ab-angle->ABCF B

Percentage Accurate: 54.2% → 60.1%

Time: 27.9s

Alternatives: 14

Speedup: 5.5×

• 32.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 60.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := 2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\\ t_1 := \pi \cdot \frac{angle_m}{180}\\ t_2 := {b}^{2} - {a}^{2}\\ angle_s \cdot \begin{array}{l} \mathbf{if}\;t_2 \leq 10^{+306}:\\ \;\;\;\;\left(t_0 \cdot \sin t_1\right) \cdot \cos \left(\frac{angle_m}{180} \cdot {\left({\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}\right)}^{2}\right)\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;{\left(\sqrt{\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle_m\right)\right)} \cdot \left(b \cdot \sqrt{2}\right)\right)}^{2} \cdot \cos t_1\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 \cdot \sin \left(\frac{angle_m}{180} \cdot \sqrt[3]{{\pi}^{3}}\right)\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle_m \cdot 0.005555555555555556\right)\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* 2.0 (* (+ b a) (- b a))))
        (t_1 (* PI (/ angle_m 180.0)))
        (t_2 (- (pow b 2.0) (pow a 2.0))))
   (*
    angle_s
    (if (<= t_2 1e+306)
      (*
       (* t_0 (sin t_1))
       (cos (* (/ angle_m 180.0) (pow (pow (cbrt (sqrt PI)) 3.0) 2.0))))
      (if (<= t_2 INFINITY)
        (*
         (pow
          (*
           (sqrt (sin (* 0.005555555555555556 (* PI angle_m))))
           (* b (sqrt 2.0)))
          2.0)
         (cos t_1))
        (*
         (* t_0 (sin (* (/ angle_m 180.0) (cbrt (pow PI 3.0)))))
         (cos (expm1 (log1p (* PI (* angle_m 0.005555555555555556)))))))))))

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 = 2.0 * ((b + a) * (b - a));
	double t_1 = ((double) M_PI) * (angle_m / 180.0);
	double t_2 = pow(b, 2.0) - pow(a, 2.0);
	double tmp;
	if (t_2 <= 1e+306) {
		tmp = (t_0 * sin(t_1)) * cos(((angle_m / 180.0) * pow(pow(cbrt(sqrt(((double) M_PI))), 3.0), 2.0)));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = pow((sqrt(sin((0.005555555555555556 * (((double) M_PI) * angle_m)))) * (b * sqrt(2.0))), 2.0) * cos(t_1);
	} else {
		tmp = (t_0 * sin(((angle_m / 180.0) * cbrt(pow(((double) M_PI), 3.0))))) * cos(expm1(log1p((((double) M_PI) * (angle_m * 0.005555555555555556)))));
	}
	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 = 2.0 * ((b + a) * (b - a));
	double t_1 = Math.PI * (angle_m / 180.0);
	double t_2 = Math.pow(b, 2.0) - Math.pow(a, 2.0);
	double tmp;
	if (t_2 <= 1e+306) {
		tmp = (t_0 * Math.sin(t_1)) * Math.cos(((angle_m / 180.0) * Math.pow(Math.pow(Math.cbrt(Math.sqrt(Math.PI)), 3.0), 2.0)));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = Math.pow((Math.sqrt(Math.sin((0.005555555555555556 * (Math.PI * angle_m)))) * (b * Math.sqrt(2.0))), 2.0) * Math.cos(t_1);
	} else {
		tmp = (t_0 * Math.sin(((angle_m / 180.0) * Math.cbrt(Math.pow(Math.PI, 3.0))))) * Math.cos(Math.expm1(Math.log1p((Math.PI * (angle_m * 0.005555555555555556)))));
	}
	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(2.0 * Float64(Float64(b + a) * Float64(b - a)))
	t_1 = Float64(pi * Float64(angle_m / 180.0))
	t_2 = Float64((b ^ 2.0) - (a ^ 2.0))
	tmp = 0.0
	if (t_2 <= 1e+306)
		tmp = Float64(Float64(t_0 * sin(t_1)) * cos(Float64(Float64(angle_m / 180.0) * ((cbrt(sqrt(pi)) ^ 3.0) ^ 2.0))));
	elseif (t_2 <= Inf)
		tmp = Float64((Float64(sqrt(sin(Float64(0.005555555555555556 * Float64(pi * angle_m)))) * Float64(b * sqrt(2.0))) ^ 2.0) * cos(t_1));
	else
		tmp = Float64(Float64(t_0 * sin(Float64(Float64(angle_m / 180.0) * cbrt((pi ^ 3.0))))) * cos(expm1(log1p(Float64(pi * Float64(angle_m * 0.005555555555555556))))));
	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[(2.0 * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[t$95$2, 1e+306], N[(N[(t$95$0 * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[Power[N[Power[N[Power[N[Sqrt[Pi], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Power[N[(N[Sqrt[N[Sin[N[(0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(b * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(Exp[N[Log[1 + N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 b 2) (pow.f64 a 2)) < 1.00000000000000002e306

   1. Initial program 56.0%

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

      1. unpow2 56.0%

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

      2. unpow2 56.0%

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

      3. difference-of-squares 56.0%

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

   3. Applied egg-rr 56.0%

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

      1. add-sqr-sqrt 56.6%

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

      2. pow2 56.6%

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

   5. Applied egg-rr 56.6%

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

      1. add-cube-cbrt 57.0%

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

      2. pow3 58.6%

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

   7. Applied egg-rr 58.6%

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

   if 1.00000000000000002e306 < (-.f64 (pow.f64 b 2) (pow.f64 a 2)) < +inf.0

   1. Initial program 47.1%

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

      1. unpow2 47.1%

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

      2. unpow2 47.1%

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

      3. difference-of-squares 47.1%

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

   3. Applied egg-rr 47.1%

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

   4. Taylor expanded in b around inf 47.1%

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

      1. associate-*r* 47.1%

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

      2. associate-*r* 49.1%

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

      3. *-commutative 49.1%

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

      4. *-commutative 49.1%

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

      5. *-commutative 49.1%

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

   6. Simplified 49.1%

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

      1. add-sqr-sqrt 27.7%

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

      2. pow2 27.7%

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

      3. *-commutative 27.7%

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

      4. rem-log-exp 18.4%

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

      5. sqrt-prod 18.4%

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

      6. *-commutative 18.4%

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

      7. sqrt-prod 18.4%

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

      8. unpow2 18.4%

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

      9. sqrt-prod 8.7%

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

      10. add-sqr-sqrt 19.5%

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

      11. rem-log-exp 48.3%

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

   8. Applied egg-rr 48.3%

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

      1. *-commutative 48.3%

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

      2. *-commutative 48.3%

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

      3. *-commutative 48.3%

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

      4. associate-*r* 46.4%

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

   10. Simplified 46.4%

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

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

   1. Initial program 0.0%

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

      1. unpow2 0.0%

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

      2. unpow2 0.0%

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

      3. difference-of-squares 56.8%

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

   3. Applied egg-rr 56.8%

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

      1. add-cbrt-cube 69.3%

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

      2. pow3 69.3%

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

   5. Applied egg-rr 69.3%

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

      1. div-inv 69.3%

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

      2. metadata-eval 69.3%

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

      3. expm1-log1p-u 44.3%

         \[\leadsto \left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\sqrt[3]{{\pi}^{3}} \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} \]

   7. Applied egg-rr 44.3%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq 10^{+306}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot {\left({\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}\right)}^{2}\right)\\ \mathbf{elif}\;{b}^{2} - {a}^{2} \leq \infty:\\ \;\;\;\;{\left(\sqrt{\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)} \cdot \left(b \cdot \sqrt{2}\right)\right)}^{2} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \sqrt[3]{{\pi}^{3}}\right)\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\\ \end{array} \]

