ab-angle->ABCF B

Percentage Accurate: 54.8% → 67.3%

Time: 34.4s

Alternatives: 21

Speedup: 32.2×

• 33.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 54.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 67.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(angle_m \cdot \left(b_m - a_m\right)\right)\\ t_1 := \pi \cdot \left(angle_m \cdot 0.005555555555555556\right)\\ t_2 := \left(b_m - a_m\right) \cdot \left(b_m + a_m\right)\\ t_3 := t_2 \cdot \cos \left(angle_m \cdot \frac{\pi}{-180}\right)\\ angle_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle_m}{180} \leq 4 \cdot 10^{-30}:\\ \;\;\;\;\left(b_m \cdot t_0 + a_m \cdot t_0\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 5 \cdot 10^{+60}:\\ \;\;\;\;2 \cdot \left(t_3 \cdot \sqrt[3]{{\sin t_1}^{3}}\right)\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 2 \cdot 10^{+134}:\\ \;\;\;\;2 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\frac{angle_m \cdot \pi}{-180}\right)\right)\right) \cdot t_3\right)\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 5 \cdot 10^{+193}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \left(angle_m \cdot -0.005555555555555556\right)\right) \cdot \left(t_2 \cdot \sin \left(0.005555555555555556 \cdot \left(angle_m \cdot \pi\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 5 \cdot 10^{+289}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{angle_m}{180} \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right) \cdot t_3\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{angle_m}{180} \cdot \pi\right) \cdot \left(t_2 \cdot \left|\cos t_1\right|\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m (- b_m a_m))))
        (t_1 (* PI (* angle_m 0.005555555555555556)))
        (t_2 (* (- b_m a_m) (+ b_m a_m)))
        (t_3 (* t_2 (cos (* angle_m (/ PI -180.0))))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 4e-30)
      (* (+ (* b_m t_0) (* a_m t_0)) 0.011111111111111112)
      (if (<= (/ angle_m 180.0) 5e+60)
        (* 2.0 (* t_3 (cbrt (pow (sin t_1) 3.0))))
        (if (<= (/ angle_m 180.0) 2e+134)
          (* 2.0 (* (log1p (expm1 (sin (/ (* angle_m PI) -180.0)))) t_3))
          (if (<= (/ angle_m 180.0) 5e+193)
            (*
             2.0
             (*
              (cos (* PI (* angle_m -0.005555555555555556)))
              (* t_2 (sin (* 0.005555555555555556 (* angle_m PI))))))
            (if (<= (/ angle_m 180.0) 5e+289)
              (*
               2.0
               (*
                (sin (* (/ angle_m 180.0) (* (cbrt PI) (pow (cbrt PI) 2.0))))
                t_3))
              (*
               2.0
               (*
                (sin (* (/ angle_m 180.0) PI))
                (* t_2 (fabs (cos t_1)))))))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m * (b_m - a_m));
	double t_1 = ((double) M_PI) * (angle_m * 0.005555555555555556);
	double t_2 = (b_m - a_m) * (b_m + a_m);
	double t_3 = t_2 * cos((angle_m * (((double) M_PI) / -180.0)));
	double tmp;
	if ((angle_m / 180.0) <= 4e-30) {
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112;
	} else if ((angle_m / 180.0) <= 5e+60) {
		tmp = 2.0 * (t_3 * cbrt(pow(sin(t_1), 3.0)));
	} else if ((angle_m / 180.0) <= 2e+134) {
		tmp = 2.0 * (log1p(expm1(sin(((angle_m * ((double) M_PI)) / -180.0)))) * t_3);
	} else if ((angle_m / 180.0) <= 5e+193) {
		tmp = 2.0 * (cos((((double) M_PI) * (angle_m * -0.005555555555555556))) * (t_2 * sin((0.005555555555555556 * (angle_m * ((double) M_PI))))));
	} else if ((angle_m / 180.0) <= 5e+289) {
		tmp = 2.0 * (sin(((angle_m / 180.0) * (cbrt(((double) M_PI)) * pow(cbrt(((double) M_PI)), 2.0)))) * t_3);
	} else {
		tmp = 2.0 * (sin(((angle_m / 180.0) * ((double) M_PI))) * (t_2 * fabs(cos(t_1))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = Math.PI * (angle_m * (b_m - a_m));
	double t_1 = Math.PI * (angle_m * 0.005555555555555556);
	double t_2 = (b_m - a_m) * (b_m + a_m);
	double t_3 = t_2 * Math.cos((angle_m * (Math.PI / -180.0)));
	double tmp;
	if ((angle_m / 180.0) <= 4e-30) {
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112;
	} else if ((angle_m / 180.0) <= 5e+60) {
		tmp = 2.0 * (t_3 * Math.cbrt(Math.pow(Math.sin(t_1), 3.0)));
	} else if ((angle_m / 180.0) <= 2e+134) {
		tmp = 2.0 * (Math.log1p(Math.expm1(Math.sin(((angle_m * Math.PI) / -180.0)))) * t_3);
	} else if ((angle_m / 180.0) <= 5e+193) {
		tmp = 2.0 * (Math.cos((Math.PI * (angle_m * -0.005555555555555556))) * (t_2 * Math.sin((0.005555555555555556 * (angle_m * Math.PI)))));
	} else if ((angle_m / 180.0) <= 5e+289) {
		tmp = 2.0 * (Math.sin(((angle_m / 180.0) * (Math.cbrt(Math.PI) * Math.pow(Math.cbrt(Math.PI), 2.0)))) * t_3);
	} else {
		tmp = 2.0 * (Math.sin(((angle_m / 180.0) * Math.PI)) * (t_2 * Math.abs(Math.cos(t_1))));
	}
	return angle_s * tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(pi * Float64(angle_m * Float64(b_m - a_m)))
	t_1 = Float64(pi * Float64(angle_m * 0.005555555555555556))
	t_2 = Float64(Float64(b_m - a_m) * Float64(b_m + a_m))
	t_3 = Float64(t_2 * cos(Float64(angle_m * Float64(pi / -180.0))))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e-30)
		tmp = Float64(Float64(Float64(b_m * t_0) + Float64(a_m * t_0)) * 0.011111111111111112);
	elseif (Float64(angle_m / 180.0) <= 5e+60)
		tmp = Float64(2.0 * Float64(t_3 * cbrt((sin(t_1) ^ 3.0))));
	elseif (Float64(angle_m / 180.0) <= 2e+134)
		tmp = Float64(2.0 * Float64(log1p(expm1(sin(Float64(Float64(angle_m * pi) / -180.0)))) * t_3));
	elseif (Float64(angle_m / 180.0) <= 5e+193)
		tmp = Float64(2.0 * Float64(cos(Float64(pi * Float64(angle_m * -0.005555555555555556))) * Float64(t_2 * sin(Float64(0.005555555555555556 * Float64(angle_m * pi))))));
	elseif (Float64(angle_m / 180.0) <= 5e+289)
		tmp = Float64(2.0 * Float64(sin(Float64(Float64(angle_m / 180.0) * Float64(cbrt(pi) * (cbrt(pi) ^ 2.0)))) * t_3));
	else
		tmp = Float64(2.0 * Float64(sin(Float64(Float64(angle_m / 180.0) * pi)) * Float64(t_2 * abs(cos(t_1)))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(b$95$m + a$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Cos[N[(angle$95$m * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e-30], N[(N[(N[(b$95$m * t$95$0), $MachinePrecision] + N[(a$95$m * t$95$0), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+60], N[(2.0 * N[(t$95$3 * N[Power[N[Power[N[Sin[t$95$1], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+134], N[(2.0 * N[(N[Log[1 + N[(Exp[N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] / -180.0), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+193], N[(2.0 * N[(N[Cos[N[(Pi * N[(angle$95$m * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$2 * N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+289], N[(2.0 * N[(N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[(N[Power[Pi, 1/3], $MachinePrecision] * N[Power[N[Power[Pi, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(t$95$2 * N[Abs[N[Cos[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 angle 180) < 4e-30

   1. Initial program 62.5%

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

      1. associate-*l* 62.5%

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

   3. Simplified 62.5%

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

   4. Taylor expanded in angle around 0 59.4%

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

      1. unpow2 59.4%

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

      2. unpow2 59.4%

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

      3. difference-of-squares 63.8%

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

   6. Applied egg-rr 63.8%

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

   7. Taylor expanded in angle around 0 63.8%

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

      1. *-commutative 63.8%

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

      2. associate-*r* 63.9%

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

      3. *-commutative 63.9%

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

      4. associate-*l* 63.8%

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

      5. +-commutative 63.8%

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

      6. difference-of-squares 59.4%

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

      7. unpow2 59.4%

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

      8. unpow2 59.4%

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

      9. associate-*l* 59.4%

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

      10. *-commutative 59.4%

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

      11. unpow2 59.4%

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

      12. unpow2 59.4%

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

      13. difference-of-squares 63.9%

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

      14. *-commutative 63.9%

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

      15. +-commutative 63.9%

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

   9. Simplified 63.9%

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

       1. associate-*r* 75.0%

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

       2. +-commutative 75.0%

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

       3. distribute-lft-in 71.0%

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

       4. *-commutative 71.0%

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

       5. associate-*l* 71.0%

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

       6. *-commutative 71.0%

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

       7. associate-*l* 71.1%

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

   11. Applied egg-rr 71.1%

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

   if 4e-30 < (/.f64 angle 180) < 4.99999999999999975e60

   1. Initial program 70.6%

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

   3. Simplified 70.5%

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

      1. unpow2 48.1%

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

      2. unpow2 48.1%

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

      3. difference-of-squares 52.3%

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

   5. Applied egg-rr 74.7%

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

      1. add-sqr-sqrt 74.6%

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

      2. sqrt-unprod 74.7%

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

      3. associate-*r/ 78.9%

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

      4. associate-*r/ 78.1%

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

      5. frac-times 78.0%

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

      6. metadata-eval 78.0%

         \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      7. metadata-eval 78.0%

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

      8. frac-times 78.1%

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

      9. associate-*l/ 78.7%

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

      10. associate-*l/ 70.3%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 22.8%

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

      13. add-cbrt-cube 22.8%

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

      14. pow3 22.8%

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

   7. Applied egg-rr 75.3%

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

   if 4.99999999999999975e60 < (/.f64 angle 180) < 1.99999999999999984e134

   1. Initial program 19.3%

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

   3. Simplified 20.6%

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

      1. unpow2 31.3%

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

      2. unpow2 31.3%

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

      3. difference-of-squares 31.3%

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

   5. Applied egg-rr 20.6%

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

      1. add-sqr-sqrt 28.5%

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

      2. sqrt-unprod 20.6%

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

      3. associate-*r/ 20.7%

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

      4. associate-*r/ 22.1%

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

      5. frac-times 21.0%

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

      6. metadata-eval 21.0%

         \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      7. metadata-eval 21.0%

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

      8. frac-times 22.1%

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

      9. associate-*l/ 20.0%

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

      10. associate-*l/ 20.0%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 41.4%

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

      13. log1p-expm1-u 41.4%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 20.0%

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

   7. Applied egg-rr 20.6%

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

      1. metadata-eval 20.6%

         \[\leadsto 2 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      2. div-inv 20.6%

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

      3. add-sqr-sqrt 28.5%

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

      4. sqrt-unprod 20.6%

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

      5. associate-*r/ 20.7%

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

      6. associate-*r/ 22.1%

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

      7. frac-times 21.0%

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

      8. metadata-eval 21.0%

         \[\leadsto 2 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      9. metadata-eval 21.0%

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

      10. frac-times 22.1%

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

      11. associate-*l/ 20.0%

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

      12. associate-*l/ 20.0%

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

      13. sqrt-unprod 0.0%

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

      14. add-sqr-sqrt 41.4%

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

      15. associate-*l/ 39.3%

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

   9. Applied egg-rr 39.3%

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

   if 1.99999999999999984e134 < (/.f64 angle 180) < 4.99999999999999972e193

   1. Initial program 39.3%

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

   3. Simplified 44.8%

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

      1. unpow2 37.3%

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

      2. unpow2 37.3%

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

      3. difference-of-squares 37.3%

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

   5. Applied egg-rr 44.8%

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

   6. Taylor expanded in angle around inf 46.4%

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

      1. associate-*r* 47.7%

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

      2. associate-*r* 46.4%

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

      3. *-commutative 46.4%

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

      4. *-commutative 46.4%

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

      5. *-commutative 46.4%

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

      6. *-commutative 46.4%

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

      7. *-commutative 46.4%

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

      8. associate-*r* 47.7%

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

   8. Simplified 47.7%

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

   if 4.99999999999999972e193 < (/.f64 angle 180) < 5.00000000000000031e289

   1. Initial program 24.7%

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

   3. Simplified 30.5%

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

      1. unpow2 25.5%

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

      2. unpow2 25.5%

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

      3. difference-of-squares 25.5%

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

   5. Applied egg-rr 30.5%

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

      1. add-cube-cbrt 53.4%

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

      2. pow2 53.4%

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

   7. Applied egg-rr 53.4%

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

   if 5.00000000000000031e289 < (/.f64 angle 180)

   1. Initial program 22.4%

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

   3. Simplified 5.6%

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

      1. unpow2 3.1%

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

      2. unpow2 3.1%

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

      3. difference-of-squares 17.4%

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

   5. Applied egg-rr 5.6%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 0.0%

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

      3. associate-*l/ 0.0%

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

      4. associate-*l/ 0.0%

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

      5. frac-times 0.0%

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

      6. metadata-eval 0.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)\right)\right) \]

      7. metadata-eval 0.0%

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

      8. frac-times 0.0%

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

      9. associate-*r/ 0.0%

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

      10. associate-*r/ 0.0%

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

      11. sqrt-unprod 48.6%

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

      12. add-sqr-sqrt 22.4%

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

      13. add-sqr-sqrt 19.7%

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

      14. sqrt-unprod 37.2%

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

      15. pow2 37.2%

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

   7. Applied egg-rr 37.2%

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

      1. unpow2 37.2%

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

      2. rem-sqrt-square 37.2%

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

   9. Simplified 37.2%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left|\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right|}\right)\right) \]
3. Recombined 6 regimes into one program.

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{-30}:\\ \;\;\;\;\left(b \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) + a \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+60}:\\ \;\;\;\;2 \cdot \left(\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \sqrt[3]{{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{3}}\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+134}:\\ \;\;\;\;2 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\frac{angle \cdot \pi}{-180}\right)\right)\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+193}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+289}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{angle}{180} \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left|\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right|\right)\right)\\ \end{array} \]

Alternative 2: 67.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(b_m - a_m\right) \cdot \left(b_m + a_m\right)\\ t_1 := t_0 \cdot \cos \left(angle_m \cdot \frac{\pi}{-180}\right)\\ t_2 := \pi \cdot \left(angle_m \cdot \left(b_m - a_m\right)\right)\\ t_3 := \sin \left(\frac{angle_m}{180} \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right)\\ angle_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle_m}{180} \leq 10^{-14}:\\ \;\;\;\;\left(b_m \cdot t_2 + a_m \cdot t_2\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 10^{+55}:\\ \;\;\;\;\left(2 \cdot t_0\right) \cdot \left(t_3 \cdot \cos \left(\frac{angle_m}{180} \cdot \pi\right)\right)\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 2 \cdot 10^{+134}:\\ \;\;\;\;2 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\frac{angle_m \cdot \pi}{-180}\right)\right)\right) \cdot t_1\right)\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 2 \cdot 10^{+226}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left({\left(\sqrt{\pi \cdot \left(angle_m \cdot 0.005555555555555556\right)}\right)}^{2}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_3 \cdot \left(t_0 \cdot {\left(\sqrt[3]{\cos \left(\left(angle_m \cdot 0.005555555555555556\right) \cdot \sqrt[3]{{\pi}^{3}}\right)}\right)}^{3}\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* (- b_m a_m) (+ b_m a_m)))
        (t_1 (* t_0 (cos (* angle_m (/ PI -180.0)))))
        (t_2 (* PI (* angle_m (- b_m a_m))))
        (t_3 (sin (* (/ angle_m 180.0) (* (cbrt PI) (pow (cbrt PI) 2.0))))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 1e-14)
      (* (+ (* b_m t_2) (* a_m t_2)) 0.011111111111111112)
      (if (<= (/ angle_m 180.0) 1e+55)
        (* (* 2.0 t_0) (* t_3 (cos (* (/ angle_m 180.0) PI))))
        (if (<= (/ angle_m 180.0) 2e+134)
          (* 2.0 (* (log1p (expm1 (sin (/ (* angle_m PI) -180.0)))) t_1))
          (if (<= (/ angle_m 180.0) 2e+226)
            (*
             2.0
             (*
              t_1
              (log1p
               (expm1
                (sin
                 (pow (sqrt (* PI (* angle_m 0.005555555555555556))) 2.0))))))
            (*
             2.0
             (*
              t_3
              (*
               t_0
               (pow
                (cbrt
                 (cos
                  (* (* angle_m 0.005555555555555556) (cbrt (pow PI 3.0)))))
                3.0)))))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = (b_m - a_m) * (b_m + a_m);
	double t_1 = t_0 * cos((angle_m * (((double) M_PI) / -180.0)));
	double t_2 = ((double) M_PI) * (angle_m * (b_m - a_m));
	double t_3 = sin(((angle_m / 180.0) * (cbrt(((double) M_PI)) * pow(cbrt(((double) M_PI)), 2.0))));
	double tmp;
	if ((angle_m / 180.0) <= 1e-14) {
		tmp = ((b_m * t_2) + (a_m * t_2)) * 0.011111111111111112;
	} else if ((angle_m / 180.0) <= 1e+55) {
		tmp = (2.0 * t_0) * (t_3 * cos(((angle_m / 180.0) * ((double) M_PI))));
	} else if ((angle_m / 180.0) <= 2e+134) {
		tmp = 2.0 * (log1p(expm1(sin(((angle_m * ((double) M_PI)) / -180.0)))) * t_1);
	} else if ((angle_m / 180.0) <= 2e+226) {
		tmp = 2.0 * (t_1 * log1p(expm1(sin(pow(sqrt((((double) M_PI) * (angle_m * 0.005555555555555556))), 2.0)))));
	} else {
		tmp = 2.0 * (t_3 * (t_0 * pow(cbrt(cos(((angle_m * 0.005555555555555556) * cbrt(pow(((double) M_PI), 3.0))))), 3.0)));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = (b_m - a_m) * (b_m + a_m);
	double t_1 = t_0 * Math.cos((angle_m * (Math.PI / -180.0)));
	double t_2 = Math.PI * (angle_m * (b_m - a_m));
	double t_3 = Math.sin(((angle_m / 180.0) * (Math.cbrt(Math.PI) * Math.pow(Math.cbrt(Math.PI), 2.0))));
	double tmp;
	if ((angle_m / 180.0) <= 1e-14) {
		tmp = ((b_m * t_2) + (a_m * t_2)) * 0.011111111111111112;
	} else if ((angle_m / 180.0) <= 1e+55) {
		tmp = (2.0 * t_0) * (t_3 * Math.cos(((angle_m / 180.0) * Math.PI)));
	} else if ((angle_m / 180.0) <= 2e+134) {
		tmp = 2.0 * (Math.log1p(Math.expm1(Math.sin(((angle_m * Math.PI) / -180.0)))) * t_1);
	} else if ((angle_m / 180.0) <= 2e+226) {
		tmp = 2.0 * (t_1 * Math.log1p(Math.expm1(Math.sin(Math.pow(Math.sqrt((Math.PI * (angle_m * 0.005555555555555556))), 2.0)))));
	} else {
		tmp = 2.0 * (t_3 * (t_0 * Math.pow(Math.cbrt(Math.cos(((angle_m * 0.005555555555555556) * Math.cbrt(Math.pow(Math.PI, 3.0))))), 3.0)));
	}
	return angle_s * tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(Float64(b_m - a_m) * Float64(b_m + a_m))
	t_1 = Float64(t_0 * cos(Float64(angle_m * Float64(pi / -180.0))))
	t_2 = Float64(pi * Float64(angle_m * Float64(b_m - a_m)))
	t_3 = sin(Float64(Float64(angle_m / 180.0) * Float64(cbrt(pi) * (cbrt(pi) ^ 2.0))))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e-14)
		tmp = Float64(Float64(Float64(b_m * t_2) + Float64(a_m * t_2)) * 0.011111111111111112);
	elseif (Float64(angle_m / 180.0) <= 1e+55)
		tmp = Float64(Float64(2.0 * t_0) * Float64(t_3 * cos(Float64(Float64(angle_m / 180.0) * pi))));
	elseif (Float64(angle_m / 180.0) <= 2e+134)
		tmp = Float64(2.0 * Float64(log1p(expm1(sin(Float64(Float64(angle_m * pi) / -180.0)))) * t_1));
	elseif (Float64(angle_m / 180.0) <= 2e+226)
		tmp = Float64(2.0 * Float64(t_1 * log1p(expm1(sin((sqrt(Float64(pi * Float64(angle_m * 0.005555555555555556))) ^ 2.0))))));
	else
		tmp = Float64(2.0 * Float64(t_3 * Float64(t_0 * (cbrt(cos(Float64(Float64(angle_m * 0.005555555555555556) * cbrt((pi ^ 3.0))))) ^ 3.0))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(b$95$m + a$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Cos[N[(angle$95$m * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(Pi * N[(angle$95$m * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[(N[Power[Pi, 1/3], $MachinePrecision] * N[Power[N[Power[Pi, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e-14], N[(N[(N[(b$95$m * t$95$2), $MachinePrecision] + N[(a$95$m * t$95$2), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+55], N[(N[(2.0 * t$95$0), $MachinePrecision] * N[(t$95$3 * N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+134], N[(2.0 * N[(N[Log[1 + N[(Exp[N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] / -180.0), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+226], N[(2.0 * N[(t$95$1 * N[Log[1 + N[(Exp[N[Sin[N[Power[N[Sqrt[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$3 * N[(t$95$0 * N[Power[N[Power[N[Cos[N[(N[(angle$95$m * 0.005555555555555556), $MachinePrecision] * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 63.3%

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

      1. associate-*l* 63.3%

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

   3. Simplified 63.3%

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

   4. Taylor expanded in angle around 0 60.2%

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

      1. unpow2 60.2%

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

      2. unpow2 60.2%

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

      3. difference-of-squares 64.6%

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

   6. Applied egg-rr 64.6%

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

   7. Taylor expanded in angle around 0 64.6%

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

      1. *-commutative 64.6%

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

      2. associate-*r* 64.6%

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

      3. *-commutative 64.6%

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

      4. associate-*l* 64.6%

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

      5. +-commutative 64.6%

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

      6. difference-of-squares 60.2%

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

      7. unpow2 60.2%

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

      8. unpow2 60.2%

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

      9. associate-*l* 60.3%

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

      10. *-commutative 60.3%

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

      11. unpow2 60.3%

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

      12. unpow2 60.3%

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

      13. difference-of-squares 64.6%

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

      14. *-commutative 64.6%

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

      15. +-commutative 64.6%

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

   9. Simplified 64.6%

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

       1. associate-*r* 75.5%

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

       2. +-commutative 75.5%

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

       3. distribute-lft-in 71.6%

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

       4. *-commutative 71.6%

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

       5. associate-*l* 71.6%

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

       6. *-commutative 71.6%

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

       7. associate-*l* 71.7%

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

   11. Applied egg-rr 71.7%

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

   if 9.99999999999999999e-15 < (/.f64 angle 180) < 1.00000000000000001e55

   1. Initial program 65.4%

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

      1. associate-*l* 65.4%

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

   3. Simplified 65.4%

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

      1. unpow2 36.4%

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

      2. unpow2 36.4%

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

      3. difference-of-squares 42.0%

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

   5. Applied egg-rr 71.0%

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

      1. add-cube-cbrt 77.3%

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

      2. pow2 77.3%

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

   7. Applied egg-rr 77.4%

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

   if 1.00000000000000001e55 < (/.f64 angle 180) < 1.99999999999999984e134

   1. Initial program 24.3%

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

   3. Simplified 25.4%

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

      1. unpow2 33.7%

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

      2. unpow2 33.7%

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

      3. difference-of-squares 33.7%

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

   5. Applied egg-rr 25.4%

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

      1. add-sqr-sqrt 31.2%

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

      2. sqrt-unprod 25.4%

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

      3. associate-*r/ 25.5%

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

      4. associate-*r/ 25.6%

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

      5. frac-times 24.7%

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

      6. metadata-eval 24.7%

         \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      7. metadata-eval 24.7%

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

      8. frac-times 25.6%

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

      9. associate-*l/ 24.8%

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

      10. associate-*l/ 24.8%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 42.5%

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

      13. log1p-expm1-u 42.5%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 24.8%

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

   7. Applied egg-rr 25.4%

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

      1. metadata-eval 25.4%

         \[\leadsto 2 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      2. div-inv 25.4%

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

      3. add-sqr-sqrt 31.2%

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

      4. sqrt-unprod 25.4%

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

      5. associate-*r/ 25.5%

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

      6. associate-*r/ 25.6%

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

      7. frac-times 24.7%

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

      8. metadata-eval 24.7%

         \[\leadsto 2 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      9. metadata-eval 24.7%

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

      10. frac-times 25.6%

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

      11. associate-*l/ 24.8%

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

      12. associate-*l/ 24.8%

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

      13. sqrt-unprod 0.0%

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

      14. add-sqr-sqrt 42.5%

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

      15. associate-*l/ 41.6%

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

   9. Applied egg-rr 41.6%

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

   if 1.99999999999999984e134 < (/.f64 angle 180) < 1.99999999999999992e226

   1. Initial program 36.1%

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

   3. Simplified 40.4%

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

      1. unpow2 34.6%

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

      2. unpow2 34.6%

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

      3. difference-of-squares 34.6%

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

   5. Applied egg-rr 40.4%

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

      1. add-sqr-sqrt 49.6%

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

      2. sqrt-unprod 11.2%

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

      3. associate-*r/ 11.2%

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

      4. associate-*r/ 11.2%

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

      5. frac-times 11.2%

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

      6. metadata-eval 11.2%

         \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      7. metadata-eval 11.2%

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

      8. frac-times 11.2%

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

      9. associate-*l/ 11.2%

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

      10. associate-*l/ 11.2%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 28.0%

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

      13. log1p-expm1-u 28.0%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 11.2%

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

   7. Applied egg-rr 40.4%

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

      1. metadata-eval 40.4%

         \[\leadsto 2 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      2. div-inv 40.4%

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

      3. add-sqr-sqrt 49.6%

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

      4. sqrt-unprod 11.2%

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

      5. associate-*r/ 11.2%

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

      6. associate-*r/ 11.2%

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

      7. frac-times 11.2%

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

      8. metadata-eval 11.2%

         \[\leadsto 2 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      9. metadata-eval 11.2%

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

      10. frac-times 11.2%

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

      11. associate-*l/ 11.2%

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

      12. associate-*l/ 11.2%

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

      13. sqrt-unprod 0.0%

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

      14. pow2 0.0%

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

   9. Applied egg-rr 55.2%

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

   if 1.99999999999999992e226 < (/.f64 angle 180)

   1. Initial program 23.8%

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

   3. Simplified 22.5%

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

      1. unpow2 17.4%

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

      2. unpow2 17.4%

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

      3. difference-of-squares 22.6%

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

   5. Applied egg-rr 22.5%

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

      1. add-cube-cbrt 35.7%

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

      2. pow2 35.7%

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

   7. Applied egg-rr 35.7%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 0.0%

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

      3. associate-*l/ 0.0%

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

      4. associate-*l/ 0.0%

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

      5. frac-times 0.0%

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

      6. metadata-eval 0.0%

         \[\leadsto 2 \cdot \left(\sin \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)\right)\right) \]

      7. metadata-eval 0.0%

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

      8. frac-times 0.0%

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

      9. associate-*r/ 0.0%

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

      10. associate-*r/ 0.0%

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

      11. sqrt-unprod 54.3%

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

      12. add-sqr-sqrt 36.1%

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

      13. add-cube-cbrt 36.1%

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

      14. pow3 36.1%

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

   9. Applied egg-rr 30.8%

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

       1. add-cbrt-cube 49.1%

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

       2. pow3 49.1%

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

   11. Applied egg-rr 49.1%

       \[\leadsto 2 \cdot \left(\sin \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot {\left(\sqrt[3]{\cos \left(\color{blue}{\sqrt[3]{{\pi}^{3}}} \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)}^{3}\right)\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{-14}:\\ \;\;\;\;\left(b \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) + a \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+55}:\\ \;\;\;\;\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+134}:\\ \;\;\;\;2 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\frac{angle \cdot \pi}{-180}\right)\right)\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+226}:\\ \;\;\;\;2 \cdot \left(\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left({\left(\sqrt{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{2}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{angle}{180} \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot {\left(\sqrt[3]{\cos \left(\left(angle \cdot 0.005555555555555556\right) \cdot \sqrt[3]{{\pi}^{3}}\right)}\right)}^{3}\right)\right)\\ \end{array} \]

