ab-angle->ABCF B

Percentage Accurate: 54.4% → 68.2%

Time: 42.8s

Alternatives: 16

Speedup: 32.2×

• 33.4% 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: 54.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 68.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := 2 \cdot \left(\left(b - a\_m\right) \cdot \left(b + a\_m\right)\right)\\ t_1 := 2 \cdot \left(b + a\_m\right)\\ t_2 := \pi \cdot \frac{angle\_m}{180}\\ t_3 := \cos t\_2\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;t\_3 \cdot \left(\left(b - a\_m\right) \cdot \left(t\_1 \cdot \left(angle\_m \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{+155}:\\ \;\;\;\;t\_3 \cdot \left(t\_0 \cdot \sin t\_2\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+190}:\\ \;\;\;\;t\_0 \cdot \sin \left(angle\_m \cdot \frac{\pi}{180}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3 \cdot \left(t\_1 \cdot \left(\left(b - a\_m\right) \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\_m\right)\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (let* ((t_0 (* 2.0 (* (- b a_m) (+ b a_m))))
        (t_1 (* 2.0 (+ b a_m)))
        (t_2 (* PI (/ angle_m 180.0)))
        (t_3 (cos t_2)))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 5e-25)
      (* t_3 (* (- b a_m) (* t_1 (* angle_m (* PI 0.005555555555555556)))))
      (if (<= (/ angle_m 180.0) 4e+155)
        (* t_3 (* t_0 (sin t_2)))
        (if (<= (/ angle_m 180.0) 2e+190)
          (* t_0 (sin (* angle_m (/ PI 180.0))))
          (*
           t_3
           (*
            t_1
            (* (- b a_m) (sin (* 0.005555555555555556 (* PI angle_m))))))))))))

C

a_m = fabs(a);
angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = 2.0 * ((b - a_m) * (b + a_m));
	double t_1 = 2.0 * (b + a_m);
	double t_2 = ((double) M_PI) * (angle_m / 180.0);
	double t_3 = cos(t_2);
	double tmp;
	if ((angle_m / 180.0) <= 5e-25) {
		tmp = t_3 * ((b - a_m) * (t_1 * (angle_m * (((double) M_PI) * 0.005555555555555556))));
	} else if ((angle_m / 180.0) <= 4e+155) {
		tmp = t_3 * (t_0 * sin(t_2));
	} else if ((angle_m / 180.0) <= 2e+190) {
		tmp = t_0 * sin((angle_m * (((double) M_PI) / 180.0)));
	} else {
		tmp = t_3 * (t_1 * ((b - a_m) * sin((0.005555555555555556 * (((double) M_PI) * angle_m)))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = 2.0 * ((b - a_m) * (b + a_m));
	double t_1 = 2.0 * (b + a_m);
	double t_2 = Math.PI * (angle_m / 180.0);
	double t_3 = Math.cos(t_2);
	double tmp;
	if ((angle_m / 180.0) <= 5e-25) {
		tmp = t_3 * ((b - a_m) * (t_1 * (angle_m * (Math.PI * 0.005555555555555556))));
	} else if ((angle_m / 180.0) <= 4e+155) {
		tmp = t_3 * (t_0 * Math.sin(t_2));
	} else if ((angle_m / 180.0) <= 2e+190) {
		tmp = t_0 * Math.sin((angle_m * (Math.PI / 180.0)));
	} else {
		tmp = t_3 * (t_1 * ((b - a_m) * Math.sin((0.005555555555555556 * (Math.PI * angle_m)))));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	t_0 = 2.0 * ((b - a_m) * (b + a_m))
	t_1 = 2.0 * (b + a_m)
	t_2 = math.pi * (angle_m / 180.0)
	t_3 = math.cos(t_2)
	tmp = 0
	if (angle_m / 180.0) <= 5e-25:
		tmp = t_3 * ((b - a_m) * (t_1 * (angle_m * (math.pi * 0.005555555555555556))))
	elif (angle_m / 180.0) <= 4e+155:
		tmp = t_3 * (t_0 * math.sin(t_2))
	elif (angle_m / 180.0) <= 2e+190:
		tmp = t_0 * math.sin((angle_m * (math.pi / 180.0)))
	else:
		tmp = t_3 * (t_1 * ((b - a_m) * math.sin((0.005555555555555556 * (math.pi * angle_m)))))
	return angle_s * tmp

Julia

a_m = abs(a)
angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	t_0 = Float64(2.0 * Float64(Float64(b - a_m) * Float64(b + a_m)))
	t_1 = Float64(2.0 * Float64(b + a_m))
	t_2 = Float64(pi * Float64(angle_m / 180.0))
	t_3 = cos(t_2)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e-25)
		tmp = Float64(t_3 * Float64(Float64(b - a_m) * Float64(t_1 * Float64(angle_m * Float64(pi * 0.005555555555555556)))));
	elseif (Float64(angle_m / 180.0) <= 4e+155)
		tmp = Float64(t_3 * Float64(t_0 * sin(t_2)));
	elseif (Float64(angle_m / 180.0) <= 2e+190)
		tmp = Float64(t_0 * sin(Float64(angle_m * Float64(pi / 180.0))));
	else
		tmp = Float64(t_3 * Float64(t_1 * Float64(Float64(b - a_m) * sin(Float64(0.005555555555555556 * Float64(pi * angle_m))))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	t_0 = 2.0 * ((b - a_m) * (b + a_m));
	t_1 = 2.0 * (b + a_m);
	t_2 = pi * (angle_m / 180.0);
	t_3 = cos(t_2);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 5e-25)
		tmp = t_3 * ((b - a_m) * (t_1 * (angle_m * (pi * 0.005555555555555556))));
	elseif ((angle_m / 180.0) <= 4e+155)
		tmp = t_3 * (t_0 * sin(t_2));
	elseif ((angle_m / 180.0) <= 2e+190)
		tmp = t_0 * sin((angle_m * (pi / 180.0)));
	else
		tmp = t_3 * (t_1 * ((b - a_m) * sin((0.005555555555555556 * (pi * angle_m)))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $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_, angle$95$m_] := Block[{t$95$0 = N[(2.0 * N[(N[(b - a$95$m), $MachinePrecision] * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[t$95$2], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e-25], N[(t$95$3 * N[(N[(b - a$95$m), $MachinePrecision] * N[(t$95$1 * N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+155], N[(t$95$3 * N[(t$95$0 * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+190], N[(t$95$0 * N[Sin[N[(angle$95$m * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$3 * N[(t$95$1 * N[(N[(b - a$95$m), $MachinePrecision] * N[Sin[N[(0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle 180) < 4.99999999999999962e-25

   1. Initial program 60.9%

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

      1. add-cbrt-cube 41.2%

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

      2. pow1/3 29.8%

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

      3. pow3 29.8%

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

      4. *-commutative 29.8%

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

      5. div-inv 29.8%

         \[\leadsto {\left({\left(\sin \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. metadata-eval 29.8%

         \[\leadsto {\left({\left(\sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. Applied egg-rr 29.8%

      \[\leadsto \color{blue}{{\left({\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   5. Step-by-step derivation

      1. unpow1/3 40.7%

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

      2. rem-cbrt-cube 60.3%

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

      3. *-commutative 60.3%

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

      4. *-commutative 60.3%

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

      5. *-commutative 60.3%

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

      6. unpow2 60.3%

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

      7. unpow2 60.3%

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

      8. difference-of-squares 63.5%

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

      9. associate-*r* 63.5%

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

      10. associate-*l* 76.4%

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

   6. Applied egg-rr 76.4%

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

      1. associate-*r* 63.5%

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

      2. *-commutative 63.5%

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

      3. associate-*l* 76.4%

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

      4. *-commutative 76.4%

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

      5. *-commutative 76.4%

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

      6. +-commutative 76.4%

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

   8. Simplified 76.4%

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

   9. Taylor expanded in angle around 0 71.3%

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

       1. associate-*r* 71.3%

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

       2. *-commutative 71.3%

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

       3. associate-*r* 71.4%

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

       4. *-commutative 71.4%

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

       5. *-commutative 71.4%

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

   11. Simplified 71.4%

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

   if 4.99999999999999962e-25 < (/.f64 angle 180) < 4.00000000000000003e155

   1. Initial program 74.1%

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

      1. unpow2 74.1%

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

      2. unpow2 74.1%

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

      3. difference-of-squares 74.1%

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

   4. Applied egg-rr 74.1%

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

   if 4.00000000000000003e155 < (/.f64 angle 180) < 2.0000000000000001e190

   1. Initial program 19.6%

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

      1. unpow2 19.6%

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

      2. unpow2 19.6%

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

      3. difference-of-squares 19.6%

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

   4. Applied egg-rr 19.6%

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

   5. Taylor expanded in angle around 0 60.0%

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

      1. clear-num 51.7%

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

      2. un-div-inv 60.0%

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

   7. Applied egg-rr 60.0%

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

      1. associate-/r/ 68.3%

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

   9. Simplified 68.3%

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

   if 2.0000000000000001e190 < (/.f64 angle 180)