Alternative 2: 60.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := 2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\\ t_1 := {b}^{2} - {a}^{2}\\ t_2 := \pi \cdot \left(angle_m \cdot 0.005555555555555556\right)\\ angle_s \cdot \begin{array}{l} \mathbf{if}\;t_1 \leq 2 \cdot 10^{+266}:\\ \;\;\;\;\left(t_0 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin t_2\right)\right)\right) \cdot \cos \left(\frac{angle_m}{\frac{180}{\pi}}\right)\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;{\left(\sqrt{\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle_m\right)\right)} \cdot \left(b \cdot \sqrt{2}\right)\right)}^{2} \cdot \cos \left(\pi \cdot \frac{angle_m}{180}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 \cdot \sin \left(\frac{angle_m}{180} \cdot \sqrt[3]{{\pi}^{3}}\right)\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(t_2\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* 2.0 (* (+ b a) (- b a))))
        (t_1 (- (pow b 2.0) (pow a 2.0)))
        (t_2 (* PI (* angle_m 0.005555555555555556))))
   (*
    angle_s
    (if (<= t_1 2e+266)
      (* (* t_0 (expm1 (log1p (sin t_2)))) (cos (/ angle_m (/ 180.0 PI))))
      (if (<= t_1 INFINITY)
        (*
         (pow
          (*
           (sqrt (sin (* 0.005555555555555556 (* PI angle_m))))
           (* b (sqrt 2.0)))
          2.0)
         (cos (* PI (/ angle_m 180.0))))
        (*
         (* t_0 (sin (* (/ angle_m 180.0) (cbrt (pow PI 3.0)))))
         (cos (expm1 (log1p t_2)))))))))

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 = 2.0 * ((b + a) * (b - a));
	double t_1 = pow(b, 2.0) - pow(a, 2.0);
	double t_2 = ((double) M_PI) * (angle_m * 0.005555555555555556);
	double tmp;
	if (t_1 <= 2e+266) {
		tmp = (t_0 * expm1(log1p(sin(t_2)))) * cos((angle_m / (180.0 / ((double) M_PI))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = pow((sqrt(sin((0.005555555555555556 * (((double) M_PI) * angle_m)))) * (b * sqrt(2.0))), 2.0) * cos((((double) M_PI) * (angle_m / 180.0)));
	} else {
		tmp = (t_0 * sin(((angle_m / 180.0) * cbrt(pow(((double) M_PI), 3.0))))) * cos(expm1(log1p(t_2)));
	}
	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 = 2.0 * ((b + a) * (b - a));
	double t_1 = Math.pow(b, 2.0) - Math.pow(a, 2.0);
	double t_2 = Math.PI * (angle_m * 0.005555555555555556);
	double tmp;
	if (t_1 <= 2e+266) {
		tmp = (t_0 * Math.expm1(Math.log1p(Math.sin(t_2)))) * Math.cos((angle_m / (180.0 / Math.PI)));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = Math.pow((Math.sqrt(Math.sin((0.005555555555555556 * (Math.PI * angle_m)))) * (b * Math.sqrt(2.0))), 2.0) * Math.cos((Math.PI * (angle_m / 180.0)));
	} else {
		tmp = (t_0 * Math.sin(((angle_m / 180.0) * Math.cbrt(Math.pow(Math.PI, 3.0))))) * Math.cos(Math.expm1(Math.log1p(t_2)));
	}
	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(2.0 * Float64(Float64(b + a) * Float64(b - a)))
	t_1 = Float64((b ^ 2.0) - (a ^ 2.0))
	t_2 = Float64(pi * Float64(angle_m * 0.005555555555555556))
	tmp = 0.0
	if (t_1 <= 2e+266)
		tmp = Float64(Float64(t_0 * expm1(log1p(sin(t_2)))) * cos(Float64(angle_m / Float64(180.0 / pi))));
	elseif (t_1 <= Inf)
		tmp = Float64((Float64(sqrt(sin(Float64(0.005555555555555556 * Float64(pi * angle_m)))) * Float64(b * sqrt(2.0))) ^ 2.0) * cos(Float64(pi * Float64(angle_m / 180.0))));
	else
		tmp = Float64(Float64(t_0 * sin(Float64(Float64(angle_m / 180.0) * cbrt((pi ^ 3.0))))) * cos(expm1(log1p(t_2))));
	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[(2.0 * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[t$95$1, 2e+266], N[(N[(t$95$0 * N[(Exp[N[Log[1 + N[Sin[t$95$2], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(angle$95$m / N[(180.0 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[Power[N[(N[Sqrt[N[Sin[N[(0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(b * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[Cos[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(Exp[N[Log[1 + t$95$2], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 b 2) (pow.f64 a 2)) < 2.0000000000000001e266

   1. Initial program 56.4%

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

      1. unpow2 56.4%

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

      2. unpow2 56.4%

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

      3. difference-of-squares 56.4%

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

   3. Applied egg-rr 56.4%

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

      1. add-cbrt-cube 55.3%

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

      2. pow3 55.3%

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

   5. Applied egg-rr 55.3%

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

      1. associate-*r/ 54.9%

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

      2. *-commutative 54.9%

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

      3. associate-/l* 56.8%

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

   7. Applied egg-rr 56.8%

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

      1. rem-cbrt-cube 58.1%

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

      2. div-inv 58.5%

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

      3. metadata-eval 58.5%

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

      4. expm1-log1p-u 58.5%

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

   9. Applied egg-rr 58.5%

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

   if 2.0000000000000001e266 < (-.f64 (pow.f64 b 2) (pow.f64 a 2)) < +inf.0

   1. Initial program 46.9%

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

      1. unpow2 46.9%

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

      2. unpow2 46.9%

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

      3. difference-of-squares 46.9%

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

   3. Applied egg-rr 46.9%

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

   4. Taylor expanded in b around inf 47.0%

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

      1. associate-*r* 47.0%

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

      2. associate-*r* 48.7%

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

      3. *-commutative 48.7%

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

      4. *-commutative 48.7%

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

      5. *-commutative 48.7%

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

   6. Simplified 48.7%

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

      1. add-sqr-sqrt 26.0%

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

      2. pow2 26.0%

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

      3. *-commutative 26.0%

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

      4. rem-log-exp 16.2%

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

      5. sqrt-prod 16.2%

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

      6. *-commutative 16.2%

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

      7. sqrt-prod 16.2%

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

      8. unpow2 16.2%

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

      9. sqrt-prod 7.7%

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

      10. add-sqr-sqrt 17.3%

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

      11. rem-log-exp 44.0%

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

   8. Applied egg-rr 44.0%

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

      1. *-commutative 44.0%

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

      2. *-commutative 44.0%

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

      3. *-commutative 44.0%

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

      4. associate-*r* 42.7%

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

   10. Simplified 42.7%

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

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

   1. Initial program 0.0%

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

      1. unpow2 0.0%

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

      2. unpow2 0.0%

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

      3. difference-of-squares 56.8%

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

   3. Applied egg-rr 56.8%

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

      1. add-cbrt-cube 69.3%

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

      2. pow3 69.3%

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

   5. Applied egg-rr 69.3%

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

      1. div-inv 69.3%

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

      2. metadata-eval 69.3%

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

      3. expm1-log1p-u 44.3%

         \[\leadsto \left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\sqrt[3]{{\pi}^{3}} \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} \]

   7. Applied egg-rr 44.3%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq 2 \cdot 10^{+266}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right) \cdot \cos \left(\frac{angle}{\frac{180}{\pi}}\right)\\ \mathbf{elif}\;{b}^{2} - {a}^{2} \leq \infty:\\ \;\;\;\;{\left(\sqrt{\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)} \cdot \left(b \cdot \sqrt{2}\right)\right)}^{2} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \sqrt[3]{{\pi}^{3}}\right)\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\\ \end{array} \]

Alternative 3: 60.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := 2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\\ t_1 := {b}^{2} - {a}^{2}\\ t_2 := \pi \cdot \frac{angle_m}{180}\\ angle_s \cdot \begin{array}{l} \mathbf{if}\;t_1 \leq 2 \cdot 10^{+266}:\\ \;\;\;\;\left(t_0 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\pi \cdot \left(angle_m \cdot 0.005555555555555556\right)\right)\right)\right)\right) \cdot \cos \left(\frac{angle_m}{\frac{180}{\pi}}\right)\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;{\left(\sqrt{\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle_m\right)\right)} \cdot \left(b \cdot \sqrt{2}\right)\right)}^{2} \cdot \cos t_2\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 \cdot \sin t_2\right) \cdot \cos \left(\frac{angle_m}{180} \cdot {\left(\sqrt[3]{{\pi}^{1.5}}\right)}^{2}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* 2.0 (* (+ b a) (- b a))))
        (t_1 (- (pow b 2.0) (pow a 2.0)))
        (t_2 (* PI (/ angle_m 180.0))))
   (*
    angle_s
    (if (<= t_1 2e+266)
      (*
       (* t_0 (expm1 (log1p (sin (* PI (* angle_m 0.005555555555555556))))))
       (cos (/ angle_m (/ 180.0 PI))))
      (if (<= t_1 INFINITY)
        (*
         (pow
          (*
           (sqrt (sin (* 0.005555555555555556 (* PI angle_m))))
           (* b (sqrt 2.0)))
          2.0)
         (cos t_2))
        (*
         (* t_0 (sin t_2))
         (cos (* (/ angle_m 180.0) (pow (cbrt (pow PI 1.5)) 2.0)))))))))