Alternative 3: 67.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(angle_m \cdot \left(b_m - a_m\right)\right)\\ t_1 := \left(b_m - a_m\right) \cdot \left(b_m + a_m\right)\\ t_2 := t_1 \cdot \cos \left(angle_m \cdot \frac{\pi}{-180}\right)\\ t_3 := \left(2 \cdot t_1\right) \cdot \left(\sin \left(\frac{angle_m}{180} \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right) \cdot \cos \left(\frac{angle_m}{180} \cdot \pi\right)\right)\\ angle_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle_m}{180} \leq 10^{-14}:\\ \;\;\;\;\left(b_m \cdot t_0 + a_m \cdot t_0\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 10^{+55}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 2 \cdot 10^{+134}:\\ \;\;\;\;2 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\frac{angle_m \cdot \pi}{-180}\right)\right)\right) \cdot t_2\right)\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 4 \cdot 10^{+190}:\\ \;\;\;\;2 \cdot \left(t_2 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left({\left(\sqrt{\pi \cdot \left(angle_m \cdot 0.005555555555555556\right)}\right)}^{2}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m (- b_m a_m))))
        (t_1 (* (- b_m a_m) (+ b_m a_m)))
        (t_2 (* t_1 (cos (* angle_m (/ PI -180.0)))))
        (t_3
         (*
          (* 2.0 t_1)
          (*
           (sin (* (/ angle_m 180.0) (* (cbrt PI) (pow (cbrt PI) 2.0))))
           (cos (* (/ angle_m 180.0) PI))))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 1e-14)
      (* (+ (* b_m t_0) (* a_m t_0)) 0.011111111111111112)
      (if (<= (/ angle_m 180.0) 1e+55)
        t_3
        (if (<= (/ angle_m 180.0) 2e+134)
          (* 2.0 (* (log1p (expm1 (sin (/ (* angle_m PI) -180.0)))) t_2))
          (if (<= (/ angle_m 180.0) 4e+190)
            (*
             2.0
             (*
              t_2
              (log1p
               (expm1
                (sin
                 (pow (sqrt (* PI (* angle_m 0.005555555555555556))) 2.0))))))
            t_3)))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m * (b_m - a_m));
	double t_1 = (b_m - a_m) * (b_m + a_m);
	double t_2 = t_1 * cos((angle_m * (((double) M_PI) / -180.0)));
	double t_3 = (2.0 * t_1) * (sin(((angle_m / 180.0) * (cbrt(((double) M_PI)) * pow(cbrt(((double) M_PI)), 2.0)))) * cos(((angle_m / 180.0) * ((double) M_PI))));
	double tmp;
	if ((angle_m / 180.0) <= 1e-14) {
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112;
	} else if ((angle_m / 180.0) <= 1e+55) {
		tmp = t_3;
	} else if ((angle_m / 180.0) <= 2e+134) {
		tmp = 2.0 * (log1p(expm1(sin(((angle_m * ((double) M_PI)) / -180.0)))) * t_2);
	} else if ((angle_m / 180.0) <= 4e+190) {
		tmp = 2.0 * (t_2 * log1p(expm1(sin(pow(sqrt((((double) M_PI) * (angle_m * 0.005555555555555556))), 2.0)))));
	} else {
		tmp = t_3;
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = Math.PI * (angle_m * (b_m - a_m));
	double t_1 = (b_m - a_m) * (b_m + a_m);
	double t_2 = t_1 * Math.cos((angle_m * (Math.PI / -180.0)));
	double t_3 = (2.0 * t_1) * (Math.sin(((angle_m / 180.0) * (Math.cbrt(Math.PI) * Math.pow(Math.cbrt(Math.PI), 2.0)))) * Math.cos(((angle_m / 180.0) * Math.PI)));
	double tmp;
	if ((angle_m / 180.0) <= 1e-14) {
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112;
	} else if ((angle_m / 180.0) <= 1e+55) {
		tmp = t_3;
	} else if ((angle_m / 180.0) <= 2e+134) {
		tmp = 2.0 * (Math.log1p(Math.expm1(Math.sin(((angle_m * Math.PI) / -180.0)))) * t_2);
	} else if ((angle_m / 180.0) <= 4e+190) {
		tmp = 2.0 * (t_2 * Math.log1p(Math.expm1(Math.sin(Math.pow(Math.sqrt((Math.PI * (angle_m * 0.005555555555555556))), 2.0)))));
	} else {
		tmp = t_3;
	}
	return angle_s * tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(pi * Float64(angle_m * Float64(b_m - a_m)))
	t_1 = Float64(Float64(b_m - a_m) * Float64(b_m + a_m))
	t_2 = Float64(t_1 * cos(Float64(angle_m * Float64(pi / -180.0))))
	t_3 = Float64(Float64(2.0 * t_1) * Float64(sin(Float64(Float64(angle_m / 180.0) * Float64(cbrt(pi) * (cbrt(pi) ^ 2.0)))) * cos(Float64(Float64(angle_m / 180.0) * pi))))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e-14)
		tmp = Float64(Float64(Float64(b_m * t_0) + Float64(a_m * t_0)) * 0.011111111111111112);
	elseif (Float64(angle_m / 180.0) <= 1e+55)
		tmp = t_3;
	elseif (Float64(angle_m / 180.0) <= 2e+134)
		tmp = Float64(2.0 * Float64(log1p(expm1(sin(Float64(Float64(angle_m * pi) / -180.0)))) * t_2));
	elseif (Float64(angle_m / 180.0) <= 4e+190)
		tmp = Float64(2.0 * Float64(t_2 * log1p(expm1(sin((sqrt(Float64(pi * Float64(angle_m * 0.005555555555555556))) ^ 2.0))))));
	else
		tmp = t_3;
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(b$95$m + a$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Cos[N[(angle$95$m * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * t$95$1), $MachinePrecision] * N[(N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[(N[Power[Pi, 1/3], $MachinePrecision] * N[Power[N[Power[Pi, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e-14], N[(N[(N[(b$95$m * t$95$0), $MachinePrecision] + N[(a$95$m * t$95$0), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+55], t$95$3, If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+134], N[(2.0 * N[(N[Log[1 + N[(Exp[N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] / -180.0), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+190], N[(2.0 * N[(t$95$2 * N[Log[1 + N[(Exp[N[Sin[N[Power[N[Sqrt[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 63.3%

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

      1. associate-*l* 63.3%

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

   3. Simplified 63.3%

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

   4. Taylor expanded in angle around 0 60.2%

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

      1. unpow2 60.2%

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

      2. unpow2 60.2%

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

      3. difference-of-squares 64.6%

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

   6. Applied egg-rr 64.6%

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

   7. Taylor expanded in angle around 0 64.6%

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

      1. *-commutative 64.6%

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

      2. associate-*r* 64.6%

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

      3. *-commutative 64.6%

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

      4. associate-*l* 64.6%

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

      5. +-commutative 64.6%

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

      6. difference-of-squares 60.2%

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

      7. unpow2 60.2%

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

      8. unpow2 60.2%

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

      9. associate-*l* 60.3%

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

      10. *-commutative 60.3%

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

      11. unpow2 60.3%

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

      12. unpow2 60.3%

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

      13. difference-of-squares 64.6%

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

      14. *-commutative 64.6%

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

      15. +-commutative 64.6%

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

   9. Simplified 64.6%

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

       1. associate-*r* 75.5%

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

       2. +-commutative 75.5%

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

       3. distribute-lft-in 71.6%

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

       4. *-commutative 71.6%

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

       5. associate-*l* 71.6%

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

       6. *-commutative 71.6%

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

       7. associate-*l* 71.7%

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

   11. Applied egg-rr 71.7%

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

   if 9.99999999999999999e-15 < (/.f64 angle 180) < 1.00000000000000001e55 or 4.0000000000000003e190 < (/.f64 angle 180)

   1. Initial program 41.6%

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

      1. associate-*l* 41.6%

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

   3. Simplified 41.6%

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

      1. unpow2 26.0%

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

      2. unpow2 26.0%

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

      3. difference-of-squares 30.8%

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

   5. Applied egg-rr 44.0%

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

      1. add-cube-cbrt 54.0%

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

      2. pow2 54.0%

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

   7. Applied egg-rr 54.7%

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

   if 1.00000000000000001e55 < (/.f64 angle 180) < 1.99999999999999984e134

   1. Initial program 24.3%

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

   3. Simplified 25.4%

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

      1. unpow2 33.7%

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

      2. unpow2 33.7%

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

      3. difference-of-squares 33.7%

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

   5. Applied egg-rr 25.4%

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

      1. add-sqr-sqrt 31.2%

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

      2. sqrt-unprod 25.4%

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

      3. associate-*r/ 25.5%

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

      4. associate-*r/ 25.6%

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

      5. frac-times 24.7%

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

      6. metadata-eval 24.7%

         \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      7. metadata-eval 24.7%

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

      8. frac-times 25.6%

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

      9. associate-*l/ 24.8%

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

      10. associate-*l/ 24.8%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 42.5%

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

      13. log1p-expm1-u 42.5%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 24.8%

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

   7. Applied egg-rr 25.4%

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

      1. metadata-eval 25.4%

         \[\leadsto 2 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      2. div-inv 25.4%

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

      3. add-sqr-sqrt 31.2%

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

      4. sqrt-unprod 25.4%

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

      5. associate-*r/ 25.5%

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

      6. associate-*r/ 25.6%

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

      7. frac-times 24.7%

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

      8. metadata-eval 24.7%

         \[\leadsto 2 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      9. metadata-eval 24.7%

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

      10. frac-times 25.6%

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

      11. associate-*l/ 24.8%

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

      12. associate-*l/ 24.8%

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

      13. sqrt-unprod 0.0%

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

      14. add-sqr-sqrt 42.5%

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

      15. associate-*l/ 41.6%

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

   9. Applied egg-rr 41.6%

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

   if 1.99999999999999984e134 < (/.f64 angle 180) < 4.0000000000000003e190

   1. Initial program 40.7%

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

   3. Simplified 48.3%

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

      1. unpow2 39.6%

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

      2. unpow2 39.6%

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

      3. difference-of-squares 39.6%

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

   5. Applied egg-rr 48.3%

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

      1. add-sqr-sqrt 51.9%

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

      2. sqrt-unprod 15.5%

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

      3. associate-*r/ 15.5%

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

      4. associate-*r/ 15.5%

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

      5. frac-times 15.5%

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

      6. metadata-eval 15.5%

         \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      7. metadata-eval 15.5%

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

      8. frac-times 15.5%

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

      9. associate-*l/ 15.5%

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

      10. associate-*l/ 15.5%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 15.7%

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

      13. log1p-expm1-u 15.7%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 15.5%

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

   7. Applied egg-rr 48.3%

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

      1. metadata-eval 48.3%

         \[\leadsto 2 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      2. div-inv 48.3%

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

      3. add-sqr-sqrt 51.9%

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

      4. sqrt-unprod 15.5%

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

      5. associate-*r/ 15.5%

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

      6. associate-*r/ 15.5%

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

      7. frac-times 15.5%

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

      8. metadata-eval 15.5%

         \[\leadsto 2 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      9. metadata-eval 15.5%

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

      10. frac-times 15.5%

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

      11. associate-*l/ 15.5%

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

      12. associate-*l/ 15.5%

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

      13. sqrt-unprod 0.0%

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

      14. pow2 0.0%

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

   9. Applied egg-rr 51.9%

      \[\leadsto 2 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \color{blue}{\left({\left(\sqrt{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{2}\right)}\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{-14}:\\ \;\;\;\;\left(b \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) + a \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+55}:\\ \;\;\;\;\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+134}:\\ \;\;\;\;2 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\frac{angle \cdot \pi}{-180}\right)\right)\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 4 \cdot 10^{+190}:\\ \;\;\;\;2 \cdot \left(\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left({\left(\sqrt{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{2}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)\\ \end{array} \]

Alternative 4: 67.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(angle_m \cdot \left(b_m - a_m\right)\right)\\ t_1 := \left(b_m - a_m\right) \cdot \left(b_m + a_m\right)\\ t_2 := \left(2 \cdot t_1\right) \cdot \left(\sin \left(\frac{angle_m}{180} \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right) \cdot \cos \left(\frac{angle_m}{180} \cdot \pi\right)\right)\\ angle_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle_m}{180} \leq 10^{-14}:\\ \;\;\;\;\left(b_m \cdot t_0 + a_m \cdot t_0\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 10^{+55}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 2 \cdot 10^{+134}:\\ \;\;\;\;2 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\frac{angle_m \cdot \pi}{-180}\right)\right)\right) \cdot \left(t_1 \cdot \cos \left(angle_m \cdot \frac{\pi}{-180}\right)\right)\right)\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 5 \cdot 10^{+193}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \left(angle_m \cdot -0.005555555555555556\right)\right) \cdot \left(t_1 \cdot \sin \left(0.005555555555555556 \cdot \left(angle_m \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m (- b_m a_m))))
        (t_1 (* (- b_m a_m) (+ b_m a_m)))
        (t_2
         (*
          (* 2.0 t_1)
          (*
           (sin (* (/ angle_m 180.0) (* (cbrt PI) (pow (cbrt PI) 2.0))))
           (cos (* (/ angle_m 180.0) PI))))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 1e-14)
      (* (+ (* b_m t_0) (* a_m t_0)) 0.011111111111111112)
      (if (<= (/ angle_m 180.0) 1e+55)
        t_2
        (if (<= (/ angle_m 180.0) 2e+134)
          (*
           2.0
           (*
            (log1p (expm1 (sin (/ (* angle_m PI) -180.0))))
            (* t_1 (cos (* angle_m (/ PI -180.0))))))
          (if (<= (/ angle_m 180.0) 5e+193)
            (*
             2.0
             (*
              (cos (* PI (* angle_m -0.005555555555555556)))
              (* t_1 (sin (* 0.005555555555555556 (* angle_m PI))))))
            t_2)))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m * (b_m - a_m));
	double t_1 = (b_m - a_m) * (b_m + a_m);
	double t_2 = (2.0 * t_1) * (sin(((angle_m / 180.0) * (cbrt(((double) M_PI)) * pow(cbrt(((double) M_PI)), 2.0)))) * cos(((angle_m / 180.0) * ((double) M_PI))));
	double tmp;
	if ((angle_m / 180.0) <= 1e-14) {
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112;
	} else if ((angle_m / 180.0) <= 1e+55) {
		tmp = t_2;
	} else if ((angle_m / 180.0) <= 2e+134) {
		tmp = 2.0 * (log1p(expm1(sin(((angle_m * ((double) M_PI)) / -180.0)))) * (t_1 * cos((angle_m * (((double) M_PI) / -180.0)))));
	} else if ((angle_m / 180.0) <= 5e+193) {
		tmp = 2.0 * (cos((((double) M_PI) * (angle_m * -0.005555555555555556))) * (t_1 * sin((0.005555555555555556 * (angle_m * ((double) M_PI))))));
	} else {
		tmp = t_2;
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = Math.PI * (angle_m * (b_m - a_m));
	double t_1 = (b_m - a_m) * (b_m + a_m);
	double t_2 = (2.0 * t_1) * (Math.sin(((angle_m / 180.0) * (Math.cbrt(Math.PI) * Math.pow(Math.cbrt(Math.PI), 2.0)))) * Math.cos(((angle_m / 180.0) * Math.PI)));
	double tmp;
	if ((angle_m / 180.0) <= 1e-14) {
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112;
	} else if ((angle_m / 180.0) <= 1e+55) {
		tmp = t_2;
	} else if ((angle_m / 180.0) <= 2e+134) {
		tmp = 2.0 * (Math.log1p(Math.expm1(Math.sin(((angle_m * Math.PI) / -180.0)))) * (t_1 * Math.cos((angle_m * (Math.PI / -180.0)))));
	} else if ((angle_m / 180.0) <= 5e+193) {
		tmp = 2.0 * (Math.cos((Math.PI * (angle_m * -0.005555555555555556))) * (t_1 * Math.sin((0.005555555555555556 * (angle_m * Math.PI)))));
	} else {
		tmp = t_2;
	}
	return angle_s * tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(pi * Float64(angle_m * Float64(b_m - a_m)))
	t_1 = Float64(Float64(b_m - a_m) * Float64(b_m + a_m))
	t_2 = Float64(Float64(2.0 * t_1) * Float64(sin(Float64(Float64(angle_m / 180.0) * Float64(cbrt(pi) * (cbrt(pi) ^ 2.0)))) * cos(Float64(Float64(angle_m / 180.0) * pi))))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e-14)
		tmp = Float64(Float64(Float64(b_m * t_0) + Float64(a_m * t_0)) * 0.011111111111111112);
	elseif (Float64(angle_m / 180.0) <= 1e+55)
		tmp = t_2;
	elseif (Float64(angle_m / 180.0) <= 2e+134)
		tmp = Float64(2.0 * Float64(log1p(expm1(sin(Float64(Float64(angle_m * pi) / -180.0)))) * Float64(t_1 * cos(Float64(angle_m * Float64(pi / -180.0))))));
	elseif (Float64(angle_m / 180.0) <= 5e+193)
		tmp = Float64(2.0 * Float64(cos(Float64(pi * Float64(angle_m * -0.005555555555555556))) * Float64(t_1 * sin(Float64(0.005555555555555556 * Float64(angle_m * pi))))));
	else
		tmp = t_2;
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(b$95$m + a$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * t$95$1), $MachinePrecision] * N[(N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[(N[Power[Pi, 1/3], $MachinePrecision] * N[Power[N[Power[Pi, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e-14], N[(N[(N[(b$95$m * t$95$0), $MachinePrecision] + N[(a$95$m * t$95$0), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+55], t$95$2, If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+134], N[(2.0 * N[(N[Log[1 + N[(Exp[N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] / -180.0), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[Cos[N[(angle$95$m * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+193], N[(2.0 * N[(N[Cos[N[(Pi * N[(angle$95$m * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 63.3%

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

      1. associate-*l* 63.3%

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

   3. Simplified 63.3%

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

   4. Taylor expanded in angle around 0 60.2%

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

      1. unpow2 60.2%

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

      2. unpow2 60.2%

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

      3. difference-of-squares 64.6%

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

   6. Applied egg-rr 64.6%

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

   7. Taylor expanded in angle around 0 64.6%

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

      1. *-commutative 64.6%

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

      2. associate-*r* 64.6%

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

      3. *-commutative 64.6%

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

      4. associate-*l* 64.6%

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

      5. +-commutative 64.6%

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

      6. difference-of-squares 60.2%

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

      7. unpow2 60.2%

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

      8. unpow2 60.2%

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

      9. associate-*l* 60.3%

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

      10. *-commutative 60.3%

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

      11. unpow2 60.3%

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

      12. unpow2 60.3%

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

      13. difference-of-squares 64.6%

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

      14. *-commutative 64.6%

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

      15. +-commutative 64.6%

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

   9. Simplified 64.6%

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

       1. associate-*r* 75.5%

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

       2. +-commutative 75.5%

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

       3. distribute-lft-in 71.6%

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

       4. *-commutative 71.6%

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

       5. associate-*l* 71.6%

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

       6. *-commutative 71.6%

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

       7. associate-*l* 71.7%

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

   11. Applied egg-rr 71.7%

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

   if 9.99999999999999999e-15 < (/.f64 angle 180) < 1.00000000000000001e55 or 4.99999999999999972e193 < (/.f64 angle 180)

   1. Initial program 42.2%

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

      1. associate-*l* 42.2%

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

   3. Simplified 42.2%

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

      1. unpow2 26.5%

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

      2. unpow2 26.5%

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

      3. difference-of-squares 31.4%

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

   5. Applied egg-rr 44.6%

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

      1. add-cube-cbrt 55.4%

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

      2. pow2 55.4%

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

   7. Applied egg-rr 55.6%

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

   if 1.00000000000000001e55 < (/.f64 angle 180) < 1.99999999999999984e134

   1. Initial program 24.3%

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

   3. Simplified 25.4%

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

      1. unpow2 33.7%

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

      2. unpow2 33.7%

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

      3. difference-of-squares 33.7%

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

   5. Applied egg-rr 25.4%

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

      1. add-sqr-sqrt 31.2%

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

      2. sqrt-unprod 25.4%

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

      3. associate-*r/ 25.5%

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

      4. associate-*r/ 25.6%

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

      5. frac-times 24.7%

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

      6. metadata-eval 24.7%

         \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      7. metadata-eval 24.7%

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

      8. frac-times 25.6%

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

      9. associate-*l/ 24.8%

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

      10. associate-*l/ 24.8%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 42.5%

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

      13. log1p-expm1-u 42.5%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 24.8%

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

   7. Applied egg-rr 25.4%

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

      1. metadata-eval 25.4%

         \[\leadsto 2 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      2. div-inv 25.4%

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

      3. add-sqr-sqrt 31.2%

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

      4. sqrt-unprod 25.4%

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

      5. associate-*r/ 25.5%

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

      6. associate-*r/ 25.6%

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

      7. frac-times 24.7%

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

      8. metadata-eval 24.7%

         \[\leadsto 2 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      9. metadata-eval 24.7%

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

      10. frac-times 25.6%

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

      11. associate-*l/ 24.8%

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

      12. associate-*l/ 24.8%

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

      13. sqrt-unprod 0.0%

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

      14. add-sqr-sqrt 42.5%

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

      15. associate-*l/ 41.6%

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

   9. Applied egg-rr 41.6%

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

   if 1.99999999999999984e134 < (/.f64 angle 180) < 4.99999999999999972e193

   1. Initial program 39.3%

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

   3. Simplified 44.8%

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

      1. unpow2 37.3%

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

      2. unpow2 37.3%

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

      3. difference-of-squares 37.3%

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

   5. Applied egg-rr 44.8%

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

   6. Taylor expanded in angle around inf 46.4%

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

      1. associate-*r* 47.7%

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

      2. associate-*r* 46.4%

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

      3. *-commutative 46.4%

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

      4. *-commutative 46.4%

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

      5. *-commutative 46.4%

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

      6. *-commutative 46.4%

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

      7. *-commutative 46.4%

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

      8. associate-*r* 47.7%

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

   8. Simplified 47.7%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{-14}:\\ \;\;\;\;\left(b \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) + a \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+55}:\\ \;\;\;\;\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+134}:\\ \;\;\;\;2 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\frac{angle \cdot \pi}{-180}\right)\right)\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+193}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)\\ \end{array} \]

Alternative 5: 67.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \frac{angle_m}{180} \cdot \pi\\ t_1 := \sin t_0\\ t_2 := \left(b_m - a_m\right) \cdot \left(b_m + a_m\right)\\ t_3 := \left|t_2\right|\\ t_4 := \pi \cdot \left(angle_m \cdot \left(b_m - a_m\right)\right)\\ t_5 := \cos t_0\\ angle_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle_m}{180} \leq 4 \cdot 10^{-30}:\\ \;\;\;\;\left(b_m \cdot t_4 + a_m \cdot t_4\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 4 \cdot 10^{+75}:\\ \;\;\;\;\left(2 \cdot t_2\right) \cdot \left(t_5 \cdot t_1\right)\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 5 \cdot 10^{+116}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle_m \cdot \left(\pi \cdot t_3\right)\right)\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 5 \cdot 10^{+196}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \left(angle_m \cdot -0.005555555555555556\right)\right) \cdot \left(t_2 \cdot \sin \left(0.005555555555555556 \cdot \left(angle_m \cdot \pi\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 10^{+212}:\\ \;\;\;\;\sin \left(angle_m \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left|{a_m}^{2} - {b_m}^{2}\right|\right)\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 10^{+258}:\\ \;\;\;\;t_5 \cdot \left(t_1 \cdot \left(2 \cdot t_3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \left(t_2 \cdot \left|\cos \left(\pi \cdot \left(angle_m \cdot 0.005555555555555556\right)\right)\right|\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* (/ angle_m 180.0) PI))
        (t_1 (sin t_0))
        (t_2 (* (- b_m a_m) (+ b_m a_m)))
        (t_3 (fabs t_2))
        (t_4 (* PI (* angle_m (- b_m a_m))))
        (t_5 (cos t_0)))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 4e-30)
      (* (+ (* b_m t_4) (* a_m t_4)) 0.011111111111111112)
      (if (<= (/ angle_m 180.0) 4e+75)
        (* (* 2.0 t_2) (* t_5 t_1))
        (if (<= (/ angle_m 180.0) 5e+116)
          (* 0.011111111111111112 (* angle_m (* PI t_3)))
          (if (<= (/ angle_m 180.0) 5e+196)
            (*
             2.0
             (*
              (cos (* PI (* angle_m -0.005555555555555556)))
              (* t_2 (sin (* 0.005555555555555556 (* angle_m PI))))))
            (if (<= (/ angle_m 180.0) 1e+212)
              (*
               (sin (* angle_m (/ PI -180.0)))
               (* 2.0 (fabs (- (pow a_m 2.0) (pow b_m 2.0)))))
              (if (<= (/ angle_m 180.0) 1e+258)
                (* t_5 (* t_1 (* 2.0 t_3)))
                (*
                 2.0
                 (*
                  t_1
                  (*
                   t_2
                   (fabs
                    (cos (* PI (* angle_m 0.005555555555555556))))))))))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = (angle_m / 180.0) * ((double) M_PI);
	double t_1 = sin(t_0);
	double t_2 = (b_m - a_m) * (b_m + a_m);
	double t_3 = fabs(t_2);
	double t_4 = ((double) M_PI) * (angle_m * (b_m - a_m));
	double t_5 = cos(t_0);
	double tmp;
	if ((angle_m / 180.0) <= 4e-30) {
		tmp = ((b_m * t_4) + (a_m * t_4)) * 0.011111111111111112;
	} else if ((angle_m / 180.0) <= 4e+75) {
		tmp = (2.0 * t_2) * (t_5 * t_1);
	} else if ((angle_m / 180.0) <= 5e+116) {
		tmp = 0.011111111111111112 * (angle_m * (((double) M_PI) * t_3));
	} else if ((angle_m / 180.0) <= 5e+196) {
		tmp = 2.0 * (cos((((double) M_PI) * (angle_m * -0.005555555555555556))) * (t_2 * sin((0.005555555555555556 * (angle_m * ((double) M_PI))))));
	} else if ((angle_m / 180.0) <= 1e+212) {
		tmp = sin((angle_m * (((double) M_PI) / -180.0))) * (2.0 * fabs((pow(a_m, 2.0) - pow(b_m, 2.0))));
	} else if ((angle_m / 180.0) <= 1e+258) {
		tmp = t_5 * (t_1 * (2.0 * t_3));
	} else {
		tmp = 2.0 * (t_1 * (t_2 * fabs(cos((((double) M_PI) * (angle_m * 0.005555555555555556))))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = (angle_m / 180.0) * Math.PI;
	double t_1 = Math.sin(t_0);
	double t_2 = (b_m - a_m) * (b_m + a_m);
	double t_3 = Math.abs(t_2);
	double t_4 = Math.PI * (angle_m * (b_m - a_m));
	double t_5 = Math.cos(t_0);
	double tmp;
	if ((angle_m / 180.0) <= 4e-30) {
		tmp = ((b_m * t_4) + (a_m * t_4)) * 0.011111111111111112;
	} else if ((angle_m / 180.0) <= 4e+75) {
		tmp = (2.0 * t_2) * (t_5 * t_1);
	} else if ((angle_m / 180.0) <= 5e+116) {
		tmp = 0.011111111111111112 * (angle_m * (Math.PI * t_3));
	} else if ((angle_m / 180.0) <= 5e+196) {
		tmp = 2.0 * (Math.cos((Math.PI * (angle_m * -0.005555555555555556))) * (t_2 * Math.sin((0.005555555555555556 * (angle_m * Math.PI)))));
	} else if ((angle_m / 180.0) <= 1e+212) {
		tmp = Math.sin((angle_m * (Math.PI / -180.0))) * (2.0 * Math.abs((Math.pow(a_m, 2.0) - Math.pow(b_m, 2.0))));
	} else if ((angle_m / 180.0) <= 1e+258) {
		tmp = t_5 * (t_1 * (2.0 * t_3));
	} else {
		tmp = 2.0 * (t_1 * (t_2 * Math.abs(Math.cos((Math.PI * (angle_m * 0.005555555555555556))))));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	t_0 = (angle_m / 180.0) * math.pi
	t_1 = math.sin(t_0)
	t_2 = (b_m - a_m) * (b_m + a_m)
	t_3 = math.fabs(t_2)
	t_4 = math.pi * (angle_m * (b_m - a_m))
	t_5 = math.cos(t_0)
	tmp = 0
	if (angle_m / 180.0) <= 4e-30:
		tmp = ((b_m * t_4) + (a_m * t_4)) * 0.011111111111111112
	elif (angle_m / 180.0) <= 4e+75:
		tmp = (2.0 * t_2) * (t_5 * t_1)
	elif (angle_m / 180.0) <= 5e+116:
		tmp = 0.011111111111111112 * (angle_m * (math.pi * t_3))
	elif (angle_m / 180.0) <= 5e+196:
		tmp = 2.0 * (math.cos((math.pi * (angle_m * -0.005555555555555556))) * (t_2 * math.sin((0.005555555555555556 * (angle_m * math.pi)))))
	elif (angle_m / 180.0) <= 1e+212:
		tmp = math.sin((angle_m * (math.pi / -180.0))) * (2.0 * math.fabs((math.pow(a_m, 2.0) - math.pow(b_m, 2.0))))
	elif (angle_m / 180.0) <= 1e+258:
		tmp = t_5 * (t_1 * (2.0 * t_3))
	else:
		tmp = 2.0 * (t_1 * (t_2 * math.fabs(math.cos((math.pi * (angle_m * 0.005555555555555556))))))
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(Float64(angle_m / 180.0) * pi)
	t_1 = sin(t_0)
	t_2 = Float64(Float64(b_m - a_m) * Float64(b_m + a_m))
	t_3 = abs(t_2)
	t_4 = Float64(pi * Float64(angle_m * Float64(b_m - a_m)))
	t_5 = cos(t_0)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e-30)
		tmp = Float64(Float64(Float64(b_m * t_4) + Float64(a_m * t_4)) * 0.011111111111111112);
	elseif (Float64(angle_m / 180.0) <= 4e+75)
		tmp = Float64(Float64(2.0 * t_2) * Float64(t_5 * t_1));
	elseif (Float64(angle_m / 180.0) <= 5e+116)
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * t_3)));
	elseif (Float64(angle_m / 180.0) <= 5e+196)
		tmp = Float64(2.0 * Float64(cos(Float64(pi * Float64(angle_m * -0.005555555555555556))) * Float64(t_2 * sin(Float64(0.005555555555555556 * Float64(angle_m * pi))))));
	elseif (Float64(angle_m / 180.0) <= 1e+212)
		tmp = Float64(sin(Float64(angle_m * Float64(pi / -180.0))) * Float64(2.0 * abs(Float64((a_m ^ 2.0) - (b_m ^ 2.0)))));
	elseif (Float64(angle_m / 180.0) <= 1e+258)
		tmp = Float64(t_5 * Float64(t_1 * Float64(2.0 * t_3)));
	else
		tmp = Float64(2.0 * Float64(t_1 * Float64(t_2 * abs(cos(Float64(pi * Float64(angle_m * 0.005555555555555556)))))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	t_0 = (angle_m / 180.0) * pi;
	t_1 = sin(t_0);
	t_2 = (b_m - a_m) * (b_m + a_m);
	t_3 = abs(t_2);
	t_4 = pi * (angle_m * (b_m - a_m));
	t_5 = cos(t_0);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 4e-30)
		tmp = ((b_m * t_4) + (a_m * t_4)) * 0.011111111111111112;
	elseif ((angle_m / 180.0) <= 4e+75)
		tmp = (2.0 * t_2) * (t_5 * t_1);
	elseif ((angle_m / 180.0) <= 5e+116)
		tmp = 0.011111111111111112 * (angle_m * (pi * t_3));
	elseif ((angle_m / 180.0) <= 5e+196)
		tmp = 2.0 * (cos((pi * (angle_m * -0.005555555555555556))) * (t_2 * sin((0.005555555555555556 * (angle_m * pi)))));
	elseif ((angle_m / 180.0) <= 1e+212)
		tmp = sin((angle_m * (pi / -180.0))) * (2.0 * abs(((a_m ^ 2.0) - (b_m ^ 2.0))));
	elseif ((angle_m / 180.0) <= 1e+258)
		tmp = t_5 * (t_1 * (2.0 * t_3));
	else
		tmp = 2.0 * (t_1 * (t_2 * abs(cos((pi * (angle_m * 0.005555555555555556))))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(b$95$m + a$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Abs[t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[(Pi * N[(angle$95$m * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Cos[t$95$0], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e-30], N[(N[(N[(b$95$m * t$95$4), $MachinePrecision] + N[(a$95$m * t$95$4), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+75], N[(N[(2.0 * t$95$2), $MachinePrecision] * N[(t$95$5 * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+116], N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+196], N[(2.0 * N[(N[Cos[N[(Pi * N[(angle$95$m * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$2 * N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+212], N[(N[Sin[N[(angle$95$m * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[Abs[N[(N[Power[a$95$m, 2.0], $MachinePrecision] - N[Power[b$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+258], N[(t$95$5 * N[(t$95$1 * N[(2.0 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$1 * N[(t$95$2 * N[Abs[N[Cos[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 7 regimes