   1. Initial program 38.4%

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

      1. add-cbrt-cube 34.8%

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

      2. pow1/3 26.1%

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

      3. pow3 26.1%

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

      4. *-commutative 26.1%

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

      5. div-inv 25.3%

         \[\leadsto {\left({\left(\sin \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. metadata-eval 25.3%

         \[\leadsto {\left({\left(\sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. Applied egg-rr 25.3%

      \[\leadsto \color{blue}{{\left({\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   5. Step-by-step derivation

      1. unpow1/3 34.4%

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

      2. rem-cbrt-cube 38.4%

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

      3. *-commutative 38.4%

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

      4. *-commutative 38.4%

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

      5. *-commutative 38.4%

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

      6. unpow2 38.4%

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

      7. unpow2 38.4%

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

      8. difference-of-squares 46.4%

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

      9. associate-*r* 46.4%

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

      10. associate-*l* 46.4%

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

   6. Applied egg-rr 46.4%

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

      1. +-commutative 46.4%

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

      2. associate-*r* 45.7%

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

      3. *-commutative 45.7%

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

      4. *-commutative 45.7%

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

   8. Simplified 45.7%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(b - a\right) \cdot \left(\left(2 \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 4 \cdot 10^{+155}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+190}:\\ \;\;\;\;\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 68.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := 2 \cdot \left(b + a\_m\right)\\ t_1 := \pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{+244}:\\ \;\;\;\;\sqrt[3]{{\cos t\_1}^{3}} \cdot \left(\left(b - a\_m\right) \cdot \left(t\_0 \cdot \sin t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\_m\right) \cdot \left(t\_0 \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle\_m \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle\_m}{180}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (let* ((t_0 (* 2.0 (+ b a_m))) (t_1 (* PI (* 0.005555555555555556 angle_m))))
   (*
    angle_s
    (if (<= b 1.15e+244)
      (* (cbrt (pow (cos t_1) 3.0)) (* (- b a_m) (* t_0 (sin t_1))))
      (*
       (*
        (- b a_m)
        (* t_0 (sin (expm1 (log1p (* angle_m (* PI 0.005555555555555556)))))))
       (cos (* PI (/ angle_m 180.0))))))))

C

a_m = fabs(a);
angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = 2.0 * (b + a_m);
	double t_1 = ((double) M_PI) * (0.005555555555555556 * angle_m);
	double tmp;
	if (b <= 1.15e+244) {
		tmp = cbrt(pow(cos(t_1), 3.0)) * ((b - a_m) * (t_0 * sin(t_1)));
	} else {
		tmp = ((b - a_m) * (t_0 * sin(expm1(log1p((angle_m * (((double) M_PI) * 0.005555555555555556))))))) * cos((((double) M_PI) * (angle_m / 180.0)));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = 2.0 * (b + a_m);
	double t_1 = Math.PI * (0.005555555555555556 * angle_m);
	double tmp;
	if (b <= 1.15e+244) {
		tmp = Math.cbrt(Math.pow(Math.cos(t_1), 3.0)) * ((b - a_m) * (t_0 * Math.sin(t_1)));
	} else {
		tmp = ((b - a_m) * (t_0 * Math.sin(Math.expm1(Math.log1p((angle_m * (Math.PI * 0.005555555555555556))))))) * Math.cos((Math.PI * (angle_m / 180.0)));
	}
	return angle_s * tmp;
}

Julia

a_m = abs(a)
angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	t_0 = Float64(2.0 * Float64(b + a_m))
	t_1 = Float64(pi * Float64(0.005555555555555556 * angle_m))
	tmp = 0.0
	if (b <= 1.15e+244)
		tmp = Float64(cbrt((cos(t_1) ^ 3.0)) * Float64(Float64(b - a_m) * Float64(t_0 * sin(t_1))));
	else
		tmp = Float64(Float64(Float64(b - a_m) * Float64(t_0 * sin(expm1(log1p(Float64(angle_m * Float64(pi * 0.005555555555555556))))))) * cos(Float64(pi * Float64(angle_m / 180.0))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $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_, angle$95$m_] := Block[{t$95$0 = N[(2.0 * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[b, 1.15e+244], N[(N[Power[N[Power[N[Cos[t$95$1], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[(t$95$0 * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b - a$95$m), $MachinePrecision] * N[(t$95$0 * N[Sin[N[(Exp[N[Log[1 + N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.15e244

   1. Initial program 58.9%

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

      1. add-cbrt-cube 42.3%

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

      2. pow1/3 31.2%

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

      3. pow3 31.2%

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

      4. *-commutative 31.2%

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

      5. div-inv 31.1%

         \[\leadsto {\left({\left(\sin \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. metadata-eval 31.1%

         \[\leadsto {\left({\left(\sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. Applied egg-rr 31.1%

      \[\leadsto \color{blue}{{\left({\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   5. Step-by-step derivation

      1. unpow1/3 42.7%

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

      2. rem-cbrt-cube 59.3%

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

      3. *-commutative 59.3%

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

      4. *-commutative 59.3%

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

      5. *-commutative 59.3%

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

      6. unpow2 59.3%

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

      7. unpow2 59.3%

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

      8. difference-of-squares 61.8%

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

      9. associate-*r* 61.8%

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

      10. associate-*l* 71.0%

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

   6. Applied egg-rr 71.0%

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

      1. associate-*r* 61.8%

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

      2. *-commutative 61.8%

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

      3. associate-*l* 71.0%

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

      4. *-commutative 71.0%

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

      5. *-commutative 71.0%

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

      6. +-commutative 71.0%

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

   8. Simplified 71.0%

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

      1. add-cbrt-cube 71.0%

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

      2. pow3 71.0%

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

      3. div-inv 71.6%

         \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \sqrt[3]{{\cos \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)}^{3}} \]

      4. metadata-eval 71.6%

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

   10. Applied egg-rr 71.6%

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

   if 1.15e244 < b

   1. Initial program 50.9%

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

      1. add-cbrt-cube 50.9%

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

      2. pow1/3 37.9%

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

      3. pow3 37.9%

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

      4. *-commutative 37.9%

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

      5. div-inv 37.9%

         \[\leadsto {\left({\left(\sin \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. metadata-eval 37.9%

         \[\leadsto {\left({\left(\sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. Applied egg-rr 37.9%

      \[\leadsto \color{blue}{{\left({\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   5. Step-by-step derivation

      1. unpow1/3 44.6%

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

      2. rem-cbrt-cube 44.6%

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

      3. *-commutative 44.6%

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

      4. *-commutative 44.6%

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

      5. *-commutative 44.6%

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

      6. unpow2 44.6%

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

      7. unpow2 44.6%

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

      8. difference-of-squares 57.1%

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

      9. associate-*r* 57.1%

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

      10. associate-*l* 68.8%

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

   6. Applied egg-rr 68.8%

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

      1. associate-*r* 57.1%

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

      2. *-commutative 57.1%

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

      3. associate-*l* 68.8%

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

      4. *-commutative 68.8%

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

      5. *-commutative 68.8%

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

      6. +-commutative 68.8%

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

   8. Simplified 68.8%

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

      1. *-commutative 68.8%

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

      2. associate-*r* 68.8%

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

      3. metadata-eval 68.8%

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

      4. div-inv 68.7%

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

      5. add-sqr-sqrt 37.5%

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

      6. associate-/l* 31.2%

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

      7. expm1-log1p-u 43.7%

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

      8. associate-/l* 43.8%

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

      9. add-sqr-sqrt 68.7%

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

      10. div-inv 68.8%

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

      11. *-commutative 68.8%

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

      12. metadata-eval 68.8%

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

      13. associate-*l* 68.8%

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

   10. Applied egg-rr 68.8%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{+244}:\\ \;\;\;\;\sqrt[3]{{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{3}} \cdot \left(\left(b - a\right) \cdot \left(\left(2 \cdot \left(b + a\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(2 \cdot \left(b + a\right)\right) \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\\ t_1 := \cos \left(\pi \cdot \frac{angle\_m}{180}\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq 3.7 \cdot 10^{+250}:\\ \;\;\;\;t\_1 \cdot \left(\left(b - a\_m\right) \cdot \left(\left(2 \cdot \left(b + a\_m\right)\right) \cdot \sin t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(\left(2 \cdot \left(\left(b - a\_m\right) \cdot \left(b + a\_m\right)\right)\right) \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(t\_0\right)\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* 0.005555555555555556 angle_m)))
        (t_1 (cos (* PI (/ angle_m 180.0)))))
   (*
    angle_s
    (if (<= b 3.7e+250)
      (* t_1 (* (- b a_m) (* (* 2.0 (+ b a_m)) (sin t_0))))
      (* t_1 (* (* 2.0 (* (- b a_m) (+ b a_m))) (sin (expm1 (log1p t_0)))))))))