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 = 2.0 * ((b + a) * (b - a));
	double t_1 = pow(b, 2.0) - pow(a, 2.0);
	double t_2 = ((double) M_PI) * (angle_m / 180.0);
	double tmp;
	if (t_1 <= 2e+266) {
		tmp = (t_0 * expm1(log1p(sin((((double) M_PI) * (angle_m * 0.005555555555555556)))))) * cos((angle_m / (180.0 / ((double) M_PI))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = pow((sqrt(sin((0.005555555555555556 * (((double) M_PI) * angle_m)))) * (b * sqrt(2.0))), 2.0) * cos(t_2);
	} else {
		tmp = (t_0 * sin(t_2)) * cos(((angle_m / 180.0) * pow(cbrt(pow(((double) M_PI), 1.5)), 2.0)));
	}
	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 = 2.0 * ((b + a) * (b - a));
	double t_1 = Math.pow(b, 2.0) - Math.pow(a, 2.0);
	double t_2 = Math.PI * (angle_m / 180.0);
	double tmp;
	if (t_1 <= 2e+266) {
		tmp = (t_0 * Math.expm1(Math.log1p(Math.sin((Math.PI * (angle_m * 0.005555555555555556)))))) * Math.cos((angle_m / (180.0 / Math.PI)));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = Math.pow((Math.sqrt(Math.sin((0.005555555555555556 * (Math.PI * angle_m)))) * (b * Math.sqrt(2.0))), 2.0) * Math.cos(t_2);
	} else {
		tmp = (t_0 * Math.sin(t_2)) * Math.cos(((angle_m / 180.0) * Math.pow(Math.cbrt(Math.pow(Math.PI, 1.5)), 2.0)));
	}
	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(2.0 * Float64(Float64(b + a) * Float64(b - a)))
	t_1 = Float64((b ^ 2.0) - (a ^ 2.0))
	t_2 = Float64(pi * Float64(angle_m / 180.0))
	tmp = 0.0
	if (t_1 <= 2e+266)
		tmp = Float64(Float64(t_0 * expm1(log1p(sin(Float64(pi * Float64(angle_m * 0.005555555555555556)))))) * cos(Float64(angle_m / Float64(180.0 / pi))));
	elseif (t_1 <= Inf)
		tmp = Float64((Float64(sqrt(sin(Float64(0.005555555555555556 * Float64(pi * angle_m)))) * Float64(b * sqrt(2.0))) ^ 2.0) * cos(t_2));
	else
		tmp = Float64(Float64(t_0 * sin(t_2)) * cos(Float64(Float64(angle_m / 180.0) * (cbrt((pi ^ 1.5)) ^ 2.0))));
	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[(2.0 * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[t$95$1, 2e+266], N[(N[(t$95$0 * N[(Exp[N[Log[1 + N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(angle$95$m / N[(180.0 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[Power[N[(N[Sqrt[N[Sin[N[(0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(b * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[Power[N[Power[N[Power[Pi, 1.5], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 b 2) (pow.f64 a 2)) < 2.0000000000000001e266

   1. Initial program 56.4%

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

      1. unpow2 56.4%

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

      2. unpow2 56.4%

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

      3. difference-of-squares 56.4%

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

   3. Applied egg-rr 56.4%

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

      1. add-cbrt-cube 55.3%

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

      2. pow3 55.3%

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

   5. Applied egg-rr 55.3%

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

      1. associate-*r/ 54.9%

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

      2. *-commutative 54.9%

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

      3. associate-/l* 56.8%

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

   7. Applied egg-rr 56.8%

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

      1. rem-cbrt-cube 58.1%

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

      2. div-inv 58.5%

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

      3. metadata-eval 58.5%

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

      4. expm1-log1p-u 58.5%

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

   9. Applied egg-rr 58.5%

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

   if 2.0000000000000001e266 < (-.f64 (pow.f64 b 2) (pow.f64 a 2)) < +inf.0

   1. Initial program 46.9%

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

      1. unpow2 46.9%

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

      2. unpow2 46.9%

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

      3. difference-of-squares 46.9%

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

   3. Applied egg-rr 46.9%

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

   4. Taylor expanded in b around inf 47.0%

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

      1. associate-*r* 47.0%

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

      2. associate-*r* 48.7%

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

      3. *-commutative 48.7%

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

      4. *-commutative 48.7%

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

      5. *-commutative 48.7%

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

   6. Simplified 48.7%

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

      1. add-sqr-sqrt 26.0%

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

      2. pow2 26.0%

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

      3. *-commutative 26.0%

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

      4. rem-log-exp 16.2%

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

      5. sqrt-prod 16.2%

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

      6. *-commutative 16.2%

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

      7. sqrt-prod 16.2%

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

      8. unpow2 16.2%

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

      9. sqrt-prod 7.7%

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

      10. add-sqr-sqrt 17.3%

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

      11. rem-log-exp 44.0%

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

   8. Applied egg-rr 44.0%

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

      1. *-commutative 44.0%

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

      2. *-commutative 44.0%

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

      3. *-commutative 44.0%

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

      4. associate-*r* 42.7%

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

   10. Simplified 42.7%

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

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

   1. Initial program 0.0%

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

      1. unpow2 0.0%

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

      2. unpow2 0.0%

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

      3. difference-of-squares 56.8%

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

   3. Applied egg-rr 56.8%

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

      1. add-sqr-sqrt 50.6%

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

      2. pow2 50.6%

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

   5. Applied egg-rr 50.6%

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

      1. add-cbrt-cube 50.6%

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

      2. pow1/3 50.6%

         \[\leadsto \left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left({\color{blue}{\left({\left(\left(\sqrt{\pi} \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}\right)}^{0.3333333333333333}\right)}}^{2} \cdot \frac{angle}{180}\right) \]

      3. add-sqr-sqrt 50.6%

         \[\leadsto \left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left({\left({\left(\color{blue}{\pi} \cdot \sqrt{\pi}\right)}^{0.3333333333333333}\right)}^{2} \cdot \frac{angle}{180}\right) \]

      4. pow1 50.6%

         \[\leadsto \left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left({\left({\left(\color{blue}{{\pi}^{1}} \cdot \sqrt{\pi}\right)}^{0.3333333333333333}\right)}^{2} \cdot \frac{angle}{180}\right) \]

      5. metadata-eval 50.6%

         \[\leadsto \left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left({\left({\left({\pi}^{\color{blue}{\left(3 \cdot 0.3333333333333333\right)}} \cdot \sqrt{\pi}\right)}^{0.3333333333333333}\right)}^{2} \cdot \frac{angle}{180}\right) \]

      6. pow1/2 50.6%

         \[\leadsto \left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left({\left({\left({\pi}^{\left(3 \cdot 0.3333333333333333\right)} \cdot \color{blue}{{\pi}^{0.5}}\right)}^{0.3333333333333333}\right)}^{2} \cdot \frac{angle}{180}\right) \]

      7. pow-prod-up 50.6%

         \[\leadsto \left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left({\left({\color{blue}{\left({\pi}^{\left(3 \cdot 0.3333333333333333 + 0.5\right)}\right)}}^{0.3333333333333333}\right)}^{2} \cdot \frac{angle}{180}\right) \]

      8. metadata-eval 50.6%

         \[\leadsto \left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left({\left({\left({\pi}^{\left(\color{blue}{1} + 0.5\right)}\right)}^{0.3333333333333333}\right)}^{2} \cdot \frac{angle}{180}\right) \]

      9. metadata-eval 50.6%

         \[\leadsto \left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left({\left({\left({\pi}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}\right)}^{2} \cdot \frac{angle}{180}\right) \]

   7. Applied egg-rr 50.6%

      \[\leadsto \left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left({\color{blue}{\left({\left({\pi}^{1.5}\right)}^{0.3333333333333333}\right)}}^{2} \cdot \frac{angle}{180}\right) \]
   8. Step-by-step derivation

      1. unpow1/3 75.6%

         \[\leadsto \left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left({\color{blue}{\left(\sqrt[3]{{\pi}^{1.5}}\right)}}^{2} \cdot \frac{angle}{180}\right) \]