2. if (/.f64 angle 180) < 4e-30

   1. Initial program 62.5%

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

      1. associate-*l* 62.5%

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

   3. Simplified 62.5%

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

   4. Taylor expanded in angle around 0 59.4%

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

      1. unpow2 59.4%

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

      2. unpow2 59.4%

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

      3. difference-of-squares 63.8%

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

   6. Applied egg-rr 63.8%

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

   7. Taylor expanded in angle around 0 63.8%

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

      1. *-commutative 63.8%

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

      2. associate-*r* 63.9%

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

      3. *-commutative 63.9%

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

      4. associate-*l* 63.8%

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

      5. +-commutative 63.8%

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

      6. difference-of-squares 59.4%

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

      7. unpow2 59.4%

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

      8. unpow2 59.4%

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

      9. associate-*l* 59.4%

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

      10. *-commutative 59.4%

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

      11. unpow2 59.4%

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

      12. unpow2 59.4%

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

      13. difference-of-squares 63.9%

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

      14. *-commutative 63.9%

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

      15. +-commutative 63.9%

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

   9. Simplified 63.9%

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

       1. associate-*r* 75.0%

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

       2. +-commutative 75.0%

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

       3. distribute-lft-in 71.0%

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

       4. *-commutative 71.0%

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

       5. associate-*l* 71.0%

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

       6. *-commutative 71.0%

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

       7. associate-*l* 71.1%

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

   11. Applied egg-rr 71.1%

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

   if 4e-30 < (/.f64 angle 180) < 3.99999999999999971e75

   1. Initial program 65.1%

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

      1. associate-*l* 65.1%

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

   3. Simplified 65.1%

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

      1. unpow2 44.8%

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

      2. unpow2 44.8%

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

      3. difference-of-squares 48.6%

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

   5. Applied egg-rr 69.0%

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

   if 3.99999999999999971e75 < (/.f64 angle 180) < 5.00000000000000025e116

   1. Initial program 27.2%

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

      1. associate-*l* 27.2%

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

   3. Simplified 27.2%

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

   4. Taylor expanded in angle around 0 39.6%

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

      1. add-sqr-sqrt 38.7%

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

      2. sqrt-unprod 52.5%

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

      3. pow2 52.5%

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

   6. Applied egg-rr 52.5%

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

      1. unpow2 52.5%

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

      2. rem-sqrt-square 51.5%

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

   8. Simplified 51.5%

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

      1. unpow2 39.6%

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

      2. unpow2 39.6%

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

      3. difference-of-squares 39.6%

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

   10. Applied egg-rr 51.5%

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

   if 5.00000000000000025e116 < (/.f64 angle 180) < 4.9999999999999998e196

   1. Initial program 31.7%

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

   3. Simplified 35.9%

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

      1. unpow2 33.4%

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

      2. unpow2 33.4%

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

      3. difference-of-squares 33.4%

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

   5. Applied egg-rr 35.9%

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

   6. Taylor expanded in angle around inf 42.2%

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

      1. associate-*r* 43.2%

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

      2. associate-*r* 37.0%

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

      3. *-commutative 37.0%

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

      4. *-commutative 37.0%

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

      5. *-commutative 37.0%

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

      6. *-commutative 37.0%

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

      7. *-commutative 37.0%

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

      8. associate-*r* 43.2%

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

   8. Simplified 43.2%

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

   if 4.9999999999999998e196 < (/.f64 angle 180) < 9.9999999999999991e211

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 100.0%

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

      3. pow2 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. unpow2 100.0%

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

      2. rem-sqrt-square 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in angle around 0 100.0%

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

   if 9.9999999999999991e211 < (/.f64 angle 180) < 1.00000000000000006e258

   1. Initial program 27.4%

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

      1. add-sqr-sqrt 0.1%

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

      2. sqrt-unprod 42.2%

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

      3. pow2 42.2%

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

   3. Applied egg-rr 67.8%

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

      1. unpow2 42.2%

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

      2. rem-sqrt-square 40.8%

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

   5. Simplified 71.2%

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

      1. unpow2 0.8%

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

      2. unpow2 0.8%

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

      3. difference-of-squares 0.8%

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

   7. Applied egg-rr 71.2%

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

   if 1.00000000000000006e258 < (/.f64 angle 180)

   1. Initial program 25.9%

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

   3. Simplified 18.2%

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

      1. unpow2 20.4%

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

      2. unpow2 20.4%

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

      3. difference-of-squares 26.6%

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

   5. Applied egg-rr 18.2%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 0.0%

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

      3. associate-*l/ 0.0%

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

      4. associate-*l/ 0.0%

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

      5. frac-times 0.0%

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

      6. metadata-eval 0.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)\right)\right) \]

      7. metadata-eval 0.0%

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

      8. frac-times 0.0%

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

      9. associate-*r/ 0.0%

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

      10. associate-*r/ 0.0%

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

      11. sqrt-unprod 37.2%

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

      12. add-sqr-sqrt 25.9%

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

      13. add-sqr-sqrt 23.8%

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

      14. sqrt-unprod 31.4%

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

      15. pow2 31.4%

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

   7. Applied egg-rr 31.4%

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

      1. unpow2 31.4%

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

      2. rem-sqrt-square 31.4%

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

   9. Simplified 31.4%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left|\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right|}\right)\right) \]
3. Recombined 7 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{-30}:\\ \;\;\;\;\left(b \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) + a \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle}{180} \leq 4 \cdot 10^{+75}:\\ \;\;\;\;\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+116}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left|\left(b - a\right) \cdot \left(b + a\right)\right|\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+196}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+212}:\\ \;\;\;\;\sin \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left|{a}^{2} - {b}^{2}\right|\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+258}:\\ \;\;\;\;\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \left|\left(b - a\right) \cdot \left(b + a\right)\right|\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left|\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right|\right)\right)\\ \end{array} \]

Alternative 6: 67.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(angle_m \cdot \left(b_m - a_m\right)\right)\\ t_1 := \left(b_m - a_m\right) \cdot \left(b_m + a_m\right)\\ t_2 := t_1 \cdot \cos \left(angle_m \cdot \frac{\pi}{-180}\right)\\ angle_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle_m}{180} \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\left(b_m \cdot t_0 + a_m \cdot t_0\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 5 \cdot 10^{+60}:\\ \;\;\;\;2 \cdot \left(t_2 \cdot \sin \left(\pi \cdot \left(angle_m \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 2 \cdot 10^{+134}:\\ \;\;\;\;2 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\frac{angle_m \cdot \pi}{-180}\right)\right)\right) \cdot t_2\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \left(angle_m \cdot -0.005555555555555556\right)\right) \cdot \left(t_1 \cdot \sin \left(0.005555555555555556 \cdot \left(angle_m \cdot \pi\right)\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m (- b_m a_m))))
        (t_1 (* (- b_m a_m) (+ b_m a_m)))
        (t_2 (* t_1 (cos (* angle_m (/ PI -180.0))))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 2e-42)
      (* (+ (* b_m t_0) (* a_m t_0)) 0.011111111111111112)
      (if (<= (/ angle_m 180.0) 5e+60)
        (* 2.0 (* t_2 (sin (* PI (* angle_m 0.005555555555555556)))))
        (if (<= (/ angle_m 180.0) 2e+134)
          (* 2.0 (* (log1p (expm1 (sin (/ (* angle_m PI) -180.0)))) t_2))
          (*
           2.0
           (*
            (cos (* PI (* angle_m -0.005555555555555556)))
            (* t_1 (sin (* 0.005555555555555556 (* angle_m PI))))))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m * (b_m - a_m));
	double t_1 = (b_m - a_m) * (b_m + a_m);
	double t_2 = t_1 * cos((angle_m * (((double) M_PI) / -180.0)));
	double tmp;
	if ((angle_m / 180.0) <= 2e-42) {
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112;
	} else if ((angle_m / 180.0) <= 5e+60) {
		tmp = 2.0 * (t_2 * sin((((double) M_PI) * (angle_m * 0.005555555555555556))));
	} else if ((angle_m / 180.0) <= 2e+134) {
		tmp = 2.0 * (log1p(expm1(sin(((angle_m * ((double) M_PI)) / -180.0)))) * t_2);
	} else {
		tmp = 2.0 * (cos((((double) M_PI) * (angle_m * -0.005555555555555556))) * (t_1 * sin((0.005555555555555556 * (angle_m * ((double) M_PI))))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = Math.PI * (angle_m * (b_m - a_m));
	double t_1 = (b_m - a_m) * (b_m + a_m);
	double t_2 = t_1 * Math.cos((angle_m * (Math.PI / -180.0)));
	double tmp;
	if ((angle_m / 180.0) <= 2e-42) {
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112;
	} else if ((angle_m / 180.0) <= 5e+60) {
		tmp = 2.0 * (t_2 * Math.sin((Math.PI * (angle_m * 0.005555555555555556))));
	} else if ((angle_m / 180.0) <= 2e+134) {
		tmp = 2.0 * (Math.log1p(Math.expm1(Math.sin(((angle_m * Math.PI) / -180.0)))) * t_2);
	} else {
		tmp = 2.0 * (Math.cos((Math.PI * (angle_m * -0.005555555555555556))) * (t_1 * Math.sin((0.005555555555555556 * (angle_m * Math.PI)))));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	t_0 = math.pi * (angle_m * (b_m - a_m))
	t_1 = (b_m - a_m) * (b_m + a_m)
	t_2 = t_1 * math.cos((angle_m * (math.pi / -180.0)))
	tmp = 0
	if (angle_m / 180.0) <= 2e-42:
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112
	elif (angle_m / 180.0) <= 5e+60:
		tmp = 2.0 * (t_2 * math.sin((math.pi * (angle_m * 0.005555555555555556))))
	elif (angle_m / 180.0) <= 2e+134:
		tmp = 2.0 * (math.log1p(math.expm1(math.sin(((angle_m * math.pi) / -180.0)))) * t_2)
	else:
		tmp = 2.0 * (math.cos((math.pi * (angle_m * -0.005555555555555556))) * (t_1 * math.sin((0.005555555555555556 * (angle_m * math.pi)))))
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(pi * Float64(angle_m * Float64(b_m - a_m)))
	t_1 = Float64(Float64(b_m - a_m) * Float64(b_m + a_m))
	t_2 = Float64(t_1 * cos(Float64(angle_m * Float64(pi / -180.0))))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e-42)
		tmp = Float64(Float64(Float64(b_m * t_0) + Float64(a_m * t_0)) * 0.011111111111111112);
	elseif (Float64(angle_m / 180.0) <= 5e+60)
		tmp = Float64(2.0 * Float64(t_2 * sin(Float64(pi * Float64(angle_m * 0.005555555555555556)))));
	elseif (Float64(angle_m / 180.0) <= 2e+134)
		tmp = Float64(2.0 * Float64(log1p(expm1(sin(Float64(Float64(angle_m * pi) / -180.0)))) * t_2));
	else
		tmp = Float64(2.0 * Float64(cos(Float64(pi * Float64(angle_m * -0.005555555555555556))) * Float64(t_1 * sin(Float64(0.005555555555555556 * Float64(angle_m * pi))))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(b$95$m + a$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Cos[N[(angle$95$m * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e-42], N[(N[(N[(b$95$m * t$95$0), $MachinePrecision] + N[(a$95$m * t$95$0), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+60], N[(2.0 * N[(t$95$2 * N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+134], N[(2.0 * N[(N[Log[1 + N[(Exp[N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] / -180.0), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Cos[N[(Pi * N[(angle$95$m * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle 180) < 2.00000000000000008e-42

   1. Initial program 61.9%

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

      1. associate-*l* 61.9%

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

   3. Simplified 61.9%

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

   4. Taylor expanded in angle around 0 58.7%

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

      1. unpow2 58.7%

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

      2. unpow2 58.7%

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

      3. difference-of-squares 63.2%

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

   6. Applied egg-rr 63.2%

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

   7. Taylor expanded in angle around 0 63.2%

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

      1. *-commutative 63.2%

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

      2. associate-*r* 63.3%

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

      3. *-commutative 63.3%

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

      4. associate-*l* 63.2%

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

      5. +-commutative 63.2%

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

      6. difference-of-squares 58.7%

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

      7. unpow2 58.7%

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

      8. unpow2 58.7%

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

      9. associate-*l* 58.7%

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

      10. *-commutative 58.7%

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

      11. unpow2 58.7%

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

      12. unpow2 58.7%

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

      13. difference-of-squares 63.3%

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

      14. *-commutative 63.3%

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

      15. +-commutative 63.3%

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

   9. Simplified 63.3%

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

       1. associate-*r* 74.6%

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

       2. +-commutative 74.6%

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

       3. distribute-lft-in 70.5%

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

       4. *-commutative 70.5%

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

       5. associate-*l* 70.5%

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

       6. *-commutative 70.5%

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

       7. associate-*l* 70.6%

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

   11. Applied egg-rr 70.6%

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

   if 2.00000000000000008e-42 < (/.f64 angle 180) < 4.99999999999999975e60

   1. Initial program 73.8%

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

   3. Simplified 73.7%

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

      1. unpow2 53.8%

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

      2. unpow2 53.8%

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

      3. difference-of-squares 57.5%

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

   5. Applied egg-rr 77.4%

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

   6. Taylor expanded in angle around inf 76.7%

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

      1. associate-*r* 78.0%

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

      2. *-commutative 78.0%

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

      3. *-commutative 78.0%

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

   8. Simplified 78.0%

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

   if 4.99999999999999975e60 < (/.f64 angle 180) < 1.99999999999999984e134

   1. Initial program 19.3%

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

   3. Simplified 20.6%

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

      1. unpow2 31.3%

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

      2. unpow2 31.3%

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

      3. difference-of-squares 31.3%

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

   5. Applied egg-rr 20.6%

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

      1. add-sqr-sqrt 28.5%

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

      2. sqrt-unprod 20.6%

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

      3. associate-*r/ 20.7%

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

      4. associate-*r/ 22.1%

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

      5. frac-times 21.0%

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

      6. metadata-eval 21.0%

         \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      7. metadata-eval 21.0%

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

      8. frac-times 22.1%

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

      9. associate-*l/ 20.0%

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

      10. associate-*l/ 20.0%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 41.4%

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

      13. log1p-expm1-u 41.4%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 20.0%

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

   7. Applied egg-rr 20.6%

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

      1. metadata-eval 20.6%

         \[\leadsto 2 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      2. div-inv 20.6%

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

      3. add-sqr-sqrt 28.5%

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

      4. sqrt-unprod 20.6%

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

      5. associate-*r/ 20.7%

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

      6. associate-*r/ 22.1%

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

      7. frac-times 21.0%

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

      8. metadata-eval 21.0%

         \[\leadsto 2 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      9. metadata-eval 21.0%

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

      10. frac-times 22.1%

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

      11. associate-*l/ 20.0%

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

      12. associate-*l/ 20.0%

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

      13. sqrt-unprod 0.0%

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

      14. add-sqr-sqrt 41.4%

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

      15. associate-*l/ 39.3%

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

   9. Applied egg-rr 39.3%

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

   if 1.99999999999999984e134 < (/.f64 angle 180)

   1. Initial program 29.8%

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

   3. Simplified 31.2%

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

      1. unpow2 25.8%

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

      2. unpow2 25.8%

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

      3. difference-of-squares 28.5%

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

   5. Applied egg-rr 31.2%

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

   6. Taylor expanded in angle around inf 34.2%

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

      1. associate-*r* 38.0%

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

      2. associate-*r* 32.1%

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

      3. *-commutative 32.1%

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

      4. *-commutative 32.1%

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

      5. *-commutative 32.1%

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

      6. *-commutative 32.1%

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

      7. *-commutative 32.1%

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

      8. associate-*r* 38.0%

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

   8. Simplified 38.0%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\left(b \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) + a \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+60}:\\ \;\;\;\;2 \cdot \left(\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+134}:\\ \;\;\;\;2 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\frac{angle \cdot \pi}{-180}\right)\right)\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \end{array} \]

Alternative 7: 67.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(angle_m \cdot \left(b_m - a_m\right)\right)\\ t_1 := \left(b_m - a_m\right) \cdot \left(b_m + a_m\right)\\ t_2 := t_1 \cdot \cos \left(angle_m \cdot \frac{\pi}{-180}\right)\\ angle_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle_m}{180} \leq 4 \cdot 10^{-30}:\\ \;\;\;\;\left(b_m \cdot t_0 + a_m \cdot t_0\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 5 \cdot 10^{+60}:\\ \;\;\;\;2 \cdot \left(t_2 \cdot \sqrt[3]{{\sin \left(\pi \cdot \left(angle_m \cdot 0.005555555555555556\right)\right)}^{3}}\right)\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 2 \cdot 10^{+134}:\\ \;\;\;\;2 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\frac{angle_m \cdot \pi}{-180}\right)\right)\right) \cdot t_2\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \left(angle_m \cdot -0.005555555555555556\right)\right) \cdot \left(t_1 \cdot \sin \left(0.005555555555555556 \cdot \left(angle_m \cdot \pi\right)\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m (- b_m a_m))))
        (t_1 (* (- b_m a_m) (+ b_m a_m)))
        (t_2 (* t_1 (cos (* angle_m (/ PI -180.0))))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 4e-30)
      (* (+ (* b_m t_0) (* a_m t_0)) 0.011111111111111112)
      (if (<= (/ angle_m 180.0) 5e+60)
        (*
         2.0
         (*
          t_2
          (cbrt (pow (sin (* PI (* angle_m 0.005555555555555556))) 3.0))))
        (if (<= (/ angle_m 180.0) 2e+134)
          (* 2.0 (* (log1p (expm1 (sin (/ (* angle_m PI) -180.0)))) t_2))
          (*
           2.0
           (*
            (cos (* PI (* angle_m -0.005555555555555556)))
            (* t_1 (sin (* 0.005555555555555556 (* angle_m PI))))))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m * (b_m - a_m));
	double t_1 = (b_m - a_m) * (b_m + a_m);
	double t_2 = t_1 * cos((angle_m * (((double) M_PI) / -180.0)));
	double tmp;
	if ((angle_m / 180.0) <= 4e-30) {
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112;
	} else if ((angle_m / 180.0) <= 5e+60) {
		tmp = 2.0 * (t_2 * cbrt(pow(sin((((double) M_PI) * (angle_m * 0.005555555555555556))), 3.0)));
	} else if ((angle_m / 180.0) <= 2e+134) {
		tmp = 2.0 * (log1p(expm1(sin(((angle_m * ((double) M_PI)) / -180.0)))) * t_2);
	} else {
		tmp = 2.0 * (cos((((double) M_PI) * (angle_m * -0.005555555555555556))) * (t_1 * sin((0.005555555555555556 * (angle_m * ((double) M_PI))))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = Math.PI * (angle_m * (b_m - a_m));
	double t_1 = (b_m - a_m) * (b_m + a_m);
	double t_2 = t_1 * Math.cos((angle_m * (Math.PI / -180.0)));
	double tmp;
	if ((angle_m / 180.0) <= 4e-30) {
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112;
	} else if ((angle_m / 180.0) <= 5e+60) {
		tmp = 2.0 * (t_2 * Math.cbrt(Math.pow(Math.sin((Math.PI * (angle_m * 0.005555555555555556))), 3.0)));
	} else if ((angle_m / 180.0) <= 2e+134) {
		tmp = 2.0 * (Math.log1p(Math.expm1(Math.sin(((angle_m * Math.PI) / -180.0)))) * t_2);
	} else {
		tmp = 2.0 * (Math.cos((Math.PI * (angle_m * -0.005555555555555556))) * (t_1 * Math.sin((0.005555555555555556 * (angle_m * Math.PI)))));
	}
	return angle_s * tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(pi * Float64(angle_m * Float64(b_m - a_m)))
	t_1 = Float64(Float64(b_m - a_m) * Float64(b_m + a_m))
	t_2 = Float64(t_1 * cos(Float64(angle_m * Float64(pi / -180.0))))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e-30)
		tmp = Float64(Float64(Float64(b_m * t_0) + Float64(a_m * t_0)) * 0.011111111111111112);
	elseif (Float64(angle_m / 180.0) <= 5e+60)
		tmp = Float64(2.0 * Float64(t_2 * cbrt((sin(Float64(pi * Float64(angle_m * 0.005555555555555556))) ^ 3.0))));
	elseif (Float64(angle_m / 180.0) <= 2e+134)
		tmp = Float64(2.0 * Float64(log1p(expm1(sin(Float64(Float64(angle_m * pi) / -180.0)))) * t_2));
	else
		tmp = Float64(2.0 * Float64(cos(Float64(pi * Float64(angle_m * -0.005555555555555556))) * Float64(t_1 * sin(Float64(0.005555555555555556 * Float64(angle_m * pi))))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(b$95$m + a$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Cos[N[(angle$95$m * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e-30], N[(N[(N[(b$95$m * t$95$0), $MachinePrecision] + N[(a$95$m * t$95$0), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+60], N[(2.0 * N[(t$95$2 * N[Power[N[Power[N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+134], N[(2.0 * N[(N[Log[1 + N[(Exp[N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] / -180.0), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Cos[N[(Pi * N[(angle$95$m * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle 180) < 4e-30

   1. Initial program 62.5%

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

      1. associate-*l* 62.5%

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

   3. Simplified 62.5%

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

   4. Taylor expanded in angle around 0 59.4%

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

      1. unpow2 59.4%

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

      2. unpow2 59.4%

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

      3. difference-of-squares 63.8%

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

   6. Applied egg-rr 63.8%

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

   7. Taylor expanded in angle around 0 63.8%

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

      1. *-commutative 63.8%

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

      2. associate-*r* 63.9%

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

      3. *-commutative 63.9%

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

      4. associate-*l* 63.8%

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

      5. +-commutative 63.8%

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

      6. difference-of-squares 59.4%

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

      7. unpow2 59.4%

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

      8. unpow2 59.4%

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

      9. associate-*l* 59.4%

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

      10. *-commutative 59.4%

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

      11. unpow2 59.4%

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

      12. unpow2 59.4%

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

      13. difference-of-squares 63.9%

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

      14. *-commutative 63.9%

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

      15. +-commutative 63.9%

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

   9. Simplified 63.9%

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

       1. associate-*r* 75.0%

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

       2. +-commutative 75.0%

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

       3. distribute-lft-in 71.0%

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

       4. *-commutative 71.0%

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

       5. associate-*l* 71.0%

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

       6. *-commutative 71.0%

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

       7. associate-*l* 71.1%

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

   11. Applied egg-rr 71.1%

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

   if 4e-30 < (/.f64 angle 180) < 4.99999999999999975e60

   1. Initial program 70.6%

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

   3. Simplified 70.5%

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

      1. unpow2 48.1%

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

      2. unpow2 48.1%

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

      3. difference-of-squares 52.3%

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

   5. Applied egg-rr 74.7%

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

      1. add-sqr-sqrt 74.6%

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

      2. sqrt-unprod 74.7%

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

      3. associate-*r/ 78.9%

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

      4. associate-*r/ 78.1%

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

      5. frac-times 78.0%

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

      6. metadata-eval 78.0%

         \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      7. metadata-eval 78.0%

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

      8. frac-times 78.1%

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

      9. associate-*l/ 78.7%

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

      10. associate-*l/ 70.3%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 22.8%

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

      13. add-cbrt-cube 22.8%

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

      14. pow3 22.8%

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

   7. Applied egg-rr 75.3%

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

   if 4.99999999999999975e60 < (/.f64 angle 180) < 1.99999999999999984e134

   1. Initial program 19.3%

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

   3. Simplified 20.6%

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

      1. unpow2 31.3%

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

      2. unpow2 31.3%

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

      3. difference-of-squares 31.3%

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

   5. Applied egg-rr 20.6%

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

      1. add-sqr-sqrt 28.5%

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

      2. sqrt-unprod 20.6%

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

      3. associate-*r/ 20.7%

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

      4. associate-*r/ 22.1%

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

      5. frac-times 21.0%

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

      6. metadata-eval 21.0%

         \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      7. metadata-eval 21.0%

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

      8. frac-times 22.1%

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

      9. associate-*l/ 20.0%

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

      10. associate-*l/ 20.0%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 41.4%

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

      13. log1p-expm1-u 41.4%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 20.0%

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

   7. Applied egg-rr 20.6%

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

      1. metadata-eval 20.6%

         \[\leadsto 2 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      2. div-inv 20.6%

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

      3. add-sqr-sqrt 28.5%

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

      4. sqrt-unprod 20.6%

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

      5. associate-*r/ 20.7%

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

      6. associate-*r/ 22.1%

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

      7. frac-times 21.0%

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

      8. metadata-eval 21.0%

         \[\leadsto 2 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      9. metadata-eval 21.0%

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

      10. frac-times 22.1%

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

      11. associate-*l/ 20.0%

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

      12. associate-*l/ 20.0%

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

      13. sqrt-unprod 0.0%

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

      14. add-sqr-sqrt 41.4%

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

      15. associate-*l/ 39.3%

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

   9. Applied egg-rr 39.3%

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

   if 1.99999999999999984e134 < (/.f64 angle 180)

   1. Initial program 29.8%

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

   3. Simplified 31.2%

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

      1. unpow2 25.8%

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

      2. unpow2 25.8%

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

      3. difference-of-squares 28.5%

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

   5. Applied egg-rr 31.2%

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

   6. Taylor expanded in angle around inf 34.2%

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

      1. associate-*r* 38.0%

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

      2. associate-*r* 32.1%

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

      3. *-commutative 32.1%

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

      4. *-commutative 32.1%

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

      5. *-commutative 32.1%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \color{blue}{\left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right) \]

      6. *-commutative 32.1%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \color{blue}{\left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)}\right)\right) \]

      7. *-commutative 32.1%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\color{blue}{\left(0.005555555555555556 \cdot angle\right)} \cdot \pi\right)\right)\right) \]

      8. associate-*r* 38.0%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right) \]

   8. Simplified 38.0%

      \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{-30}:\\ \;\;\;\;\left(b \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) + a \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+60}:\\ \;\;\;\;2 \cdot \left(\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \sqrt[3]{{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{3}}\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+134}:\\ \;\;\;\;2 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\frac{angle \cdot \pi}{-180}\right)\right)\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \end{array} \]

