      (* 2.0 (* (+ a b) (* (* angle_m 0.005555555555555556) (* (- b a) PI))))
      (if (<= (/ angle_m 180.0) 5e+95)
        (*
         2.0
         (*
          t_0
          (* (sin (/ (* angle_m PI) 180.0)) (cos (/ (* angle_m PI) -180.0)))))
        (* 2.0 (* t_0 (sin (* PI (/ angle_m 180.0))))))))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (a + b) * (b - a);
	double tmp;
	if ((angle_m / 180.0) <= 1e-14) {
		tmp = 2.0 * ((a + b) * ((angle_m * 0.005555555555555556) * ((b - a) * ((double) M_PI))));
	} else if ((angle_m / 180.0) <= 5e+95) {
		tmp = 2.0 * (t_0 * (sin(((angle_m * ((double) M_PI)) / 180.0)) * cos(((angle_m * ((double) M_PI)) / -180.0))));
	} else {
		tmp = 2.0 * (t_0 * sin((((double) M_PI) * (angle_m / 180.0))));
	}
	return angle_s * tmp;
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (a + b) * (b - a);
	double tmp;
	if ((angle_m / 180.0) <= 1e-14) {
		tmp = 2.0 * ((a + b) * ((angle_m * 0.005555555555555556) * ((b - a) * Math.PI)));
	} else if ((angle_m / 180.0) <= 5e+95) {
		tmp = 2.0 * (t_0 * (Math.sin(((angle_m * Math.PI) / 180.0)) * Math.cos(((angle_m * Math.PI) / -180.0))));
	} else {
		tmp = 2.0 * (t_0 * Math.sin((Math.PI * (angle_m / 180.0))));
	}
	return angle_s * tmp;
}

Python

angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = (a + b) * (b - a)
	tmp = 0
	if (angle_m / 180.0) <= 1e-14:
		tmp = 2.0 * ((a + b) * ((angle_m * 0.005555555555555556) * ((b - a) * math.pi)))
	elif (angle_m / 180.0) <= 5e+95:
		tmp = 2.0 * (t_0 * (math.sin(((angle_m * math.pi) / 180.0)) * math.cos(((angle_m * math.pi) / -180.0))))
	else:
		tmp = 2.0 * (t_0 * math.sin((math.pi * (angle_m / 180.0))))
	return angle_s * tmp

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(Float64(a + b) * Float64(b - a))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e-14)
		tmp = Float64(2.0 * Float64(Float64(a + b) * Float64(Float64(angle_m * 0.005555555555555556) * Float64(Float64(b - a) * pi))));
	elseif (Float64(angle_m / 180.0) <= 5e+95)
		tmp = Float64(2.0 * Float64(t_0 * Float64(sin(Float64(Float64(angle_m * pi) / 180.0)) * cos(Float64(Float64(angle_m * pi) / -180.0)))));
	else
		tmp = Float64(2.0 * Float64(t_0 * sin(Float64(pi * Float64(angle_m / 180.0)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = (a + b) * (b - a);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 1e-14)
		tmp = 2.0 * ((a + b) * ((angle_m * 0.005555555555555556) * ((b - a) * pi)));
	elseif ((angle_m / 180.0) <= 5e+95)
		tmp = 2.0 * (t_0 * (sin(((angle_m * pi) / 180.0)) * cos(((angle_m * pi) / -180.0))));
	else
		tmp = 2.0 * (t_0 * sin((pi * (angle_m / 180.0))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e-14], N[(2.0 * N[(N[(a + b), $MachinePrecision] * N[(N[(angle$95$m * 0.005555555555555556), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+95], N[(2.0 * N[(t$95$0 * N[(N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(angle$95$m * Pi), $MachinePrecision] / -180.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$0 * N[Sin[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < 9.99999999999999999e-15

   1. Initial program 60.2%

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

   3. Simplified 61.6%

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

      1. unpow2 61.6%

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

      2. unpow2 61.6%

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

      3. difference-of-squares 65.5%

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

   6. Applied egg-rr 65.5%

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

      1. add-log-exp 38.7%

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

      2. associate-*l* 38.7%

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

      3. exp-prod 36.4%

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

      4. *-commutative 36.4%

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

   8. Applied egg-rr 35.9%

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

      1. log-pow 36.0%

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

      2. +-commutative 36.0%

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

      3. rem-log-exp 78.4%

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

      4. *-commutative 78.4%

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

      5. associate-*r* 77.6%

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

      6. *-commutative 77.6%

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

      7. associate-*r* 78.0%

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

      8. associate-*r* 78.7%

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

      9. *-commutative 78.7%

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

      10. associate-*r* 78.5%

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

   10. Simplified 78.5%

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

       1. associate-*r* 78.7%

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

       2. metadata-eval 78.7%

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

       3. div-inv 78.6%

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

   12. Applied egg-rr 78.6%

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

   13. Taylor expanded in angle around 0 69.1%

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

       1. associate-*r* 69.1%

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

   15. Simplified 69.1%

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

   if 9.99999999999999999e-15 < (/.f64 angle 180) < 5.00000000000000025e95

   1. Initial program 46.2%

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

   3. Simplified 43.0%

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

      1. unpow2 43.0%

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

      2. unpow2 43.0%

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

      3. difference-of-squares 43.0%

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

   6. Applied egg-rr 43.0%

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

   7. Taylor expanded in angle around inf 45.9%

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

      1. *-commutative 45.9%

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

      2. associate-*l* 46.9%

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

   9. Simplified 46.9%

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

       1. associate-*r* 46.1%

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

       2. metadata-eval 46.1%

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

       3. div-inv 50.9%

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

   11. Applied egg-rr 51.0%

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

   if 5.00000000000000025e95 < (/.f64 angle 180)