   9. Simplified 75.6%

      \[\leadsto \left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left({\color{blue}{\left(\sqrt[3]{{\pi}^{1.5}}\right)}}^{2} \cdot \frac{angle}{180}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq 2 \cdot 10^{+266}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right) \cdot \cos \left(\frac{angle}{\frac{180}{\pi}}\right)\\ \mathbf{elif}\;{b}^{2} - {a}^{2} \leq \infty:\\ \;\;\;\;{\left(\sqrt{\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)} \cdot \left(b \cdot \sqrt{2}\right)\right)}^{2} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot {\left(\sqrt[3]{{\pi}^{1.5}}\right)}^{2}\right)\\ \end{array} \]

Alternative 4: 59.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := 2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\\ t_1 := {b}^{2} - {a}^{2}\\ t_2 := \pi \cdot \frac{angle_m}{180}\\ angle_s \cdot \begin{array}{l} \mathbf{if}\;t_1 \leq 2 \cdot 10^{+266}:\\ \;\;\;\;\left(t_0 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\pi \cdot \left(angle_m \cdot 0.005555555555555556\right)\right)\right)\right)\right) \cdot \cos \left(\frac{angle_m}{\frac{180}{\pi}}\right)\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;{\left(\sqrt{\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle_m\right)\right)} \cdot \left(b \cdot \sqrt{2}\right)\right)}^{2} \cdot \cos t_2\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 \cdot \sin t_2\right) \cdot \cos \left(\frac{angle_m}{180} \cdot \sqrt[3]{{\pi}^{3}}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* 2.0 (* (+ b a) (- b a))))
        (t_1 (- (pow b 2.0) (pow a 2.0)))
        (t_2 (* PI (/ angle_m 180.0))))
   (*
    angle_s
    (if (<= t_1 2e+266)
      (*
       (* t_0 (expm1 (log1p (sin (* PI (* angle_m 0.005555555555555556))))))
       (cos (/ angle_m (/ 180.0 PI))))
      (if (<= t_1 INFINITY)
        (*
         (pow
          (*
           (sqrt (sin (* 0.005555555555555556 (* PI angle_m))))
           (* b (sqrt 2.0)))
          2.0)
         (cos t_2))
        (*
         (* t_0 (sin t_2))
         (cos (* (/ angle_m 180.0) (cbrt (pow PI 3.0))))))))))

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 = 2.0 * ((b + a) * (b - a));
	double t_1 = pow(b, 2.0) - pow(a, 2.0);
	double t_2 = ((double) M_PI) * (angle_m / 180.0);
	double tmp;
	if (t_1 <= 2e+266) {
		tmp = (t_0 * expm1(log1p(sin((((double) M_PI) * (angle_m * 0.005555555555555556)))))) * cos((angle_m / (180.0 / ((double) M_PI))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = pow((sqrt(sin((0.005555555555555556 * (((double) M_PI) * angle_m)))) * (b * sqrt(2.0))), 2.0) * cos(t_2);
	} else {
		tmp = (t_0 * sin(t_2)) * cos(((angle_m / 180.0) * cbrt(pow(((double) M_PI), 3.0))));
	}
	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 = 2.0 * ((b + a) * (b - a));
	double t_1 = Math.pow(b, 2.0) - Math.pow(a, 2.0);
	double t_2 = Math.PI * (angle_m / 180.0);
	double tmp;
	if (t_1 <= 2e+266) {
		tmp = (t_0 * Math.expm1(Math.log1p(Math.sin((Math.PI * (angle_m * 0.005555555555555556)))))) * Math.cos((angle_m / (180.0 / Math.PI)));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = Math.pow((Math.sqrt(Math.sin((0.005555555555555556 * (Math.PI * angle_m)))) * (b * Math.sqrt(2.0))), 2.0) * Math.cos(t_2);
	} else {
		tmp = (t_0 * Math.sin(t_2)) * Math.cos(((angle_m / 180.0) * Math.cbrt(Math.pow(Math.PI, 3.0))));
	}
	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(2.0 * Float64(Float64(b + a) * Float64(b - a)))
	t_1 = Float64((b ^ 2.0) - (a ^ 2.0))
	t_2 = Float64(pi * Float64(angle_m / 180.0))
	tmp = 0.0
	if (t_1 <= 2e+266)
		tmp = Float64(Float64(t_0 * expm1(log1p(sin(Float64(pi * Float64(angle_m * 0.005555555555555556)))))) * cos(Float64(angle_m / Float64(180.0 / pi))));
	elseif (t_1 <= Inf)
		tmp = Float64((Float64(sqrt(sin(Float64(0.005555555555555556 * Float64(pi * angle_m)))) * Float64(b * sqrt(2.0))) ^ 2.0) * cos(t_2));
	else
		tmp = Float64(Float64(t_0 * sin(t_2)) * cos(Float64(Float64(angle_m / 180.0) * cbrt((pi ^ 3.0)))));
	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[(2.0 * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[t$95$1, 2e+266], N[(N[(t$95$0 * N[(Exp[N[Log[1 + N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(angle$95$m / N[(180.0 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[Power[N[(N[Sqrt[N[Sin[N[(0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(b * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 b 2) (pow.f64 a 2)) < 2.0000000000000001e266

   1. Initial program 56.4%

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

      1. unpow2 56.4%

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

      2. unpow2 56.4%

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

      3. difference-of-squares 56.4%

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

   3. Applied egg-rr 56.4%

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

      1. add-cbrt-cube 55.3%

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

      2. pow3 55.3%

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

   5. Applied egg-rr 55.3%

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

      1. associate-*r/ 54.9%

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

      2. *-commutative 54.9%

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

      3. associate-/l* 56.8%

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

   7. Applied egg-rr 56.8%

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

      1. rem-cbrt-cube 58.1%

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

      2. div-inv 58.5%

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

      3. metadata-eval 58.5%

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

      4. expm1-log1p-u 58.5%

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

   9. Applied egg-rr 58.5%

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

   if 2.0000000000000001e266 < (-.f64 (pow.f64 b 2) (pow.f64 a 2)) < +inf.0

   1. Initial program 46.9%

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

      1. unpow2 46.9%

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

      2. unpow2 46.9%

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

      3. difference-of-squares 46.9%

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

   3. Applied egg-rr 46.9%

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

   4. Taylor expanded in b around inf 47.0%

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

      1. associate-*r* 47.0%

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

      2. associate-*r* 48.7%

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

      3. *-commutative 48.7%

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

      4. *-commutative 48.7%

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

      5. *-commutative 48.7%

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

   6. Simplified 48.7%

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

      1. add-sqr-sqrt 26.0%

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

      2. pow2 26.0%

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

      3. *-commutative 26.0%

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

      4. rem-log-exp 16.2%

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

      5. sqrt-prod 16.2%

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

      6. *-commutative 16.2%

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

      7. sqrt-prod 16.2%

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

      8. unpow2 16.2%

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

      9. sqrt-prod 7.7%

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

      10. add-sqr-sqrt 17.3%

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

      11. rem-log-exp 44.0%

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

   8. Applied egg-rr 44.0%

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

      1. *-commutative 44.0%

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

      2. *-commutative 44.0%

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

      3. *-commutative 44.0%

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

      4. associate-*r* 42.7%

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

   10. Simplified 42.7%

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

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

   1. Initial program 0.0%

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

      1. unpow2 0.0%

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

      2. unpow2 0.0%

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

      3. difference-of-squares 56.8%

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

   3. Applied egg-rr 56.8%

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

      1. add-cbrt-cube 69.3%

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

      2. pow3 69.3%

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

   5. Applied egg-rr 69.3%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq 2 \cdot 10^{+266}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right) \cdot \cos \left(\frac{angle}{\frac{180}{\pi}}\right)\\ \mathbf{elif}\;{b}^{2} - {a}^{2} \leq \infty:\\ \;\;\;\;{\left(\sqrt{\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)} \cdot \left(b \cdot \sqrt{2}\right)\right)}^{2} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \sqrt[3]{{\pi}^{3}}\right)\\ \end{array} \]