Alternative 8: 67.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(angle_m \cdot \left(b_m - a_m\right)\right)\\ t_1 := \pi \cdot \left(angle_m \cdot 0.005555555555555556\right)\\ t_2 := \sin t_1\\ t_3 := \left(b_m - a_m\right) \cdot \left(b_m + a_m\right)\\ t_4 := angle_m \cdot \left(\pi \cdot -0.005555555555555556\right)\\ angle_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle_m}{180} \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\left(b_m \cdot t_0 + a_m \cdot t_0\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 5 \cdot 10^{+60}:\\ \;\;\;\;2 \cdot \left(\left(t_3 \cdot \cos \left(angle_m \cdot \frac{\pi}{-180}\right)\right) \cdot t_2\right)\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 5 \cdot 10^{+84}:\\ \;\;\;\;2 \cdot \left(\left(\cos t_4 \cdot \left({b_m}^{2} - {a_m}^{2}\right)\right) \cdot \sin t_4\right)\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 10^{+140}:\\ \;\;\;\;2 \cdot \left(t_3 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(t_2\right)\right)\right)\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 2 \cdot 10^{+226}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \left(angle_m \cdot -0.005555555555555556\right)\right) \cdot \left(t_3 \cdot \sin \left(0.005555555555555556 \cdot \left(angle_m \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{angle_m}{180} \cdot \pi\right) \cdot \left(t_3 \cdot \left|\cos t_1\right|\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m (- b_m a_m))))
        (t_1 (* PI (* angle_m 0.005555555555555556)))
        (t_2 (sin t_1))
        (t_3 (* (- b_m a_m) (+ b_m a_m)))
        (t_4 (* angle_m (* PI -0.005555555555555556))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 2e-42)
      (* (+ (* b_m t_0) (* a_m t_0)) 0.011111111111111112)
      (if (<= (/ angle_m 180.0) 5e+60)
        (* 2.0 (* (* t_3 (cos (* angle_m (/ PI -180.0)))) t_2))
        (if (<= (/ angle_m 180.0) 5e+84)
          (* 2.0 (* (* (cos t_4) (- (pow b_m 2.0) (pow a_m 2.0))) (sin t_4)))
          (if (<= (/ angle_m 180.0) 1e+140)
            (* 2.0 (* t_3 (expm1 (log1p t_2))))
            (if (<= (/ angle_m 180.0) 2e+226)
              (*
               2.0
               (*
                (cos (* PI (* angle_m -0.005555555555555556)))
                (* t_3 (sin (* 0.005555555555555556 (* angle_m PI))))))
              (*
               2.0
               (*
                (sin (* (/ angle_m 180.0) PI))
                (* t_3 (fabs (cos t_1)))))))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m * (b_m - a_m));
	double t_1 = ((double) M_PI) * (angle_m * 0.005555555555555556);
	double t_2 = sin(t_1);
	double t_3 = (b_m - a_m) * (b_m + a_m);
	double t_4 = angle_m * (((double) M_PI) * -0.005555555555555556);
	double tmp;
	if ((angle_m / 180.0) <= 2e-42) {
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112;
	} else if ((angle_m / 180.0) <= 5e+60) {
		tmp = 2.0 * ((t_3 * cos((angle_m * (((double) M_PI) / -180.0)))) * t_2);
	} else if ((angle_m / 180.0) <= 5e+84) {
		tmp = 2.0 * ((cos(t_4) * (pow(b_m, 2.0) - pow(a_m, 2.0))) * sin(t_4));
	} else if ((angle_m / 180.0) <= 1e+140) {
		tmp = 2.0 * (t_3 * expm1(log1p(t_2)));
	} else if ((angle_m / 180.0) <= 2e+226) {
		tmp = 2.0 * (cos((((double) M_PI) * (angle_m * -0.005555555555555556))) * (t_3 * sin((0.005555555555555556 * (angle_m * ((double) M_PI))))));
	} else {
		tmp = 2.0 * (sin(((angle_m / 180.0) * ((double) M_PI))) * (t_3 * fabs(cos(t_1))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = Math.PI * (angle_m * (b_m - a_m));
	double t_1 = Math.PI * (angle_m * 0.005555555555555556);
	double t_2 = Math.sin(t_1);
	double t_3 = (b_m - a_m) * (b_m + a_m);
	double t_4 = angle_m * (Math.PI * -0.005555555555555556);
	double tmp;
	if ((angle_m / 180.0) <= 2e-42) {
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112;
	} else if ((angle_m / 180.0) <= 5e+60) {
		tmp = 2.0 * ((t_3 * Math.cos((angle_m * (Math.PI / -180.0)))) * t_2);
	} else if ((angle_m / 180.0) <= 5e+84) {
		tmp = 2.0 * ((Math.cos(t_4) * (Math.pow(b_m, 2.0) - Math.pow(a_m, 2.0))) * Math.sin(t_4));
	} else if ((angle_m / 180.0) <= 1e+140) {
		tmp = 2.0 * (t_3 * Math.expm1(Math.log1p(t_2)));
	} else if ((angle_m / 180.0) <= 2e+226) {
		tmp = 2.0 * (Math.cos((Math.PI * (angle_m * -0.005555555555555556))) * (t_3 * Math.sin((0.005555555555555556 * (angle_m * Math.PI)))));
	} else {
		tmp = 2.0 * (Math.sin(((angle_m / 180.0) * Math.PI)) * (t_3 * Math.abs(Math.cos(t_1))));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	t_0 = math.pi * (angle_m * (b_m - a_m))
	t_1 = math.pi * (angle_m * 0.005555555555555556)
	t_2 = math.sin(t_1)
	t_3 = (b_m - a_m) * (b_m + a_m)
	t_4 = angle_m * (math.pi * -0.005555555555555556)
	tmp = 0
	if (angle_m / 180.0) <= 2e-42:
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112
	elif (angle_m / 180.0) <= 5e+60:
		tmp = 2.0 * ((t_3 * math.cos((angle_m * (math.pi / -180.0)))) * t_2)
	elif (angle_m / 180.0) <= 5e+84:
		tmp = 2.0 * ((math.cos(t_4) * (math.pow(b_m, 2.0) - math.pow(a_m, 2.0))) * math.sin(t_4))
	elif (angle_m / 180.0) <= 1e+140:
		tmp = 2.0 * (t_3 * math.expm1(math.log1p(t_2)))
	elif (angle_m / 180.0) <= 2e+226:
		tmp = 2.0 * (math.cos((math.pi * (angle_m * -0.005555555555555556))) * (t_3 * math.sin((0.005555555555555556 * (angle_m * math.pi)))))
	else:
		tmp = 2.0 * (math.sin(((angle_m / 180.0) * math.pi)) * (t_3 * math.fabs(math.cos(t_1))))
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(pi * Float64(angle_m * Float64(b_m - a_m)))
	t_1 = Float64(pi * Float64(angle_m * 0.005555555555555556))
	t_2 = sin(t_1)
	t_3 = Float64(Float64(b_m - a_m) * Float64(b_m + a_m))
	t_4 = Float64(angle_m * Float64(pi * -0.005555555555555556))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e-42)
		tmp = Float64(Float64(Float64(b_m * t_0) + Float64(a_m * t_0)) * 0.011111111111111112);
	elseif (Float64(angle_m / 180.0) <= 5e+60)
		tmp = Float64(2.0 * Float64(Float64(t_3 * cos(Float64(angle_m * Float64(pi / -180.0)))) * t_2));
	elseif (Float64(angle_m / 180.0) <= 5e+84)
		tmp = Float64(2.0 * Float64(Float64(cos(t_4) * Float64((b_m ^ 2.0) - (a_m ^ 2.0))) * sin(t_4)));
	elseif (Float64(angle_m / 180.0) <= 1e+140)
		tmp = Float64(2.0 * Float64(t_3 * expm1(log1p(t_2))));
	elseif (Float64(angle_m / 180.0) <= 2e+226)
		tmp = Float64(2.0 * Float64(cos(Float64(pi * Float64(angle_m * -0.005555555555555556))) * Float64(t_3 * sin(Float64(0.005555555555555556 * Float64(angle_m * pi))))));
	else
		tmp = Float64(2.0 * Float64(sin(Float64(Float64(angle_m / 180.0) * pi)) * Float64(t_3 * abs(cos(t_1)))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(b$95$m + a$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(angle$95$m * N[(Pi * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e-42], N[(N[(N[(b$95$m * t$95$0), $MachinePrecision] + N[(a$95$m * t$95$0), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+60], N[(2.0 * N[(N[(t$95$3 * N[Cos[N[(angle$95$m * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+84], N[(2.0 * N[(N[(N[Cos[t$95$4], $MachinePrecision] * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+140], N[(2.0 * N[(t$95$3 * N[(Exp[N[Log[1 + t$95$2], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+226], N[(2.0 * N[(N[Cos[N[(Pi * N[(angle$95$m * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$3 * N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(t$95$3 * N[Abs[N[Cos[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 angle 180) < 2.00000000000000008e-42

   1. Initial program 61.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 61.9%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   3. Simplified 61.9%

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0 58.7%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. unpow2 58.7%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 58.7%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 63.2%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   6. Applied egg-rr 63.2%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   7. Taylor expanded in angle around 0 63.2%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative 63.2%

         \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112} \]

      2. associate-*r* 63.3%

         \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot 0.011111111111111112 \]

      3. *-commutative 63.3%

         \[\leadsto \left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

      4. associate-*l* 63.2%

         \[\leadsto \color{blue}{\left(\pi \cdot angle\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\right)} \]

      5. +-commutative 63.2%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\right) \]

      6. difference-of-squares 58.7%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\color{blue}{\left(b \cdot b - a \cdot a\right)} \cdot 0.011111111111111112\right) \]

      7. unpow2 58.7%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\color{blue}{{b}^{2}} - a \cdot a\right) \cdot 0.011111111111111112\right) \]

      8. unpow2 58.7%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left({b}^{2} - \color{blue}{{a}^{2}}\right) \cdot 0.011111111111111112\right) \]

      9. associate-*l* 58.7%

         \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot 0.011111111111111112} \]

      10. *-commutative 58.7%

          \[\leadsto \left(\color{blue}{\left(angle \cdot \pi\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot 0.011111111111111112 \]

      11. unpow2 58.7%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot 0.011111111111111112 \]

      12. unpow2 58.7%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot 0.011111111111111112 \]

      13. difference-of-squares 63.3%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot 0.011111111111111112 \]

      14. *-commutative 63.3%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(b + a\right)\right)}\right) \cdot 0.011111111111111112 \]

      15. +-commutative 63.3%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(a + b\right)}\right)\right) \cdot 0.011111111111111112 \]

   9. Simplified 63.3%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112} \]
   10. Step-by-step derivation

       1. associate-*r* 74.6%

          \[\leadsto \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot \left(a + b\right)\right)} \cdot 0.011111111111111112 \]

       2. +-commutative 74.6%

          \[\leadsto \left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot 0.011111111111111112 \]

       3. distribute-lft-in 70.5%

          \[\leadsto \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right)} \cdot 0.011111111111111112 \]

       4. *-commutative 70.5%

          \[\leadsto \left(\left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(b - a\right)\right) \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       5. associate-*l* 70.5%

          \[\leadsto \left(\color{blue}{\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)} \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       6. *-commutative 70.5%

          \[\leadsto \left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       7. associate-*l* 70.6%

          \[\leadsto \left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \color{blue}{\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)} \cdot a\right) \cdot 0.011111111111111112 \]

   11. Applied egg-rr 70.6%

       \[\leadsto \color{blue}{\left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot a\right)} \cdot 0.011111111111111112 \]

   if 2.00000000000000008e-42 < (/.f64 angle 180) < 4.99999999999999975e60

   1. Initial program 73.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 73.7%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 53.8%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 53.8%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 57.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   5. Applied egg-rr 77.4%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   6. Taylor expanded in angle around inf 76.7%

      \[\leadsto 2 \cdot \left(\color{blue}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r* 78.0%

         \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      2. *-commutative 78.0%

         \[\leadsto 2 \cdot \left(\sin \left(\color{blue}{\left(angle \cdot 0.005555555555555556\right)} \cdot \pi\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      3. *-commutative 78.0%

         \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   8. Simplified 78.0%

      \[\leadsto 2 \cdot \left(\color{blue}{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   if 4.99999999999999975e60 < (/.f64 angle 180) < 5.0000000000000001e84

   1. Initial program 0.0%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 0.0%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 3.3%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 3.3%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 3.3%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   5. Applied egg-rr 0.0%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]
   6. Step-by-step derivation

      1. add-cube-cbrt 33.4%

         \[\leadsto 2 \cdot \left(\sin \left(\color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)} \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      2. pow2 33.4%

         \[\leadsto 2 \cdot \left(\sin \left(\left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2}} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   7. Applied egg-rr 33.4%

      \[\leadsto 2 \cdot \left(\sin \left(\color{blue}{\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right)} \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

         \[\leadsto 2 \cdot \left(\sin \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\sqrt{\frac{\pi}{-180} \cdot angle} \cdot \sqrt{\frac{\pi}{-180} \cdot angle}\right)}\right)\right) \]

      2. sqrt-unprod 33.4%

         \[\leadsto 2 \cdot \left(\sin \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\sqrt{\left(\frac{\pi}{-180} \cdot angle\right) \cdot \left(\frac{\pi}{-180} \cdot angle\right)}\right)}\right)\right) \]

      3. associate-*l/ 33.4%

         \[\leadsto 2 \cdot \left(\sin \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\sqrt{\color{blue}{\frac{\pi \cdot angle}{-180}} \cdot \left(\frac{\pi}{-180} \cdot angle\right)}\right)\right)\right) \]

      4. associate-*l/ 5.5%

         \[\leadsto 2 \cdot \left(\sin \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\sqrt{\frac{\pi \cdot angle}{-180} \cdot \color{blue}{\frac{\pi \cdot angle}{-180}}}\right)\right)\right) \]

      5. frac-times 38.8%

         \[\leadsto 2 \cdot \left(\sin \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\sqrt{\color{blue}{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{-180 \cdot -180}}}\right)\right)\right) \]

      6. metadata-eval 38.8%

         \[\leadsto 2 \cdot \left(\sin \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)\right)\right) \]

      7. metadata-eval 38.8%

         \[\leadsto 2 \cdot \left(\sin \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{180 \cdot 180}}}\right)\right)\right) \]

      8. frac-times 5.5%

         \[\leadsto 2 \cdot \left(\sin \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\sqrt{\color{blue}{\frac{\pi \cdot angle}{180} \cdot \frac{\pi \cdot angle}{180}}}\right)\right)\right) \]

      9. associate-*r/ 33.4%

         \[\leadsto 2 \cdot \left(\sin \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\sqrt{\color{blue}{\left(\pi \cdot \frac{angle}{180}\right)} \cdot \frac{\pi \cdot angle}{180}}\right)\right)\right) \]

      10. associate-*r/ 33.4%

          \[\leadsto 2 \cdot \left(\sin \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\sqrt{\left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\pi \cdot \frac{angle}{180}\right)}}\right)\right)\right) \]

      11. sqrt-unprod 38.8%

          \[\leadsto 2 \cdot \left(\sin \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\sqrt{\pi \cdot \frac{angle}{180}} \cdot \sqrt{\pi \cdot \frac{angle}{180}}\right)}\right)\right) \]

      12. add-sqr-sqrt 33.4%

          \[\leadsto 2 \cdot \left(\sin \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\pi \cdot \frac{angle}{180}\right)}\right)\right) \]

      13. add-cube-cbrt 33.4%

          \[\leadsto 2 \cdot \left(\sin \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\cos \left(\pi \cdot \frac{angle}{180}\right)} \cdot \sqrt[3]{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \sqrt[3]{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right)}\right)\right) \]

      14. pow3 33.4%

          \[\leadsto 2 \cdot \left(\sin \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{{\left(\sqrt[3]{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right)}^{3}}\right)\right) \]

   9. Applied egg-rr 33.4%

      \[\leadsto 2 \cdot \left(\sin \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{{\left(\sqrt[3]{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)}^{3}}\right)\right) \]

   11. Applied egg-rr 0.0%

       \[\leadsto 2 \cdot \color{blue}{e^{\log \left(\cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) + \log \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}} \]
   12. Step-by-step derivation

       1. exp-sum 0.0%

          \[\leadsto 2 \cdot \color{blue}{\left(e^{\log \left(\cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot e^{\log \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}\right)} \]

       2. rem-exp-log 6.4%

          \[\leadsto 2 \cdot \left(\color{blue}{\left(\cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot e^{\log \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}\right) \]

       3. rem-exp-log 73.1%

          \[\leadsto 2 \cdot \left(\left(\cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}\right) \]

   13. Simplified 73.1%

       \[\leadsto 2 \cdot \color{blue}{\left(\left(\cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)} \]

   if 5.0000000000000001e84 < (/.f64 angle 180) < 1.00000000000000006e140

   1. Initial program 28.6%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 30.0%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 41.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 41.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 41.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   5. Applied egg-rr 30.0%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 38.2%

         \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(\sqrt{\pi \cdot \frac{angle}{180}} \cdot \sqrt{\pi \cdot \frac{angle}{180}}\right)} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      2. sqrt-unprod 30.0%

         \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(\sqrt{\left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\pi \cdot \frac{angle}{180}\right)}\right)} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      3. associate-*r/ 30.1%

         \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\color{blue}{\frac{\pi \cdot angle}{180}} \cdot \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      4. associate-*r/ 30.1%

         \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\frac{\pi \cdot angle}{180} \cdot \color{blue}{\frac{\pi \cdot angle}{180}}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      5. frac-times 28.9%

         \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\color{blue}{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{180 \cdot 180}}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      6. metadata-eval 28.9%

         \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      7. metadata-eval 28.9%

         \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{-180 \cdot -180}}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      8. frac-times 30.1%

         \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\color{blue}{\frac{\pi \cdot angle}{-180} \cdot \frac{\pi \cdot angle}{-180}}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      9. associate-*l/ 29.3%

         \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\color{blue}{\left(\frac{\pi}{-180} \cdot angle\right)} \cdot \frac{\pi \cdot angle}{-180}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      10. associate-*l/ 29.3%

          \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\left(\frac{\pi}{-180} \cdot angle\right) \cdot \color{blue}{\left(\frac{\pi}{-180} \cdot angle\right)}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      11. sqrt-unprod 0.0%

          \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(\sqrt{\frac{\pi}{-180} \cdot angle} \cdot \sqrt{\frac{\pi}{-180} \cdot angle}\right)} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      12. add-sqr-sqrt 37.2%

          \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(\frac{\pi}{-180} \cdot angle\right)} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      13. expm1-log1p-u 37.2%

          \[\leadsto 2 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{\pi}{-180} \cdot angle\right)\right)\right)} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      14. add-sqr-sqrt 0.0%

          \[\leadsto 2 \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \color{blue}{\left(\sqrt{\frac{\pi}{-180} \cdot angle} \cdot \sqrt{\frac{\pi}{-180} \cdot angle}\right)}\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      15. sqrt-unprod 29.3%

          \[\leadsto 2 \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \color{blue}{\left(\sqrt{\left(\frac{\pi}{-180} \cdot angle\right) \cdot \left(\frac{\pi}{-180} \cdot angle\right)}\right)}\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   7. Applied egg-rr 30.0%

      \[\leadsto 2 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   8. Taylor expanded in angle around 0 39.7%

      \[\leadsto 2 \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{1}\right)\right) \]

   if 1.00000000000000006e140 < (/.f64 angle 180) < 1.99999999999999992e226

   1. Initial program 34.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 39.2%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 32.4%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 32.4%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 32.4%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   5. Applied egg-rr 39.2%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   6. Taylor expanded in angle around inf 46.8%

      \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 47.9%

         \[\leadsto 2 \cdot \left(\cos \color{blue}{\left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right)} \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      2. associate-*r* 40.5%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\sin \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      3. *-commutative 40.5%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\sin \left(\color{blue}{\left(angle \cdot 0.005555555555555556\right)} \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      4. *-commutative 40.5%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\sin \color{blue}{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      5. *-commutative 40.5%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \color{blue}{\left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right) \]

      6. *-commutative 40.5%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \color{blue}{\left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)}\right)\right) \]

      7. *-commutative 40.5%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\color{blue}{\left(0.005555555555555556 \cdot angle\right)} \cdot \pi\right)\right)\right) \]

      8. associate-*r* 47.9%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right) \]

   8. Simplified 47.9%

      \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} \]

   if 1.99999999999999992e226 < (/.f64 angle 180)

   1. Initial program 23.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 22.5%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 17.4%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 17.4%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 22.6%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   5. Applied egg-rr 22.5%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\sqrt{\frac{\pi}{-180} \cdot angle} \cdot \sqrt{\frac{\pi}{-180} \cdot angle}\right)}\right)\right) \]

      2. sqrt-unprod 0.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\sqrt{\left(\frac{\pi}{-180} \cdot angle\right) \cdot \left(\frac{\pi}{-180} \cdot angle\right)}\right)}\right)\right) \]

      3. associate-*l/ 0.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\sqrt{\color{blue}{\frac{\pi \cdot angle}{-180}} \cdot \left(\frac{\pi}{-180} \cdot angle\right)}\right)\right)\right) \]

      4. associate-*l/ 0.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\sqrt{\frac{\pi \cdot angle}{-180} \cdot \color{blue}{\frac{\pi \cdot angle}{-180}}}\right)\right)\right) \]

      5. frac-times 0.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\sqrt{\color{blue}{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{-180 \cdot -180}}}\right)\right)\right) \]

      6. metadata-eval 0.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)\right)\right) \]

      7. metadata-eval 0.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{180 \cdot 180}}}\right)\right)\right) \]

      8. frac-times 0.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\sqrt{\color{blue}{\frac{\pi \cdot angle}{180} \cdot \frac{\pi \cdot angle}{180}}}\right)\right)\right) \]

      9. associate-*r/ 0.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\sqrt{\color{blue}{\left(\pi \cdot \frac{angle}{180}\right)} \cdot \frac{\pi \cdot angle}{180}}\right)\right)\right) \]

      10. associate-*r/ 0.0%

          \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\sqrt{\left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\pi \cdot \frac{angle}{180}\right)}}\right)\right)\right) \]

      11. sqrt-unprod 38.5%

          \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\sqrt{\pi \cdot \frac{angle}{180}} \cdot \sqrt{\pi \cdot \frac{angle}{180}}\right)}\right)\right) \]

      12. add-sqr-sqrt 23.8%

          \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\pi \cdot \frac{angle}{180}\right)}\right)\right) \]

      13. add-sqr-sqrt 20.0%

          \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(\sqrt{\cos \left(\pi \cdot \frac{angle}{180}\right)} \cdot \sqrt{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right)}\right)\right) \]

      14. sqrt-unprod 33.2%

          \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\sqrt{\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)}}\right)\right) \]

      15. pow2 33.2%

          \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sqrt{\color{blue}{{\cos \left(\pi \cdot \frac{angle}{180}\right)}^{2}}}\right)\right) \]

   7. Applied egg-rr 33.2%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\sqrt{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}}}\right)\right) \]
   8. Step-by-step derivation

      1. unpow2 33.2%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sqrt{\color{blue}{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}}\right)\right) \]

      2. rem-sqrt-square 33.2%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left|\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right|}\right)\right) \]

   9. Simplified 33.2%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left|\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right|}\right)\right) \]
3. Recombined 6 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\left(b \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) + a \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+60}:\\ \;\;\;\;2 \cdot \left(\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+84}:\\ \;\;\;\;2 \cdot \left(\left(\cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+140}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+226}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left|\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right|\right)\right)\\ \end{array} \]

Alternative 9: 67.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(angle_m \cdot 0.005555555555555556\right)\\ t_1 := \sin t_0\\ t_2 := \left(b_m - a_m\right) \cdot \left(b_m + a_m\right)\\ t_3 := \pi \cdot \left(angle_m \cdot \left(b_m - a_m\right)\right)\\ t_4 := \cos \left(angle_m \cdot \frac{\pi}{-180}\right)\\ angle_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle_m}{180} \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\left(b_m \cdot t_3 + a_m \cdot t_3\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 5 \cdot 10^{+60}:\\ \;\;\;\;2 \cdot \left(\left(t_2 \cdot t_4\right) \cdot t_1\right)\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 5 \cdot 10^{+84}:\\ \;\;\;\;t_4 \cdot \left(\left({a_m}^{2} - {b_m}^{2}\right) \cdot \left(2 \cdot t_1\right)\right)\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 10^{+140}:\\ \;\;\;\;2 \cdot \left(t_2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(t_1\right)\right)\right)\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 2 \cdot 10^{+226}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \left(angle_m \cdot -0.005555555555555556\right)\right) \cdot \left(t_2 \cdot \sin \left(0.005555555555555556 \cdot \left(angle_m \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{angle_m}{180} \cdot \pi\right) \cdot \left(t_2 \cdot \left|\cos t_0\right|\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m 0.005555555555555556)))
        (t_1 (sin t_0))
        (t_2 (* (- b_m a_m) (+ b_m a_m)))
        (t_3 (* PI (* angle_m (- b_m a_m))))
        (t_4 (cos (* angle_m (/ PI -180.0)))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 2e-42)
      (* (+ (* b_m t_3) (* a_m t_3)) 0.011111111111111112)
      (if (<= (/ angle_m 180.0) 5e+60)
        (* 2.0 (* (* t_2 t_4) t_1))
        (if (<= (/ angle_m 180.0) 5e+84)
          (* t_4 (* (- (pow a_m 2.0) (pow b_m 2.0)) (* 2.0 t_1)))
          (if (<= (/ angle_m 180.0) 1e+140)
            (* 2.0 (* t_2 (expm1 (log1p t_1))))
            (if (<= (/ angle_m 180.0) 2e+226)
              (*
               2.0
               (*
                (cos (* PI (* angle_m -0.005555555555555556)))
                (* t_2 (sin (* 0.005555555555555556 (* angle_m PI))))))
              (*
               2.0
               (*
                (sin (* (/ angle_m 180.0) PI))
                (* t_2 (fabs (cos t_0)))))))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m * 0.005555555555555556);
	double t_1 = sin(t_0);
	double t_2 = (b_m - a_m) * (b_m + a_m);
	double t_3 = ((double) M_PI) * (angle_m * (b_m - a_m));
	double t_4 = cos((angle_m * (((double) M_PI) / -180.0)));
	double tmp;
	if ((angle_m / 180.0) <= 2e-42) {
		tmp = ((b_m * t_3) + (a_m * t_3)) * 0.011111111111111112;
	} else if ((angle_m / 180.0) <= 5e+60) {
		tmp = 2.0 * ((t_2 * t_4) * t_1);
	} else if ((angle_m / 180.0) <= 5e+84) {
		tmp = t_4 * ((pow(a_m, 2.0) - pow(b_m, 2.0)) * (2.0 * t_1));
	} else if ((angle_m / 180.0) <= 1e+140) {
		tmp = 2.0 * (t_2 * expm1(log1p(t_1)));
	} else if ((angle_m / 180.0) <= 2e+226) {
		tmp = 2.0 * (cos((((double) M_PI) * (angle_m * -0.005555555555555556))) * (t_2 * sin((0.005555555555555556 * (angle_m * ((double) M_PI))))));
	} else {
		tmp = 2.0 * (sin(((angle_m / 180.0) * ((double) M_PI))) * (t_2 * fabs(cos(t_0))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = Math.PI * (angle_m * 0.005555555555555556);
	double t_1 = Math.sin(t_0);
	double t_2 = (b_m - a_m) * (b_m + a_m);
	double t_3 = Math.PI * (angle_m * (b_m - a_m));
	double t_4 = Math.cos((angle_m * (Math.PI / -180.0)));
	double tmp;
	if ((angle_m / 180.0) <= 2e-42) {
		tmp = ((b_m * t_3) + (a_m * t_3)) * 0.011111111111111112;
	} else if ((angle_m / 180.0) <= 5e+60) {
		tmp = 2.0 * ((t_2 * t_4) * t_1);
	} else if ((angle_m / 180.0) <= 5e+84) {
		tmp = t_4 * ((Math.pow(a_m, 2.0) - Math.pow(b_m, 2.0)) * (2.0 * t_1));
	} else if ((angle_m / 180.0) <= 1e+140) {
		tmp = 2.0 * (t_2 * Math.expm1(Math.log1p(t_1)));
	} else if ((angle_m / 180.0) <= 2e+226) {
		tmp = 2.0 * (Math.cos((Math.PI * (angle_m * -0.005555555555555556))) * (t_2 * Math.sin((0.005555555555555556 * (angle_m * Math.PI)))));
	} else {
		tmp = 2.0 * (Math.sin(((angle_m / 180.0) * Math.PI)) * (t_2 * Math.abs(Math.cos(t_0))));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	t_0 = math.pi * (angle_m * 0.005555555555555556)
	t_1 = math.sin(t_0)
	t_2 = (b_m - a_m) * (b_m + a_m)
	t_3 = math.pi * (angle_m * (b_m - a_m))
	t_4 = math.cos((angle_m * (math.pi / -180.0)))
	tmp = 0
	if (angle_m / 180.0) <= 2e-42:
		tmp = ((b_m * t_3) + (a_m * t_3)) * 0.011111111111111112
	elif (angle_m / 180.0) <= 5e+60:
		tmp = 2.0 * ((t_2 * t_4) * t_1)
	elif (angle_m / 180.0) <= 5e+84:
		tmp = t_4 * ((math.pow(a_m, 2.0) - math.pow(b_m, 2.0)) * (2.0 * t_1))
	elif (angle_m / 180.0) <= 1e+140:
		tmp = 2.0 * (t_2 * math.expm1(math.log1p(t_1)))
	elif (angle_m / 180.0) <= 2e+226:
		tmp = 2.0 * (math.cos((math.pi * (angle_m * -0.005555555555555556))) * (t_2 * math.sin((0.005555555555555556 * (angle_m * math.pi)))))
	else:
		tmp = 2.0 * (math.sin(((angle_m / 180.0) * math.pi)) * (t_2 * math.fabs(math.cos(t_0))))
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(pi * Float64(angle_m * 0.005555555555555556))
	t_1 = sin(t_0)
	t_2 = Float64(Float64(b_m - a_m) * Float64(b_m + a_m))
	t_3 = Float64(pi * Float64(angle_m * Float64(b_m - a_m)))
	t_4 = cos(Float64(angle_m * Float64(pi / -180.0)))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e-42)
		tmp = Float64(Float64(Float64(b_m * t_3) + Float64(a_m * t_3)) * 0.011111111111111112);
	elseif (Float64(angle_m / 180.0) <= 5e+60)
		tmp = Float64(2.0 * Float64(Float64(t_2 * t_4) * t_1));
	elseif (Float64(angle_m / 180.0) <= 5e+84)
		tmp = Float64(t_4 * Float64(Float64((a_m ^ 2.0) - (b_m ^ 2.0)) * Float64(2.0 * t_1)));
	elseif (Float64(angle_m / 180.0) <= 1e+140)
		tmp = Float64(2.0 * Float64(t_2 * expm1(log1p(t_1))));
	elseif (Float64(angle_m / 180.0) <= 2e+226)
		tmp = Float64(2.0 * Float64(cos(Float64(pi * Float64(angle_m * -0.005555555555555556))) * Float64(t_2 * sin(Float64(0.005555555555555556 * Float64(angle_m * pi))))));
	else
		tmp = Float64(2.0 * Float64(sin(Float64(Float64(angle_m / 180.0) * pi)) * Float64(t_2 * abs(cos(t_0)))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(b$95$m + a$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(Pi * N[(angle$95$m * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Cos[N[(angle$95$m * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e-42], N[(N[(N[(b$95$m * t$95$3), $MachinePrecision] + N[(a$95$m * t$95$3), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+60], N[(2.0 * N[(N[(t$95$2 * t$95$4), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+84], N[(t$95$4 * N[(N[(N[Power[a$95$m, 2.0], $MachinePrecision] - N[Power[b$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+140], N[(2.0 * N[(t$95$2 * N[(Exp[N[Log[1 + t$95$1], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+226], N[(2.0 * N[(N[Cos[N[(Pi * N[(angle$95$m * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$2 * N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(t$95$2 * N[Abs[N[Cos[t$95$0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 angle 180) < 2.00000000000000008e-42