C

a_m = fabs(a);
angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (0.005555555555555556 * angle_m);
	double t_1 = cos((((double) M_PI) * (angle_m / 180.0)));
	double tmp;
	if (b <= 3.7e+250) {
		tmp = t_1 * ((b - a_m) * ((2.0 * (b + a_m)) * sin(t_0)));
	} else {
		tmp = t_1 * ((2.0 * ((b - a_m) * (b + a_m))) * sin(expm1(log1p(t_0))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = Math.PI * (0.005555555555555556 * angle_m);
	double t_1 = Math.cos((Math.PI * (angle_m / 180.0)));
	double tmp;
	if (b <= 3.7e+250) {
		tmp = t_1 * ((b - a_m) * ((2.0 * (b + a_m)) * Math.sin(t_0)));
	} else {
		tmp = t_1 * ((2.0 * ((b - a_m) * (b + a_m))) * Math.sin(Math.expm1(Math.log1p(t_0))));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	t_0 = math.pi * (0.005555555555555556 * angle_m)
	t_1 = math.cos((math.pi * (angle_m / 180.0)))
	tmp = 0
	if b <= 3.7e+250:
		tmp = t_1 * ((b - a_m) * ((2.0 * (b + a_m)) * math.sin(t_0)))
	else:
		tmp = t_1 * ((2.0 * ((b - a_m) * (b + a_m))) * math.sin(math.expm1(math.log1p(t_0))))
	return angle_s * tmp

Julia

a_m = abs(a)
angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	t_0 = Float64(pi * Float64(0.005555555555555556 * angle_m))
	t_1 = cos(Float64(pi * Float64(angle_m / 180.0)))
	tmp = 0.0
	if (b <= 3.7e+250)
		tmp = Float64(t_1 * Float64(Float64(b - a_m) * Float64(Float64(2.0 * Float64(b + a_m)) * sin(t_0))));
	else
		tmp = Float64(t_1 * Float64(Float64(2.0 * Float64(Float64(b - a_m) * Float64(b + a_m))) * sin(expm1(log1p(t_0)))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $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_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[b, 3.7e+250], N[(t$95$1 * N[(N[(b - a$95$m), $MachinePrecision] * N[(N[(2.0 * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(2.0 * N[(N[(b - a$95$m), $MachinePrecision] * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Exp[N[Log[1 + t$95$0], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 3.70000000000000003e250

   1. Initial program 59.1%

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

      1. add-cbrt-cube 42.5%

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

      2. pow1/3 31.5%

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

      3. pow3 31.5%

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

      4. *-commutative 31.5%

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

      5. div-inv 31.4%

         \[\leadsto {\left({\left(\sin \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. metadata-eval 31.4%

         \[\leadsto {\left({\left(\sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. Applied egg-rr 31.4%

      \[\leadsto \color{blue}{{\left({\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   5. Step-by-step derivation

      1. unpow1/3 43.0%

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

      2. rem-cbrt-cube 59.4%

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

      3. *-commutative 59.4%

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

      4. *-commutative 59.4%

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

      5. *-commutative 59.4%

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

      6. unpow2 59.4%

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

      7. unpow2 59.4%

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

      8. difference-of-squares 62.0%

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

      9. associate-*r* 62.0%

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

      10. associate-*l* 71.2%

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

   6. Applied egg-rr 71.2%

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

      1. associate-*r* 62.0%

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

      2. *-commutative 62.0%

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

      3. associate-*l* 71.2%

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

      4. *-commutative 71.2%

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

      5. *-commutative 71.2%

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

      6. +-commutative 71.2%

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

   8. Simplified 71.2%

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

   if 3.70000000000000003e250 < b

   1. Initial program 47.6%

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

      1. unpow2 47.6%

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

      2. unpow2 47.6%

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

      3. difference-of-squares 54.3%

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

   4. Applied egg-rr 54.3%

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

      1. div-inv 54.3%

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

      2. metadata-eval 54.3%

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

      3. expm1-log1p-u 61.0%

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

   6. Applied egg-rr 61.0%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.7 \cdot 10^{+250}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(b - a\right) \cdot \left(\left(2 \cdot \left(b + a\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := 2 \cdot \left(b + a\_m\right)\\ t_1 := \cos \left(\pi \cdot \frac{angle\_m}{180}\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq 5.8 \cdot 10^{+243}:\\ \;\;\;\;t\_1 \cdot \left(\left(b - a\_m\right) \cdot \left(t\_0 \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\_m\right) \cdot \left(t\_0 \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle\_m \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)\right) \cdot t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (let* ((t_0 (* 2.0 (+ b a_m))) (t_1 (cos (* PI (/ angle_m 180.0)))))
   (*
    angle_s
    (if (<= b 5.8e+243)
      (*
       t_1
       (* (- b a_m) (* t_0 (sin (* PI (* 0.005555555555555556 angle_m))))))
      (*
       (*
        (- b a_m)
        (* t_0 (sin (expm1 (log1p (* angle_m (* PI 0.005555555555555556)))))))
       t_1)))))

C

a_m = fabs(a);
angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = 2.0 * (b + a_m);
	double t_1 = cos((((double) M_PI) * (angle_m / 180.0)));
	double tmp;
	if (b <= 5.8e+243) {
		tmp = t_1 * ((b - a_m) * (t_0 * sin((((double) M_PI) * (0.005555555555555556 * angle_m)))));
	} else {
		tmp = ((b - a_m) * (t_0 * sin(expm1(log1p((angle_m * (((double) M_PI) * 0.005555555555555556))))))) * t_1;
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = 2.0 * (b + a_m);
	double t_1 = Math.cos((Math.PI * (angle_m / 180.0)));
	double tmp;
	if (b <= 5.8e+243) {
		tmp = t_1 * ((b - a_m) * (t_0 * Math.sin((Math.PI * (0.005555555555555556 * angle_m)))));
	} else {
		tmp = ((b - a_m) * (t_0 * Math.sin(Math.expm1(Math.log1p((angle_m * (Math.PI * 0.005555555555555556))))))) * t_1;
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	t_0 = 2.0 * (b + a_m)
	t_1 = math.cos((math.pi * (angle_m / 180.0)))
	tmp = 0
	if b <= 5.8e+243:
		tmp = t_1 * ((b - a_m) * (t_0 * math.sin((math.pi * (0.005555555555555556 * angle_m)))))
	else:
		tmp = ((b - a_m) * (t_0 * math.sin(math.expm1(math.log1p((angle_m * (math.pi * 0.005555555555555556))))))) * t_1
	return angle_s * tmp

Julia

a_m = abs(a)
angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	t_0 = Float64(2.0 * Float64(b + a_m))
	t_1 = cos(Float64(pi * Float64(angle_m / 180.0)))
	tmp = 0.0
	if (b <= 5.8e+243)
		tmp = Float64(t_1 * Float64(Float64(b - a_m) * Float64(t_0 * sin(Float64(pi * Float64(0.005555555555555556 * angle_m))))));
	else
		tmp = Float64(Float64(Float64(b - a_m) * Float64(t_0 * sin(expm1(log1p(Float64(angle_m * Float64(pi * 0.005555555555555556))))))) * t_1);
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $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_, angle$95$m_] := Block[{t$95$0 = N[(2.0 * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[b, 5.8e+243], N[(t$95$1 * N[(N[(b - a$95$m), $MachinePrecision] * N[(t$95$0 * N[Sin[N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b - a$95$m), $MachinePrecision] * N[(t$95$0 * N[Sin[N[(Exp[N[Log[1 + N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 5.80000000000000013e243

   1. Initial program 58.9%

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

      1. add-cbrt-cube 42.3%

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

      2. pow1/3 31.2%

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

      3. pow3 31.2%

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

      4. *-commutative 31.2%

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

      5. div-inv 31.1%

         \[\leadsto {\left({\left(\sin \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. metadata-eval 31.1%

         \[\leadsto {\left({\left(\sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. Applied egg-rr 31.1%

      \[\leadsto \color{blue}{{\left({\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   5. Step-by-step derivation

      1. unpow1/3 42.7%

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

      2. rem-cbrt-cube 59.3%

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

      3. *-commutative 59.3%

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

      4. *-commutative 59.3%

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

      5. *-commutative 59.3%

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

      6. unpow2 59.3%

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

      7. unpow2 59.3%

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

      8. difference-of-squares 61.8%

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

      9. associate-*r* 61.8%

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

      10. associate-*l* 71.0%

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

   6. Applied egg-rr 71.0%

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

      1. associate-*r* 61.8%

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

      2. *-commutative 61.8%

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

      3. associate-*l* 71.0%

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

      4. *-commutative 71.0%

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

      5. *-commutative 71.0%

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

      6. +-commutative 71.0%

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

   8. Simplified 71.0%

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

   if 5.80000000000000013e243 < b

   1. Initial program 50.9%

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

      1. add-cbrt-cube 50.9%

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

      2. pow1/3 37.9%

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

      3. pow3 37.9%

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

      4. *-commutative 37.9%

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

      5. div-inv 37.9%

         \[\leadsto {\left({\left(\sin \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. metadata-eval 37.9%

         \[\leadsto {\left({\left(\sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. Applied egg-rr 37.9%

      \[\leadsto \color{blue}{{\left({\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   5. Step-by-step derivation