   1. Initial program 23.3%

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

   3. Simplified 24.3%

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

      1. unpow2 24.3%

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

      2. unpow2 24.3%

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

      3. difference-of-squares 24.3%

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

   6. Applied egg-rr 24.3%

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

   7. Taylor expanded in angle around 0 32.2%

      \[\leadsto 2 \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{1}\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{-14}:\\ \;\;\;\;2 \cdot \left(\left(a + b\right) \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+95}:\\ \;\;\;\;2 \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(\sin \left(\frac{angle \cdot \pi}{180}\right) \cdot \cos \left(\frac{angle \cdot \pi}{-180}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := angle\_m \cdot \left(\pi \cdot 0.005555555555555556\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 3.2 \cdot 10^{-57}:\\ \;\;\;\;2 \cdot \left(\left(a + b\right) \cdot \left(b \cdot \left(\cos t\_0 \cdot \sin t\_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{angle\_m \cdot \pi}{180}\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* angle_m (* PI 0.005555555555555556))))
   (*
    angle_s
    (if (<= a 3.2e-57)
      (* 2.0 (* (+ a b) (* b (* (cos t_0) (sin t_0)))))
      (* 2.0 (* (+ a b) (* (- b a) (sin (/ (* angle_m PI) 180.0)))))))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = angle_m * (((double) M_PI) * 0.005555555555555556);
	double tmp;
	if (a <= 3.2e-57) {
		tmp = 2.0 * ((a + b) * (b * (cos(t_0) * sin(t_0))));
	} else {
		tmp = 2.0 * ((a + b) * ((b - a) * sin(((angle_m * ((double) M_PI)) / 180.0))));
	}
	return angle_s * tmp;
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = angle_m * (Math.PI * 0.005555555555555556);
	double tmp;
	if (a <= 3.2e-57) {
		tmp = 2.0 * ((a + b) * (b * (Math.cos(t_0) * Math.sin(t_0))));
	} else {
		tmp = 2.0 * ((a + b) * ((b - a) * Math.sin(((angle_m * Math.PI) / 180.0))));
	}
	return angle_s * tmp;
}

Python

angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = angle_m * (math.pi * 0.005555555555555556)
	tmp = 0
	if a <= 3.2e-57:
		tmp = 2.0 * ((a + b) * (b * (math.cos(t_0) * math.sin(t_0))))
	else:
		tmp = 2.0 * ((a + b) * ((b - a) * math.sin(((angle_m * math.pi) / 180.0))))
	return angle_s * tmp

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(angle_m * Float64(pi * 0.005555555555555556))
	tmp = 0.0
	if (a <= 3.2e-57)
		tmp = Float64(2.0 * Float64(Float64(a + b) * Float64(b * Float64(cos(t_0) * sin(t_0)))));
	else
		tmp = Float64(2.0 * Float64(Float64(a + b) * Float64(Float64(b - a) * sin(Float64(Float64(angle_m * pi) / 180.0)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = angle_m * (pi * 0.005555555555555556);
	tmp = 0.0;
	if (a <= 3.2e-57)
		tmp = 2.0 * ((a + b) * (b * (cos(t_0) * sin(t_0))));
	else
		tmp = 2.0 * ((a + b) * ((b - a) * sin(((angle_m * pi) / 180.0))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[a, 3.2e-57], N[(2.0 * N[(N[(a + b), $MachinePrecision] * N[(b * N[(N[Cos[t$95$0], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.2000000000000001e-57