   1. Initial program 61.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 61.9%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   3. Simplified 61.9%

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0 58.7%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. unpow2 58.7%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 58.7%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 63.2%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   6. Applied egg-rr 63.2%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   7. Taylor expanded in angle around 0 63.2%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative 63.2%

         \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112} \]

      2. associate-*r* 63.3%

         \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot 0.011111111111111112 \]

      3. *-commutative 63.3%

         \[\leadsto \left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

      4. associate-*l* 63.2%

         \[\leadsto \color{blue}{\left(\pi \cdot angle\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\right)} \]

      5. +-commutative 63.2%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\right) \]

      6. difference-of-squares 58.7%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\color{blue}{\left(b \cdot b - a \cdot a\right)} \cdot 0.011111111111111112\right) \]

      7. unpow2 58.7%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\color{blue}{{b}^{2}} - a \cdot a\right) \cdot 0.011111111111111112\right) \]

      8. unpow2 58.7%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left({b}^{2} - \color{blue}{{a}^{2}}\right) \cdot 0.011111111111111112\right) \]

      9. associate-*l* 58.7%

         \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot 0.011111111111111112} \]

      10. *-commutative 58.7%

          \[\leadsto \left(\color{blue}{\left(angle \cdot \pi\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot 0.011111111111111112 \]

      11. unpow2 58.7%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot 0.011111111111111112 \]

      12. unpow2 58.7%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot 0.011111111111111112 \]

      13. difference-of-squares 63.3%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot 0.011111111111111112 \]

      14. *-commutative 63.3%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(b + a\right)\right)}\right) \cdot 0.011111111111111112 \]

      15. +-commutative 63.3%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(a + b\right)}\right)\right) \cdot 0.011111111111111112 \]

   9. Simplified 63.3%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112} \]
   10. Step-by-step derivation

       1. associate-*r* 74.6%

          \[\leadsto \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot \left(a + b\right)\right)} \cdot 0.011111111111111112 \]

       2. +-commutative 74.6%

          \[\leadsto \left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot 0.011111111111111112 \]

       3. distribute-lft-in 70.5%

          \[\leadsto \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right)} \cdot 0.011111111111111112 \]

       4. *-commutative 70.5%

          \[\leadsto \left(\left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(b - a\right)\right) \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       5. associate-*l* 70.5%

          \[\leadsto \left(\color{blue}{\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)} \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       6. *-commutative 70.5%

          \[\leadsto \left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       7. associate-*l* 70.6%

          \[\leadsto \left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \color{blue}{\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)} \cdot a\right) \cdot 0.011111111111111112 \]

   11. Applied egg-rr 70.6%

       \[\leadsto \color{blue}{\left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot a\right)} \cdot 0.011111111111111112 \]

   if 2.00000000000000008e-42 < (/.f64 angle 180) < 4.99999999999999975e60

   1. Initial program 73.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 73.7%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 53.8%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 53.8%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 57.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   5. Applied egg-rr 77.4%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   6. Taylor expanded in angle around inf 76.7%

      \[\leadsto 2 \cdot \left(\color{blue}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r* 78.0%

         \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      2. *-commutative 78.0%

         \[\leadsto 2 \cdot \left(\sin \left(\color{blue}{\left(angle \cdot 0.005555555555555556\right)} \cdot \pi\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      3. *-commutative 78.0%

         \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   8. Simplified 78.0%

      \[\leadsto 2 \cdot \left(\color{blue}{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   if 4.99999999999999975e60 < (/.f64 angle 180) < 5.0000000000000001e84

   1. Initial program 0.0%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 0.0%

      \[\leadsto \color{blue}{\cos \left(\frac{\pi}{-180} \cdot angle\right) \cdot \left(\sin \left(\frac{\pi}{-180} \cdot angle\right) \cdot \left(2 \cdot \left({a}^{2} - {b}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 0.0%

         \[\leadsto \cos \left(\frac{\pi}{-180} \cdot angle\right) \cdot \color{blue}{\left(\left(\sin \left(\frac{\pi}{-180} \cdot angle\right) \cdot 2\right) \cdot \left({a}^{2} - {b}^{2}\right)\right)} \]

      2. sub-neg 0.0%

         \[\leadsto \cos \left(\frac{\pi}{-180} \cdot angle\right) \cdot \left(\left(\sin \left(\frac{\pi}{-180} \cdot angle\right) \cdot 2\right) \cdot \color{blue}{\left({a}^{2} + \left(-{b}^{2}\right)\right)}\right) \]

      3. distribute-lft-in 0.0%

         \[\leadsto \cos \left(\frac{\pi}{-180} \cdot angle\right) \cdot \color{blue}{\left(\left(\sin \left(\frac{\pi}{-180} \cdot angle\right) \cdot 2\right) \cdot {a}^{2} + \left(\sin \left(\frac{\pi}{-180} \cdot angle\right) \cdot 2\right) \cdot \left(-{b}^{2}\right)\right)} \]

   5. Applied egg-rr 73.1%

      \[\leadsto \cos \left(\frac{\pi}{-180} \cdot angle\right) \cdot \color{blue}{\left(\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right) \cdot {a}^{2} + \left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right) \cdot \left(-{b}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-out 73.1%

         \[\leadsto \cos \left(\frac{\pi}{-180} \cdot angle\right) \cdot \color{blue}{\left(\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right) \cdot \left({a}^{2} + \left(-{b}^{2}\right)\right)\right)} \]

      2. sub-neg 73.1%

         \[\leadsto \cos \left(\frac{\pi}{-180} \cdot angle\right) \cdot \left(\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right) \cdot \color{blue}{\left({a}^{2} - {b}^{2}\right)}\right) \]

      3. *-commutative 73.1%

         \[\leadsto \cos \left(\frac{\pi}{-180} \cdot angle\right) \cdot \color{blue}{\left(\left({a}^{2} - {b}^{2}\right) \cdot \left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right)} \]

      4. *-commutative 73.1%

         \[\leadsto \cos \left(\frac{\pi}{-180} \cdot angle\right) \cdot \left(\left({a}^{2} - {b}^{2}\right) \cdot \color{blue}{\left(2 \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right) \]

   7. Simplified 73.1%

      \[\leadsto \cos \left(\frac{\pi}{-180} \cdot angle\right) \cdot \color{blue}{\left(\left({a}^{2} - {b}^{2}\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} \]

   if 5.0000000000000001e84 < (/.f64 angle 180) < 1.00000000000000006e140

   1. Initial program 28.6%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 30.0%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 41.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 41.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 41.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   5. Applied egg-rr 30.0%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 38.2%

         \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(\sqrt{\pi \cdot \frac{angle}{180}} \cdot \sqrt{\pi \cdot \frac{angle}{180}}\right)} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      2. sqrt-unprod 30.0%

         \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(\sqrt{\left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\pi \cdot \frac{angle}{180}\right)}\right)} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      3. associate-*r/ 30.1%

         \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\color{blue}{\frac{\pi \cdot angle}{180}} \cdot \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      4. associate-*r/ 30.1%

         \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\frac{\pi \cdot angle}{180} \cdot \color{blue}{\frac{\pi \cdot angle}{180}}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      5. frac-times 28.9%

         \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\color{blue}{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{180 \cdot 180}}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      6. metadata-eval 28.9%

         \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      7. metadata-eval 28.9%

         \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{-180 \cdot -180}}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      8. frac-times 30.1%

         \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\color{blue}{\frac{\pi \cdot angle}{-180} \cdot \frac{\pi \cdot angle}{-180}}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      9. associate-*l/ 29.3%

         \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\color{blue}{\left(\frac{\pi}{-180} \cdot angle\right)} \cdot \frac{\pi \cdot angle}{-180}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      10. associate-*l/ 29.3%

          \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\left(\frac{\pi}{-180} \cdot angle\right) \cdot \color{blue}{\left(\frac{\pi}{-180} \cdot angle\right)}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      11. sqrt-unprod 0.0%

          \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(\sqrt{\frac{\pi}{-180} \cdot angle} \cdot \sqrt{\frac{\pi}{-180} \cdot angle}\right)} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      12. add-sqr-sqrt 37.2%

          \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(\frac{\pi}{-180} \cdot angle\right)} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      13. expm1-log1p-u 37.2%

          \[\leadsto 2 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{\pi}{-180} \cdot angle\right)\right)\right)} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      14. add-sqr-sqrt 0.0%

          \[\leadsto 2 \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \color{blue}{\left(\sqrt{\frac{\pi}{-180} \cdot angle} \cdot \sqrt{\frac{\pi}{-180} \cdot angle}\right)}\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      15. sqrt-unprod 29.3%

          \[\leadsto 2 \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \color{blue}{\left(\sqrt{\left(\frac{\pi}{-180} \cdot angle\right) \cdot \left(\frac{\pi}{-180} \cdot angle\right)}\right)}\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   7. Applied egg-rr 30.0%

      \[\leadsto 2 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   8. Taylor expanded in angle around 0 39.7%

      \[\leadsto 2 \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{1}\right)\right) \]

   if 1.00000000000000006e140 < (/.f64 angle 180) < 1.99999999999999992e226

   1. Initial program 34.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 39.2%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 32.4%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 32.4%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 32.4%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   5. Applied egg-rr 39.2%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   6. Taylor expanded in angle around inf 46.8%

      \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 47.9%

         \[\leadsto 2 \cdot \left(\cos \color{blue}{\left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right)} \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      2. associate-*r* 40.5%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\sin \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      3. *-commutative 40.5%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\sin \left(\color{blue}{\left(angle \cdot 0.005555555555555556\right)} \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      4. *-commutative 40.5%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\sin \color{blue}{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      5. *-commutative 40.5%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \color{blue}{\left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right) \]

      6. *-commutative 40.5%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \color{blue}{\left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)}\right)\right) \]

      7. *-commutative 40.5%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\color{blue}{\left(0.005555555555555556 \cdot angle\right)} \cdot \pi\right)\right)\right) \]

      8. associate-*r* 47.9%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right) \]

   8. Simplified 47.9%

      \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} \]

   if 1.99999999999999992e226 < (/.f64 angle 180)

   1. Initial program 23.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 22.5%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 17.4%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 17.4%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 22.6%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   5. Applied egg-rr 22.5%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\sqrt{\frac{\pi}{-180} \cdot angle} \cdot \sqrt{\frac{\pi}{-180} \cdot angle}\right)}\right)\right) \]

      2. sqrt-unprod 0.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\sqrt{\left(\frac{\pi}{-180} \cdot angle\right) \cdot \left(\frac{\pi}{-180} \cdot angle\right)}\right)}\right)\right) \]

      3. associate-*l/ 0.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\sqrt{\color{blue}{\frac{\pi \cdot angle}{-180}} \cdot \left(\frac{\pi}{-180} \cdot angle\right)}\right)\right)\right) \]

      4. associate-*l/ 0.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\sqrt{\frac{\pi \cdot angle}{-180} \cdot \color{blue}{\frac{\pi \cdot angle}{-180}}}\right)\right)\right) \]

      5. frac-times 0.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\sqrt{\color{blue}{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{-180 \cdot -180}}}\right)\right)\right) \]

      6. metadata-eval 0.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)\right)\right) \]

      7. metadata-eval 0.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{180 \cdot 180}}}\right)\right)\right) \]

      8. frac-times 0.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\sqrt{\color{blue}{\frac{\pi \cdot angle}{180} \cdot \frac{\pi \cdot angle}{180}}}\right)\right)\right) \]

      9. associate-*r/ 0.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\sqrt{\color{blue}{\left(\pi \cdot \frac{angle}{180}\right)} \cdot \frac{\pi \cdot angle}{180}}\right)\right)\right) \]

      10. associate-*r/ 0.0%

          \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\sqrt{\left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\pi \cdot \frac{angle}{180}\right)}}\right)\right)\right) \]

      11. sqrt-unprod 38.5%

          \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\sqrt{\pi \cdot \frac{angle}{180}} \cdot \sqrt{\pi \cdot \frac{angle}{180}}\right)}\right)\right) \]

      12. add-sqr-sqrt 23.8%

          \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\pi \cdot \frac{angle}{180}\right)}\right)\right) \]

      13. add-sqr-sqrt 20.0%

          \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(\sqrt{\cos \left(\pi \cdot \frac{angle}{180}\right)} \cdot \sqrt{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right)}\right)\right) \]

      14. sqrt-unprod 33.2%

          \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\sqrt{\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)}}\right)\right) \]

      15. pow2 33.2%

          \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sqrt{\color{blue}{{\cos \left(\pi \cdot \frac{angle}{180}\right)}^{2}}}\right)\right) \]

   7. Applied egg-rr 33.2%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\sqrt{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}}}\right)\right) \]
   8. Step-by-step derivation

      1. unpow2 33.2%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sqrt{\color{blue}{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}}\right)\right) \]

      2. rem-sqrt-square 33.2%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left|\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right|}\right)\right) \]

   9. Simplified 33.2%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left|\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right|}\right)\right) \]
3. Recombined 6 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\left(b \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) + a \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+60}:\\ \;\;\;\;2 \cdot \left(\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+84}:\\ \;\;\;\;\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\left({a}^{2} - {b}^{2}\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+140}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+226}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left|\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right|\right)\right)\\ \end{array} \]

Alternative 10: 67.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(angle_m \cdot \left(b_m - a_m\right)\right)\\ t_1 := \frac{angle_m}{180} \cdot \pi\\ t_2 := \sin t_1\\ t_3 := \left(b_m - a_m\right) \cdot \left(b_m + a_m\right)\\ angle_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle_m}{180} \leq 4 \cdot 10^{-30}:\\ \;\;\;\;\left(b_m \cdot t_0 + a_m \cdot t_0\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 4 \cdot 10^{+75}:\\ \;\;\;\;\left(2 \cdot t_3\right) \cdot \left(\cos t_1 \cdot t_2\right)\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 5 \cdot 10^{+116}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle_m \cdot \left(\pi \cdot \left|t_3\right|\right)\right)\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 10^{+258}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \left(angle_m \cdot -0.005555555555555556\right)\right) \cdot \left(t_3 \cdot \sin \left(0.005555555555555556 \cdot \left(angle_m \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_2 \cdot \left(t_3 \cdot \left|\cos \left(\pi \cdot \left(angle_m \cdot 0.005555555555555556\right)\right)\right|\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m (- b_m a_m))))
        (t_1 (* (/ angle_m 180.0) PI))
        (t_2 (sin t_1))
        (t_3 (* (- b_m a_m) (+ b_m a_m))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 4e-30)
      (* (+ (* b_m t_0) (* a_m t_0)) 0.011111111111111112)
      (if (<= (/ angle_m 180.0) 4e+75)
        (* (* 2.0 t_3) (* (cos t_1) t_2))
        (if (<= (/ angle_m 180.0) 5e+116)
          (* 0.011111111111111112 (* angle_m (* PI (fabs t_3))))
          (if (<= (/ angle_m 180.0) 1e+258)
            (*
             2.0
             (*
              (cos (* PI (* angle_m -0.005555555555555556)))
              (* t_3 (sin (* 0.005555555555555556 (* angle_m PI))))))
            (*
             2.0
             (*
              t_2
              (*
               t_3
               (fabs (cos (* PI (* angle_m 0.005555555555555556))))))))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m * (b_m - a_m));
	double t_1 = (angle_m / 180.0) * ((double) M_PI);
	double t_2 = sin(t_1);
	double t_3 = (b_m - a_m) * (b_m + a_m);
	double tmp;
	if ((angle_m / 180.0) <= 4e-30) {
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112;
	} else if ((angle_m / 180.0) <= 4e+75) {
		tmp = (2.0 * t_3) * (cos(t_1) * t_2);
	} else if ((angle_m / 180.0) <= 5e+116) {
		tmp = 0.011111111111111112 * (angle_m * (((double) M_PI) * fabs(t_3)));
	} else if ((angle_m / 180.0) <= 1e+258) {
		tmp = 2.0 * (cos((((double) M_PI) * (angle_m * -0.005555555555555556))) * (t_3 * sin((0.005555555555555556 * (angle_m * ((double) M_PI))))));
	} else {
		tmp = 2.0 * (t_2 * (t_3 * fabs(cos((((double) M_PI) * (angle_m * 0.005555555555555556))))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = Math.PI * (angle_m * (b_m - a_m));
	double t_1 = (angle_m / 180.0) * Math.PI;
	double t_2 = Math.sin(t_1);
	double t_3 = (b_m - a_m) * (b_m + a_m);
	double tmp;
	if ((angle_m / 180.0) <= 4e-30) {
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112;
	} else if ((angle_m / 180.0) <= 4e+75) {
		tmp = (2.0 * t_3) * (Math.cos(t_1) * t_2);
	} else if ((angle_m / 180.0) <= 5e+116) {
		tmp = 0.011111111111111112 * (angle_m * (Math.PI * Math.abs(t_3)));
	} else if ((angle_m / 180.0) <= 1e+258) {
		tmp = 2.0 * (Math.cos((Math.PI * (angle_m * -0.005555555555555556))) * (t_3 * Math.sin((0.005555555555555556 * (angle_m * Math.PI)))));
	} else {
		tmp = 2.0 * (t_2 * (t_3 * Math.abs(Math.cos((Math.PI * (angle_m * 0.005555555555555556))))));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	t_0 = math.pi * (angle_m * (b_m - a_m))
	t_1 = (angle_m / 180.0) * math.pi
	t_2 = math.sin(t_1)
	t_3 = (b_m - a_m) * (b_m + a_m)
	tmp = 0
	if (angle_m / 180.0) <= 4e-30:
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112
	elif (angle_m / 180.0) <= 4e+75:
		tmp = (2.0 * t_3) * (math.cos(t_1) * t_2)
	elif (angle_m / 180.0) <= 5e+116:
		tmp = 0.011111111111111112 * (angle_m * (math.pi * math.fabs(t_3)))
	elif (angle_m / 180.0) <= 1e+258:
		tmp = 2.0 * (math.cos((math.pi * (angle_m * -0.005555555555555556))) * (t_3 * math.sin((0.005555555555555556 * (angle_m * math.pi)))))
	else:
		tmp = 2.0 * (t_2 * (t_3 * math.fabs(math.cos((math.pi * (angle_m * 0.005555555555555556))))))
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(pi * Float64(angle_m * Float64(b_m - a_m)))
	t_1 = Float64(Float64(angle_m / 180.0) * pi)
	t_2 = sin(t_1)
	t_3 = Float64(Float64(b_m - a_m) * Float64(b_m + a_m))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e-30)
		tmp = Float64(Float64(Float64(b_m * t_0) + Float64(a_m * t_0)) * 0.011111111111111112);
	elseif (Float64(angle_m / 180.0) <= 4e+75)
		tmp = Float64(Float64(2.0 * t_3) * Float64(cos(t_1) * t_2));
	elseif (Float64(angle_m / 180.0) <= 5e+116)
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * abs(t_3))));
	elseif (Float64(angle_m / 180.0) <= 1e+258)
		tmp = Float64(2.0 * Float64(cos(Float64(pi * Float64(angle_m * -0.005555555555555556))) * Float64(t_3 * sin(Float64(0.005555555555555556 * Float64(angle_m * pi))))));
	else
		tmp = Float64(2.0 * Float64(t_2 * Float64(t_3 * abs(cos(Float64(pi * Float64(angle_m * 0.005555555555555556)))))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	t_0 = pi * (angle_m * (b_m - a_m));
	t_1 = (angle_m / 180.0) * pi;
	t_2 = sin(t_1);
	t_3 = (b_m - a_m) * (b_m + a_m);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 4e-30)
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112;
	elseif ((angle_m / 180.0) <= 4e+75)
		tmp = (2.0 * t_3) * (cos(t_1) * t_2);
	elseif ((angle_m / 180.0) <= 5e+116)
		tmp = 0.011111111111111112 * (angle_m * (pi * abs(t_3)));
	elseif ((angle_m / 180.0) <= 1e+258)
		tmp = 2.0 * (cos((pi * (angle_m * -0.005555555555555556))) * (t_3 * sin((0.005555555555555556 * (angle_m * pi)))));
	else
		tmp = 2.0 * (t_2 * (t_3 * abs(cos((pi * (angle_m * 0.005555555555555556))))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(b$95$m + a$95$m), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e-30], N[(N[(N[(b$95$m * t$95$0), $MachinePrecision] + N[(a$95$m * t$95$0), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+75], N[(N[(2.0 * t$95$3), $MachinePrecision] * N[(N[Cos[t$95$1], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+116], N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[Abs[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+258], N[(2.0 * N[(N[Cos[N[(Pi * N[(angle$95$m * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$3 * N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$2 * N[(t$95$3 * N[Abs[N[Cos[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 angle 180) < 4e-30

   1. Initial program 62.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 62.5%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   3. Simplified 62.5%

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0 59.4%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. unpow2 59.4%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 59.4%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 63.8%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   6. Applied egg-rr 63.8%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   7. Taylor expanded in angle around 0 63.8%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative 63.8%

         \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112} \]

      2. associate-*r* 63.9%

         \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot 0.011111111111111112 \]

      3. *-commutative 63.9%

         \[\leadsto \left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

      4. associate-*l* 63.8%

         \[\leadsto \color{blue}{\left(\pi \cdot angle\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\right)} \]

      5. +-commutative 63.8%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\right) \]

      6. difference-of-squares 59.4%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\color{blue}{\left(b \cdot b - a \cdot a\right)} \cdot 0.011111111111111112\right) \]

      7. unpow2 59.4%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\color{blue}{{b}^{2}} - a \cdot a\right) \cdot 0.011111111111111112\right) \]

      8. unpow2 59.4%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left({b}^{2} - \color{blue}{{a}^{2}}\right) \cdot 0.011111111111111112\right) \]

      9. associate-*l* 59.4%

         \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot 0.011111111111111112} \]

      10. *-commutative 59.4%

          \[\leadsto \left(\color{blue}{\left(angle \cdot \pi\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot 0.011111111111111112 \]

      11. unpow2 59.4%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot 0.011111111111111112 \]

      12. unpow2 59.4%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot 0.011111111111111112 \]

      13. difference-of-squares 63.9%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot 0.011111111111111112 \]

      14. *-commutative 63.9%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(b + a\right)\right)}\right) \cdot 0.011111111111111112 \]

      15. +-commutative 63.9%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(a + b\right)}\right)\right) \cdot 0.011111111111111112 \]

   9. Simplified 63.9%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112} \]
   10. Step-by-step derivation

       1. associate-*r* 75.0%

          \[\leadsto \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot \left(a + b\right)\right)} \cdot 0.011111111111111112 \]

       2. +-commutative 75.0%

          \[\leadsto \left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot 0.011111111111111112 \]

       3. distribute-lft-in 71.0%

          \[\leadsto \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right)} \cdot 0.011111111111111112 \]

       4. *-commutative 71.0%

          \[\leadsto \left(\left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(b - a\right)\right) \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       5. associate-*l* 71.0%

          \[\leadsto \left(\color{blue}{\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)} \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       6. *-commutative 71.0%

          \[\leadsto \left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       7. associate-*l* 71.1%

          \[\leadsto \left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \color{blue}{\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)} \cdot a\right) \cdot 0.011111111111111112 \]

   11. Applied egg-rr 71.1%

       \[\leadsto \color{blue}{\left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot a\right)} \cdot 0.011111111111111112 \]

   if 4e-30 < (/.f64 angle 180) < 3.99999999999999971e75

   1. Initial program 65.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 65.1%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   3. Simplified 65.1%

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 44.8%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 44.8%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 48.6%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   5. Applied egg-rr 69.0%

      \[\leadsto \left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   if 3.99999999999999971e75 < (/.f64 angle 180) < 5.00000000000000025e116

   1. Initial program 27.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 27.2%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   3. Simplified 27.2%

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0 39.6%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 38.7%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\sqrt{{b}^{2} - {a}^{2}} \cdot \sqrt{{b}^{2} - {a}^{2}}\right)}\right)\right) \]

      2. sqrt-unprod 52.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\sqrt{\left({b}^{2} - {a}^{2}\right) \cdot \left({b}^{2} - {a}^{2}\right)}}\right)\right) \]

      3. pow2 52.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \sqrt{\color{blue}{{\left({b}^{2} - {a}^{2}\right)}^{2}}}\right)\right) \]

   6. Applied egg-rr 52.5%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\sqrt{{\left({b}^{2} - {a}^{2}\right)}^{2}}}\right)\right) \]
   7. Step-by-step derivation

      1. unpow2 52.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \sqrt{\color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left({b}^{2} - {a}^{2}\right)}}\right)\right) \]

      2. rem-sqrt-square 51.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left|{b}^{2} - {a}^{2}\right|}\right)\right) \]

   8. Simplified 51.5%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left|{b}^{2} - {a}^{2}\right|}\right)\right) \]
   9. Step-by-step derivation

      1. unpow2 39.6%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 39.6%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 39.6%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   10. Applied egg-rr 51.5%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left|\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}\right|\right)\right) \]

   if 5.00000000000000025e116 < (/.f64 angle 180) < 1.00000000000000006e258

   1. Initial program 29.6%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 36.8%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 29.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 29.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 29.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   5. Applied egg-rr 36.8%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   6. Taylor expanded in angle around inf 37.6%

      \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 42.3%

         \[\leadsto 2 \cdot \left(\cos \color{blue}{\left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right)} \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      2. associate-*r* 33.6%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\sin \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      3. *-commutative 33.6%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\sin \left(\color{blue}{\left(angle \cdot 0.005555555555555556\right)} \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      4. *-commutative 33.6%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\sin \color{blue}{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      5. *-commutative 33.6%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \color{blue}{\left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right) \]

      6. *-commutative 33.6%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \color{blue}{\left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)}\right)\right) \]

      7. *-commutative 33.6%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\color{blue}{\left(0.005555555555555556 \cdot angle\right)} \cdot \pi\right)\right)\right) \]

      8. associate-*r* 42.3%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right) \]

   8. Simplified 42.3%

      \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} \]

   if 1.00000000000000006e258 < (/.f64 angle 180)

   1. Initial program 25.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 18.2%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 20.4%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 20.4%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 26.6%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   5. Applied egg-rr 18.2%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\sqrt{\frac{\pi}{-180} \cdot angle} \cdot \sqrt{\frac{\pi}{-180} \cdot angle}\right)}\right)\right) \]

      2. sqrt-unprod 0.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\sqrt{\left(\frac{\pi}{-180} \cdot angle\right) \cdot \left(\frac{\pi}{-180} \cdot angle\right)}\right)}\right)\right) \]

      3. associate-*l/ 0.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\sqrt{\color{blue}{\frac{\pi \cdot angle}{-180}} \cdot \left(\frac{\pi}{-180} \cdot angle\right)}\right)\right)\right) \]

      4. associate-*l/ 0.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\sqrt{\frac{\pi \cdot angle}{-180} \cdot \color{blue}{\frac{\pi \cdot angle}{-180}}}\right)\right)\right) \]

      5. frac-times 0.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\sqrt{\color{blue}{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{-180 \cdot -180}}}\right)\right)\right) \]

      6. metadata-eval 0.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)\right)\right) \]

      7. metadata-eval 0.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{180 \cdot 180}}}\right)\right)\right) \]

      8. frac-times 0.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\sqrt{\color{blue}{\frac{\pi \cdot angle}{180} \cdot \frac{\pi \cdot angle}{180}}}\right)\right)\right) \]

      9. associate-*r/ 0.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\sqrt{\color{blue}{\left(\pi \cdot \frac{angle}{180}\right)} \cdot \frac{\pi \cdot angle}{180}}\right)\right)\right) \]

      10. associate-*r/ 0.0%

          \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\sqrt{\left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\pi \cdot \frac{angle}{180}\right)}}\right)\right)\right) \]

      11. sqrt-unprod 37.2%

          \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\sqrt{\pi \cdot \frac{angle}{180}} \cdot \sqrt{\pi \cdot \frac{angle}{180}}\right)}\right)\right) \]

      12. add-sqr-sqrt 25.9%

          \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\pi \cdot \frac{angle}{180}\right)}\right)\right) \]

      13. add-sqr-sqrt 23.8%

          \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(\sqrt{\cos \left(\pi \cdot \frac{angle}{180}\right)} \cdot \sqrt{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right)}\right)\right) \]

      14. sqrt-unprod 31.4%

          \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\sqrt{\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)}}\right)\right) \]

      15. pow2 31.4%

          \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sqrt{\color{blue}{{\cos \left(\pi \cdot \frac{angle}{180}\right)}^{2}}}\right)\right) \]

   7. Applied egg-rr 31.4%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\sqrt{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}}}\right)\right) \]
   8. Step-by-step derivation

      1. unpow2 31.4%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sqrt{\color{blue}{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}}\right)\right) \]

      2. rem-sqrt-square 31.4%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left|\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right|}\right)\right) \]