      1. unpow1/3 44.6%

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

      2. rem-cbrt-cube 44.6%

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

      3. *-commutative 44.6%

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

      4. *-commutative 44.6%

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

      5. *-commutative 44.6%

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

      6. unpow2 44.6%

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

      7. unpow2 44.6%

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

      8. difference-of-squares 57.1%

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

      9. associate-*r* 57.1%

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

      10. associate-*l* 68.8%

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

   6. Applied egg-rr 68.8%

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

      1. associate-*r* 57.1%

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

      2. *-commutative 57.1%

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

      3. associate-*l* 68.8%

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

      4. *-commutative 68.8%

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

      5. *-commutative 68.8%

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

      6. +-commutative 68.8%

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

   8. Simplified 68.8%

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

      1. *-commutative 68.8%

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

      2. associate-*r* 68.8%

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

      3. metadata-eval 68.8%

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

      4. div-inv 68.7%

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

      5. add-sqr-sqrt 37.5%

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

      6. associate-/l* 31.2%

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

      7. expm1-log1p-u 43.7%

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

      8. associate-/l* 43.8%

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

      9. add-sqr-sqrt 68.7%

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

      10. div-inv 68.8%

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

      11. *-commutative 68.8%

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

      12. metadata-eval 68.8%

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

      13. associate-*l* 68.8%

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

   10. Applied egg-rr 68.8%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.8 \cdot 10^{+243}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(b - a\right) \cdot \left(\left(2 \cdot \left(b + a\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(2 \cdot \left(b + a\right)\right) \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := 2 \cdot \left(\left(b - a\_m\right) \cdot \left(b + a\_m\right)\right)\\ t_1 := \cos \left(\pi \cdot \frac{angle\_m}{180}\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;t\_1 \cdot \left(\left(b - a\_m\right) \cdot \left(\left(2 \cdot \left(b + a\_m\right)\right) \cdot \left(angle\_m \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{+155} \lor \neg \left(\frac{angle\_m}{180} \leq 10^{+198}\right):\\ \;\;\;\;t\_1 \cdot \left(t\_0 \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sin \left(angle\_m \cdot \frac{\pi}{180}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (let* ((t_0 (* 2.0 (* (- b a_m) (+ b a_m))))
        (t_1 (cos (* PI (/ angle_m 180.0)))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 5e-25)
      (*
       t_1
       (*
        (- b a_m)
        (* (* 2.0 (+ b a_m)) (* angle_m (* PI 0.005555555555555556)))))
      (if (or (<= (/ angle_m 180.0) 4e+155)
              (not (<= (/ angle_m 180.0) 1e+198)))
        (* t_1 (* t_0 (sin (* PI (* 0.005555555555555556 angle_m)))))
        (* t_0 (sin (* angle_m (/ PI 180.0)))))))))

C

a_m = fabs(a);
angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = 2.0 * ((b - a_m) * (b + a_m));
	double t_1 = cos((((double) M_PI) * (angle_m / 180.0)));
	double tmp;
	if ((angle_m / 180.0) <= 5e-25) {
		tmp = t_1 * ((b - a_m) * ((2.0 * (b + a_m)) * (angle_m * (((double) M_PI) * 0.005555555555555556))));
	} else if (((angle_m / 180.0) <= 4e+155) || !((angle_m / 180.0) <= 1e+198)) {
		tmp = t_1 * (t_0 * sin((((double) M_PI) * (0.005555555555555556 * angle_m))));
	} else {
		tmp = t_0 * sin((angle_m * (((double) M_PI) / 180.0)));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = 2.0 * ((b - a_m) * (b + a_m));
	double t_1 = Math.cos((Math.PI * (angle_m / 180.0)));
	double tmp;
	if ((angle_m / 180.0) <= 5e-25) {
		tmp = t_1 * ((b - a_m) * ((2.0 * (b + a_m)) * (angle_m * (Math.PI * 0.005555555555555556))));
	} else if (((angle_m / 180.0) <= 4e+155) || !((angle_m / 180.0) <= 1e+198)) {
		tmp = t_1 * (t_0 * Math.sin((Math.PI * (0.005555555555555556 * angle_m))));
	} else {
		tmp = t_0 * Math.sin((angle_m * (Math.PI / 180.0)));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	t_0 = 2.0 * ((b - a_m) * (b + a_m))
	t_1 = math.cos((math.pi * (angle_m / 180.0)))
	tmp = 0
	if (angle_m / 180.0) <= 5e-25:
		tmp = t_1 * ((b - a_m) * ((2.0 * (b + a_m)) * (angle_m * (math.pi * 0.005555555555555556))))
	elif ((angle_m / 180.0) <= 4e+155) or not ((angle_m / 180.0) <= 1e+198):
		tmp = t_1 * (t_0 * math.sin((math.pi * (0.005555555555555556 * angle_m))))
	else:
		tmp = t_0 * math.sin((angle_m * (math.pi / 180.0)))
	return angle_s * tmp

Julia

a_m = abs(a)
angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	t_0 = Float64(2.0 * Float64(Float64(b - a_m) * Float64(b + a_m)))
	t_1 = cos(Float64(pi * Float64(angle_m / 180.0)))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e-25)
		tmp = Float64(t_1 * Float64(Float64(b - a_m) * Float64(Float64(2.0 * Float64(b + a_m)) * Float64(angle_m * Float64(pi * 0.005555555555555556)))));
	elseif ((Float64(angle_m / 180.0) <= 4e+155) || !(Float64(angle_m / 180.0) <= 1e+198))
		tmp = Float64(t_1 * Float64(t_0 * sin(Float64(pi * Float64(0.005555555555555556 * angle_m)))));
	else
		tmp = Float64(t_0 * sin(Float64(angle_m * Float64(pi / 180.0))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	t_0 = 2.0 * ((b - a_m) * (b + a_m));
	t_1 = cos((pi * (angle_m / 180.0)));
	tmp = 0.0;
	if ((angle_m / 180.0) <= 5e-25)
		tmp = t_1 * ((b - a_m) * ((2.0 * (b + a_m)) * (angle_m * (pi * 0.005555555555555556))));
	elseif (((angle_m / 180.0) <= 4e+155) || ~(((angle_m / 180.0) <= 1e+198)))
		tmp = t_1 * (t_0 * sin((pi * (0.005555555555555556 * angle_m))));
	else
		tmp = t_0 * sin((angle_m * (pi / 180.0)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $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_, angle$95$m_] := Block[{t$95$0 = N[(2.0 * N[(N[(b - a$95$m), $MachinePrecision] * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e-25], N[(t$95$1 * N[(N[(b - a$95$m), $MachinePrecision] * N[(N[(2.0 * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision] * N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+155], N[Not[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+198]], $MachinePrecision]], N[(t$95$1 * N[(t$95$0 * N[Sin[N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Sin[N[(angle$95$m * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < 4.99999999999999962e-25

   1. Initial program 60.9%

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

      1. add-cbrt-cube 41.2%

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

      2. pow1/3 29.8%

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

      3. pow3 29.8%

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

      4. *-commutative 29.8%

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

      5. div-inv 29.8%

         \[\leadsto {\left({\left(\sin \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. metadata-eval 29.8%

         \[\leadsto {\left({\left(\sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. Applied egg-rr 29.8%

      \[\leadsto \color{blue}{{\left({\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   5. Step-by-step derivation

      1. unpow1/3 40.7%

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

      2. rem-cbrt-cube 60.3%

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

      3. *-commutative 60.3%

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

      4. *-commutative 60.3%

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

      5. *-commutative 60.3%

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

      6. unpow2 60.3%

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

      7. unpow2 60.3%

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

      8. difference-of-squares 63.5%

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

      9. associate-*r* 63.5%

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

      10. associate-*l* 76.4%

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

   6. Applied egg-rr 76.4%

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

      1. associate-*r* 63.5%

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

      2. *-commutative 63.5%

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

      3. associate-*l* 76.4%

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

      4. *-commutative 76.4%

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

      5. *-commutative 76.4%

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

      6. +-commutative 76.4%

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

   8. Simplified 76.4%

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

   9. Taylor expanded in angle around 0 71.3%

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

       1. associate-*r* 71.3%

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

       2. *-commutative 71.3%

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

       3. associate-*r* 71.4%

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

       4. *-commutative 71.4%

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

       5. *-commutative 71.4%

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

   11. Simplified 71.4%

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

   if 4.99999999999999962e-25 < (/.f64 angle 180) < 4.00000000000000003e155 or 1.00000000000000002e198 < (/.f64 angle 180)