   1. Initial program 50.4%

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

   3. Simplified 50.4%

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

      1. unpow2 50.4%

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

      2. unpow2 50.4%

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

      3. difference-of-squares 52.1%

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

   6. Applied egg-rr 52.1%

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

      1. add-log-exp 30.6%

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

      2. associate-*l* 30.6%

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

      3. exp-prod 30.0%

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

      4. *-commutative 30.0%

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

   8. Applied egg-rr 29.9%

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

      1. log-pow 30.1%

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

      2. +-commutative 30.1%

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

      3. rem-log-exp 60.3%

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

      4. *-commutative 60.3%

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

      5. associate-*r* 58.7%

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

      6. *-commutative 58.7%

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

      7. associate-*r* 60.6%

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

      8. associate-*r* 59.5%

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

      9. *-commutative 59.5%

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

      10. associate-*r* 59.9%

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

   10. Simplified 59.9%

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

   11. Taylor expanded in b around inf 46.7%

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

       1. *-commutative 46.7%

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

       2. associate-*r* 47.2%

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

       3. *-commutative 47.2%

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

       4. associate-*r* 47.9%

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

   13. Simplified 47.9%

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

   if 3.2000000000000001e-57 < a

   1. Initial program 55.6%

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

   3. Simplified 58.2%

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

      1. unpow2 58.2%

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

      2. unpow2 58.2%

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

      3. difference-of-squares 63.8%

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

   6. Applied egg-rr 63.8%

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

      1. add-log-exp 40.4%

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

      2. associate-*l* 40.4%

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

      3. exp-prod 36.2%

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

      4. *-commutative 36.2%

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

   8. Applied egg-rr 34.8%

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

      1. log-pow 34.8%

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

      2. +-commutative 34.8%

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

      3. rem-log-exp 73.8%

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

      4. *-commutative 73.8%

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

      5. associate-*r* 74.2%

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

      6. *-commutative 74.2%

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

      7. associate-*r* 74.3%

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

      8. associate-*r* 77.1%

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

      9. *-commutative 77.1%

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

      10. associate-*r* 75.9%

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

   10. Simplified 75.9%

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

       1. associate-*r* 77.1%

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

       2. metadata-eval 77.1%

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

       3. div-inv 75.7%

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

   12. Applied egg-rr 75.7%

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

   13. Taylor expanded in angle around 0 73.3%

       \[\leadsto 2 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\color{blue}{1} \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.2 \cdot 10^{-57}:\\ \;\;\;\;2 \cdot \left(\left(a + b\right) \cdot \left(b \cdot \left(\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 2.4 \cdot 10^{-57}:\\ \;\;\;\;2 \cdot \left(\left(a + b\right) \cdot \left(b \cdot \left(\sin t\_0 \cdot \cos t\_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{angle\_m \cdot \pi}{180}\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m 0.005555555555555556))))
   (*
    angle_s
    (if (<= a 2.4e-57)
      (* 2.0 (* (+ a b) (* b (* (sin t_0) (cos t_0)))))
      (* 2.0 (* (+ a b) (* (- b a) (sin (/ (* angle_m PI) 180.0)))))))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m * 0.005555555555555556);
	double tmp;
	if (a <= 2.4e-57) {
		tmp = 2.0 * ((a + b) * (b * (sin(t_0) * cos(t_0))));
	} else {
		tmp = 2.0 * ((a + b) * ((b - a) * sin(((angle_m * ((double) M_PI)) / 180.0))));
	}
	return angle_s * tmp;
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.PI * (angle_m * 0.005555555555555556);
	double tmp;
	if (a <= 2.4e-57) {
		tmp = 2.0 * ((a + b) * (b * (Math.sin(t_0) * Math.cos(t_0))));
	} else {
		tmp = 2.0 * ((a + b) * ((b - a) * Math.sin(((angle_m * Math.PI) / 180.0))));
	}
	return angle_s * tmp;
}

Python

angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = math.pi * (angle_m * 0.005555555555555556)
	tmp = 0
	if a <= 2.4e-57:
		tmp = 2.0 * ((a + b) * (b * (math.sin(t_0) * math.cos(t_0))))
	else:
		tmp = 2.0 * ((a + b) * ((b - a) * math.sin(((angle_m * math.pi) / 180.0))))
	return angle_s * tmp

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(pi * Float64(angle_m * 0.005555555555555556))
	tmp = 0.0
	if (a <= 2.4e-57)
		tmp = Float64(2.0 * Float64(Float64(a + b) * Float64(b * Float64(sin(t_0) * cos(t_0)))));
	else
		tmp = Float64(2.0 * Float64(Float64(a + b) * Float64(Float64(b - a) * sin(Float64(Float64(angle_m * pi) / 180.0)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = pi * (angle_m * 0.005555555555555556);
	tmp = 0.0;
	if (a <= 2.4e-57)
		tmp = 2.0 * ((a + b) * (b * (sin(t_0) * cos(t_0))));
	else
		tmp = 2.0 * ((a + b) * ((b - a) * sin(((angle_m * pi) / 180.0))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[a, 2.4e-57], N[(2.0 * N[(N[(a + b), $MachinePrecision] * N[(b * N[(N[Sin[t$95$0], $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.40000000000000006e-57