   9. Simplified 31.4%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left|\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right|}\right)\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{-30}:\\ \;\;\;\;\left(b \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) + a \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle}{180} \leq 4 \cdot 10^{+75}:\\ \;\;\;\;\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+116}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left|\left(b - a\right) \cdot \left(b + a\right)\right|\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+258}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left|\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right|\right)\right)\\ \end{array} \]

Alternative 11: 66.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(angle_m \cdot \left(b_m - a_m\right)\right)\\ t_1 := \left(b_m - a_m\right) \cdot \left(b_m + a_m\right)\\ t_2 := 2 \cdot \left(\left(t_1 \cdot \sin \left(0.005555555555555556 \cdot \left(angle_m \cdot \pi\right)\right)\right) \cdot \cos \left(-0.005555555555555556 \cdot \left(angle_m \cdot \pi\right)\right)\right)\\ angle_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle_m}{180} \leq 5 \cdot 10^{-81}:\\ \;\;\;\;\left(b_m \cdot t_0 + a_m \cdot t_0\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 5 \cdot 10^{+60}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 5 \cdot 10^{+116}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle_m \cdot \left(\pi \cdot \left|t_1\right|\right)\right)\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 2 \cdot 10^{+221}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \sin \left(\frac{angle_m}{180} \cdot \pi\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m (- b_m a_m))))
        (t_1 (* (- b_m a_m) (+ b_m a_m)))
        (t_2
         (*
          2.0
          (*
           (* t_1 (sin (* 0.005555555555555556 (* angle_m PI))))
           (cos (* -0.005555555555555556 (* angle_m PI)))))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 5e-81)
      (* (+ (* b_m t_0) (* a_m t_0)) 0.011111111111111112)
      (if (<= (/ angle_m 180.0) 5e+60)
        t_2
        (if (<= (/ angle_m 180.0) 5e+116)
          (* 0.011111111111111112 (* angle_m (* PI (fabs t_1))))
          (if (<= (/ angle_m 180.0) 2e+221)
            t_2
            (* 2.0 (* t_1 (sin (* (/ angle_m 180.0) PI)))))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m * (b_m - a_m));
	double t_1 = (b_m - a_m) * (b_m + a_m);
	double t_2 = 2.0 * ((t_1 * sin((0.005555555555555556 * (angle_m * ((double) M_PI))))) * cos((-0.005555555555555556 * (angle_m * ((double) M_PI)))));
	double tmp;
	if ((angle_m / 180.0) <= 5e-81) {
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112;
	} else if ((angle_m / 180.0) <= 5e+60) {
		tmp = t_2;
	} else if ((angle_m / 180.0) <= 5e+116) {
		tmp = 0.011111111111111112 * (angle_m * (((double) M_PI) * fabs(t_1)));
	} else if ((angle_m / 180.0) <= 2e+221) {
		tmp = t_2;
	} else {
		tmp = 2.0 * (t_1 * sin(((angle_m / 180.0) * ((double) M_PI))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = Math.PI * (angle_m * (b_m - a_m));
	double t_1 = (b_m - a_m) * (b_m + a_m);
	double t_2 = 2.0 * ((t_1 * Math.sin((0.005555555555555556 * (angle_m * Math.PI)))) * Math.cos((-0.005555555555555556 * (angle_m * Math.PI))));
	double tmp;
	if ((angle_m / 180.0) <= 5e-81) {
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112;
	} else if ((angle_m / 180.0) <= 5e+60) {
		tmp = t_2;
	} else if ((angle_m / 180.0) <= 5e+116) {
		tmp = 0.011111111111111112 * (angle_m * (Math.PI * Math.abs(t_1)));
	} else if ((angle_m / 180.0) <= 2e+221) {
		tmp = t_2;
	} else {
		tmp = 2.0 * (t_1 * Math.sin(((angle_m / 180.0) * Math.PI)));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	t_0 = math.pi * (angle_m * (b_m - a_m))
	t_1 = (b_m - a_m) * (b_m + a_m)
	t_2 = 2.0 * ((t_1 * math.sin((0.005555555555555556 * (angle_m * math.pi)))) * math.cos((-0.005555555555555556 * (angle_m * math.pi))))
	tmp = 0
	if (angle_m / 180.0) <= 5e-81:
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112
	elif (angle_m / 180.0) <= 5e+60:
		tmp = t_2
	elif (angle_m / 180.0) <= 5e+116:
		tmp = 0.011111111111111112 * (angle_m * (math.pi * math.fabs(t_1)))
	elif (angle_m / 180.0) <= 2e+221:
		tmp = t_2
	else:
		tmp = 2.0 * (t_1 * math.sin(((angle_m / 180.0) * math.pi)))
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(pi * Float64(angle_m * Float64(b_m - a_m)))
	t_1 = Float64(Float64(b_m - a_m) * Float64(b_m + a_m))
	t_2 = Float64(2.0 * Float64(Float64(t_1 * sin(Float64(0.005555555555555556 * Float64(angle_m * pi)))) * cos(Float64(-0.005555555555555556 * Float64(angle_m * pi)))))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e-81)
		tmp = Float64(Float64(Float64(b_m * t_0) + Float64(a_m * t_0)) * 0.011111111111111112);
	elseif (Float64(angle_m / 180.0) <= 5e+60)
		tmp = t_2;
	elseif (Float64(angle_m / 180.0) <= 5e+116)
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * abs(t_1))));
	elseif (Float64(angle_m / 180.0) <= 2e+221)
		tmp = t_2;
	else
		tmp = Float64(2.0 * Float64(t_1 * sin(Float64(Float64(angle_m / 180.0) * pi))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	t_0 = pi * (angle_m * (b_m - a_m));
	t_1 = (b_m - a_m) * (b_m + a_m);
	t_2 = 2.0 * ((t_1 * sin((0.005555555555555556 * (angle_m * pi)))) * cos((-0.005555555555555556 * (angle_m * pi))));
	tmp = 0.0;
	if ((angle_m / 180.0) <= 5e-81)
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112;
	elseif ((angle_m / 180.0) <= 5e+60)
		tmp = t_2;
	elseif ((angle_m / 180.0) <= 5e+116)
		tmp = 0.011111111111111112 * (angle_m * (pi * abs(t_1)));
	elseif ((angle_m / 180.0) <= 2e+221)
		tmp = t_2;
	else
		tmp = 2.0 * (t_1 * sin(((angle_m / 180.0) * pi)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(b$95$m + a$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[(t$95$1 * N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(-0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e-81], N[(N[(N[(b$95$m * t$95$0), $MachinePrecision] + N[(a$95$m * t$95$0), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+60], t$95$2, If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+116], N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[Abs[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+221], t$95$2, N[(2.0 * N[(t$95$1 * N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle 180) < 4.99999999999999981e-81

   1. Initial program 60.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 60.3%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   3. Simplified 60.3%

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0 57.0%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. unpow2 57.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 57.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 61.7%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   6. Applied egg-rr 61.7%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   7. Taylor expanded in angle around 0 61.7%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative 61.7%

         \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112} \]

      2. associate-*r* 61.8%

         \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot 0.011111111111111112 \]

      3. *-commutative 61.8%

         \[\leadsto \left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

      4. associate-*l* 61.7%

         \[\leadsto \color{blue}{\left(\pi \cdot angle\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\right)} \]

      5. +-commutative 61.7%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\right) \]

      6. difference-of-squares 57.0%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\color{blue}{\left(b \cdot b - a \cdot a\right)} \cdot 0.011111111111111112\right) \]

      7. unpow2 57.0%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\color{blue}{{b}^{2}} - a \cdot a\right) \cdot 0.011111111111111112\right) \]

      8. unpow2 57.0%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left({b}^{2} - \color{blue}{{a}^{2}}\right) \cdot 0.011111111111111112\right) \]

      9. associate-*l* 57.0%

         \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot 0.011111111111111112} \]

      10. *-commutative 57.0%

          \[\leadsto \left(\color{blue}{\left(angle \cdot \pi\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot 0.011111111111111112 \]

      11. unpow2 57.0%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot 0.011111111111111112 \]

      12. unpow2 57.0%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot 0.011111111111111112 \]

      13. difference-of-squares 61.8%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot 0.011111111111111112 \]

      14. *-commutative 61.8%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(b + a\right)\right)}\right) \cdot 0.011111111111111112 \]

      15. +-commutative 61.8%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(a + b\right)}\right)\right) \cdot 0.011111111111111112 \]

   9. Simplified 61.8%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112} \]
   10. Step-by-step derivation

       1. associate-*r* 73.5%

          \[\leadsto \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot \left(a + b\right)\right)} \cdot 0.011111111111111112 \]

       2. +-commutative 73.5%

          \[\leadsto \left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot 0.011111111111111112 \]

       3. distribute-lft-in 69.4%

          \[\leadsto \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right)} \cdot 0.011111111111111112 \]

       4. *-commutative 69.4%

          \[\leadsto \left(\left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(b - a\right)\right) \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       5. associate-*l* 69.4%

          \[\leadsto \left(\color{blue}{\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)} \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       6. *-commutative 69.4%

          \[\leadsto \left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       7. associate-*l* 69.4%

          \[\leadsto \left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \color{blue}{\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)} \cdot a\right) \cdot 0.011111111111111112 \]

   11. Applied egg-rr 69.4%

       \[\leadsto \color{blue}{\left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot a\right)} \cdot 0.011111111111111112 \]

   if 4.99999999999999981e-81 < (/.f64 angle 180) < 4.99999999999999975e60 or 5.00000000000000025e116 < (/.f64 angle 180) < 2.0000000000000001e221

   1. Initial program 61.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 63.1%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 52.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 52.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 54.3%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   5. Applied egg-rr 64.9%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   6. Taylor expanded in angle around inf 65.3%

      \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]

   if 4.99999999999999975e60 < (/.f64 angle 180) < 5.00000000000000025e116

   1. Initial program 21.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 21.7%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   3. Simplified 21.7%

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0 32.6%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 30.9%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\sqrt{{b}^{2} - {a}^{2}} \cdot \sqrt{{b}^{2} - {a}^{2}}\right)}\right)\right) \]

      2. sqrt-unprod 52.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\sqrt{\left({b}^{2} - {a}^{2}\right) \cdot \left({b}^{2} - {a}^{2}\right)}}\right)\right) \]

      3. pow2 52.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \sqrt{\color{blue}{{\left({b}^{2} - {a}^{2}\right)}^{2}}}\right)\right) \]

   6. Applied egg-rr 52.0%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\sqrt{{\left({b}^{2} - {a}^{2}\right)}^{2}}}\right)\right) \]
   7. Step-by-step derivation

      1. unpow2 52.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \sqrt{\color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left({b}^{2} - {a}^{2}\right)}}\right)\right) \]

      2. rem-sqrt-square 51.2%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left|{b}^{2} - {a}^{2}\right|}\right)\right) \]

   8. Simplified 51.2%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left|{b}^{2} - {a}^{2}\right|}\right)\right) \]
   9. Step-by-step derivation

      1. unpow2 32.6%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 32.6%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 32.6%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   10. Applied egg-rr 51.2%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left|\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}\right|\right)\right) \]

   if 2.0000000000000001e221 < (/.f64 angle 180)

   1. Initial program 22.6%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 21.4%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 16.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 16.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 21.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   5. Applied egg-rr 21.4%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   6. Taylor expanded in angle around 0 30.5%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{1}\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-81}:\\ \;\;\;\;\left(b \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) + a \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+60}:\\ \;\;\;\;2 \cdot \left(\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+116}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left|\left(b - a\right) \cdot \left(b + a\right)\right|\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+221}:\\ \;\;\;\;2 \cdot \left(\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)\\ \end{array} \]

Alternative 12: 66.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(angle_m \cdot \left(b_m - a_m\right)\right)\\ t_1 := \left(b_m - a_m\right) \cdot \left(b_m + a_m\right)\\ t_2 := t_1 \cdot \sin \left(0.005555555555555556 \cdot \left(angle_m \cdot \pi\right)\right)\\ angle_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle_m}{180} \leq 5 \cdot 10^{-81}:\\ \;\;\;\;\left(b_m \cdot t_0 + a_m \cdot t_0\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 5 \cdot 10^{+60}:\\ \;\;\;\;2 \cdot \left(t_2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle_m \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 5 \cdot 10^{+116}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle_m \cdot \left(\pi \cdot \left|t_1\right|\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \left(angle_m \cdot -0.005555555555555556\right)\right) \cdot t_2\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m (- b_m a_m))))
        (t_1 (* (- b_m a_m) (+ b_m a_m)))
        (t_2 (* t_1 (sin (* 0.005555555555555556 (* angle_m PI))))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 5e-81)
      (* (+ (* b_m t_0) (* a_m t_0)) 0.011111111111111112)
      (if (<= (/ angle_m 180.0) 5e+60)
        (* 2.0 (* t_2 (cos (* -0.005555555555555556 (* angle_m PI)))))
        (if (<= (/ angle_m 180.0) 5e+116)
          (* 0.011111111111111112 (* angle_m (* PI (fabs t_1))))
          (* 2.0 (* (cos (* PI (* angle_m -0.005555555555555556))) t_2))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m * (b_m - a_m));
	double t_1 = (b_m - a_m) * (b_m + a_m);
	double t_2 = t_1 * sin((0.005555555555555556 * (angle_m * ((double) M_PI))));
	double tmp;
	if ((angle_m / 180.0) <= 5e-81) {
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112;
	} else if ((angle_m / 180.0) <= 5e+60) {
		tmp = 2.0 * (t_2 * cos((-0.005555555555555556 * (angle_m * ((double) M_PI)))));
	} else if ((angle_m / 180.0) <= 5e+116) {
		tmp = 0.011111111111111112 * (angle_m * (((double) M_PI) * fabs(t_1)));
	} else {
		tmp = 2.0 * (cos((((double) M_PI) * (angle_m * -0.005555555555555556))) * t_2);
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = Math.PI * (angle_m * (b_m - a_m));
	double t_1 = (b_m - a_m) * (b_m + a_m);
	double t_2 = t_1 * Math.sin((0.005555555555555556 * (angle_m * Math.PI)));
	double tmp;
	if ((angle_m / 180.0) <= 5e-81) {
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112;
	} else if ((angle_m / 180.0) <= 5e+60) {
		tmp = 2.0 * (t_2 * Math.cos((-0.005555555555555556 * (angle_m * Math.PI))));
	} else if ((angle_m / 180.0) <= 5e+116) {
		tmp = 0.011111111111111112 * (angle_m * (Math.PI * Math.abs(t_1)));
	} else {
		tmp = 2.0 * (Math.cos((Math.PI * (angle_m * -0.005555555555555556))) * t_2);
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	t_0 = math.pi * (angle_m * (b_m - a_m))
	t_1 = (b_m - a_m) * (b_m + a_m)
	t_2 = t_1 * math.sin((0.005555555555555556 * (angle_m * math.pi)))
	tmp = 0
	if (angle_m / 180.0) <= 5e-81:
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112
	elif (angle_m / 180.0) <= 5e+60:
		tmp = 2.0 * (t_2 * math.cos((-0.005555555555555556 * (angle_m * math.pi))))
	elif (angle_m / 180.0) <= 5e+116:
		tmp = 0.011111111111111112 * (angle_m * (math.pi * math.fabs(t_1)))
	else:
		tmp = 2.0 * (math.cos((math.pi * (angle_m * -0.005555555555555556))) * t_2)
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(pi * Float64(angle_m * Float64(b_m - a_m)))
	t_1 = Float64(Float64(b_m - a_m) * Float64(b_m + a_m))
	t_2 = Float64(t_1 * sin(Float64(0.005555555555555556 * Float64(angle_m * pi))))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e-81)
		tmp = Float64(Float64(Float64(b_m * t_0) + Float64(a_m * t_0)) * 0.011111111111111112);
	elseif (Float64(angle_m / 180.0) <= 5e+60)
		tmp = Float64(2.0 * Float64(t_2 * cos(Float64(-0.005555555555555556 * Float64(angle_m * pi)))));
	elseif (Float64(angle_m / 180.0) <= 5e+116)
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * abs(t_1))));
	else
		tmp = Float64(2.0 * Float64(cos(Float64(pi * Float64(angle_m * -0.005555555555555556))) * t_2));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	t_0 = pi * (angle_m * (b_m - a_m));
	t_1 = (b_m - a_m) * (b_m + a_m);
	t_2 = t_1 * sin((0.005555555555555556 * (angle_m * pi)));
	tmp = 0.0;
	if ((angle_m / 180.0) <= 5e-81)
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112;
	elseif ((angle_m / 180.0) <= 5e+60)
		tmp = 2.0 * (t_2 * cos((-0.005555555555555556 * (angle_m * pi))));
	elseif ((angle_m / 180.0) <= 5e+116)
		tmp = 0.011111111111111112 * (angle_m * (pi * abs(t_1)));
	else
		tmp = 2.0 * (cos((pi * (angle_m * -0.005555555555555556))) * t_2);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(b$95$m + a$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e-81], N[(N[(N[(b$95$m * t$95$0), $MachinePrecision] + N[(a$95$m * t$95$0), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+60], N[(2.0 * N[(t$95$2 * N[Cos[N[(-0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+116], N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[Abs[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Cos[N[(Pi * N[(angle$95$m * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle 180) < 4.99999999999999981e-81

   1. Initial program 60.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 60.3%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   3. Simplified 60.3%

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0 57.0%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. unpow2 57.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 57.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 61.7%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   6. Applied egg-rr 61.7%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   7. Taylor expanded in angle around 0 61.7%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative 61.7%

         \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112} \]

      2. associate-*r* 61.8%

         \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot 0.011111111111111112 \]

      3. *-commutative 61.8%

         \[\leadsto \left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

      4. associate-*l* 61.7%

         \[\leadsto \color{blue}{\left(\pi \cdot angle\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\right)} \]

      5. +-commutative 61.7%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\right) \]

      6. difference-of-squares 57.0%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\color{blue}{\left(b \cdot b - a \cdot a\right)} \cdot 0.011111111111111112\right) \]

      7. unpow2 57.0%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\color{blue}{{b}^{2}} - a \cdot a\right) \cdot 0.011111111111111112\right) \]

      8. unpow2 57.0%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left({b}^{2} - \color{blue}{{a}^{2}}\right) \cdot 0.011111111111111112\right) \]

      9. associate-*l* 57.0%

         \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot 0.011111111111111112} \]

      10. *-commutative 57.0%

          \[\leadsto \left(\color{blue}{\left(angle \cdot \pi\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot 0.011111111111111112 \]

      11. unpow2 57.0%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot 0.011111111111111112 \]

      12. unpow2 57.0%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot 0.011111111111111112 \]

      13. difference-of-squares 61.8%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot 0.011111111111111112 \]

      14. *-commutative 61.8%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(b + a\right)\right)}\right) \cdot 0.011111111111111112 \]

      15. +-commutative 61.8%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(a + b\right)}\right)\right) \cdot 0.011111111111111112 \]

   9. Simplified 61.8%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112} \]
   10. Step-by-step derivation

       1. associate-*r* 73.5%

          \[\leadsto \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot \left(a + b\right)\right)} \cdot 0.011111111111111112 \]

       2. +-commutative 73.5%

          \[\leadsto \left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot 0.011111111111111112 \]

       3. distribute-lft-in 69.4%

          \[\leadsto \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right)} \cdot 0.011111111111111112 \]

       4. *-commutative 69.4%

          \[\leadsto \left(\left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(b - a\right)\right) \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       5. associate-*l* 69.4%

          \[\leadsto \left(\color{blue}{\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)} \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       6. *-commutative 69.4%

          \[\leadsto \left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       7. associate-*l* 69.4%

          \[\leadsto \left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \color{blue}{\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)} \cdot a\right) \cdot 0.011111111111111112 \]

   11. Applied egg-rr 69.4%

       \[\leadsto \color{blue}{\left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot a\right)} \cdot 0.011111111111111112 \]

   if 4.99999999999999981e-81 < (/.f64 angle 180) < 4.99999999999999975e60

   1. Initial program 79.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 79.1%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 63.3%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 63.3%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 66.2%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   5. Applied egg-rr 82.0%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   6. Taylor expanded in angle around inf 79.1%

      \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]

   if 4.99999999999999975e60 < (/.f64 angle 180) < 5.00000000000000025e116

   1. Initial program 21.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 21.7%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   3. Simplified 21.7%

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0 32.6%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 30.9%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\sqrt{{b}^{2} - {a}^{2}} \cdot \sqrt{{b}^{2} - {a}^{2}}\right)}\right)\right) \]

      2. sqrt-unprod 52.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\sqrt{\left({b}^{2} - {a}^{2}\right) \cdot \left({b}^{2} - {a}^{2}\right)}}\right)\right) \]

      3. pow2 52.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \sqrt{\color{blue}{{\left({b}^{2} - {a}^{2}\right)}^{2}}}\right)\right) \]

   6. Applied egg-rr 52.0%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\sqrt{{\left({b}^{2} - {a}^{2}\right)}^{2}}}\right)\right) \]
   7. Step-by-step derivation

      1. unpow2 52.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \sqrt{\color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left({b}^{2} - {a}^{2}\right)}}\right)\right) \]

      2. rem-sqrt-square 51.2%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left|{b}^{2} - {a}^{2}\right|}\right)\right) \]

   8. Simplified 51.2%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left|{b}^{2} - {a}^{2}\right|}\right)\right) \]
   9. Step-by-step derivation

      1. unpow2 32.6%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 32.6%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 32.6%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   10. Applied egg-rr 51.2%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left|\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}\right|\right)\right) \]

   if 5.00000000000000025e116 < (/.f64 angle 180)

   1. Initial program 28.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 29.5%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 26.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 26.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 28.4%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   5. Applied egg-rr 29.5%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   6. Taylor expanded in angle around inf 32.1%

      \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 35.6%

         \[\leadsto 2 \cdot \left(\cos \color{blue}{\left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right)} \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      2. associate-*r* 30.2%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\sin \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      3. *-commutative 30.2%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\sin \left(\color{blue}{\left(angle \cdot 0.005555555555555556\right)} \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      4. *-commutative 30.2%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\sin \color{blue}{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      5. *-commutative 30.2%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \color{blue}{\left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right) \]

      6. *-commutative 30.2%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \color{blue}{\left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)}\right)\right) \]

      7. *-commutative 30.2%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\color{blue}{\left(0.005555555555555556 \cdot angle\right)} \cdot \pi\right)\right)\right) \]

      8. associate-*r* 35.6%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right) \]

   8. Simplified 35.6%

      \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-81}:\\ \;\;\;\;\left(b \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) + a \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+60}:\\ \;\;\;\;2 \cdot \left(\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+116}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left|\left(b - a\right) \cdot \left(b + a\right)\right|\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \end{array} \]

Alternative 13: 66.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(angle_m \cdot \left(b_m - a_m\right)\right)\\ t_1 := \left(b_m - a_m\right) \cdot \left(b_m + a_m\right)\\ angle_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle_m}{180} \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\left(b_m \cdot t_0 + a_m \cdot t_0\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 5 \cdot 10^{+60}:\\ \;\;\;\;2 \cdot \left(\left(t_1 \cdot \cos \left(angle_m \cdot \frac{\pi}{-180}\right)\right) \cdot \sin \left(\pi \cdot \left(angle_m \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 5 \cdot 10^{+116}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle_m \cdot \left(\pi \cdot \left|t_1\right|\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \left(angle_m \cdot -0.005555555555555556\right)\right) \cdot \left(t_1 \cdot \sin \left(0.005555555555555556 \cdot \left(angle_m \cdot \pi\right)\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m (- b_m a_m)))) (t_1 (* (- b_m a_m) (+ b_m a_m))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 2e-42)
      (* (+ (* b_m t_0) (* a_m t_0)) 0.011111111111111112)
      (if (<= (/ angle_m 180.0) 5e+60)
        (*
         2.0
         (*
          (* t_1 (cos (* angle_m (/ PI -180.0))))
          (sin (* PI (* angle_m 0.005555555555555556)))))
        (if (<= (/ angle_m 180.0) 5e+116)
          (* 0.011111111111111112 (* angle_m (* PI (fabs t_1))))
          (*
           2.0
           (*
            (cos (* PI (* angle_m -0.005555555555555556)))
            (* t_1 (sin (* 0.005555555555555556 (* angle_m PI))))))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m * (b_m - a_m));
	double t_1 = (b_m - a_m) * (b_m + a_m);
	double tmp;
	if ((angle_m / 180.0) <= 2e-42) {
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112;
	} else if ((angle_m / 180.0) <= 5e+60) {
		tmp = 2.0 * ((t_1 * cos((angle_m * (((double) M_PI) / -180.0)))) * sin((((double) M_PI) * (angle_m * 0.005555555555555556))));
	} else if ((angle_m / 180.0) <= 5e+116) {
		tmp = 0.011111111111111112 * (angle_m * (((double) M_PI) * fabs(t_1)));
	} else {
		tmp = 2.0 * (cos((((double) M_PI) * (angle_m * -0.005555555555555556))) * (t_1 * sin((0.005555555555555556 * (angle_m * ((double) M_PI))))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = Math.PI * (angle_m * (b_m - a_m));
	double t_1 = (b_m - a_m) * (b_m + a_m);
	double tmp;
	if ((angle_m / 180.0) <= 2e-42) {
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112;
	} else if ((angle_m / 180.0) <= 5e+60) {
		tmp = 2.0 * ((t_1 * Math.cos((angle_m * (Math.PI / -180.0)))) * Math.sin((Math.PI * (angle_m * 0.005555555555555556))));
	} else if ((angle_m / 180.0) <= 5e+116) {
		tmp = 0.011111111111111112 * (angle_m * (Math.PI * Math.abs(t_1)));
	} else {
		tmp = 2.0 * (Math.cos((Math.PI * (angle_m * -0.005555555555555556))) * (t_1 * Math.sin((0.005555555555555556 * (angle_m * Math.PI)))));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	t_0 = math.pi * (angle_m * (b_m - a_m))
	t_1 = (b_m - a_m) * (b_m + a_m)
	tmp = 0
	if (angle_m / 180.0) <= 2e-42:
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112
	elif (angle_m / 180.0) <= 5e+60:
		tmp = 2.0 * ((t_1 * math.cos((angle_m * (math.pi / -180.0)))) * math.sin((math.pi * (angle_m * 0.005555555555555556))))
	elif (angle_m / 180.0) <= 5e+116:
		tmp = 0.011111111111111112 * (angle_m * (math.pi * math.fabs(t_1)))
	else:
		tmp = 2.0 * (math.cos((math.pi * (angle_m * -0.005555555555555556))) * (t_1 * math.sin((0.005555555555555556 * (angle_m * math.pi)))))
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(pi * Float64(angle_m * Float64(b_m - a_m)))
	t_1 = Float64(Float64(b_m - a_m) * Float64(b_m + a_m))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e-42)
		tmp = Float64(Float64(Float64(b_m * t_0) + Float64(a_m * t_0)) * 0.011111111111111112);
	elseif (Float64(angle_m / 180.0) <= 5e+60)
		tmp = Float64(2.0 * Float64(Float64(t_1 * cos(Float64(angle_m * Float64(pi / -180.0)))) * sin(Float64(pi * Float64(angle_m * 0.005555555555555556)))));
	elseif (Float64(angle_m / 180.0) <= 5e+116)
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * abs(t_1))));
	else
		tmp = Float64(2.0 * Float64(cos(Float64(pi * Float64(angle_m * -0.005555555555555556))) * Float64(t_1 * sin(Float64(0.005555555555555556 * Float64(angle_m * pi))))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	t_0 = pi * (angle_m * (b_m - a_m));
	t_1 = (b_m - a_m) * (b_m + a_m);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 2e-42)
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112;
	elseif ((angle_m / 180.0) <= 5e+60)
		tmp = 2.0 * ((t_1 * cos((angle_m * (pi / -180.0)))) * sin((pi * (angle_m * 0.005555555555555556))));
	elseif ((angle_m / 180.0) <= 5e+116)
		tmp = 0.011111111111111112 * (angle_m * (pi * abs(t_1)));
	else
		tmp = 2.0 * (cos((pi * (angle_m * -0.005555555555555556))) * (t_1 * sin((0.005555555555555556 * (angle_m * pi)))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(b$95$m + a$95$m), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e-42], N[(N[(N[(b$95$m * t$95$0), $MachinePrecision] + N[(a$95$m * t$95$0), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+60], N[(2.0 * N[(N[(t$95$1 * N[Cos[N[(angle$95$m * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+116], N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[Abs[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Cos[N[(Pi * N[(angle$95$m * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle 180) < 2.00000000000000008e-42

   1. Initial program 61.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 61.9%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   3. Simplified 61.9%

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0 58.7%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. unpow2 58.7%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 58.7%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 63.2%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   6. Applied egg-rr 63.2%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   7. Taylor expanded in angle around 0 63.2%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative 63.2%

         \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112} \]

      2. associate-*r* 63.3%

         \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot 0.011111111111111112 \]

      3. *-commutative 63.3%

         \[\leadsto \left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

      4. associate-*l* 63.2%

         \[\leadsto \color{blue}{\left(\pi \cdot angle\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\right)} \]

      5. +-commutative 63.2%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\right) \]

      6. difference-of-squares 58.7%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\color{blue}{\left(b \cdot b - a \cdot a\right)} \cdot 0.011111111111111112\right) \]

      7. unpow2 58.7%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\color{blue}{{b}^{2}} - a \cdot a\right) \cdot 0.011111111111111112\right) \]

      8. unpow2 58.7%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left({b}^{2} - \color{blue}{{a}^{2}}\right) \cdot 0.011111111111111112\right) \]

      9. associate-*l* 58.7%

         \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot 0.011111111111111112} \]

      10. *-commutative 58.7%

          \[\leadsto \left(\color{blue}{\left(angle \cdot \pi\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot 0.011111111111111112 \]