   1. Initial program 61.4%

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

      1. unpow2 61.4%

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

      2. unpow2 61.4%

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

      3. difference-of-squares 63.2%

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

   4. Applied egg-rr 63.2%

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

   5. Taylor expanded in angle around inf 51.6%

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

      1. *-commutative 51.6%

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

      2. *-commutative 51.6%

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

      3. associate-*r* 61.4%

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

   7. Simplified 61.4%

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

   if 4.00000000000000003e155 < (/.f64 angle 180) < 1.00000000000000002e198

   1. Initial program 16.9%

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

      1. unpow2 16.9%

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

      2. unpow2 16.9%

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

      3. difference-of-squares 16.9%

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

   4. Applied egg-rr 16.9%

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

   5. Taylor expanded in angle around 0 49.2%

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

      1. clear-num 41.4%

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

      2. un-div-inv 61.4%

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

   7. Applied egg-rr 61.4%

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

      1. associate-/r/ 69.2%

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

   9. Simplified 69.2%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(b - a\right) \cdot \left(\left(2 \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 4 \cdot 10^{+155} \lor \neg \left(\frac{angle}{180} \leq 10^{+198}\right):\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := 2 \cdot \left(\left(b - a\_m\right) \cdot \left(b + a\_m\right)\right)\\ t_1 := \pi \cdot \frac{angle\_m}{180}\\ t_2 := \cos t\_1\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;t\_2 \cdot \left(\left(b - a\_m\right) \cdot \left(\left(2 \cdot \left(b + a\_m\right)\right) \cdot \left(angle\_m \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{+155}:\\ \;\;\;\;t\_2 \cdot \left(t\_0 \cdot \sin t\_1\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 10^{+198}:\\ \;\;\;\;t\_0 \cdot \sin \left(angle\_m \cdot \frac{\pi}{180}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(t\_0 \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (let* ((t_0 (* 2.0 (* (- b a_m) (+ b a_m))))
        (t_1 (* PI (/ angle_m 180.0)))
        (t_2 (cos t_1)))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 5e-25)
      (*
       t_2
       (*
        (- b a_m)
        (* (* 2.0 (+ b a_m)) (* angle_m (* PI 0.005555555555555556)))))
      (if (<= (/ angle_m 180.0) 4e+155)
        (* t_2 (* t_0 (sin t_1)))
        (if (<= (/ angle_m 180.0) 1e+198)
          (* t_0 (sin (* angle_m (/ PI 180.0))))
          (* t_2 (* t_0 (sin (* PI (* 0.005555555555555556 angle_m)))))))))))

C

a_m = fabs(a);
angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = 2.0 * ((b - a_m) * (b + a_m));
	double t_1 = ((double) M_PI) * (angle_m / 180.0);
	double t_2 = cos(t_1);
	double tmp;
	if ((angle_m / 180.0) <= 5e-25) {
		tmp = t_2 * ((b - a_m) * ((2.0 * (b + a_m)) * (angle_m * (((double) M_PI) * 0.005555555555555556))));
	} else if ((angle_m / 180.0) <= 4e+155) {
		tmp = t_2 * (t_0 * sin(t_1));
	} else if ((angle_m / 180.0) <= 1e+198) {
		tmp = t_0 * sin((angle_m * (((double) M_PI) / 180.0)));
	} else {
		tmp = t_2 * (t_0 * sin((((double) M_PI) * (0.005555555555555556 * angle_m))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = 2.0 * ((b - a_m) * (b + a_m));
	double t_1 = Math.PI * (angle_m / 180.0);
	double t_2 = Math.cos(t_1);
	double tmp;
	if ((angle_m / 180.0) <= 5e-25) {
		tmp = t_2 * ((b - a_m) * ((2.0 * (b + a_m)) * (angle_m * (Math.PI * 0.005555555555555556))));
	} else if ((angle_m / 180.0) <= 4e+155) {
		tmp = t_2 * (t_0 * Math.sin(t_1));
	} else if ((angle_m / 180.0) <= 1e+198) {
		tmp = t_0 * Math.sin((angle_m * (Math.PI / 180.0)));
	} else {
		tmp = t_2 * (t_0 * Math.sin((Math.PI * (0.005555555555555556 * angle_m))));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	t_0 = 2.0 * ((b - a_m) * (b + a_m))
	t_1 = math.pi * (angle_m / 180.0)
	t_2 = math.cos(t_1)
	tmp = 0
	if (angle_m / 180.0) <= 5e-25:
		tmp = t_2 * ((b - a_m) * ((2.0 * (b + a_m)) * (angle_m * (math.pi * 0.005555555555555556))))
	elif (angle_m / 180.0) <= 4e+155:
		tmp = t_2 * (t_0 * math.sin(t_1))
	elif (angle_m / 180.0) <= 1e+198:
		tmp = t_0 * math.sin((angle_m * (math.pi / 180.0)))
	else:
		tmp = t_2 * (t_0 * math.sin((math.pi * (0.005555555555555556 * angle_m))))
	return angle_s * tmp

Julia

a_m = abs(a)
angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	t_0 = Float64(2.0 * Float64(Float64(b - a_m) * Float64(b + a_m)))
	t_1 = Float64(pi * Float64(angle_m / 180.0))
	t_2 = cos(t_1)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e-25)
		tmp = Float64(t_2 * Float64(Float64(b - a_m) * Float64(Float64(2.0 * Float64(b + a_m)) * Float64(angle_m * Float64(pi * 0.005555555555555556)))));
	elseif (Float64(angle_m / 180.0) <= 4e+155)
		tmp = Float64(t_2 * Float64(t_0 * sin(t_1)));
	elseif (Float64(angle_m / 180.0) <= 1e+198)
		tmp = Float64(t_0 * sin(Float64(angle_m * Float64(pi / 180.0))));
	else
		tmp = Float64(t_2 * Float64(t_0 * sin(Float64(pi * Float64(0.005555555555555556 * angle_m)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	t_0 = 2.0 * ((b - a_m) * (b + a_m));
	t_1 = pi * (angle_m / 180.0);
	t_2 = cos(t_1);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 5e-25)
		tmp = t_2 * ((b - a_m) * ((2.0 * (b + a_m)) * (angle_m * (pi * 0.005555555555555556))));
	elseif ((angle_m / 180.0) <= 4e+155)
		tmp = t_2 * (t_0 * sin(t_1));
	elseif ((angle_m / 180.0) <= 1e+198)
		tmp = t_0 * sin((angle_m * (pi / 180.0)));
	else
		tmp = t_2 * (t_0 * sin((pi * (0.005555555555555556 * angle_m))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $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_, angle$95$m_] := Block[{t$95$0 = N[(2.0 * N[(N[(b - a$95$m), $MachinePrecision] * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$1], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e-25], N[(t$95$2 * N[(N[(b - a$95$m), $MachinePrecision] * N[(N[(2.0 * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision] * N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+155], N[(t$95$2 * N[(t$95$0 * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+198], N[(t$95$0 * N[Sin[N[(angle$95$m * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(t$95$0 * N[Sin[N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle 180) < 4.99999999999999962e-25

   1. Initial program 60.9%

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

      1. add-cbrt-cube 41.2%

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

      2. pow1/3 29.8%

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

      3. pow3 29.8%

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

      4. *-commutative 29.8%

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

      5. div-inv 29.8%

         \[\leadsto {\left({\left(\sin \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. metadata-eval 29.8%

         \[\leadsto {\left({\left(\sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. Applied egg-rr 29.8%

      \[\leadsto \color{blue}{{\left({\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   5. Step-by-step derivation

      1. unpow1/3 40.7%

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

      2. rem-cbrt-cube 60.3%

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

      3. *-commutative 60.3%

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

      4. *-commutative 60.3%

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

      5. *-commutative 60.3%

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

      6. unpow2 60.3%

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

      7. unpow2 60.3%

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

      8. difference-of-squares 63.5%

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

      9. associate-*r* 63.5%

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

      10. associate-*l* 76.4%

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

   6. Applied egg-rr 76.4%

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

      1. associate-*r* 63.5%

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

      2. *-commutative 63.5%

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

      3. associate-*l* 76.4%

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

      4. *-commutative 76.4%

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

      5. *-commutative 76.4%

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

      6. +-commutative 76.4%

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

   8. Simplified 76.4%

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

   9. Taylor expanded in angle around 0 71.3%

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

       1. associate-*r* 71.3%

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

       2. *-commutative 71.3%

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

       3. associate-*r* 71.4%

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

       4. *-commutative 71.4%

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

       5. *-commutative 71.4%

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

   11. Simplified 71.4%

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

   if 4.99999999999999962e-25 < (/.f64 angle 180) < 4.00000000000000003e155

   1. Initial program 74.1%

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

      1. unpow2 74.1%

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

      2. unpow2 74.1%

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

      3. difference-of-squares 74.1%

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

   4. Applied egg-rr 74.1%

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

   if 4.00000000000000003e155 < (/.f64 angle 180) < 1.00000000000000002e198

   1. Initial program 16.9%

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

      1. unpow2 16.9%

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

      2. unpow2 16.9%

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

      3. difference-of-squares 16.9%

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

   4. Applied egg-rr 16.9%

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

   5. Taylor expanded in angle around 0 49.2%

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

      1. clear-num 41.4%

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

      2. un-div-inv 61.4%

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

   7. Applied egg-rr 61.4%

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

      1. associate-/r/ 69.2%

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

   9. Simplified 69.2%

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

   if 1.00000000000000002e198 < (/.f64 angle 180)