   1. Initial program 50.4%

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

   3. Simplified 50.4%

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

      1. unpow2 50.4%

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

      2. unpow2 50.4%

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

      3. difference-of-squares 52.1%

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

   6. Applied egg-rr 52.1%

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

      1. add-log-exp 30.6%

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

      2. associate-*l* 30.6%

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

      3. exp-prod 30.0%

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

      4. *-commutative 30.0%

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

   8. Applied egg-rr 29.9%

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

      1. log-pow 30.1%

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

      2. +-commutative 30.1%

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

      3. rem-log-exp 60.3%

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

      4. *-commutative 60.3%

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

      5. associate-*r* 58.7%

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

      6. *-commutative 58.7%

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

      7. associate-*r* 60.6%

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

      8. associate-*r* 59.5%

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

      9. *-commutative 59.5%

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

      10. associate-*r* 59.9%

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

   10. Simplified 59.9%

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

       1. associate-*r* 59.5%

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

       2. metadata-eval 59.5%

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

       3. div-inv 61.9%

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

   12. Applied egg-rr 61.9%

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

   13. Taylor expanded in b around inf 46.7%

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

       1. *-commutative 46.7%

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

       2. associate-*r* 46.9%

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

       3. *-commutative 46.9%

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

       4. associate-*r* 48.6%

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

       5. *-commutative 48.6%

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

   15. Simplified 48.6%

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

   if 2.40000000000000006e-57 < a

   1. Initial program 55.6%

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

   3. Simplified 58.2%

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

      1. unpow2 58.2%

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

      2. unpow2 58.2%

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

      3. difference-of-squares 63.8%

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

   6. Applied egg-rr 63.8%

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

      1. add-log-exp 40.4%

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

      2. associate-*l* 40.4%

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

      3. exp-prod 36.2%

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

      4. *-commutative 36.2%

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

   8. Applied egg-rr 34.8%

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

      1. log-pow 34.8%

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

      2. +-commutative 34.8%

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

      3. rem-log-exp 73.8%

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

      4. *-commutative 73.8%

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

      5. associate-*r* 74.2%

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

      6. *-commutative 74.2%

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

      7. associate-*r* 74.3%

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

      8. associate-*r* 77.1%

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

      9. *-commutative 77.1%

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

      10. associate-*r* 75.9%

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

   10. Simplified 75.9%

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

       1. associate-*r* 77.1%

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

       2. metadata-eval 77.1%

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

       3. div-inv 75.7%

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

   12. Applied egg-rr 75.7%

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

   13. Taylor expanded in angle around 0 73.3%

       \[\leadsto 2 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\color{blue}{1} \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.4 \cdot 10^{-57}:\\ \;\;\;\;2 \cdot \left(\left(a + b\right) \cdot \left(b \cdot \left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(2 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\cos \left(angle\_m \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot \sin \left(\frac{angle\_m \cdot \pi}{180}\right)\right)\right)\right)\right) \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (*
   2.0
   (*
    (+ a b)
    (*
     (- b a)
     (*
      (cos (* angle_m (* PI 0.005555555555555556)))
      (sin (/ (* angle_m PI) 180.0))))))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (2.0 * ((a + b) * ((b - a) * (cos((angle_m * (((double) M_PI) * 0.005555555555555556))) * sin(((angle_m * ((double) M_PI)) / 180.0))))));
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (2.0 * ((a + b) * ((b - a) * (Math.cos((angle_m * (Math.PI * 0.005555555555555556))) * Math.sin(((angle_m * Math.PI) / 180.0))))));
}

Python

angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * (2.0 * ((a + b) * ((b - a) * (math.cos((angle_m * (math.pi * 0.005555555555555556))) * math.sin(((angle_m * math.pi) / 180.0))))))

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(2.0 * Float64(Float64(a + b) * Float64(Float64(b - a) * Float64(cos(Float64(angle_m * Float64(pi * 0.005555555555555556))) * sin(Float64(Float64(angle_m * pi) / 180.0)))))))
end

MATLAB

angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * (2.0 * ((a + b) * ((b - a) * (cos((angle_m * (pi * 0.005555555555555556))) * sin(((angle_m * pi) / 180.0))))));
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(2.0 * N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(N[Cos[N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.9%

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

3. Simplified 52.7%

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

   1. unpow2 52.7%

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

   2. unpow2 52.7%

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

   3. difference-of-squares 55.5%

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

6. Applied egg-rr 55.5%

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

   1. add-log-exp 33.4%

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

   2. associate-*l* 33.4%

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

   3. exp-prod 31.8%

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

   4. *-commutative 31.8%

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

8. Applied egg-rr 31.3%

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

   1. log-pow 31.4%

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

   2. +-commutative 31.4%

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

   3. rem-log-exp 64.2%

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

   4. *-commutative 64.2%

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

   5. associate-*r* 63.2%

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

   6. *-commutative 63.2%

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

   7. associate-*r* 64.5%

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

   8. associate-*r* 64.6%

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

   9. *-commutative 64.6%

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

   10. associate-*r* 64.5%

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

10. Simplified 64.5%

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

    1. associate-*r* 64.6%

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

    2. metadata-eval 64.6%

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

    3. div-inv 65.9%

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

12. Applied egg-rr 65.9%

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

13. Final simplification 65.9%

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

Alternative 11: 67.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 500:\\ \;\;\;\;2 \cdot \left(\left(a + b\right) \cdot \left(\left(angle\_m \cdot 0.005555555555555556\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\pi \cdot \frac{angle\_m}{180}\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 500.0)
    (* 2.0 (* (+ a b) (* (* angle_m 0.005555555555555556) (* (- b a) PI))))
    (* 2.0 (* (* (+ a b) (- b a)) (sin (* PI (/ angle_m 180.0))))))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 500.0) {
		tmp = 2.0 * ((a + b) * ((angle_m * 0.005555555555555556) * ((b - a) * ((double) M_PI))));
	} else {
		tmp = 2.0 * (((a + b) * (b - a)) * sin((((double) M_PI) * (angle_m / 180.0))));
	}
	return angle_s * tmp;
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 500.0) {
		tmp = 2.0 * ((a + b) * ((angle_m * 0.005555555555555556) * ((b - a) * Math.PI)));
	} else {
		tmp = 2.0 * (((a + b) * (b - a)) * Math.sin((Math.PI * (angle_m / 180.0))));
	}
	return angle_s * tmp;
}

Python

angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 500.0:
		tmp = 2.0 * ((a + b) * ((angle_m * 0.005555555555555556) * ((b - a) * math.pi)))
	else:
		tmp = 2.0 * (((a + b) * (b - a)) * math.sin((math.pi * (angle_m / 180.0))))
	return angle_s * tmp