      11. unpow2 58.7%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot 0.011111111111111112 \]

      12. unpow2 58.7%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot 0.011111111111111112 \]

      13. difference-of-squares 63.3%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot 0.011111111111111112 \]

      14. *-commutative 63.3%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(b + a\right)\right)}\right) \cdot 0.011111111111111112 \]

      15. +-commutative 63.3%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(a + b\right)}\right)\right) \cdot 0.011111111111111112 \]

   9. Simplified 63.3%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112} \]
   10. Step-by-step derivation

       1. associate-*r* 74.6%

          \[\leadsto \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot \left(a + b\right)\right)} \cdot 0.011111111111111112 \]

       2. +-commutative 74.6%

          \[\leadsto \left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot 0.011111111111111112 \]

       3. distribute-lft-in 70.5%

          \[\leadsto \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right)} \cdot 0.011111111111111112 \]

       4. *-commutative 70.5%

          \[\leadsto \left(\left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(b - a\right)\right) \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       5. associate-*l* 70.5%

          \[\leadsto \left(\color{blue}{\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)} \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       6. *-commutative 70.5%

          \[\leadsto \left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       7. associate-*l* 70.6%

          \[\leadsto \left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \color{blue}{\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)} \cdot a\right) \cdot 0.011111111111111112 \]

   11. Applied egg-rr 70.6%

       \[\leadsto \color{blue}{\left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot a\right)} \cdot 0.011111111111111112 \]

   if 2.00000000000000008e-42 < (/.f64 angle 180) < 4.99999999999999975e60

   1. Initial program 73.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 73.7%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 53.8%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 53.8%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 57.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   5. Applied egg-rr 77.4%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   6. Taylor expanded in angle around inf 76.7%

      \[\leadsto 2 \cdot \left(\color{blue}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r* 78.0%

         \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      2. *-commutative 78.0%

         \[\leadsto 2 \cdot \left(\sin \left(\color{blue}{\left(angle \cdot 0.005555555555555556\right)} \cdot \pi\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      3. *-commutative 78.0%

         \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   8. Simplified 78.0%

      \[\leadsto 2 \cdot \left(\color{blue}{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   if 4.99999999999999975e60 < (/.f64 angle 180) < 5.00000000000000025e116

   1. Initial program 21.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 21.7%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   3. Simplified 21.7%

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0 32.6%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 30.9%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\sqrt{{b}^{2} - {a}^{2}} \cdot \sqrt{{b}^{2} - {a}^{2}}\right)}\right)\right) \]

      2. sqrt-unprod 52.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\sqrt{\left({b}^{2} - {a}^{2}\right) \cdot \left({b}^{2} - {a}^{2}\right)}}\right)\right) \]

      3. pow2 52.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \sqrt{\color{blue}{{\left({b}^{2} - {a}^{2}\right)}^{2}}}\right)\right) \]

   6. Applied egg-rr 52.0%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\sqrt{{\left({b}^{2} - {a}^{2}\right)}^{2}}}\right)\right) \]
   7. Step-by-step derivation

      1. unpow2 52.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \sqrt{\color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left({b}^{2} - {a}^{2}\right)}}\right)\right) \]

      2. rem-sqrt-square 51.2%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left|{b}^{2} - {a}^{2}\right|}\right)\right) \]

   8. Simplified 51.2%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left|{b}^{2} - {a}^{2}\right|}\right)\right) \]
   9. Step-by-step derivation

      1. unpow2 32.6%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 32.6%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 32.6%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   10. Applied egg-rr 51.2%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left|\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}\right|\right)\right) \]

   if 5.00000000000000025e116 < (/.f64 angle 180)

   1. Initial program 28.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 29.5%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 26.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 26.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 28.4%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   5. Applied egg-rr 29.5%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   6. Taylor expanded in angle around inf 32.1%

      \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 35.6%

         \[\leadsto 2 \cdot \left(\cos \color{blue}{\left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right)} \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      2. associate-*r* 30.2%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\sin \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      3. *-commutative 30.2%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\sin \left(\color{blue}{\left(angle \cdot 0.005555555555555556\right)} \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      4. *-commutative 30.2%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\sin \color{blue}{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      5. *-commutative 30.2%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \color{blue}{\left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right) \]

      6. *-commutative 30.2%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \color{blue}{\left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)}\right)\right) \]

      7. *-commutative 30.2%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\color{blue}{\left(0.005555555555555556 \cdot angle\right)} \cdot \pi\right)\right)\right) \]

      8. associate-*r* 35.6%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right) \]

   8. Simplified 35.6%

      \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\left(b \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) + a \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+60}:\\ \;\;\;\;2 \cdot \left(\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+116}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left|\left(b - a\right) \cdot \left(b + a\right)\right|\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \end{array} \]

Alternative 14: 67.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(angle_m \cdot \left(b_m - a_m\right)\right)\\ t_1 := \left(b_m - a_m\right) \cdot \left(b_m + a_m\right)\\ angle_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle_m}{180} \leq 4 \cdot 10^{-30}:\\ \;\;\;\;\left(b_m \cdot t_0 + a_m \cdot t_0\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 2 \cdot 10^{+78}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{angle_m}{180} \cdot \pi\right) \cdot \left(t_1 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle_m \cdot \pi\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 5 \cdot 10^{+116}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle_m \cdot \left(\pi \cdot \left|t_1\right|\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \left(angle_m \cdot -0.005555555555555556\right)\right) \cdot \left(t_1 \cdot \sin \left(0.005555555555555556 \cdot \left(angle_m \cdot \pi\right)\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m (- b_m a_m)))) (t_1 (* (- b_m a_m) (+ b_m a_m))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 4e-30)
      (* (+ (* b_m t_0) (* a_m t_0)) 0.011111111111111112)
      (if (<= (/ angle_m 180.0) 2e+78)
        (*
         2.0
         (*
          (sin (* (/ angle_m 180.0) PI))
          (* t_1 (cos (* -0.005555555555555556 (* angle_m PI))))))
        (if (<= (/ angle_m 180.0) 5e+116)
          (* 0.011111111111111112 (* angle_m (* PI (fabs t_1))))
          (*
           2.0
           (*
            (cos (* PI (* angle_m -0.005555555555555556)))
            (* t_1 (sin (* 0.005555555555555556 (* angle_m PI))))))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m * (b_m - a_m));
	double t_1 = (b_m - a_m) * (b_m + a_m);
	double tmp;
	if ((angle_m / 180.0) <= 4e-30) {
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112;
	} else if ((angle_m / 180.0) <= 2e+78) {
		tmp = 2.0 * (sin(((angle_m / 180.0) * ((double) M_PI))) * (t_1 * cos((-0.005555555555555556 * (angle_m * ((double) M_PI))))));
	} else if ((angle_m / 180.0) <= 5e+116) {
		tmp = 0.011111111111111112 * (angle_m * (((double) M_PI) * fabs(t_1)));
	} else {
		tmp = 2.0 * (cos((((double) M_PI) * (angle_m * -0.005555555555555556))) * (t_1 * sin((0.005555555555555556 * (angle_m * ((double) M_PI))))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = Math.PI * (angle_m * (b_m - a_m));
	double t_1 = (b_m - a_m) * (b_m + a_m);
	double tmp;
	if ((angle_m / 180.0) <= 4e-30) {
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112;
	} else if ((angle_m / 180.0) <= 2e+78) {
		tmp = 2.0 * (Math.sin(((angle_m / 180.0) * Math.PI)) * (t_1 * Math.cos((-0.005555555555555556 * (angle_m * Math.PI)))));
	} else if ((angle_m / 180.0) <= 5e+116) {
		tmp = 0.011111111111111112 * (angle_m * (Math.PI * Math.abs(t_1)));
	} else {
		tmp = 2.0 * (Math.cos((Math.PI * (angle_m * -0.005555555555555556))) * (t_1 * Math.sin((0.005555555555555556 * (angle_m * Math.PI)))));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	t_0 = math.pi * (angle_m * (b_m - a_m))
	t_1 = (b_m - a_m) * (b_m + a_m)
	tmp = 0
	if (angle_m / 180.0) <= 4e-30:
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112
	elif (angle_m / 180.0) <= 2e+78:
		tmp = 2.0 * (math.sin(((angle_m / 180.0) * math.pi)) * (t_1 * math.cos((-0.005555555555555556 * (angle_m * math.pi)))))
	elif (angle_m / 180.0) <= 5e+116:
		tmp = 0.011111111111111112 * (angle_m * (math.pi * math.fabs(t_1)))
	else:
		tmp = 2.0 * (math.cos((math.pi * (angle_m * -0.005555555555555556))) * (t_1 * math.sin((0.005555555555555556 * (angle_m * math.pi)))))
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(pi * Float64(angle_m * Float64(b_m - a_m)))
	t_1 = Float64(Float64(b_m - a_m) * Float64(b_m + a_m))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e-30)
		tmp = Float64(Float64(Float64(b_m * t_0) + Float64(a_m * t_0)) * 0.011111111111111112);
	elseif (Float64(angle_m / 180.0) <= 2e+78)
		tmp = Float64(2.0 * Float64(sin(Float64(Float64(angle_m / 180.0) * pi)) * Float64(t_1 * cos(Float64(-0.005555555555555556 * Float64(angle_m * pi))))));
	elseif (Float64(angle_m / 180.0) <= 5e+116)
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * abs(t_1))));
	else
		tmp = Float64(2.0 * Float64(cos(Float64(pi * Float64(angle_m * -0.005555555555555556))) * Float64(t_1 * sin(Float64(0.005555555555555556 * Float64(angle_m * pi))))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	t_0 = pi * (angle_m * (b_m - a_m));
	t_1 = (b_m - a_m) * (b_m + a_m);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 4e-30)
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112;
	elseif ((angle_m / 180.0) <= 2e+78)
		tmp = 2.0 * (sin(((angle_m / 180.0) * pi)) * (t_1 * cos((-0.005555555555555556 * (angle_m * pi)))));
	elseif ((angle_m / 180.0) <= 5e+116)
		tmp = 0.011111111111111112 * (angle_m * (pi * abs(t_1)));
	else
		tmp = 2.0 * (cos((pi * (angle_m * -0.005555555555555556))) * (t_1 * sin((0.005555555555555556 * (angle_m * pi)))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(b$95$m + a$95$m), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e-30], N[(N[(N[(b$95$m * t$95$0), $MachinePrecision] + N[(a$95$m * t$95$0), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+78], N[(2.0 * N[(N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[Cos[N[(-0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+116], N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[Abs[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Cos[N[(Pi * N[(angle$95$m * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle 180) < 4e-30

   1. Initial program 62.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 62.5%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   3. Simplified 62.5%

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0 59.4%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. unpow2 59.4%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 59.4%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 63.8%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   6. Applied egg-rr 63.8%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   7. Taylor expanded in angle around 0 63.8%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative 63.8%

         \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112} \]

      2. associate-*r* 63.9%

         \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot 0.011111111111111112 \]

      3. *-commutative 63.9%

         \[\leadsto \left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

      4. associate-*l* 63.8%

         \[\leadsto \color{blue}{\left(\pi \cdot angle\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\right)} \]

      5. +-commutative 63.8%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\right) \]

      6. difference-of-squares 59.4%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\color{blue}{\left(b \cdot b - a \cdot a\right)} \cdot 0.011111111111111112\right) \]

      7. unpow2 59.4%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\color{blue}{{b}^{2}} - a \cdot a\right) \cdot 0.011111111111111112\right) \]

      8. unpow2 59.4%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left({b}^{2} - \color{blue}{{a}^{2}}\right) \cdot 0.011111111111111112\right) \]

      9. associate-*l* 59.4%

         \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot 0.011111111111111112} \]

      10. *-commutative 59.4%

          \[\leadsto \left(\color{blue}{\left(angle \cdot \pi\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot 0.011111111111111112 \]

      11. unpow2 59.4%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot 0.011111111111111112 \]

      12. unpow2 59.4%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot 0.011111111111111112 \]

      13. difference-of-squares 63.9%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot 0.011111111111111112 \]

      14. *-commutative 63.9%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(b + a\right)\right)}\right) \cdot 0.011111111111111112 \]

      15. +-commutative 63.9%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(a + b\right)}\right)\right) \cdot 0.011111111111111112 \]

   9. Simplified 63.9%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112} \]
   10. Step-by-step derivation

       1. associate-*r* 75.0%

          \[\leadsto \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot \left(a + b\right)\right)} \cdot 0.011111111111111112 \]

       2. +-commutative 75.0%

          \[\leadsto \left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot 0.011111111111111112 \]

       3. distribute-lft-in 71.0%

          \[\leadsto \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right)} \cdot 0.011111111111111112 \]

       4. *-commutative 71.0%

          \[\leadsto \left(\left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(b - a\right)\right) \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       5. associate-*l* 71.0%

          \[\leadsto \left(\color{blue}{\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)} \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       6. *-commutative 71.0%

          \[\leadsto \left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       7. associate-*l* 71.1%

          \[\leadsto \left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \color{blue}{\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)} \cdot a\right) \cdot 0.011111111111111112 \]

   11. Applied egg-rr 71.1%

       \[\leadsto \color{blue}{\left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot a\right)} \cdot 0.011111111111111112 \]

   if 4e-30 < (/.f64 angle 180) < 2.00000000000000002e78

   1. Initial program 65.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 65.1%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 44.8%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 44.8%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 48.6%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   5. Applied egg-rr 68.9%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   6. Taylor expanded in angle around inf 72.2%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right) \]

   if 2.00000000000000002e78 < (/.f64 angle 180) < 5.00000000000000025e116

   1. Initial program 27.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 27.2%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   3. Simplified 27.2%

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0 39.6%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 38.7%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\sqrt{{b}^{2} - {a}^{2}} \cdot \sqrt{{b}^{2} - {a}^{2}}\right)}\right)\right) \]

      2. sqrt-unprod 52.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\sqrt{\left({b}^{2} - {a}^{2}\right) \cdot \left({b}^{2} - {a}^{2}\right)}}\right)\right) \]

      3. pow2 52.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \sqrt{\color{blue}{{\left({b}^{2} - {a}^{2}\right)}^{2}}}\right)\right) \]

   6. Applied egg-rr 52.5%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\sqrt{{\left({b}^{2} - {a}^{2}\right)}^{2}}}\right)\right) \]
   7. Step-by-step derivation

      1. unpow2 52.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \sqrt{\color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left({b}^{2} - {a}^{2}\right)}}\right)\right) \]

      2. rem-sqrt-square 51.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left|{b}^{2} - {a}^{2}\right|}\right)\right) \]

   8. Simplified 51.5%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left|{b}^{2} - {a}^{2}\right|}\right)\right) \]
   9. Step-by-step derivation

      1. unpow2 39.6%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 39.6%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 39.6%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   10. Applied egg-rr 51.5%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left|\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}\right|\right)\right) \]

   if 5.00000000000000025e116 < (/.f64 angle 180)

   1. Initial program 28.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 29.5%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 26.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 26.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 28.4%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   5. Applied egg-rr 29.5%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   6. Taylor expanded in angle around inf 32.1%

      \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 35.6%

         \[\leadsto 2 \cdot \left(\cos \color{blue}{\left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right)} \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      2. associate-*r* 30.2%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\sin \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      3. *-commutative 30.2%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\sin \left(\color{blue}{\left(angle \cdot 0.005555555555555556\right)} \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      4. *-commutative 30.2%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\sin \color{blue}{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      5. *-commutative 30.2%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \color{blue}{\left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right) \]

      6. *-commutative 30.2%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \color{blue}{\left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)}\right)\right) \]

      7. *-commutative 30.2%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\color{blue}{\left(0.005555555555555556 \cdot angle\right)} \cdot \pi\right)\right)\right) \]

      8. associate-*r* 35.6%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right) \]

   8. Simplified 35.6%

      \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{-30}:\\ \;\;\;\;\left(b \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) + a \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+78}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+116}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left|\left(b - a\right) \cdot \left(b + a\right)\right|\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \end{array} \]

Alternative 15: 67.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(angle_m \cdot \left(b_m - a_m\right)\right)\\ t_1 := 0.005555555555555556 \cdot \left(angle_m \cdot \pi\right)\\ t_2 := \left(b_m - a_m\right) \cdot \left(b_m + a_m\right)\\ angle_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle_m}{180} \leq 4 \cdot 10^{-30}:\\ \;\;\;\;\left(b_m \cdot t_0 + a_m \cdot t_0\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 2 \cdot 10^{+78}:\\ \;\;\;\;\left(2 \cdot t_2\right) \cdot \left(\sin \left(\frac{angle_m}{180} \cdot \pi\right) \cdot \cos t_1\right)\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 5 \cdot 10^{+116}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle_m \cdot \left(\pi \cdot \left|t_2\right|\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \left(angle_m \cdot -0.005555555555555556\right)\right) \cdot \left(t_2 \cdot \sin t_1\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m (- b_m a_m))))
        (t_1 (* 0.005555555555555556 (* angle_m PI)))
        (t_2 (* (- b_m a_m) (+ b_m a_m))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 4e-30)
      (* (+ (* b_m t_0) (* a_m t_0)) 0.011111111111111112)
      (if (<= (/ angle_m 180.0) 2e+78)
        (* (* 2.0 t_2) (* (sin (* (/ angle_m 180.0) PI)) (cos t_1)))
        (if (<= (/ angle_m 180.0) 5e+116)
          (* 0.011111111111111112 (* angle_m (* PI (fabs t_2))))
          (*
           2.0
           (*
            (cos (* PI (* angle_m -0.005555555555555556)))
            (* t_2 (sin t_1))))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m * (b_m - a_m));
	double t_1 = 0.005555555555555556 * (angle_m * ((double) M_PI));
	double t_2 = (b_m - a_m) * (b_m + a_m);
	double tmp;
	if ((angle_m / 180.0) <= 4e-30) {
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112;
	} else if ((angle_m / 180.0) <= 2e+78) {
		tmp = (2.0 * t_2) * (sin(((angle_m / 180.0) * ((double) M_PI))) * cos(t_1));
	} else if ((angle_m / 180.0) <= 5e+116) {
		tmp = 0.011111111111111112 * (angle_m * (((double) M_PI) * fabs(t_2)));
	} else {
		tmp = 2.0 * (cos((((double) M_PI) * (angle_m * -0.005555555555555556))) * (t_2 * sin(t_1)));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = Math.PI * (angle_m * (b_m - a_m));
	double t_1 = 0.005555555555555556 * (angle_m * Math.PI);
	double t_2 = (b_m - a_m) * (b_m + a_m);
	double tmp;
	if ((angle_m / 180.0) <= 4e-30) {
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112;
	} else if ((angle_m / 180.0) <= 2e+78) {
		tmp = (2.0 * t_2) * (Math.sin(((angle_m / 180.0) * Math.PI)) * Math.cos(t_1));
	} else if ((angle_m / 180.0) <= 5e+116) {
		tmp = 0.011111111111111112 * (angle_m * (Math.PI * Math.abs(t_2)));
	} else {
		tmp = 2.0 * (Math.cos((Math.PI * (angle_m * -0.005555555555555556))) * (t_2 * Math.sin(t_1)));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	t_0 = math.pi * (angle_m * (b_m - a_m))
	t_1 = 0.005555555555555556 * (angle_m * math.pi)
	t_2 = (b_m - a_m) * (b_m + a_m)
	tmp = 0
	if (angle_m / 180.0) <= 4e-30:
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112
	elif (angle_m / 180.0) <= 2e+78:
		tmp = (2.0 * t_2) * (math.sin(((angle_m / 180.0) * math.pi)) * math.cos(t_1))
	elif (angle_m / 180.0) <= 5e+116:
		tmp = 0.011111111111111112 * (angle_m * (math.pi * math.fabs(t_2)))
	else:
		tmp = 2.0 * (math.cos((math.pi * (angle_m * -0.005555555555555556))) * (t_2 * math.sin(t_1)))
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(pi * Float64(angle_m * Float64(b_m - a_m)))
	t_1 = Float64(0.005555555555555556 * Float64(angle_m * pi))
	t_2 = Float64(Float64(b_m - a_m) * Float64(b_m + a_m))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e-30)
		tmp = Float64(Float64(Float64(b_m * t_0) + Float64(a_m * t_0)) * 0.011111111111111112);
	elseif (Float64(angle_m / 180.0) <= 2e+78)
		tmp = Float64(Float64(2.0 * t_2) * Float64(sin(Float64(Float64(angle_m / 180.0) * pi)) * cos(t_1)));
	elseif (Float64(angle_m / 180.0) <= 5e+116)
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * abs(t_2))));
	else
		tmp = Float64(2.0 * Float64(cos(Float64(pi * Float64(angle_m * -0.005555555555555556))) * Float64(t_2 * sin(t_1))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	t_0 = pi * (angle_m * (b_m - a_m));
	t_1 = 0.005555555555555556 * (angle_m * pi);
	t_2 = (b_m - a_m) * (b_m + a_m);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 4e-30)
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112;
	elseif ((angle_m / 180.0) <= 2e+78)
		tmp = (2.0 * t_2) * (sin(((angle_m / 180.0) * pi)) * cos(t_1));
	elseif ((angle_m / 180.0) <= 5e+116)
		tmp = 0.011111111111111112 * (angle_m * (pi * abs(t_2)));
	else
		tmp = 2.0 * (cos((pi * (angle_m * -0.005555555555555556))) * (t_2 * sin(t_1)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(b$95$m + a$95$m), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e-30], N[(N[(N[(b$95$m * t$95$0), $MachinePrecision] + N[(a$95$m * t$95$0), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+78], N[(N[(2.0 * t$95$2), $MachinePrecision] * N[(N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+116], N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[Abs[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Cos[N[(Pi * N[(angle$95$m * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$2 * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle 180) < 4e-30

   1. Initial program 62.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 62.5%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   3. Simplified 62.5%

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0 59.4%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. unpow2 59.4%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 59.4%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 63.8%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   6. Applied egg-rr 63.8%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   7. Taylor expanded in angle around 0 63.8%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative 63.8%

         \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112} \]

      2. associate-*r* 63.9%

         \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot 0.011111111111111112 \]

      3. *-commutative 63.9%

         \[\leadsto \left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

      4. associate-*l* 63.8%

         \[\leadsto \color{blue}{\left(\pi \cdot angle\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\right)} \]

      5. +-commutative 63.8%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\right) \]

      6. difference-of-squares 59.4%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\color{blue}{\left(b \cdot b - a \cdot a\right)} \cdot 0.011111111111111112\right) \]

      7. unpow2 59.4%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\color{blue}{{b}^{2}} - a \cdot a\right) \cdot 0.011111111111111112\right) \]

      8. unpow2 59.4%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left({b}^{2} - \color{blue}{{a}^{2}}\right) \cdot 0.011111111111111112\right) \]

      9. associate-*l* 59.4%

         \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot 0.011111111111111112} \]

      10. *-commutative 59.4%

          \[\leadsto \left(\color{blue}{\left(angle \cdot \pi\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot 0.011111111111111112 \]

      11. unpow2 59.4%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot 0.011111111111111112 \]

      12. unpow2 59.4%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot 0.011111111111111112 \]

      13. difference-of-squares 63.9%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot 0.011111111111111112 \]

      14. *-commutative 63.9%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(b + a\right)\right)}\right) \cdot 0.011111111111111112 \]

      15. +-commutative 63.9%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(a + b\right)}\right)\right) \cdot 0.011111111111111112 \]

   9. Simplified 63.9%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112} \]
   10. Step-by-step derivation

       1. associate-*r* 75.0%

          \[\leadsto \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot \left(a + b\right)\right)} \cdot 0.011111111111111112 \]

       2. +-commutative 75.0%

          \[\leadsto \left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot 0.011111111111111112 \]

       3. distribute-lft-in 71.0%

          \[\leadsto \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right)} \cdot 0.011111111111111112 \]

       4. *-commutative 71.0%

          \[\leadsto \left(\left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(b - a\right)\right) \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       5. associate-*l* 71.0%

          \[\leadsto \left(\color{blue}{\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)} \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       6. *-commutative 71.0%

          \[\leadsto \left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       7. associate-*l* 71.1%

          \[\leadsto \left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \color{blue}{\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)} \cdot a\right) \cdot 0.011111111111111112 \]

   11. Applied egg-rr 71.1%

       \[\leadsto \color{blue}{\left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot a\right)} \cdot 0.011111111111111112 \]

   if 4e-30 < (/.f64 angle 180) < 2.00000000000000002e78

   1. Initial program 65.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 65.1%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   3. Simplified 65.1%

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 44.8%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 44.8%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 48.6%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   5. Applied egg-rr 69.0%

      \[\leadsto \left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   6. Taylor expanded in angle around inf 72.2%

      \[\leadsto \left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right) \]

   if 2.00000000000000002e78 < (/.f64 angle 180) < 5.00000000000000025e116

   1. Initial program 27.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 27.2%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   3. Simplified 27.2%

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0 39.6%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 38.7%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\sqrt{{b}^{2} - {a}^{2}} \cdot \sqrt{{b}^{2} - {a}^{2}}\right)}\right)\right) \]

      2. sqrt-unprod 52.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\sqrt{\left({b}^{2} - {a}^{2}\right) \cdot \left({b}^{2} - {a}^{2}\right)}}\right)\right) \]

      3. pow2 52.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \sqrt{\color{blue}{{\left({b}^{2} - {a}^{2}\right)}^{2}}}\right)\right) \]

   6. Applied egg-rr 52.5%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\sqrt{{\left({b}^{2} - {a}^{2}\right)}^{2}}}\right)\right) \]
   7. Step-by-step derivation

      1. unpow2 52.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \sqrt{\color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left({b}^{2} - {a}^{2}\right)}}\right)\right) \]

      2. rem-sqrt-square 51.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left|{b}^{2} - {a}^{2}\right|}\right)\right) \]

   8. Simplified 51.5%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left|{b}^{2} - {a}^{2}\right|}\right)\right) \]
   9. Step-by-step derivation

      1. unpow2 39.6%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 39.6%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 39.6%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   10. Applied egg-rr 51.5%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left|\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}\right|\right)\right) \]

   if 5.00000000000000025e116 < (/.f64 angle 180)

   1. Initial program 28.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 29.5%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 26.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 26.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 28.4%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   5. Applied egg-rr 29.5%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   6. Taylor expanded in angle around inf 32.1%

      \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 35.6%

         \[\leadsto 2 \cdot \left(\cos \color{blue}{\left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right)} \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      2. associate-*r* 30.2%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\sin \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      3. *-commutative 30.2%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\sin \left(\color{blue}{\left(angle \cdot 0.005555555555555556\right)} \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      4. *-commutative 30.2%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\sin \color{blue}{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      5. *-commutative 30.2%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \color{blue}{\left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right) \]

      6. *-commutative 30.2%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \color{blue}{\left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)}\right)\right) \]

      7. *-commutative 30.2%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\color{blue}{\left(0.005555555555555556 \cdot angle\right)} \cdot \pi\right)\right)\right) \]

      8. associate-*r* 35.6%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right) \]

   8. Simplified 35.6%

      \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{-30}:\\ \;\;\;\;\left(b \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) + a \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+78}:\\ \;\;\;\;\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+116}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left|\left(b - a\right) \cdot \left(b + a\right)\right|\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \end{array} \]