   1. Initial program 42.9%

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

      1. unpow2 42.9%

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

      2. unpow2 42.9%

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

      3. difference-of-squares 47.4%

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

   4. Applied egg-rr 47.4%

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

   5. Taylor expanded in angle around inf 42.1%

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

      1. *-commutative 42.1%

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

      2. *-commutative 42.1%

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

      3. associate-*r* 47.4%

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

   7. Simplified 47.4%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(b - a\right) \cdot \left(\left(2 \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 4 \cdot 10^{+155}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+198}:\\ \;\;\;\;\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

a_m = N[Abs[a], $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_, angle$95$m_] := N[(angle$95$s * N[(N[Cos[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[(N[(2.0 * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.4%

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

   1. add-cbrt-cube 42.8%

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

   2. pow1/3 31.6%

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

   3. pow3 31.6%

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

   4. *-commutative 31.6%

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

   5. div-inv 31.6%

      \[\leadsto {\left({\left(\sin \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. metadata-eval 31.6%

      \[\leadsto {\left({\left(\sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

4. Applied egg-rr 31.6%

   \[\leadsto \color{blue}{{\left({\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
5. Step-by-step derivation

   1. unpow1/3 42.8%

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

   2. rem-cbrt-cube 58.3%

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

   3. *-commutative 58.3%

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

   4. *-commutative 58.3%

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

   5. *-commutative 58.3%

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

   6. unpow2 58.3%

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

   7. unpow2 58.3%

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

   8. difference-of-squares 61.5%

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

   9. associate-*r* 61.5%

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

   10. associate-*l* 70.9%

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

6. Applied egg-rr 70.9%

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

   1. associate-*r* 61.5%

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

   2. *-commutative 61.5%

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

   3. associate-*l* 70.9%

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

   4. *-commutative 70.9%

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

   5. *-commutative 70.9%

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

   6. +-commutative 70.9%

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

8. Simplified 70.9%

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

9. Final simplification 70.9%

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

Alternative 8: 65.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(b - a\_m\right) \cdot \left(b + a\_m\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\left(b - a\_m\right) \cdot \left(\left(2 \cdot \left(b + a\_m\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+116}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot t\_0\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 6 \cdot 10^{+197}:\\ \;\;\;\;\left(2 \cdot t\_0\right) \cdot \sin \left(angle\_m \cdot \frac{\pi}{180}\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\left(b - a\_m\right) \cdot \left(\pi \cdot \left(b + a\_m\right)\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (let* ((t_0 (* (- b a_m) (+ b a_m))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 2e+14)
      (*
       (- b a_m)
       (* (* 2.0 (+ b a_m)) (sin (* PI (* 0.005555555555555556 angle_m)))))
      (if (<= (/ angle_m 180.0) 5e+116)
        (* 0.011111111111111112 (* angle_m (* PI t_0)))
        (if (<= (/ angle_m 180.0) 6e+197)
          (* (* 2.0 t_0) (sin (* angle_m (/ PI 180.0))))
          (*
           0.011111111111111112
           (* angle_m (* (- b a_m) (* PI (+ b a_m)))))))))))

C

a_m = fabs(a);
angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = (b - a_m) * (b + a_m);
	double tmp;
	if ((angle_m / 180.0) <= 2e+14) {
		tmp = (b - a_m) * ((2.0 * (b + a_m)) * sin((((double) M_PI) * (0.005555555555555556 * angle_m))));
	} else if ((angle_m / 180.0) <= 5e+116) {
		tmp = 0.011111111111111112 * (angle_m * (((double) M_PI) * t_0));
	} else if ((angle_m / 180.0) <= 6e+197) {
		tmp = (2.0 * t_0) * sin((angle_m * (((double) M_PI) / 180.0)));
	} else {
		tmp = 0.011111111111111112 * (angle_m * ((b - a_m) * (((double) M_PI) * (b + a_m))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = (b - a_m) * (b + a_m);
	double tmp;
	if ((angle_m / 180.0) <= 2e+14) {
		tmp = (b - a_m) * ((2.0 * (b + a_m)) * Math.sin((Math.PI * (0.005555555555555556 * angle_m))));
	} else if ((angle_m / 180.0) <= 5e+116) {
		tmp = 0.011111111111111112 * (angle_m * (Math.PI * t_0));
	} else if ((angle_m / 180.0) <= 6e+197) {
		tmp = (2.0 * t_0) * Math.sin((angle_m * (Math.PI / 180.0)));
	} else {
		tmp = 0.011111111111111112 * (angle_m * ((b - a_m) * (Math.PI * (b + a_m))));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	t_0 = (b - a_m) * (b + a_m)
	tmp = 0
	if (angle_m / 180.0) <= 2e+14:
		tmp = (b - a_m) * ((2.0 * (b + a_m)) * math.sin((math.pi * (0.005555555555555556 * angle_m))))
	elif (angle_m / 180.0) <= 5e+116:
		tmp = 0.011111111111111112 * (angle_m * (math.pi * t_0))
	elif (angle_m / 180.0) <= 6e+197:
		tmp = (2.0 * t_0) * math.sin((angle_m * (math.pi / 180.0)))
	else:
		tmp = 0.011111111111111112 * (angle_m * ((b - a_m) * (math.pi * (b + a_m))))
	return angle_s * tmp

Julia

a_m = abs(a)
angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	t_0 = Float64(Float64(b - a_m) * Float64(b + a_m))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e+14)
		tmp = Float64(Float64(b - a_m) * Float64(Float64(2.0 * Float64(b + a_m)) * sin(Float64(pi * Float64(0.005555555555555556 * angle_m)))));
	elseif (Float64(angle_m / 180.0) <= 5e+116)
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * t_0)));
	elseif (Float64(angle_m / 180.0) <= 6e+197)
		tmp = Float64(Float64(2.0 * t_0) * sin(Float64(angle_m * Float64(pi / 180.0))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(Float64(b - a_m) * Float64(pi * Float64(b + a_m)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	t_0 = (b - a_m) * (b + a_m);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 2e+14)
		tmp = (b - a_m) * ((2.0 * (b + a_m)) * sin((pi * (0.005555555555555556 * angle_m))));
	elseif ((angle_m / 180.0) <= 5e+116)
		tmp = 0.011111111111111112 * (angle_m * (pi * t_0));
	elseif ((angle_m / 180.0) <= 6e+197)
		tmp = (2.0 * t_0) * sin((angle_m * (pi / 180.0)));
	else
		tmp = 0.011111111111111112 * (angle_m * ((b - a_m) * (pi * (b + a_m))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $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_, angle$95$m_] := Block[{t$95$0 = N[(N[(b - a$95$m), $MachinePrecision] * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+14], N[(N[(b - a$95$m), $MachinePrecision] * N[(N[(2.0 * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(0.005555555555555556 * angle$95$m), $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 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 6e+197], N[(N[(2.0 * t$95$0), $MachinePrecision] * N[Sin[N[(angle$95$m * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle$95$m * N[(N[(b - a$95$m), $MachinePrecision] * N[(Pi * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle 180) < 2e14

   1. Initial program 62.8%

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

      1. add-cbrt-cube 43.6%

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

      2. pow1/3 30.4%

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

      3. pow3 30.4%

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

      4. *-commutative 30.4%

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

      5. div-inv 30.4%

         \[\leadsto {\left({\left(\sin \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. metadata-eval 30.4%

         \[\leadsto {\left({\left(\sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. Applied egg-rr 30.4%

      \[\leadsto \color{blue}{{\left({\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   5. Step-by-step derivation