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 500.0)
		tmp = Float64(2.0 * Float64(Float64(a + b) * Float64(Float64(angle_m * 0.005555555555555556) * Float64(Float64(b - a) * pi))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(a + b) * Float64(b - a)) * sin(Float64(pi * Float64(angle_m / 180.0)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if ((angle_m / 180.0) <= 500.0)
		tmp = 2.0 * ((a + b) * ((angle_m * 0.005555555555555556) * ((b - a) * pi)));
	else
		tmp = 2.0 * (((a + b) * (b - a)) * sin((pi * (angle_m / 180.0))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 500.0], N[(2.0 * N[(N[(a + b), $MachinePrecision] * N[(N[(angle$95$m * 0.005555555555555556), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < 500

   1. Initial program 61.1%

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

   3. Simplified 62.4%

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

      1. unpow2 62.4%

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

      2. unpow2 62.4%

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

      3. difference-of-squares 66.2%

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

   6. Applied egg-rr 66.2%

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

      1. add-log-exp 39.0%

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

      2. associate-*l* 39.0%

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

      3. exp-prod 36.8%

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

      4. *-commutative 36.8%

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

   8. Applied egg-rr 36.3%

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

      1. log-pow 36.4%

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

      2. +-commutative 36.4%

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

      3. rem-log-exp 78.9%

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

      4. *-commutative 78.9%

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

      5. associate-*r* 78.1%

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

      6. *-commutative 78.1%

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

      7. associate-*r* 78.4%

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

      8. associate-*r* 79.1%

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

      9. *-commutative 79.1%

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

      10. associate-*r* 78.9%

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

   10. Simplified 78.9%

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

       1. associate-*r* 79.1%

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

       2. metadata-eval 79.1%

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

       3. div-inv 79.0%

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

   12. Applied egg-rr 79.0%

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

   13. Taylor expanded in angle around 0 69.2%

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

       1. associate-*r* 69.3%

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

   15. Simplified 69.3%

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

   if 500 < (/.f64 angle 180)

   1. Initial program 27.5%

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

   3. Simplified 26.9%

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

      1. unpow2 26.9%

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

      2. unpow2 26.9%

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

      3. difference-of-squares 26.9%

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

   6. Applied egg-rr 26.9%

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

   7. Taylor expanded in angle around 0 28.6%

      \[\leadsto 2 \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{1}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 500:\\ \;\;\;\;2 \cdot \left(\left(a + b\right) \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 205000:\\ \;\;\;\;2 \cdot \left(\left(a + b\right) \cdot \left(\left(angle\_m \cdot 0.005555555555555556\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(angle\_m \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 205000.0)
    (* 2.0 (* (+ a b) (* (* angle_m 0.005555555555555556) (* (- b a) PI))))
    (*
     2.0
     (* (sin (* angle_m (* PI 0.005555555555555556))) (* (+ a b) (- b a)))))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 205000.0) {
		tmp = 2.0 * ((a + b) * ((angle_m * 0.005555555555555556) * ((b - a) * ((double) M_PI))));
	} else {
		tmp = 2.0 * (sin((angle_m * (((double) M_PI) * 0.005555555555555556))) * ((a + b) * (b - a)));
	}
	return angle_s * tmp;
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 205000.0) {
		tmp = 2.0 * ((a + b) * ((angle_m * 0.005555555555555556) * ((b - a) * Math.PI)));
	} else {
		tmp = 2.0 * (Math.sin((angle_m * (Math.PI * 0.005555555555555556))) * ((a + b) * (b - a)));
	}
	return angle_s * tmp;
}

Python

angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if angle_m <= 205000.0:
		tmp = 2.0 * ((a + b) * ((angle_m * 0.005555555555555556) * ((b - a) * math.pi)))
	else:
		tmp = 2.0 * (math.sin((angle_m * (math.pi * 0.005555555555555556))) * ((a + b) * (b - a)))
	return angle_s * tmp

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (angle_m <= 205000.0)
		tmp = Float64(2.0 * Float64(Float64(a + b) * Float64(Float64(angle_m * 0.005555555555555556) * Float64(Float64(b - a) * pi))));
	else
		tmp = Float64(2.0 * Float64(sin(Float64(angle_m * Float64(pi * 0.005555555555555556))) * Float64(Float64(a + b) * Float64(b - a))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (angle_m <= 205000.0)
		tmp = 2.0 * ((a + b) * ((angle_m * 0.005555555555555556) * ((b - a) * pi)));
	else
		tmp = 2.0 * (sin((angle_m * (pi * 0.005555555555555556))) * ((a + b) * (b - a)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 205000.0], N[(2.0 * N[(N[(a + b), $MachinePrecision] * N[(N[(angle$95$m * 0.005555555555555556), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sin[N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 205000