Alternative 5: 57.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := 2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\\ t_1 := \pi \cdot \left(angle_m \cdot 0.005555555555555556\right)\\ angle_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle_m}{180} \leq 10^{+49}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle_m}{180}\right) \cdot \left(t_0 \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle_m\right)\right)\right)\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 2 \cdot 10^{+190}:\\ \;\;\;\;t_0 \cdot \sin \left({\left(\sqrt[3]{t_1}\right)}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin t_1\right)\right)\right) \cdot \cos \left(\frac{angle_m}{\frac{180}{\pi}}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* 2.0 (* (+ b a) (- b a))))
        (t_1 (* PI (* angle_m 0.005555555555555556))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 1e+49)
      (*
       (cos (* PI (/ angle_m 180.0)))
       (* t_0 (sin (* 0.005555555555555556 (* PI angle_m)))))
      (if (<= (/ angle_m 180.0) 2e+190)
        (* t_0 (sin (pow (cbrt t_1) 3.0)))
        (*
         (* t_0 (expm1 (log1p (sin t_1))))
         (cos (/ angle_m (/ 180.0 PI)))))))))

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 = 2.0 * ((b + a) * (b - a));
	double t_1 = ((double) M_PI) * (angle_m * 0.005555555555555556);
	double tmp;
	if ((angle_m / 180.0) <= 1e+49) {
		tmp = cos((((double) M_PI) * (angle_m / 180.0))) * (t_0 * sin((0.005555555555555556 * (((double) M_PI) * angle_m))));
	} else if ((angle_m / 180.0) <= 2e+190) {
		tmp = t_0 * sin(pow(cbrt(t_1), 3.0));
	} else {
		tmp = (t_0 * expm1(log1p(sin(t_1)))) * cos((angle_m / (180.0 / ((double) M_PI))));
	}
	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 = 2.0 * ((b + a) * (b - a));
	double t_1 = Math.PI * (angle_m * 0.005555555555555556);
	double tmp;
	if ((angle_m / 180.0) <= 1e+49) {
		tmp = Math.cos((Math.PI * (angle_m / 180.0))) * (t_0 * Math.sin((0.005555555555555556 * (Math.PI * angle_m))));
	} else if ((angle_m / 180.0) <= 2e+190) {
		tmp = t_0 * Math.sin(Math.pow(Math.cbrt(t_1), 3.0));
	} else {
		tmp = (t_0 * Math.expm1(Math.log1p(Math.sin(t_1)))) * Math.cos((angle_m / (180.0 / Math.PI)));
	}
	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(2.0 * Float64(Float64(b + a) * Float64(b - a)))
	t_1 = Float64(pi * Float64(angle_m * 0.005555555555555556))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e+49)
		tmp = Float64(cos(Float64(pi * Float64(angle_m / 180.0))) * Float64(t_0 * sin(Float64(0.005555555555555556 * Float64(pi * angle_m)))));
	elseif (Float64(angle_m / 180.0) <= 2e+190)
		tmp = Float64(t_0 * sin((cbrt(t_1) ^ 3.0)));
	else
		tmp = Float64(Float64(t_0 * expm1(log1p(sin(t_1)))) * cos(Float64(angle_m / Float64(180.0 / pi))));
	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[(2.0 * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+49], N[(N[Cos[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[Sin[N[(0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+190], N[(t$95$0 * N[Sin[N[Power[N[Power[t$95$1, 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(Exp[N[Log[1 + N[Sin[t$95$1], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(angle$95$m / N[(180.0 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < 9.99999999999999946e48

   1. Initial program 53.8%

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

      1. unpow2 53.8%

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

      2. unpow2 53.8%

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

      3. difference-of-squares 58.0%

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

   3. Applied egg-rr 58.0%

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

   4. Taylor expanded in angle around inf 58.4%

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

   if 9.99999999999999946e48 < (/.f64 angle 180) < 2.0000000000000001e190

   1. Initial program 35.8%

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

      1. unpow2 35.8%

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

      2. unpow2 35.8%

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

      3. difference-of-squares 35.8%

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

   3. Applied egg-rr 35.8%

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

   4. Taylor expanded in angle around 0 28.4%

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

      1. *-commutative 28.4%

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

      2. div-inv 33.4%

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

      3. metadata-eval 33.4%

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

      4. *-commutative 33.4%

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

      5. associate-*r* 29.3%

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

      6. add-cube-cbrt 46.0%

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

      7. pow3 50.1%

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

      8. *-commutative 50.1%

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

      9. *-commutative 50.1%

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

      10. associate-*r* 45.2%

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

   6. Applied egg-rr 45.2%

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

   if 2.0000000000000001e190 < (/.f64 angle 180)

   1. Initial program 31.6%

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

      1. unpow2 31.6%

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

      2. unpow2 31.6%

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

      3. difference-of-squares 31.6%

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

   3. Applied egg-rr 31.6%

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

      1. add-cbrt-cube 30.7%

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

      2. pow3 30.7%

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

   5. Applied egg-rr 30.7%

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

      1. associate-*r/ 36.1%

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

      2. *-commutative 36.1%

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

      3. associate-/l* 36.1%

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

   7. Applied egg-rr 36.1%

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

      1. rem-cbrt-cube 47.0%

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

      2. div-inv 51.3%

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

      3. metadata-eval 51.3%

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

      4. expm1-log1p-u 51.3%

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

   9. Applied egg-rr 51.3%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{+49}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+190}:\\ \;\;\;\;\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left({\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right) \cdot \cos \left(\frac{angle}{\frac{180}{\pi}}\right)\\ \end{array} \]

Alternative 6: 56.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := 2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\\ t_1 := \pi \cdot \frac{angle_m}{180}\\ angle_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle_m}{180} \leq 5 \cdot 10^{+83}:\\ \;\;\;\;\cos t_1 \cdot \left(t_0 \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot angle_m\right)\right)\right)\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 5 \cdot 10^{+204}:\\ \;\;\;\;t_0 \cdot \left|\sin \left(angle_m \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 \cdot \sin t_1\right) \cdot \cos \left(\frac{angle_m}{\frac{180}{\pi}}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* 2.0 (* (+ b a) (- b a)))) (t_1 (* PI (/ angle_m 180.0))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 5e+83)
      (* (cos t_1) (* t_0 (* 0.005555555555555556 (* PI angle_m))))
      (if (<= (/ angle_m 180.0) 5e+204)
        (* t_0 (fabs (sin (* angle_m (* PI 0.005555555555555556)))))
        (* (* t_0 (sin t_1)) (cos (/ angle_m (/ 180.0 PI)))))))))

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 = 2.0 * ((b + a) * (b - a));
	double t_1 = ((double) M_PI) * (angle_m / 180.0);
	double tmp;
	if ((angle_m / 180.0) <= 5e+83) {
		tmp = cos(t_1) * (t_0 * (0.005555555555555556 * (((double) M_PI) * angle_m)));
	} else if ((angle_m / 180.0) <= 5e+204) {
		tmp = t_0 * fabs(sin((angle_m * (((double) M_PI) * 0.005555555555555556))));
	} else {
		tmp = (t_0 * sin(t_1)) * cos((angle_m / (180.0 / ((double) M_PI))));
	}
	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 = 2.0 * ((b + a) * (b - a));
	double t_1 = Math.PI * (angle_m / 180.0);
	double tmp;
	if ((angle_m / 180.0) <= 5e+83) {
		tmp = Math.cos(t_1) * (t_0 * (0.005555555555555556 * (Math.PI * angle_m)));
	} else if ((angle_m / 180.0) <= 5e+204) {
		tmp = t_0 * Math.abs(Math.sin((angle_m * (Math.PI * 0.005555555555555556))));
	} else {
		tmp = (t_0 * Math.sin(t_1)) * Math.cos((angle_m / (180.0 / Math.PI)));
	}
	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 = 2.0 * ((b + a) * (b - a))
	t_1 = math.pi * (angle_m / 180.0)
	tmp = 0
	if (angle_m / 180.0) <= 5e+83:
		tmp = math.cos(t_1) * (t_0 * (0.005555555555555556 * (math.pi * angle_m)))
	elif (angle_m / 180.0) <= 5e+204:
		tmp = t_0 * math.fabs(math.sin((angle_m * (math.pi * 0.005555555555555556))))
	else:
		tmp = (t_0 * math.sin(t_1)) * math.cos((angle_m / (180.0 / math.pi)))
	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(2.0 * Float64(Float64(b + a) * Float64(b - a)))
	t_1 = Float64(pi * Float64(angle_m / 180.0))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e+83)
		tmp = Float64(cos(t_1) * Float64(t_0 * Float64(0.005555555555555556 * Float64(pi * angle_m))));
	elseif (Float64(angle_m / 180.0) <= 5e+204)
		tmp = Float64(t_0 * abs(sin(Float64(angle_m * Float64(pi * 0.005555555555555556)))));
	else
		tmp = Float64(Float64(t_0 * sin(t_1)) * cos(Float64(angle_m / Float64(180.0 / pi))));
	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 = 2.0 * ((b + a) * (b - a));
	t_1 = pi * (angle_m / 180.0);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 5e+83)
		tmp = cos(t_1) * (t_0 * (0.005555555555555556 * (pi * angle_m)));
	elseif ((angle_m / 180.0) <= 5e+204)
		tmp = t_0 * abs(sin((angle_m * (pi * 0.005555555555555556))));
	else
		tmp = (t_0 * sin(t_1)) * cos((angle_m / (180.0 / pi)));
	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[(2.0 * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+83], N[(N[Cos[t$95$1], $MachinePrecision] * N[(t$95$0 * N[(0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+204], N[(t$95$0 * N[Abs[N[Sin[N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(angle$95$m / N[(180.0 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < 5.00000000000000029e83