      1. unpow1/3 43.2%

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

      2. rem-cbrt-cube 62.3%

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

      3. *-commutative 62.3%

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

      4. *-commutative 62.3%

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

      5. *-commutative 62.3%

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

      6. unpow2 62.3%

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

      7. unpow2 62.3%

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

      8. difference-of-squares 65.3%

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

      9. associate-*r* 65.3%

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

      10. associate-*l* 77.4%

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

   6. Applied egg-rr 77.4%

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

      1. associate-*r* 65.3%

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

      2. *-commutative 65.3%

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

      3. associate-*l* 77.4%

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

      4. *-commutative 77.4%

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

      5. *-commutative 77.4%

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

      6. +-commutative 77.4%

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

   8. Simplified 77.4%

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

   9. Taylor expanded in angle around 0 72.3%

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

   if 2e14 < (/.f64 angle 180) < 5.00000000000000025e116

   1. Initial program 62.2%

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

      1. unpow2 62.2%

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

      2. unpow2 62.2%

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

      3. difference-of-squares 62.2%

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

   4. Applied egg-rr 62.2%

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

   5. Taylor expanded in angle around 0 25.2%

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

   6. Taylor expanded in angle around 0 67.3%

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

   if 5.00000000000000025e116 < (/.f64 angle 180) < 6.0000000000000004e197

   1. Initial program 32.8%

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

      1. unpow2 32.8%

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

      2. unpow2 32.8%

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

      3. difference-of-squares 32.8%

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

   4. Applied egg-rr 32.8%

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

   5. Taylor expanded in angle around 0 51.2%

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

      1. clear-num 45.7%

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

      2. un-div-inv 50.4%

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

   7. Applied egg-rr 50.4%

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

      1. associate-/r/ 56.1%

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

   9. Simplified 56.1%

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

   if 6.0000000000000004e197 < (/.f64 angle 180)

   1. Initial program 41.0%

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

      1. unpow2 41.0%

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

      2. unpow2 41.0%

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

      3. difference-of-squares 45.3%

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

   4. Applied egg-rr 45.3%

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

   5. Taylor expanded in angle around 0 17.0%

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

   6. Taylor expanded in angle around 0 36.5%

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

      1. associate-*r* 36.5%

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

      2. sub-neg 36.5%

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

      3. distribute-lft-in 32.1%

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

   8. Applied egg-rr 32.1%

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

      1. distribute-lft-out 36.5%

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

      2. sub-neg 36.5%

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

      3. +-commutative 36.5%

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

   10. Simplified 36.5%

       \[\leadsto \left(0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right)\right) \cdot 1 \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(2 \cdot \left(b + a\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+116}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 6 \cdot 10^{+197}:\\ \;\;\;\;\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(b - a\_m\right) \cdot \left(b + a\_m\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2000000:\\ \;\;\;\;\left(b - a\_m\right) \cdot \left(\left(2 \cdot \left(b + a\_m\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+116}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle\_m}{180}\right) \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot t\_0\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 6 \cdot 10^{+197}:\\ \;\;\;\;\left(2 \cdot t\_0\right) \cdot \sin \left(angle\_m \cdot \frac{\pi}{180}\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\left(b - a\_m\right) \cdot \left(\pi \cdot \left(b + a\_m\right)\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (let* ((t_0 (* (- b a_m) (+ b a_m))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 2000000.0)
      (*
       (- b a_m)
       (* (* 2.0 (+ b a_m)) (sin (* PI (* 0.005555555555555556 angle_m)))))
      (if (<= (/ angle_m 180.0) 5e+116)
        (*
         (cos (* PI (/ angle_m 180.0)))
         (* 0.011111111111111112 (* angle_m (* PI t_0))))
        (if (<= (/ angle_m 180.0) 6e+197)
          (* (* 2.0 t_0) (sin (* angle_m (/ PI 180.0))))
          (*
           0.011111111111111112
           (* angle_m (* (- b a_m) (* PI (+ b a_m)))))))))))

C

a_m = fabs(a);
angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = (b - a_m) * (b + a_m);
	double tmp;
	if ((angle_m / 180.0) <= 2000000.0) {
		tmp = (b - a_m) * ((2.0 * (b + a_m)) * sin((((double) M_PI) * (0.005555555555555556 * angle_m))));
	} else if ((angle_m / 180.0) <= 5e+116) {
		tmp = cos((((double) M_PI) * (angle_m / 180.0))) * (0.011111111111111112 * (angle_m * (((double) M_PI) * t_0)));
	} else if ((angle_m / 180.0) <= 6e+197) {
		tmp = (2.0 * t_0) * sin((angle_m * (((double) M_PI) / 180.0)));
	} else {
		tmp = 0.011111111111111112 * (angle_m * ((b - a_m) * (((double) M_PI) * (b + a_m))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = (b - a_m) * (b + a_m);
	double tmp;
	if ((angle_m / 180.0) <= 2000000.0) {
		tmp = (b - a_m) * ((2.0 * (b + a_m)) * Math.sin((Math.PI * (0.005555555555555556 * angle_m))));
	} else if ((angle_m / 180.0) <= 5e+116) {
		tmp = Math.cos((Math.PI * (angle_m / 180.0))) * (0.011111111111111112 * (angle_m * (Math.PI * t_0)));
	} else if ((angle_m / 180.0) <= 6e+197) {
		tmp = (2.0 * t_0) * Math.sin((angle_m * (Math.PI / 180.0)));
	} else {
		tmp = 0.011111111111111112 * (angle_m * ((b - a_m) * (Math.PI * (b + a_m))));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	t_0 = (b - a_m) * (b + a_m)
	tmp = 0
	if (angle_m / 180.0) <= 2000000.0:
		tmp = (b - a_m) * ((2.0 * (b + a_m)) * math.sin((math.pi * (0.005555555555555556 * angle_m))))
	elif (angle_m / 180.0) <= 5e+116:
		tmp = math.cos((math.pi * (angle_m / 180.0))) * (0.011111111111111112 * (angle_m * (math.pi * t_0)))
	elif (angle_m / 180.0) <= 6e+197:
		tmp = (2.0 * t_0) * math.sin((angle_m * (math.pi / 180.0)))
	else:
		tmp = 0.011111111111111112 * (angle_m * ((b - a_m) * (math.pi * (b + a_m))))
	return angle_s * tmp

Julia

a_m = abs(a)
angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	t_0 = Float64(Float64(b - a_m) * Float64(b + a_m))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2000000.0)
		tmp = Float64(Float64(b - a_m) * Float64(Float64(2.0 * Float64(b + a_m)) * sin(Float64(pi * Float64(0.005555555555555556 * angle_m)))));
	elseif (Float64(angle_m / 180.0) <= 5e+116)
		tmp = Float64(cos(Float64(pi * Float64(angle_m / 180.0))) * Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * t_0))));
	elseif (Float64(angle_m / 180.0) <= 6e+197)
		tmp = Float64(Float64(2.0 * t_0) * sin(Float64(angle_m * Float64(pi / 180.0))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(Float64(b - a_m) * Float64(pi * Float64(b + a_m)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	t_0 = (b - a_m) * (b + a_m);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 2000000.0)
		tmp = (b - a_m) * ((2.0 * (b + a_m)) * sin((pi * (0.005555555555555556 * angle_m))));
	elseif ((angle_m / 180.0) <= 5e+116)
		tmp = cos((pi * (angle_m / 180.0))) * (0.011111111111111112 * (angle_m * (pi * t_0)));
	elseif ((angle_m / 180.0) <= 6e+197)
		tmp = (2.0 * t_0) * sin((angle_m * (pi / 180.0)));
	else
		tmp = 0.011111111111111112 * (angle_m * ((b - a_m) * (pi * (b + a_m))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $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_, angle$95$m_] := Block[{t$95$0 = N[(N[(b - a$95$m), $MachinePrecision] * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2000000.0], N[(N[(b - a$95$m), $MachinePrecision] * N[(N[(2.0 * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+116], N[(N[Cos[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 6e+197], N[(N[(2.0 * t$95$0), $MachinePrecision] * N[Sin[N[(angle$95$m * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle$95$m * N[(N[(b - a$95$m), $MachinePrecision] * N[(Pi * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle 180) < 2e6

   1. Initial program 62.4%

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

      1. add-cbrt-cube 42.9%

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

      2. pow1/3 30.0%

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

      3. pow3 30.0%

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

      4. *-commutative 30.0%

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

      5. div-inv 30.0%

         \[\leadsto {\left({\left(\sin \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. metadata-eval 30.0%

         \[\leadsto {\left({\left(\sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. Applied egg-rr 30.0%

      \[\leadsto \color{blue}{{\left({\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   5. Step-by-step derivation