Alternative 16: 67.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(angle_m \cdot \left(b_m - a_m\right)\right)\\ t_1 := \frac{angle_m}{180} \cdot \pi\\ t_2 := \left(b_m - a_m\right) \cdot \left(b_m + a_m\right)\\ angle_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle_m}{180} \leq 4 \cdot 10^{-30}:\\ \;\;\;\;\left(b_m \cdot t_0 + a_m \cdot t_0\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 4 \cdot 10^{+75}:\\ \;\;\;\;\left(2 \cdot t_2\right) \cdot \left(\cos t_1 \cdot \sin t_1\right)\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 5 \cdot 10^{+116}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle_m \cdot \left(\pi \cdot \left|t_2\right|\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \left(angle_m \cdot -0.005555555555555556\right)\right) \cdot \left(t_2 \cdot \sin \left(0.005555555555555556 \cdot \left(angle_m \cdot \pi\right)\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m (- b_m a_m))))
        (t_1 (* (/ angle_m 180.0) PI))
        (t_2 (* (- b_m a_m) (+ b_m a_m))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 4e-30)
      (* (+ (* b_m t_0) (* a_m t_0)) 0.011111111111111112)
      (if (<= (/ angle_m 180.0) 4e+75)
        (* (* 2.0 t_2) (* (cos t_1) (sin t_1)))
        (if (<= (/ angle_m 180.0) 5e+116)
          (* 0.011111111111111112 (* angle_m (* PI (fabs t_2))))
          (*
           2.0
           (*
            (cos (* PI (* angle_m -0.005555555555555556)))
            (* t_2 (sin (* 0.005555555555555556 (* angle_m PI))))))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m * (b_m - a_m));
	double t_1 = (angle_m / 180.0) * ((double) M_PI);
	double t_2 = (b_m - a_m) * (b_m + a_m);
	double tmp;
	if ((angle_m / 180.0) <= 4e-30) {
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112;
	} else if ((angle_m / 180.0) <= 4e+75) {
		tmp = (2.0 * t_2) * (cos(t_1) * sin(t_1));
	} else if ((angle_m / 180.0) <= 5e+116) {
		tmp = 0.011111111111111112 * (angle_m * (((double) M_PI) * fabs(t_2)));
	} else {
		tmp = 2.0 * (cos((((double) M_PI) * (angle_m * -0.005555555555555556))) * (t_2 * sin((0.005555555555555556 * (angle_m * ((double) M_PI))))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = Math.PI * (angle_m * (b_m - a_m));
	double t_1 = (angle_m / 180.0) * Math.PI;
	double t_2 = (b_m - a_m) * (b_m + a_m);
	double tmp;
	if ((angle_m / 180.0) <= 4e-30) {
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112;
	} else if ((angle_m / 180.0) <= 4e+75) {
		tmp = (2.0 * t_2) * (Math.cos(t_1) * Math.sin(t_1));
	} else if ((angle_m / 180.0) <= 5e+116) {
		tmp = 0.011111111111111112 * (angle_m * (Math.PI * Math.abs(t_2)));
	} else {
		tmp = 2.0 * (Math.cos((Math.PI * (angle_m * -0.005555555555555556))) * (t_2 * Math.sin((0.005555555555555556 * (angle_m * Math.PI)))));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	t_0 = math.pi * (angle_m * (b_m - a_m))
	t_1 = (angle_m / 180.0) * math.pi
	t_2 = (b_m - a_m) * (b_m + a_m)
	tmp = 0
	if (angle_m / 180.0) <= 4e-30:
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112
	elif (angle_m / 180.0) <= 4e+75:
		tmp = (2.0 * t_2) * (math.cos(t_1) * math.sin(t_1))
	elif (angle_m / 180.0) <= 5e+116:
		tmp = 0.011111111111111112 * (angle_m * (math.pi * math.fabs(t_2)))
	else:
		tmp = 2.0 * (math.cos((math.pi * (angle_m * -0.005555555555555556))) * (t_2 * math.sin((0.005555555555555556 * (angle_m * math.pi)))))
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(pi * Float64(angle_m * Float64(b_m - a_m)))
	t_1 = Float64(Float64(angle_m / 180.0) * pi)
	t_2 = Float64(Float64(b_m - a_m) * Float64(b_m + a_m))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e-30)
		tmp = Float64(Float64(Float64(b_m * t_0) + Float64(a_m * t_0)) * 0.011111111111111112);
	elseif (Float64(angle_m / 180.0) <= 4e+75)
		tmp = Float64(Float64(2.0 * t_2) * Float64(cos(t_1) * sin(t_1)));
	elseif (Float64(angle_m / 180.0) <= 5e+116)
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * abs(t_2))));
	else
		tmp = Float64(2.0 * Float64(cos(Float64(pi * Float64(angle_m * -0.005555555555555556))) * Float64(t_2 * sin(Float64(0.005555555555555556 * Float64(angle_m * pi))))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	t_0 = pi * (angle_m * (b_m - a_m));
	t_1 = (angle_m / 180.0) * pi;
	t_2 = (b_m - a_m) * (b_m + a_m);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 4e-30)
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112;
	elseif ((angle_m / 180.0) <= 4e+75)
		tmp = (2.0 * t_2) * (cos(t_1) * sin(t_1));
	elseif ((angle_m / 180.0) <= 5e+116)
		tmp = 0.011111111111111112 * (angle_m * (pi * abs(t_2)));
	else
		tmp = 2.0 * (cos((pi * (angle_m * -0.005555555555555556))) * (t_2 * sin((0.005555555555555556 * (angle_m * pi)))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(b$95$m + a$95$m), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e-30], N[(N[(N[(b$95$m * t$95$0), $MachinePrecision] + N[(a$95$m * t$95$0), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+75], N[(N[(2.0 * t$95$2), $MachinePrecision] * N[(N[Cos[t$95$1], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+116], N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[Abs[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Cos[N[(Pi * N[(angle$95$m * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$2 * N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle 180) < 4e-30

   1. Initial program 62.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 62.5%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   3. Simplified 62.5%

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0 59.4%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. unpow2 59.4%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 59.4%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 63.8%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   6. Applied egg-rr 63.8%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   7. Taylor expanded in angle around 0 63.8%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative 63.8%

         \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112} \]

      2. associate-*r* 63.9%

         \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot 0.011111111111111112 \]

      3. *-commutative 63.9%

         \[\leadsto \left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

      4. associate-*l* 63.8%

         \[\leadsto \color{blue}{\left(\pi \cdot angle\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\right)} \]

      5. +-commutative 63.8%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\right) \]

      6. difference-of-squares 59.4%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\color{blue}{\left(b \cdot b - a \cdot a\right)} \cdot 0.011111111111111112\right) \]

      7. unpow2 59.4%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\color{blue}{{b}^{2}} - a \cdot a\right) \cdot 0.011111111111111112\right) \]

      8. unpow2 59.4%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left({b}^{2} - \color{blue}{{a}^{2}}\right) \cdot 0.011111111111111112\right) \]

      9. associate-*l* 59.4%

         \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot 0.011111111111111112} \]

      10. *-commutative 59.4%

          \[\leadsto \left(\color{blue}{\left(angle \cdot \pi\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot 0.011111111111111112 \]

      11. unpow2 59.4%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot 0.011111111111111112 \]

      12. unpow2 59.4%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot 0.011111111111111112 \]

      13. difference-of-squares 63.9%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot 0.011111111111111112 \]

      14. *-commutative 63.9%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(b + a\right)\right)}\right) \cdot 0.011111111111111112 \]

      15. +-commutative 63.9%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(a + b\right)}\right)\right) \cdot 0.011111111111111112 \]

   9. Simplified 63.9%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112} \]
   10. Step-by-step derivation

       1. associate-*r* 75.0%

          \[\leadsto \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot \left(a + b\right)\right)} \cdot 0.011111111111111112 \]

       2. +-commutative 75.0%

          \[\leadsto \left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot 0.011111111111111112 \]

       3. distribute-lft-in 71.0%

          \[\leadsto \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right)} \cdot 0.011111111111111112 \]

       4. *-commutative 71.0%

          \[\leadsto \left(\left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(b - a\right)\right) \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       5. associate-*l* 71.0%

          \[\leadsto \left(\color{blue}{\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)} \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       6. *-commutative 71.0%

          \[\leadsto \left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       7. associate-*l* 71.1%

          \[\leadsto \left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \color{blue}{\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)} \cdot a\right) \cdot 0.011111111111111112 \]

   11. Applied egg-rr 71.1%

       \[\leadsto \color{blue}{\left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot a\right)} \cdot 0.011111111111111112 \]

   if 4e-30 < (/.f64 angle 180) < 3.99999999999999971e75

   1. Initial program 65.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 65.1%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   3. Simplified 65.1%

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 44.8%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 44.8%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 48.6%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   5. Applied egg-rr 69.0%

      \[\leadsto \left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   if 3.99999999999999971e75 < (/.f64 angle 180) < 5.00000000000000025e116

   1. Initial program 27.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 27.2%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   3. Simplified 27.2%

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0 39.6%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 38.7%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\sqrt{{b}^{2} - {a}^{2}} \cdot \sqrt{{b}^{2} - {a}^{2}}\right)}\right)\right) \]

      2. sqrt-unprod 52.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\sqrt{\left({b}^{2} - {a}^{2}\right) \cdot \left({b}^{2} - {a}^{2}\right)}}\right)\right) \]

      3. pow2 52.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \sqrt{\color{blue}{{\left({b}^{2} - {a}^{2}\right)}^{2}}}\right)\right) \]

   6. Applied egg-rr 52.5%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\sqrt{{\left({b}^{2} - {a}^{2}\right)}^{2}}}\right)\right) \]
   7. Step-by-step derivation

      1. unpow2 52.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \sqrt{\color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left({b}^{2} - {a}^{2}\right)}}\right)\right) \]

      2. rem-sqrt-square 51.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left|{b}^{2} - {a}^{2}\right|}\right)\right) \]

   8. Simplified 51.5%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left|{b}^{2} - {a}^{2}\right|}\right)\right) \]
   9. Step-by-step derivation

      1. unpow2 39.6%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 39.6%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 39.6%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   10. Applied egg-rr 51.5%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left|\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}\right|\right)\right) \]

   if 5.00000000000000025e116 < (/.f64 angle 180)

   1. Initial program 28.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 29.5%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 26.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 26.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 28.4%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   5. Applied egg-rr 29.5%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   6. Taylor expanded in angle around inf 32.1%

      \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 35.6%

         \[\leadsto 2 \cdot \left(\cos \color{blue}{\left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right)} \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      2. associate-*r* 30.2%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\sin \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      3. *-commutative 30.2%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\sin \left(\color{blue}{\left(angle \cdot 0.005555555555555556\right)} \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      4. *-commutative 30.2%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\sin \color{blue}{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      5. *-commutative 30.2%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \color{blue}{\left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right) \]

      6. *-commutative 30.2%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \color{blue}{\left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)}\right)\right) \]

      7. *-commutative 30.2%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\color{blue}{\left(0.005555555555555556 \cdot angle\right)} \cdot \pi\right)\right)\right) \]

      8. associate-*r* 35.6%

         \[\leadsto 2 \cdot \left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right) \]

   8. Simplified 35.6%

      \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{-30}:\\ \;\;\;\;\left(b \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) + a \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle}{180} \leq 4 \cdot 10^{+75}:\\ \;\;\;\;\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+116}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left|\left(b - a\right) \cdot \left(b + a\right)\right|\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \end{array} \]

Alternative 17: 66.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(angle_m \cdot \left(b_m - a_m\right)\right)\\ angle_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle_m}{180} \leq 100:\\ \;\;\;\;\left(b_m \cdot t_0 + a_m \cdot t_0\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(b_m - a_m\right) \cdot \left(b_m + a_m\right)\right) \cdot \sin \left(\frac{angle_m}{180} \cdot \pi\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m (- b_m a_m)))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 100.0)
      (* (+ (* b_m t_0) (* a_m t_0)) 0.011111111111111112)
      (*
       2.0
       (* (* (- b_m a_m) (+ b_m a_m)) (sin (* (/ angle_m 180.0) PI))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m * (b_m - a_m));
	double tmp;
	if ((angle_m / 180.0) <= 100.0) {
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112;
	} else {
		tmp = 2.0 * (((b_m - a_m) * (b_m + a_m)) * sin(((angle_m / 180.0) * ((double) M_PI))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = Math.PI * (angle_m * (b_m - a_m));
	double tmp;
	if ((angle_m / 180.0) <= 100.0) {
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112;
	} else {
		tmp = 2.0 * (((b_m - a_m) * (b_m + a_m)) * Math.sin(((angle_m / 180.0) * Math.PI)));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	t_0 = math.pi * (angle_m * (b_m - a_m))
	tmp = 0
	if (angle_m / 180.0) <= 100.0:
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112
	else:
		tmp = 2.0 * (((b_m - a_m) * (b_m + a_m)) * math.sin(((angle_m / 180.0) * math.pi)))
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(pi * Float64(angle_m * Float64(b_m - a_m)))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 100.0)
		tmp = Float64(Float64(Float64(b_m * t_0) + Float64(a_m * t_0)) * 0.011111111111111112);
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(b_m - a_m) * Float64(b_m + a_m)) * sin(Float64(Float64(angle_m / 180.0) * pi))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	t_0 = pi * (angle_m * (b_m - a_m));
	tmp = 0.0;
	if ((angle_m / 180.0) <= 100.0)
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112;
	else
		tmp = 2.0 * (((b_m - a_m) * (b_m + a_m)) * sin(((angle_m / 180.0) * pi)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 100.0], N[(N[(N[(b$95$m * t$95$0), $MachinePrecision] + N[(a$95$m * t$95$0), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], N[(2.0 * N[(N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(b$95$m + a$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < 100

   1. Initial program 63.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 63.9%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   3. Simplified 63.9%

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0 60.6%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. unpow2 60.6%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 60.6%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 65.4%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   6. Applied egg-rr 65.4%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   7. Taylor expanded in angle around 0 65.4%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative 65.4%

         \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112} \]

      2. associate-*r* 65.4%

         \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot 0.011111111111111112 \]

      3. *-commutative 65.4%

         \[\leadsto \left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

      4. associate-*l* 65.4%

         \[\leadsto \color{blue}{\left(\pi \cdot angle\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\right)} \]

      5. +-commutative 65.4%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\right) \]

      6. difference-of-squares 60.6%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\color{blue}{\left(b \cdot b - a \cdot a\right)} \cdot 0.011111111111111112\right) \]

      7. unpow2 60.6%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\color{blue}{{b}^{2}} - a \cdot a\right) \cdot 0.011111111111111112\right) \]

      8. unpow2 60.6%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left({b}^{2} - \color{blue}{{a}^{2}}\right) \cdot 0.011111111111111112\right) \]

      9. associate-*l* 60.7%

         \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot 0.011111111111111112} \]

      10. *-commutative 60.7%

          \[\leadsto \left(\color{blue}{\left(angle \cdot \pi\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot 0.011111111111111112 \]

      11. unpow2 60.7%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot 0.011111111111111112 \]

      12. unpow2 60.7%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot 0.011111111111111112 \]

      13. difference-of-squares 65.4%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot 0.011111111111111112 \]

      14. *-commutative 65.4%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(b + a\right)\right)}\right) \cdot 0.011111111111111112 \]

      15. +-commutative 65.4%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(a + b\right)}\right)\right) \cdot 0.011111111111111112 \]

   9. Simplified 65.4%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112} \]
   10. Step-by-step derivation

       1. associate-*r* 75.9%

          \[\leadsto \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot \left(a + b\right)\right)} \cdot 0.011111111111111112 \]

       2. +-commutative 75.9%

          \[\leadsto \left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot 0.011111111111111112 \]

       3. distribute-lft-in 71.7%

          \[\leadsto \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right)} \cdot 0.011111111111111112 \]

       4. *-commutative 71.7%

          \[\leadsto \left(\left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(b - a\right)\right) \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       5. associate-*l* 71.7%

          \[\leadsto \left(\color{blue}{\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)} \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       6. *-commutative 71.7%

          \[\leadsto \left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       7. associate-*l* 71.7%

          \[\leadsto \left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \color{blue}{\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)} \cdot a\right) \cdot 0.011111111111111112 \]

   11. Applied egg-rr 71.7%

       \[\leadsto \color{blue}{\left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot a\right)} \cdot 0.011111111111111112 \]

   if 100 < (/.f64 angle 180)

   1. Initial program 33.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 34.5%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 26.3%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 26.3%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 27.9%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   5. Applied egg-rr 34.5%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   6. Taylor expanded in angle around 0 33.5%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{1}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 100:\\ \;\;\;\;\left(b \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) + a \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)\\ \end{array} \]

Alternative 18: 63.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(angle_m \cdot \left(b_m - a_m\right)\right)\\ angle_s \cdot \begin{array}{l} \mathbf{if}\;angle_m \leq 10^{+38}:\\ \;\;\;\;\left(b_m \cdot t_0 + a_m \cdot t_0\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle_m \cdot \left(\pi \cdot \left|\left(b_m - a_m\right) \cdot \left(b_m + a_m\right)\right|\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m (- b_m a_m)))))
   (*
    angle_s
    (if (<= angle_m 1e+38)
      (* (+ (* b_m t_0) (* a_m t_0)) 0.011111111111111112)
      (*
       0.011111111111111112
       (* angle_m (* PI (fabs (* (- b_m a_m) (+ b_m a_m))))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m * (b_m - a_m));
	double tmp;
	if (angle_m <= 1e+38) {
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112;
	} else {
		tmp = 0.011111111111111112 * (angle_m * (((double) M_PI) * fabs(((b_m - a_m) * (b_m + a_m)))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = Math.PI * (angle_m * (b_m - a_m));
	double tmp;
	if (angle_m <= 1e+38) {
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112;
	} else {
		tmp = 0.011111111111111112 * (angle_m * (Math.PI * Math.abs(((b_m - a_m) * (b_m + a_m)))));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	t_0 = math.pi * (angle_m * (b_m - a_m))
	tmp = 0
	if angle_m <= 1e+38:
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112
	else:
		tmp = 0.011111111111111112 * (angle_m * (math.pi * math.fabs(((b_m - a_m) * (b_m + a_m)))))
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(pi * Float64(angle_m * Float64(b_m - a_m)))
	tmp = 0.0
	if (angle_m <= 1e+38)
		tmp = Float64(Float64(Float64(b_m * t_0) + Float64(a_m * t_0)) * 0.011111111111111112);
	else
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * abs(Float64(Float64(b_m - a_m) * Float64(b_m + a_m))))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	t_0 = pi * (angle_m * (b_m - a_m));
	tmp = 0.0;
	if (angle_m <= 1e+38)
		tmp = ((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112;
	else
		tmp = 0.011111111111111112 * (angle_m * (pi * abs(((b_m - a_m) * (b_m + a_m)))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[angle$95$m, 1e+38], N[(N[(N[(b$95$m * t$95$0), $MachinePrecision] + N[(a$95$m * t$95$0), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[Abs[N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(b$95$m + a$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 9.99999999999999977e37

   1. Initial program 63.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 63.3%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   3. Simplified 63.3%

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0 59.4%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. unpow2 59.4%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 59.4%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 64.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   6. Applied egg-rr 64.0%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   7. Taylor expanded in angle around 0 64.0%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative 64.0%

         \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112} \]

      2. associate-*r* 64.0%

         \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot 0.011111111111111112 \]

      3. *-commutative 64.0%

         \[\leadsto \left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

      4. associate-*l* 64.0%

         \[\leadsto \color{blue}{\left(\pi \cdot angle\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\right)} \]

      5. +-commutative 64.0%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\right) \]

      6. difference-of-squares 59.4%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\color{blue}{\left(b \cdot b - a \cdot a\right)} \cdot 0.011111111111111112\right) \]

      7. unpow2 59.4%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\color{blue}{{b}^{2}} - a \cdot a\right) \cdot 0.011111111111111112\right) \]

      8. unpow2 59.4%

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left({b}^{2} - \color{blue}{{a}^{2}}\right) \cdot 0.011111111111111112\right) \]

      9. associate-*l* 59.4%

         \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot 0.011111111111111112} \]

      10. *-commutative 59.4%

          \[\leadsto \left(\color{blue}{\left(angle \cdot \pi\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot 0.011111111111111112 \]

      11. unpow2 59.4%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot 0.011111111111111112 \]

      12. unpow2 59.4%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot 0.011111111111111112 \]

      13. difference-of-squares 64.0%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot 0.011111111111111112 \]

      14. *-commutative 64.0%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(b + a\right)\right)}\right) \cdot 0.011111111111111112 \]

      15. +-commutative 64.0%

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(a + b\right)}\right)\right) \cdot 0.011111111111111112 \]

   9. Simplified 64.0%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112} \]
   10. Step-by-step derivation

       1. associate-*r* 74.2%

          \[\leadsto \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot \left(a + b\right)\right)} \cdot 0.011111111111111112 \]

       2. +-commutative 74.2%

          \[\leadsto \left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot 0.011111111111111112 \]

       3. distribute-lft-in 70.1%

          \[\leadsto \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right)} \cdot 0.011111111111111112 \]

       4. *-commutative 70.1%

          \[\leadsto \left(\left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(b - a\right)\right) \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       5. associate-*l* 70.1%

          \[\leadsto \left(\color{blue}{\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)} \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       6. *-commutative 70.1%

          \[\leadsto \left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       7. associate-*l* 70.1%

          \[\leadsto \left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \color{blue}{\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)} \cdot a\right) \cdot 0.011111111111111112 \]

   11. Applied egg-rr 70.1%

       \[\leadsto \color{blue}{\left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot a\right)} \cdot 0.011111111111111112 \]

   if 9.99999999999999977e37 < angle

   1. Initial program 32.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 32.4%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   3. Simplified 32.4%

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0 27.0%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 14.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\sqrt{{b}^{2} - {a}^{2}} \cdot \sqrt{{b}^{2} - {a}^{2}}\right)}\right)\right) \]

      2. sqrt-unprod 30.7%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\sqrt{\left({b}^{2} - {a}^{2}\right) \cdot \left({b}^{2} - {a}^{2}\right)}}\right)\right) \]

      3. pow2 30.7%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \sqrt{\color{blue}{{\left({b}^{2} - {a}^{2}\right)}^{2}}}\right)\right) \]

   6. Applied egg-rr 30.7%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\sqrt{{\left({b}^{2} - {a}^{2}\right)}^{2}}}\right)\right) \]
   7. Step-by-step derivation

      1. unpow2 30.7%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \sqrt{\color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left({b}^{2} - {a}^{2}\right)}}\right)\right) \]

      2. rem-sqrt-square 30.4%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left|{b}^{2} - {a}^{2}\right|}\right)\right) \]

   8. Simplified 30.4%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left|{b}^{2} - {a}^{2}\right|}\right)\right) \]
   9. Step-by-step derivation

      1. unpow2 27.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

      2. unpow2 27.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. difference-of-squares 28.7%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   10. Applied egg-rr 32.1%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left|\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}\right|\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 10^{+38}:\\ \;\;\;\;\left(b \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) + a \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left|\left(b - a\right) \cdot \left(b + a\right)\right|\right)\right)\\ \end{array} \]

Alternative 19: 61.8% accurate, 20.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(angle_m \cdot \left(b_m - a_m\right)\right)\\ angle_s \cdot \left(\left(b_m \cdot t_0 + a_m \cdot t_0\right) \cdot 0.011111111111111112\right) \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m (- b_m a_m)))))
   (* angle_s (* (+ (* b_m t_0) (* a_m t_0)) 0.011111111111111112))))

C

a_m = fabs(a);
b_m = fabs(b);
angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m * (b_m - a_m));
	return angle_s * (((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112);
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = Math.PI * (angle_m * (b_m - a_m));
	return angle_s * (((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112);
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	t_0 = math.pi * (angle_m * (b_m - a_m))
	return angle_s * (((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112)

Julia

a_m = abs(a)
b_m = abs(b)
angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(pi * Float64(angle_m * Float64(b_m - a_m)))
	return Float64(angle_s * Float64(Float64(Float64(b_m * t_0) + Float64(a_m * t_0)) * 0.011111111111111112))
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a_m, b_m, angle_m)
	t_0 = pi * (angle_m * (b_m - a_m));
	tmp = angle_s * (((b_m * t_0) + (a_m * t_0)) * 0.011111111111111112);
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * N[(N[(N[(b$95$m * t$95$0), $MachinePrecision] + N[(a$95$m * t$95$0), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 56.2%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Step-by-step derivation

   1. associate-*l* 56.2%

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

3. Simplified 56.2%

   \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

4. Taylor expanded in angle around 0 51.9%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
5. Step-by-step derivation

   1. unpow2 51.9%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

   2. unpow2 51.9%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

   3. difference-of-squares 55.9%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

6. Applied egg-rr 55.9%

   \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

7. Taylor expanded in angle around 0 55.9%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
8. Step-by-step derivation

   1. *-commutative 55.9%

      \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112} \]

   2. associate-*r* 55.9%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot 0.011111111111111112 \]

   3. *-commutative 55.9%

      \[\leadsto \left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

   4. associate-*l* 55.8%

      \[\leadsto \color{blue}{\left(\pi \cdot angle\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\right)} \]

   5. +-commutative 55.8%

      \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\right) \]

   6. difference-of-squares 51.9%

      \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\color{blue}{\left(b \cdot b - a \cdot a\right)} \cdot 0.011111111111111112\right) \]

   7. unpow2 51.9%

      \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\color{blue}{{b}^{2}} - a \cdot a\right) \cdot 0.011111111111111112\right) \]

   8. unpow2 51.9%

      \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left({b}^{2} - \color{blue}{{a}^{2}}\right) \cdot 0.011111111111111112\right) \]

   9. associate-*l* 51.9%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot 0.011111111111111112} \]

   10. *-commutative 51.9%

       \[\leadsto \left(\color{blue}{\left(angle \cdot \pi\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot 0.011111111111111112 \]

   11. unpow2 51.9%

       \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot 0.011111111111111112 \]

   12. unpow2 51.9%

       \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot 0.011111111111111112 \]

   13. difference-of-squares 55.9%

       \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot 0.011111111111111112 \]

   14. *-commutative 55.9%

       \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(b + a\right)\right)}\right) \cdot 0.011111111111111112 \]

   15. +-commutative 55.9%

       \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(a + b\right)}\right)\right) \cdot 0.011111111111111112 \]

9. Simplified 55.9%

   \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112} \]
10. Step-by-step derivation

    1. associate-*r* 63.4%

       \[\leadsto \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot \left(a + b\right)\right)} \cdot 0.011111111111111112 \]

    2. +-commutative 63.4%

       \[\leadsto \left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot 0.011111111111111112 \]

    3. distribute-lft-in 57.8%

       \[\leadsto \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right)} \cdot 0.011111111111111112 \]

    4. *-commutative 57.8%

       \[\leadsto \left(\left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(b - a\right)\right) \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

    5. associate-*l* 57.8%

       \[\leadsto \left(\color{blue}{\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)} \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

    6. *-commutative 57.8%

       \[\leadsto \left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

    7. associate-*l* 57.8%

       \[\leadsto \left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \color{blue}{\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)} \cdot a\right) \cdot 0.011111111111111112 \]

11. Applied egg-rr 57.8%

    \[\leadsto \color{blue}{\left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot a\right)} \cdot 0.011111111111111112 \]

12. Final simplification 57.8%

    \[\leadsto \left(b \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) + a \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112 \]

Alternative 20: 54.8% accurate, 32.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ angle_s \cdot \left(0.011111111111111112 \cdot \left(angle_m \cdot \left(\pi \cdot \left(\left(b_m - a_m\right) \cdot \left(b_m + a_m\right)\right)\right)\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (* 0.011111111111111112 (* angle_m (* PI (* (- b_m a_m) (+ b_m a_m)))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	return angle_s * (0.011111111111111112 * (angle_m * (((double) M_PI) * ((b_m - a_m) * (b_m + a_m)))));
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	return angle_s * (0.011111111111111112 * (angle_m * (Math.PI * ((b_m - a_m) * (b_m + a_m)))));
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	return angle_s * (0.011111111111111112 * (angle_m * (math.pi * ((b_m - a_m) * (b_m + a_m)))))

Julia

a_m = abs(a)
b_m = abs(b)
angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	return Float64(angle_s * Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(Float64(b_m - a_m) * Float64(b_m + a_m))))))
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a_m, b_m, angle_m)
	tmp = angle_s * (0.011111111111111112 * (angle_m * (pi * ((b_m - a_m) * (b_m + a_m)))));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(b$95$m + a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.2%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Step-by-step derivation

   1. associate-*l* 56.2%

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

3. Simplified 56.2%

   \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

4. Taylor expanded in angle around 0 51.9%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
5. Step-by-step derivation

   1. unpow2 51.9%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

   2. unpow2 51.9%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

   3. difference-of-squares 55.9%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

6. Applied egg-rr 55.9%

   \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

7. Final simplification 55.9%

   \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right) \]

Alternative 21: 54.8% accurate, 32.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ angle_s \cdot \left(0.011111111111111112 \cdot \left(\left(\left(b_m - a_m\right) \cdot \left(b_m + a_m\right)\right) \cdot \left(angle_m \cdot \pi\right)\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (* 0.011111111111111112 (* (* (- b_m a_m) (+ b_m a_m)) (* angle_m PI)))))

C

a_m = fabs(a);
b_m = fabs(b);
angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	return angle_s * (0.011111111111111112 * (((b_m - a_m) * (b_m + a_m)) * (angle_m * ((double) M_PI))));
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	return angle_s * (0.011111111111111112 * (((b_m - a_m) * (b_m + a_m)) * (angle_m * Math.PI)));
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	return angle_s * (0.011111111111111112 * (((b_m - a_m) * (b_m + a_m)) * (angle_m * math.pi)))

Julia

a_m = abs(a)
b_m = abs(b)
angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	return Float64(angle_s * Float64(0.011111111111111112 * Float64(Float64(Float64(b_m - a_m) * Float64(b_m + a_m)) * Float64(angle_m * pi))))
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a_m, b_m, angle_m)
	tmp = angle_s * (0.011111111111111112 * (((b_m - a_m) * (b_m + a_m)) * (angle_m * pi)));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * N[(0.011111111111111112 * N[(N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(b$95$m + a$95$m), $MachinePrecision]), $MachinePrecision] * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.2%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Step-by-step derivation

   1. associate-*l* 56.2%

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

3. Simplified 56.2%

   \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

4. Taylor expanded in angle around 0 51.9%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
5. Step-by-step derivation

   1. unpow2 51.9%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

   2. unpow2 51.9%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

   3. difference-of-squares 55.9%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

6. Applied egg-rr 55.9%

   \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

7. Taylor expanded in angle around 0 55.9%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
8. Step-by-step derivation

   1. *-commutative 55.9%

      \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112} \]

   2. associate-*r* 55.9%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot 0.011111111111111112 \]

   3. *-commutative 55.9%

      \[\leadsto \left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

   4. associate-*l* 55.8%

      \[\leadsto \color{blue}{\left(\pi \cdot angle\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\right)} \]

   5. +-commutative 55.8%

      \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\right) \]

   6. difference-of-squares 51.9%

      \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\color{blue}{\left(b \cdot b - a \cdot a\right)} \cdot 0.011111111111111112\right) \]

   7. unpow2 51.9%

      \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\color{blue}{{b}^{2}} - a \cdot a\right) \cdot 0.011111111111111112\right) \]

   8. unpow2 51.9%

      \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left({b}^{2} - \color{blue}{{a}^{2}}\right) \cdot 0.011111111111111112\right) \]

   9. associate-*l* 51.9%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot 0.011111111111111112} \]

   10. *-commutative 51.9%

       \[\leadsto \left(\color{blue}{\left(angle \cdot \pi\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot 0.011111111111111112 \]

   11. unpow2 51.9%

       \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot 0.011111111111111112 \]

   12. unpow2 51.9%

       \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot 0.011111111111111112 \]

   13. difference-of-squares 55.9%

       \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot 0.011111111111111112 \]

   14. *-commutative 55.9%

       \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(b + a\right)\right)}\right) \cdot 0.011111111111111112 \]

   15. +-commutative 55.9%

       \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(a + b\right)}\right)\right) \cdot 0.011111111111111112 \]

9. Simplified 55.9%

   \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112} \]

10. Final simplification 55.9%

    \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \pi\right)\right) \]

Reproduce

herbie shell --seed 2023336 
(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)))))