   1. Initial program 53.4%

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

      1. unpow2 53.4%

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

      2. unpow2 53.4%

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

      3. difference-of-squares 57.5%

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

   3. Applied egg-rr 57.5%

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

   4. Taylor expanded in angle around 0 57.0%

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

   if 5.00000000000000029e83 < (/.f64 angle 180) < 5.00000000000000008e204

   1. Initial program 32.2%

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

      1. unpow2 32.2%

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

      2. unpow2 32.2%

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

      3. difference-of-squares 32.2%

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

   3. Applied egg-rr 32.2%

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

   4. Taylor expanded in angle around 0 24.0%

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

      1. *-commutative 24.0%

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

      2. div-inv 24.0%

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

      3. metadata-eval 24.0%

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

      4. *-commutative 24.0%

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

      5. associate-*r* 24.8%

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

      6. add-sqr-sqrt 31.0%

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

      7. pow2 31.7%

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

      8. *-commutative 31.7%

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

      9. *-commutative 31.7%

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

      10. associate-*r* 30.9%

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

   6. Applied egg-rr 30.9%

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

      1. rem-square-sqrt 15.5%

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

      2. sqrt-unprod 49.6%

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

      3. pow2 49.6%

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

      4. unpow2 49.6%

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

      5. add-sqr-sqrt 49.7%

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

      6. *-commutative 49.7%

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

      7. associate-*r* 49.8%

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

      8. metadata-eval 49.8%

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

      9. div-inv 49.8%

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

      10. *-commutative 49.8%

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

      11. div-inv 49.8%

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

      12. metadata-eval 49.8%

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

   8. Applied egg-rr 49.8%

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

      1. unpow2 49.8%

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

      2. rem-sqrt-square 49.8%

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

   10. Simplified 49.8%

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

   if 5.00000000000000008e204 < (/.f64 angle 180)

   1. Initial program 41.2%

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

      1. unpow2 41.2%

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

      2. unpow2 41.2%

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

      3. difference-of-squares 41.2%

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

   3. Applied egg-rr 41.2%

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

      1. add-cbrt-cube 34.3%

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

      2. pow3 34.3%

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

   5. Applied egg-rr 34.3%

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

      1. associate-*r/ 34.5%

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

      2. *-commutative 34.5%

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

      3. associate-/l* 34.5%

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

   7. Applied egg-rr 34.5%

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

      1. rem-cbrt-cube 54.9%

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

      2. expm1-log1p-u 54.9%

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

      3. expm1-udef 54.9%

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

   9. Applied egg-rr 54.9%

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

       1. expm1-def 54.9%

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

       2. expm1-log1p 54.9%

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

   11. Simplified 54.9%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+83}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+204}:\\ \;\;\;\;\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \left|\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\frac{angle}{\frac{180}{\pi}}\right)\\ \end{array} \]

Alternative 7: 57.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := 2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\\ t_1 := \pi \cdot \left(angle_m \cdot 0.005555555555555556\right)\\ angle_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle_m}{180} \leq 10^{+49}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle_m}{180}\right) \cdot \left(t_0 \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle_m\right)\right)\right)\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 4 \cdot 10^{+152}:\\ \;\;\;\;t_0 \cdot \sin \left({\left(\sqrt[3]{t_1}\right)}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(t_1\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* 2.0 (* (+ b a) (- b a))))
        (t_1 (* PI (* angle_m 0.005555555555555556))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 1e+49)
      (*
       (cos (* PI (/ angle_m 180.0)))
       (* t_0 (sin (* 0.005555555555555556 (* PI angle_m)))))
      (if (<= (/ angle_m 180.0) 4e+152)
        (* t_0 (sin (pow (cbrt t_1) 3.0)))
        (* t_0 (sin (expm1 (log1p t_1)))))))))

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 = 2.0 * ((b + a) * (b - a));
	double t_1 = ((double) M_PI) * (angle_m * 0.005555555555555556);
	double tmp;
	if ((angle_m / 180.0) <= 1e+49) {
		tmp = cos((((double) M_PI) * (angle_m / 180.0))) * (t_0 * sin((0.005555555555555556 * (((double) M_PI) * angle_m))));
	} else if ((angle_m / 180.0) <= 4e+152) {
		tmp = t_0 * sin(pow(cbrt(t_1), 3.0));
	} else {
		tmp = t_0 * sin(expm1(log1p(t_1)));
	}
	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 = 2.0 * ((b + a) * (b - a));
	double t_1 = Math.PI * (angle_m * 0.005555555555555556);
	double tmp;
	if ((angle_m / 180.0) <= 1e+49) {
		tmp = Math.cos((Math.PI * (angle_m / 180.0))) * (t_0 * Math.sin((0.005555555555555556 * (Math.PI * angle_m))));
	} else if ((angle_m / 180.0) <= 4e+152) {
		tmp = t_0 * Math.sin(Math.pow(Math.cbrt(t_1), 3.0));
	} else {
		tmp = t_0 * Math.sin(Math.expm1(Math.log1p(t_1)));
	}
	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(2.0 * Float64(Float64(b + a) * Float64(b - a)))
	t_1 = Float64(pi * Float64(angle_m * 0.005555555555555556))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e+49)
		tmp = Float64(cos(Float64(pi * Float64(angle_m / 180.0))) * Float64(t_0 * sin(Float64(0.005555555555555556 * Float64(pi * angle_m)))));
	elseif (Float64(angle_m / 180.0) <= 4e+152)
		tmp = Float64(t_0 * sin((cbrt(t_1) ^ 3.0)));
	else
		tmp = Float64(t_0 * sin(expm1(log1p(t_1))));
	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[(2.0 * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+49], N[(N[Cos[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[Sin[N[(0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+152], N[(t$95$0 * N[Sin[N[Power[N[Power[t$95$1, 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Sin[N[(Exp[N[Log[1 + t$95$1], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < 9.99999999999999946e48

   1. Initial program 53.8%

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

      1. unpow2 53.8%

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

      2. unpow2 53.8%

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

      3. difference-of-squares 58.0%

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

   3. Applied egg-rr 58.0%

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

   4. Taylor expanded in angle around inf 58.4%

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

   if 9.99999999999999946e48 < (/.f64 angle 180) < 4.0000000000000002e152

   1. Initial program 42.1%

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

      1. unpow2 42.1%

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

      2. unpow2 42.1%

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

      3. difference-of-squares 42.1%

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

   3. Applied egg-rr 42.1%

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

   4. Taylor expanded in angle around 0 33.4%

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

      1. *-commutative 33.4%

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

      2. div-inv 33.4%

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

      3. metadata-eval 33.4%

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

      4. *-commutative 33.4%

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

      5. associate-*r* 34.5%

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

      6. add-cube-cbrt 42.4%

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

      7. pow3 47.2%

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

      8. *-commutative 47.2%

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

      9. *-commutative 47.2%

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

      10. associate-*r* 47.3%

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

   6. Applied egg-rr 47.3%

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

   if 4.0000000000000002e152 < (/.f64 angle 180)

   1. Initial program 27.3%

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

      1. unpow2 27.3%

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

      2. unpow2 27.3%

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

      3. difference-of-squares 27.3%

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

   3. Applied egg-rr 27.3%

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

   4. Taylor expanded in angle around 0 24.4%

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

      1. div-inv 27.6%

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

      2. metadata-eval 27.6%

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

      3. expm1-log1p-u 40.5%

         \[\leadsto \left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\sqrt[3]{{\pi}^{3}} \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} \]

   6. Applied egg-rr 32.2%

      \[\leadsto \left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}\right) \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{+49}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 4 \cdot 10^{+152}:\\ \;\;\;\;\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left({\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\\ \end{array} \]