   1. Initial program 61.1%

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

   3. Simplified 62.4%

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

      1. unpow2 62.4%

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

      2. unpow2 62.4%

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

      3. difference-of-squares 66.2%

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

   6. Applied egg-rr 66.2%

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

      1. add-log-exp 39.0%

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

      2. associate-*l* 39.0%

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

      3. exp-prod 36.8%

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

      4. *-commutative 36.8%

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

   8. Applied egg-rr 36.3%

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

      1. log-pow 36.4%

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

      2. +-commutative 36.4%

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

      3. rem-log-exp 78.9%

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

      4. *-commutative 78.9%

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

      5. associate-*r* 78.1%

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

      6. *-commutative 78.1%

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

      7. associate-*r* 78.4%

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

      8. associate-*r* 79.1%

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

      9. *-commutative 79.1%

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

      10. associate-*r* 78.9%

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

   10. Simplified 78.9%

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

       1. associate-*r* 79.1%

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

       2. metadata-eval 79.1%

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

       3. div-inv 79.0%

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

   12. Applied egg-rr 79.0%

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

   13. Taylor expanded in angle around 0 69.2%

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

       1. associate-*r* 69.3%

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

   15. Simplified 69.3%

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

   if 205000 < angle

   1. Initial program 27.5%

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

   3. Simplified 26.9%

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

      1. unpow2 26.9%

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

      2. unpow2 26.9%

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

      3. difference-of-squares 26.9%

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

   6. Applied egg-rr 26.9%

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

   7. Taylor expanded in angle around inf 23.6%

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

      1. *-commutative 23.6%

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

      2. associate-*l* 23.6%

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

   9. Simplified 23.6%

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

   10. Taylor expanded in angle around 0 25.3%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 205000:\\ \;\;\;\;2 \cdot \left(\left(a + b\right) \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 66.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(2 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{angle\_m \cdot \pi}{180}\right)\right)\right)\right) \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (* angle_s (* 2.0 (* (+ a b) (* (- b a) (sin (/ (* angle_m PI) 180.0)))))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (2.0 * ((a + b) * ((b - a) * sin(((angle_m * ((double) M_PI)) / 180.0)))));
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (2.0 * ((a + b) * ((b - a) * Math.sin(((angle_m * Math.PI) / 180.0)))));
}

Python

angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * (2.0 * ((a + b) * ((b - a) * math.sin(((angle_m * math.pi) / 180.0)))))

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(2.0 * Float64(Float64(a + b) * Float64(Float64(b - a) * sin(Float64(Float64(angle_m * pi) / 180.0))))))
end

MATLAB

angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * (2.0 * ((a + b) * ((b - a) * sin(((angle_m * pi) / 180.0)))));
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(2.0 * N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.9%

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

3. Simplified 52.7%

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

   1. unpow2 52.7%

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

   2. unpow2 52.7%

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

   3. difference-of-squares 55.5%

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

6. Applied egg-rr 55.5%

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

   1. add-log-exp 33.4%

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

   2. associate-*l* 33.4%

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

   3. exp-prod 31.8%

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

   4. *-commutative 31.8%

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

8. Applied egg-rr 31.3%

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

   1. log-pow 31.4%

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

   2. +-commutative 31.4%

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

   3. rem-log-exp 64.2%

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

   4. *-commutative 64.2%

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

   5. associate-*r* 63.2%

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

   6. *-commutative 63.2%

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

   7. associate-*r* 64.5%

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

   8. associate-*r* 64.6%

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

   9. *-commutative 64.6%

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

   10. associate-*r* 64.5%

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

10. Simplified 64.5%

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

    1. associate-*r* 64.6%

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

    2. metadata-eval 64.6%

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

    3. div-inv 65.9%

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

12. Applied egg-rr 65.9%

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

13. Taylor expanded in angle around 0 61.2%

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

14. Final simplification 61.2%

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

Alternative 14: 54.7% accurate, 27.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(2 \cdot \left(0.005555555555555556 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (* 2.0 (* 0.005555555555555556 (* angle_m (* PI (* (+ a b) (- b a))))))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (2.0 * (0.005555555555555556 * (angle_m * (((double) M_PI) * ((a + b) * (b - a))))));
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (2.0 * (0.005555555555555556 * (angle_m * (Math.PI * ((a + b) * (b - a))))));
}

Python

angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * (2.0 * (0.005555555555555556 * (angle_m * (math.pi * ((a + b) * (b - a))))))

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(2.0 * Float64(0.005555555555555556 * Float64(angle_m * Float64(pi * Float64(Float64(a + b) * Float64(b - a)))))))
end

MATLAB

angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * (2.0 * (0.005555555555555556 * (angle_m * (pi * ((a + b) * (b - a))))));
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(2.0 * N[(0.005555555555555556 * N[(angle$95$m * N[(Pi * N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.9%

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

3. Simplified 52.7%

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

5. Taylor expanded in angle around 0 44.4%

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

   1. unpow2 52.7%

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

   2. unpow2 52.7%

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

   3. difference-of-squares 55.5%

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

7. Applied egg-rr 47.6%

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

8. Final simplification 47.6%

   \[\leadsto 2 \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right) \]
9. Add Preprocessing

Alternative 15: 54.8% accurate, 27.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(2 \cdot \left(0.005555555555555556 \cdot \left(\left(angle\_m \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right) \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (* 2.0 (* 0.005555555555555556 (* (* angle_m PI) (* (+ a b) (- b a)))))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (2.0 * (0.005555555555555556 * ((angle_m * ((double) M_PI)) * ((a + b) * (b - a)))));
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (2.0 * (0.005555555555555556 * ((angle_m * Math.PI) * ((a + b) * (b - a)))));
}

Python

angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * (2.0 * (0.005555555555555556 * ((angle_m * math.pi) * ((a + b) * (b - a)))))

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(2.0 * Float64(0.005555555555555556 * Float64(Float64(angle_m * pi) * Float64(Float64(a + b) * Float64(b - a))))))
end

MATLAB

angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * (2.0 * (0.005555555555555556 * ((angle_m * pi) * ((a + b) * (b - a)))));
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(2.0 * N[(0.005555555555555556 * N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.9%

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

3. Simplified 52.7%

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

   1. unpow2 52.7%

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

   2. unpow2 52.7%

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

   3. difference-of-squares 55.5%

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

6. Applied egg-rr 55.5%

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

7. Taylor expanded in angle around 0 47.6%

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

   1. associate-*r* 47.7%

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

9. Simplified 47.7%

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

10. Final simplification 47.7%

    \[\leadsto 2 \cdot \left(0.005555555555555556 \cdot \left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]
11. Add Preprocessing

Alternative 16: 63.1% accurate, 27.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(2 \cdot \left(\left(a + b\right) \cdot \left(0.005555555555555556 \cdot \left(angle\_m \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)\right)\right) \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (* 2.0 (* (+ a b) (* 0.005555555555555556 (* angle_m (* (- b a) PI)))))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (2.0 * ((a + b) * (0.005555555555555556 * (angle_m * ((b - a) * ((double) M_PI))))));
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (2.0 * ((a + b) * (0.005555555555555556 * (angle_m * ((b - a) * Math.PI)))));
}

Python

angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * (2.0 * ((a + b) * (0.005555555555555556 * (angle_m * ((b - a) * math.pi)))))

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(2.0 * Float64(Float64(a + b) * Float64(0.005555555555555556 * Float64(angle_m * Float64(Float64(b - a) * pi))))))
end

MATLAB

angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * (2.0 * ((a + b) * (0.005555555555555556 * (angle_m * ((b - a) * pi)))));
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(2.0 * N[(N[(a + b), $MachinePrecision] * N[(0.005555555555555556 * N[(angle$95$m * N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.9%

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

3. Simplified 52.7%

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

   1. unpow2 52.7%

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

   2. unpow2 52.7%

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

   3. difference-of-squares 55.5%

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

6. Applied egg-rr 55.5%

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

   1. add-log-exp 33.4%

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

   2. associate-*l* 33.4%

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

   3. exp-prod 31.8%

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

   4. *-commutative 31.8%

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

8. Applied egg-rr 31.3%

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

   1. log-pow 31.4%

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

   2. +-commutative 31.4%

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

   3. rem-log-exp 64.2%

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

   4. *-commutative 64.2%

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

   5. associate-*r* 63.2%

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

   6. *-commutative 63.2%

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

   7. associate-*r* 64.5%

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

   8. associate-*r* 64.6%

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

   9. *-commutative 64.6%

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

   10. associate-*r* 64.5%

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

10. Simplified 64.5%

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

11. Taylor expanded in angle around 0 56.7%

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

12. Final simplification 56.7%

    \[\leadsto 2 \cdot \left(\left(a + b\right) \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)\right) \]
13. Add Preprocessing

Alternative 17: 63.0% accurate, 27.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(2 \cdot \left(\left(a + b\right) \cdot \left(\left(angle\_m \cdot 0.005555555555555556\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)\right) \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (* 2.0 (* (+ a b) (* (* angle_m 0.005555555555555556) (* (- b a) PI))))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (2.0 * ((a + b) * ((angle_m * 0.005555555555555556) * ((b - a) * ((double) M_PI)))));
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (2.0 * ((a + b) * ((angle_m * 0.005555555555555556) * ((b - a) * Math.PI))));
}

Python

angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * (2.0 * ((a + b) * ((angle_m * 0.005555555555555556) * ((b - a) * math.pi))))

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(2.0 * Float64(Float64(a + b) * Float64(Float64(angle_m * 0.005555555555555556) * Float64(Float64(b - a) * pi)))))
end

MATLAB

angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * (2.0 * ((a + b) * ((angle_m * 0.005555555555555556) * ((b - a) * pi))));
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(2.0 * N[(N[(a + b), $MachinePrecision] * N[(N[(angle$95$m * 0.005555555555555556), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.9%

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

3. Simplified 52.7%

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

   1. unpow2 52.7%

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

   2. unpow2 52.7%

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

   3. difference-of-squares 55.5%

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

6. Applied egg-rr 55.5%

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

   1. add-log-exp 33.4%

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

   2. associate-*l* 33.4%

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

   3. exp-prod 31.8%

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

   4. *-commutative 31.8%

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

8. Applied egg-rr 31.3%

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

   1. log-pow 31.4%

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

   2. +-commutative 31.4%

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

   3. rem-log-exp 64.2%

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

   4. *-commutative 64.2%

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

   5. associate-*r* 63.2%

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

   6. *-commutative 63.2%

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

   7. associate-*r* 64.5%

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

   8. associate-*r* 64.6%

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

   9. *-commutative 64.6%

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

   10. associate-*r* 64.5%

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

10. Simplified 64.5%

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

    1. associate-*r* 64.6%

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

    2. metadata-eval 64.6%

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

    3. div-inv 65.9%

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

12. Applied egg-rr 65.9%

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

13. Taylor expanded in angle around 0 56.7%

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

    1. associate-*r* 56.8%

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

15. Simplified 56.8%

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

16. Final simplification 56.8%

    \[\leadsto 2 \cdot \left(\left(a + b\right) \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right) \]
17. Add Preprocessing

Alternative 18: 13.9% accurate, 59.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(2 \cdot \left(\left(a + b\right) \cdot 0\right)\right) \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (* angle_s (* 2.0 (* (+ a b) 0.0))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (2.0 * ((a + b) * 0.0));
}