Alternative 8: 55.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(angle_m \cdot 0.005555555555555556\right)\\ t_1 := 2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\\ angle_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle_m}{180} \leq 4 \cdot 10^{+152}:\\ \;\;\;\;t_1 \cdot \sin \left({\left(\sqrt[3]{t_0}\right)}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(t_0\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

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)))
        (t_1 (* 2.0 (* (+ b a) (- b a)))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 4e+152)
      (* t_1 (sin (pow (cbrt t_0) 3.0)))
      (* t_1 (sin (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 = ((double) M_PI) * (angle_m * 0.005555555555555556);
	double t_1 = 2.0 * ((b + a) * (b - a));
	double tmp;
	if ((angle_m / 180.0) <= 4e+152) {
		tmp = t_1 * sin(pow(cbrt(t_0), 3.0));
	} else {
		tmp = t_1 * sin(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 = Math.PI * (angle_m * 0.005555555555555556);
	double t_1 = 2.0 * ((b + a) * (b - a));
	double tmp;
	if ((angle_m / 180.0) <= 4e+152) {
		tmp = t_1 * Math.sin(Math.pow(Math.cbrt(t_0), 3.0));
	} else {
		tmp = t_1 * Math.sin(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(pi * Float64(angle_m * 0.005555555555555556))
	t_1 = Float64(2.0 * Float64(Float64(b + a) * Float64(b - a)))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e+152)
		tmp = Float64(t_1 * sin((cbrt(t_0) ^ 3.0)));
	else
		tmp = Float64(t_1 * sin(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[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+152], N[(t$95$1 * N[Sin[N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Sin[N[(Exp[N[Log[1 + t$95$0], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < 4.0000000000000002e152

   1. Initial program 53.0%

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

      1. unpow2 53.0%

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

      2. unpow2 53.0%

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

      3. difference-of-squares 56.9%

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

   3. Applied egg-rr 56.9%

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

   4. Taylor expanded in angle around 0 55.2%

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

      1. *-commutative 55.2%

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

      2. div-inv 55.2%

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

      3. metadata-eval 55.2%

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

      4. *-commutative 55.2%

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

      5. associate-*r* 55.4%

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

      6. add-cube-cbrt 54.5%

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

      7. pow3 55.9%

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

      8. *-commutative 55.9%

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

      9. *-commutative 55.9%

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

      10. associate-*r* 56.5%

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

   6. Applied egg-rr 56.5%

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

   if 4.0000000000000002e152 < (/.f64 angle 180)

   1. Initial program 27.3%

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

      1. unpow2 27.3%

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

      2. unpow2 27.3%

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

      3. difference-of-squares 27.3%

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

   3. Applied egg-rr 27.3%

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

   4. Taylor expanded in angle around 0 24.4%

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

      1. div-inv 27.6%

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

      2. metadata-eval 27.6%

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

      3. expm1-log1p-u 40.5%

         \[\leadsto \left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\sqrt[3]{{\pi}^{3}} \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} \]

   6. Applied egg-rr 32.2%

      \[\leadsto \left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}\right) \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{+152}:\\ \;\;\;\;\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left({\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\\ \end{array} \]

Alternative 9: 56.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := 2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\\ angle_s \cdot \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 10^{+290}:\\ \;\;\;\;t_0 \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\pi \cdot \left(angle_m \cdot 0.005555555555555556\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* 2.0 (* (+ b a) (- b a)))))
   (*
    angle_s
    (if (<= (pow b 2.0) 1e+290)
      (* t_0 (sin (* 0.005555555555555556 (* PI angle_m))))
      (* t_0 (* PI (* angle_m 0.005555555555555556)))))))

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 = 2.0 * ((b + a) * (b - a));
	double tmp;
	if (pow(b, 2.0) <= 1e+290) {
		tmp = t_0 * sin((0.005555555555555556 * (((double) M_PI) * angle_m)));
	} else {
		tmp = t_0 * (((double) M_PI) * (angle_m * 0.005555555555555556));
	}
	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 = 2.0 * ((b + a) * (b - a));
	double tmp;
	if (Math.pow(b, 2.0) <= 1e+290) {
		tmp = t_0 * Math.sin((0.005555555555555556 * (Math.PI * angle_m)));
	} else {
		tmp = t_0 * (Math.PI * (angle_m * 0.005555555555555556));
	}
	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 = 2.0 * ((b + a) * (b - a))
	tmp = 0
	if math.pow(b, 2.0) <= 1e+290:
		tmp = t_0 * math.sin((0.005555555555555556 * (math.pi * angle_m)))
	else:
		tmp = t_0 * (math.pi * (angle_m * 0.005555555555555556))
	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(2.0 * Float64(Float64(b + a) * Float64(b - a)))
	tmp = 0.0
	if ((b ^ 2.0) <= 1e+290)
		tmp = Float64(t_0 * sin(Float64(0.005555555555555556 * Float64(pi * angle_m))));
	else
		tmp = Float64(t_0 * Float64(pi * Float64(angle_m * 0.005555555555555556)));
	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 = 2.0 * ((b + a) * (b - a));
	tmp = 0.0;
	if ((b ^ 2.0) <= 1e+290)
		tmp = t_0 * sin((0.005555555555555556 * (pi * angle_m)));
	else
		tmp = t_0 * (pi * (angle_m * 0.005555555555555556));
	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[(2.0 * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[Power[b, 2.0], $MachinePrecision], 1e+290], N[(t$95$0 * N[Sin[N[(0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 b 2) < 1.00000000000000006e290

   1. Initial program 56.4%

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

      1. unpow2 56.4%

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

      2. unpow2 56.4%

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

      3. difference-of-squares 56.4%

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

   3. Applied egg-rr 56.4%

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

   4. Taylor expanded in angle around 0 55.5%

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

   5. Taylor expanded in angle around inf 56.6%

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

   if 1.00000000000000006e290 < (pow.f64 b 2)

   1. Initial program 36.4%

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

      1. unpow2 36.4%

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

      2. unpow2 36.4%

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

      3. difference-of-squares 49.0%

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

   3. Applied egg-rr 49.0%

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

   4. Taylor expanded in angle around 0 44.9%

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

   5. Taylor expanded in angle around 0 51.9%

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

      1. associate-*r* 51.9%

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

      2. *-commutative 51.9%

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

      3. *-commutative 51.9%

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

      4. *-commutative 51.9%

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

   7. Simplified 51.9%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 10^{+290}:\\ \;\;\;\;\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\\ \end{array} \]

Alternative 10: 56.3% accurate, 2.0× speedup

Math

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

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 (* (+ b a) (- b a)))
   (fabs (sin (* angle_m (* PI 0.005555555555555556)))))))

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 * ((b + a) * (b - a))) * fabs(sin((angle_m * (((double) M_PI) * 0.005555555555555556)))));
}

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 * ((b + a) * (b - a))) * Math.abs(Math.sin((angle_m * (Math.PI * 0.005555555555555556)))));
}

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 * ((b + a) * (b - a))) * math.fabs(math.sin((angle_m * (math.pi * 0.005555555555555556)))))

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(Float64(2.0 * Float64(Float64(b + a) * Float64(b - a))) * abs(sin(Float64(angle_m * Float64(pi * 0.005555555555555556))))))
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 * ((b + a) * (b - a))) * abs(sin((angle_m * (pi * 0.005555555555555556)))));
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[(N[(2.0 * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[N[Sin[N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.8%

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

   1. unpow2 50.8%

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

   2. unpow2 50.8%

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

   3. difference-of-squares 54.3%

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

3. Applied egg-rr 54.3%

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

4. Taylor expanded in angle around 0 52.5%

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

   1. *-commutative 52.5%

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

   2. div-inv 52.7%

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

   3. metadata-eval 52.7%

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

   4. *-commutative 52.7%

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

   5. associate-*r* 53.3%

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

   6. add-sqr-sqrt 26.1%

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

   7. pow2 26.2%

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

   8. *-commutative 26.2%

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

   9. *-commutative 26.2%

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

   10. associate-*r* 26.1%

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

6. Applied egg-rr 26.1%

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

   1. rem-square-sqrt 22.7%

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

   2. sqrt-unprod 21.5%

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

   3. pow2 21.5%

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

   4. unpow2 21.5%

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

   5. add-sqr-sqrt 31.4%

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

   6. *-commutative 31.4%

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

   7. associate-*r* 31.4%

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

   8. metadata-eval 31.4%

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

   9. div-inv 31.4%

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

   10. *-commutative 31.4%

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

   11. div-inv 31.4%

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

   12. metadata-eval 31.4%

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

8. Applied egg-rr 31.4%

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

   1. unpow2 31.4%

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

   2. rem-sqrt-square 37.6%

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

10. Simplified 37.6%

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

11. Final simplification 37.6%

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

Alternative 11: 55.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(b + a\right) \cdot \left(b - a\right)\\ t_1 := 2 \cdot t_0\\ angle_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle_m}{180} \leq 2 \cdot 10^{+118}:\\ \;\;\;\;t_1 \cdot \sin \left(\frac{\pi \cdot angle_m}{180}\right)\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 2 \cdot 10^{+211}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle_m \cdot \left(\pi \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle_m\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* (+ b a) (- b a))) (t_1 (* 2.0 t_0)))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 2e+118)
      (* t_1 (sin (/ (* PI angle_m) 180.0)))
      (if (<= (/ angle_m 180.0) 2e+211)
        (* 0.011111111111111112 (* angle_m (* PI t_0)))
        (* t_1 (sin (* 0.005555555555555556 (* PI angle_m)))))))))