Fortran

angle_m = abs(angle)
angle_s = copysign(1.0d0, angle)
real(8) function code(angle_s, a, b, angle_m)
    real(8), intent (in) :: angle_s
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle_m
    code = angle_s * (2.0d0 * ((a + b) * 0.0d0))
end function

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (2.0 * ((a + b) * 0.0));
}

Python

angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * (2.0 * ((a + b) * 0.0))

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(2.0 * Float64(Float64(a + b) * 0.0)))
end

MATLAB

angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * (2.0 * ((a + b) * 0.0));
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(2.0 * N[(N[(a + b), $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.9%

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

3. Simplified 52.7%

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

   1. unpow2 52.7%

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

   2. unpow2 52.7%

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

   3. difference-of-squares 55.5%

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

6. Applied egg-rr 55.5%

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

   1. add-log-exp 33.4%

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

   2. associate-*l* 33.4%

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

   3. exp-prod 31.8%

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

   4. *-commutative 31.8%

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

8. Applied egg-rr 31.3%

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

   1. log-pow 31.4%

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

   2. +-commutative 31.4%

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

   3. rem-log-exp 64.2%

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

   4. *-commutative 64.2%

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

   5. associate-*r* 63.2%

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

   6. *-commutative 63.2%

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

   7. associate-*r* 64.5%

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

   8. associate-*r* 64.6%

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

   9. *-commutative 64.6%

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

   10. associate-*r* 64.5%

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

10. Simplified 64.5%

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

    1. associate-*r* 64.6%

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

    2. metadata-eval 64.6%

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

    3. div-inv 65.9%

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

12. Applied egg-rr 65.9%

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

14. Applied egg-rr 5.5%

    \[\leadsto 2 \cdot \left(\left(a + b\right) \cdot \color{blue}{\frac{\left({b}^{3} - {a}^{3}\right) \cdot \left(\sin \left(\left(angle \cdot \pi\right) \cdot 0\right) + \sin \left(\left(angle \cdot \pi\right) \cdot 0\right)\right)}{\mathsf{fma}\left(b, b, a \cdot \left(b + a\right)\right) \cdot 2}}\right) \]
15. Step-by-step derivation

    1. *-commutative 5.5%

       \[\leadsto 2 \cdot \left(\left(a + b\right) \cdot \frac{\color{blue}{\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0\right) + \sin \left(\left(angle \cdot \pi\right) \cdot 0\right)\right) \cdot \left({b}^{3} - {a}^{3}\right)}}{\mathsf{fma}\left(b, b, a \cdot \left(b + a\right)\right) \cdot 2}\right) \]

    2. associate-/l* 5.5%

       \[\leadsto 2 \cdot \left(\left(a + b\right) \cdot \color{blue}{\left(\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0\right) + \sin \left(\left(angle \cdot \pi\right) \cdot 0\right)\right) \cdot \frac{{b}^{3} - {a}^{3}}{\mathsf{fma}\left(b, b, a \cdot \left(b + a\right)\right) \cdot 2}\right)}\right) \]

    3. count-2 5.5%

       \[\leadsto 2 \cdot \left(\left(a + b\right) \cdot \left(\color{blue}{\left(2 \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0\right)\right)} \cdot \frac{{b}^{3} - {a}^{3}}{\mathsf{fma}\left(b, b, a \cdot \left(b + a\right)\right) \cdot 2}\right)\right) \]

    4. *-commutative 5.5%

       \[\leadsto 2 \cdot \left(\left(a + b\right) \cdot \left(\color{blue}{\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0\right) \cdot 2\right)} \cdot \frac{{b}^{3} - {a}^{3}}{\mathsf{fma}\left(b, b, a \cdot \left(b + a\right)\right) \cdot 2}\right)\right) \]

    5. mul0-rgt 5.5%

       \[\leadsto 2 \cdot \left(\left(a + b\right) \cdot \left(\left(\sin \color{blue}{0} \cdot 2\right) \cdot \frac{{b}^{3} - {a}^{3}}{\mathsf{fma}\left(b, b, a \cdot \left(b + a\right)\right) \cdot 2}\right)\right) \]

    6. sin-0 5.5%

       \[\leadsto 2 \cdot \left(\left(a + b\right) \cdot \left(\left(\color{blue}{0} \cdot 2\right) \cdot \frac{{b}^{3} - {a}^{3}}{\mathsf{fma}\left(b, b, a \cdot \left(b + a\right)\right) \cdot 2}\right)\right) \]

    7. metadata-eval 5.5%

       \[\leadsto 2 \cdot \left(\left(a + b\right) \cdot \left(\color{blue}{0} \cdot \frac{{b}^{3} - {a}^{3}}{\mathsf{fma}\left(b, b, a \cdot \left(b + a\right)\right) \cdot 2}\right)\right) \]

    8. mul0-lft 10.8%

       \[\leadsto 2 \cdot \left(\left(a + b\right) \cdot \color{blue}{0}\right) \]

16. Simplified 10.8%

    \[\leadsto 2 \cdot \left(\left(a + b\right) \cdot \color{blue}{0}\right) \]

17. Final simplification 10.8%

    \[\leadsto 2 \cdot \left(\left(a + b\right) \cdot 0\right) \]
18. Add Preprocessing

Reproduce

herbie shell --seed 2024044 
(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)))))