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 = (b + a) * (b - a);
	double t_1 = 2.0 * t_0;
	double tmp;
	if ((angle_m / 180.0) <= 2e+118) {
		tmp = t_1 * sin(((((double) M_PI) * angle_m) / 180.0));
	} else if ((angle_m / 180.0) <= 2e+211) {
		tmp = 0.011111111111111112 * (angle_m * (((double) M_PI) * t_0));
	} else {
		tmp = t_1 * sin((0.005555555555555556 * (((double) M_PI) * angle_m)));
	}
	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 = (b + a) * (b - a);
	double t_1 = 2.0 * t_0;
	double tmp;
	if ((angle_m / 180.0) <= 2e+118) {
		tmp = t_1 * Math.sin(((Math.PI * angle_m) / 180.0));
	} else if ((angle_m / 180.0) <= 2e+211) {
		tmp = 0.011111111111111112 * (angle_m * (Math.PI * t_0));
	} else {
		tmp = t_1 * Math.sin((0.005555555555555556 * (Math.PI * angle_m)));
	}
	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 = (b + a) * (b - a)
	t_1 = 2.0 * t_0
	tmp = 0
	if (angle_m / 180.0) <= 2e+118:
		tmp = t_1 * math.sin(((math.pi * angle_m) / 180.0))
	elif (angle_m / 180.0) <= 2e+211:
		tmp = 0.011111111111111112 * (angle_m * (math.pi * t_0))
	else:
		tmp = t_1 * math.sin((0.005555555555555556 * (math.pi * angle_m)))
	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(b + a) * Float64(b - a))
	t_1 = Float64(2.0 * t_0)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e+118)
		tmp = Float64(t_1 * sin(Float64(Float64(pi * angle_m) / 180.0)));
	elseif (Float64(angle_m / 180.0) <= 2e+211)
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * t_0)));
	else
		tmp = Float64(t_1 * sin(Float64(0.005555555555555556 * Float64(pi * angle_m))));
	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 = (b + a) * (b - a);
	t_1 = 2.0 * t_0;
	tmp = 0.0;
	if ((angle_m / 180.0) <= 2e+118)
		tmp = t_1 * sin(((pi * angle_m) / 180.0));
	elseif ((angle_m / 180.0) <= 2e+211)
		tmp = 0.011111111111111112 * (angle_m * (pi * t_0));
	else
		tmp = t_1 * sin((0.005555555555555556 * (pi * angle_m)));
	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[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * t$95$0), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+118], N[(t$95$1 * N[Sin[N[(N[(Pi * angle$95$m), $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+211], N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Sin[N[(0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < 1.99999999999999993e118

   1. Initial program 53.1%

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

      1. unpow2 53.1%

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

      2. unpow2 53.1%

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

      3. difference-of-squares 57.1%

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

   3. Applied egg-rr 57.1%

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

   4. Taylor expanded in angle around 0 56.0%

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

      1. associate-*r/ 56.7%

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

   6. Applied egg-rr 56.7%

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

   if 1.99999999999999993e118 < (/.f64 angle 180) < 1.9999999999999999e211

   1. Initial program 30.1%

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

      1. unpow2 30.1%

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

      2. unpow2 30.1%

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

      3. difference-of-squares 30.1%

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

   3. Applied egg-rr 30.1%

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

   4. Taylor expanded in angle around 0 19.3%

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

   5. Taylor expanded in angle around 0 45.5%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)} \cdot 1 \]

   if 1.9999999999999999e211 < (/.f64 angle 180)

   1. Initial program 36.7%

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

      1. unpow2 36.7%

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

      2. unpow2 36.7%

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

      3. difference-of-squares 36.7%

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

   3. Applied egg-rr 36.7%

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

   4. Taylor expanded in angle around 0 33.3%

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

   5. Taylor expanded in angle around inf 44.1%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+118}:\\ \;\;\;\;\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\frac{\pi \cdot angle}{180}\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+211}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\\ \end{array} \]

Alternative 12: 54.4% accurate, 5.4× speedup

Math

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

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 (* (+ b a) (- b a))) (* 0.005555555555555556 (* PI angle_m)))))

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 * ((b + a) * (b - a))) * (0.005555555555555556 * (((double) M_PI) * angle_m)));
}

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 * ((b + a) * (b - a))) * (0.005555555555555556 * (Math.PI * angle_m)));
}

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 * ((b + a) * (b - a))) * (0.005555555555555556 * (math.pi * angle_m)))

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(Float64(2.0 * Float64(Float64(b + a) * Float64(b - a))) * Float64(0.005555555555555556 * Float64(pi * angle_m))))
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 * ((b + a) * (b - a))) * (0.005555555555555556 * (pi * angle_m)));
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[(N[(2.0 * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.8%

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

   1. unpow2 50.8%

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

   2. unpow2 50.8%

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

   3. difference-of-squares 54.3%

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

3. Applied egg-rr 54.3%

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

4. Taylor expanded in angle around 0 52.5%

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

5. Taylor expanded in angle around 0 53.3%

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

6. Final simplification 53.3%

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

Alternative 13: 54.4% accurate, 5.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ angle_s \cdot \left(0.011111111111111112 \cdot \left(angle_m \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)\right) \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (* angle_s (* 0.011111111111111112 (* angle_m (* PI (* (+ b a) (- 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 * (0.011111111111111112 * (angle_m * (((double) M_PI) * ((b + a) * (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 * (0.011111111111111112 * (angle_m * (Math.PI * ((b + a) * (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 * (0.011111111111111112 * (angle_m * (math.pi * ((b + a) * (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(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(Float64(b + a) * 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 * (0.011111111111111112 * (angle_m * (pi * ((b + a) * (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[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.8%

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

   1. unpow2 50.8%

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

   2. unpow2 50.8%

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

   3. difference-of-squares 54.3%

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

3. Applied egg-rr 54.3%

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

4. Taylor expanded in angle around 0 52.5%

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

5. Taylor expanded in angle around 0 53.3%

   \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)} \cdot 1 \]

6. Final simplification 53.3%

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

Alternative 14: 54.4% accurate, 5.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ angle_s \cdot \left(0.011111111111111112 \cdot \left(\left(\pi \cdot angle_m\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right) \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (* angle_s (* 0.011111111111111112 (* (* PI angle_m) (* (+ b a) (- 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 * (0.011111111111111112 * ((((double) M_PI) * angle_m) * ((b + a) * (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 * (0.011111111111111112 * ((Math.PI * angle_m) * ((b + a) * (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 * (0.011111111111111112 * ((math.pi * angle_m) * ((b + a) * (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(0.011111111111111112 * Float64(Float64(pi * angle_m) * Float64(Float64(b + a) * 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 * (0.011111111111111112 * ((pi * angle_m) * ((b + a) * (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[(0.011111111111111112 * N[(N[(Pi * angle$95$m), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.8%

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

   1. unpow2 50.8%

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

   2. unpow2 50.8%

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

   3. difference-of-squares 54.3%

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

3. Applied egg-rr 54.3%

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

4. Taylor expanded in angle around 0 52.5%

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

5. Taylor expanded in angle around 0 53.3%

   \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)} \cdot 1 \]
6. Step-by-step derivation

   1. associate-*r* 53.3%

      \[\leadsto \left(0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)}\right) \cdot 1 \]

7. Simplified 53.3%

   \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \cdot 1 \]

8. Final simplification 53.3%

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

Reproduce

herbie shell --seed 2023318 
(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)))))