      1. unpow1/3 42.5%

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

      2. rem-cbrt-cube 61.9%

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

      3. *-commutative 61.9%

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

      4. *-commutative 61.9%

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

      5. *-commutative 61.9%

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

      6. unpow2 61.9%

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

      7. unpow2 61.9%

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

      8. difference-of-squares 65.0%

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

      9. associate-*r* 65.0%

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

      10. associate-*l* 77.3%

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

   6. Applied egg-rr 77.3%

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

      1. associate-*r* 65.0%

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

      2. *-commutative 65.0%

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

      3. associate-*l* 77.3%

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

      4. *-commutative 77.3%

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

      5. *-commutative 77.3%

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

      6. +-commutative 77.3%

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

   8. Simplified 77.3%

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

   9. Taylor expanded in angle around 0 73.1%

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

   if 2e6 < (/.f64 angle 180) < 5.00000000000000025e116

   1. Initial program 67.3%

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

      1. unpow2 67.3%

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

      2. unpow2 67.3%

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

      3. difference-of-squares 67.3%

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

   4. Applied egg-rr 67.3%

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

   5. Taylor expanded in angle around 0 57.8%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   if 5.00000000000000025e116 < (/.f64 angle 180) < 6.0000000000000004e197

   1. Initial program 32.8%

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

      1. unpow2 32.8%

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

      2. unpow2 32.8%

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

      3. difference-of-squares 32.8%

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

   4. Applied egg-rr 32.8%

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

   5. Taylor expanded in angle around 0 51.2%

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

      1. clear-num 45.7%

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

      2. un-div-inv 50.4%

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

   7. Applied egg-rr 50.4%

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

      1. associate-/r/ 56.1%

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

   9. Simplified 56.1%

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

   if 6.0000000000000004e197 < (/.f64 angle 180)

   1. Initial program 41.0%

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

      1. unpow2 41.0%

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

      2. unpow2 41.0%

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

      3. difference-of-squares 45.3%

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

   4. Applied egg-rr 45.3%

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

   5. Taylor expanded in angle around 0 17.0%

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

   6. Taylor expanded in angle around 0 36.5%

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

      1. associate-*r* 36.5%

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

      2. sub-neg 36.5%

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

      3. distribute-lft-in 32.1%

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

   8. Applied egg-rr 32.1%

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

      1. distribute-lft-out 36.5%

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

      2. sub-neg 36.5%

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

      3. +-commutative 36.5%

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

   10. Simplified 36.5%

       \[\leadsto \left(0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right)\right) \cdot 1 \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2000000:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(2 \cdot \left(b + a\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+116}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 6 \cdot 10^{+197}:\\ \;\;\;\;\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := 2 \cdot \left(\left(b - a\_m\right) \cdot \left(b + a\_m\right)\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{-34}:\\ \;\;\;\;\left(b - a\_m\right) \cdot \left(\left(2 \cdot \left(b + a\_m\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+139}:\\ \;\;\;\;t\_0 \cdot \sin \left(\frac{1}{\frac{180}{\pi \cdot angle\_m}}\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 6 \cdot 10^{+197}:\\ \;\;\;\;t\_0 \cdot \sin \left(angle\_m \cdot \frac{\pi}{180}\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\left(b - a\_m\right) \cdot \left(\pi \cdot \left(b + a\_m\right)\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (let* ((t_0 (* 2.0 (* (- b a_m) (+ b a_m)))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 2e-34)
      (*
       (- b a_m)
       (* (* 2.0 (+ b a_m)) (sin (* PI (* 0.005555555555555556 angle_m)))))
      (if (<= (/ angle_m 180.0) 5e+139)
        (* t_0 (sin (/ 1.0 (/ 180.0 (* PI angle_m)))))
        (if (<= (/ angle_m 180.0) 6e+197)
          (* t_0 (sin (* angle_m (/ PI 180.0))))
          (*
           0.011111111111111112
           (* angle_m (* (- b a_m) (* PI (+ b a_m)))))))))))

C

a_m = fabs(a);
angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = 2.0 * ((b - a_m) * (b + a_m));
	double tmp;
	if ((angle_m / 180.0) <= 2e-34) {
		tmp = (b - a_m) * ((2.0 * (b + a_m)) * sin((((double) M_PI) * (0.005555555555555556 * angle_m))));
	} else if ((angle_m / 180.0) <= 5e+139) {
		tmp = t_0 * sin((1.0 / (180.0 / (((double) M_PI) * angle_m))));
	} else if ((angle_m / 180.0) <= 6e+197) {
		tmp = t_0 * sin((angle_m * (((double) M_PI) / 180.0)));
	} else {
		tmp = 0.011111111111111112 * (angle_m * ((b - a_m) * (((double) M_PI) * (b + a_m))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = 2.0 * ((b - a_m) * (b + a_m));
	double tmp;
	if ((angle_m / 180.0) <= 2e-34) {
		tmp = (b - a_m) * ((2.0 * (b + a_m)) * Math.sin((Math.PI * (0.005555555555555556 * angle_m))));
	} else if ((angle_m / 180.0) <= 5e+139) {
		tmp = t_0 * Math.sin((1.0 / (180.0 / (Math.PI * angle_m))));
	} else if ((angle_m / 180.0) <= 6e+197) {
		tmp = t_0 * Math.sin((angle_m * (Math.PI / 180.0)));
	} else {
		tmp = 0.011111111111111112 * (angle_m * ((b - a_m) * (Math.PI * (b + a_m))));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	t_0 = 2.0 * ((b - a_m) * (b + a_m))
	tmp = 0
	if (angle_m / 180.0) <= 2e-34:
		tmp = (b - a_m) * ((2.0 * (b + a_m)) * math.sin((math.pi * (0.005555555555555556 * angle_m))))
	elif (angle_m / 180.0) <= 5e+139:
		tmp = t_0 * math.sin((1.0 / (180.0 / (math.pi * angle_m))))
	elif (angle_m / 180.0) <= 6e+197:
		tmp = t_0 * math.sin((angle_m * (math.pi / 180.0)))
	else:
		tmp = 0.011111111111111112 * (angle_m * ((b - a_m) * (math.pi * (b + a_m))))
	return angle_s * tmp

Julia

a_m = abs(a)
angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	t_0 = Float64(2.0 * Float64(Float64(b - a_m) * Float64(b + a_m)))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e-34)
		tmp = Float64(Float64(b - a_m) * Float64(Float64(2.0 * Float64(b + a_m)) * sin(Float64(pi * Float64(0.005555555555555556 * angle_m)))));
	elseif (Float64(angle_m / 180.0) <= 5e+139)
		tmp = Float64(t_0 * sin(Float64(1.0 / Float64(180.0 / Float64(pi * angle_m)))));
	elseif (Float64(angle_m / 180.0) <= 6e+197)
		tmp = Float64(t_0 * sin(Float64(angle_m * Float64(pi / 180.0))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(Float64(b - a_m) * Float64(pi * Float64(b + a_m)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	t_0 = 2.0 * ((b - a_m) * (b + a_m));
	tmp = 0.0;
	if ((angle_m / 180.0) <= 2e-34)
		tmp = (b - a_m) * ((2.0 * (b + a_m)) * sin((pi * (0.005555555555555556 * angle_m))));
	elseif ((angle_m / 180.0) <= 5e+139)
		tmp = t_0 * sin((1.0 / (180.0 / (pi * angle_m))));
	elseif ((angle_m / 180.0) <= 6e+197)
		tmp = t_0 * sin((angle_m * (pi / 180.0)));
	else
		tmp = 0.011111111111111112 * (angle_m * ((b - a_m) * (pi * (b + a_m))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $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_, angle$95$m_] := Block[{t$95$0 = N[(2.0 * N[(N[(b - a$95$m), $MachinePrecision] * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e-34], N[(N[(b - a$95$m), $MachinePrecision] * N[(N[(2.0 * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+139], N[(t$95$0 * N[Sin[N[(1.0 / N[(180.0 / N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 6e+197], N[(t$95$0 * N[Sin[N[(angle$95$m * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle$95$m * N[(N[(b - a$95$m), $MachinePrecision] * N[(Pi * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle 180) < 1.99999999999999986e-34

   1. Initial program 60.2%

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

      1. add-cbrt-cube 40.8%

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

      2. pow1/3 29.7%

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

      3. pow3 29.7%

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

      4. *-commutative 29.7%

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

      5. div-inv 29.7%

         \[\leadsto {\left({\left(\sin \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. metadata-eval 29.7%

         \[\leadsto {\left({\left(\sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. Applied egg-rr 29.7%

      \[\leadsto \color{blue}{{\left({\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   5. Step-by-step derivation

      1. unpow1/3 40.3%

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

      2. rem-cbrt-cube 59.6%

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

      3. *-commutative 59.6%

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

      4. *-commutative 59.6%

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

      5. *-commutative 59.6%

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

      6. unpow2 59.6%

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

      7. unpow2 59.6%

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

      8. difference-of-squares 62.9%

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

      9. associate-*r* 62.9%

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

      10. associate-*l* 76.0%

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

   6. Applied egg-rr 76.0%

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

      1. associate-*r* 62.9%

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

      2. *-commutative 62.9%

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

      3. associate-*l* 76.0%

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

      4. *-commutative 76.0%

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

      5. *-commutative 76.0%

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

      6. +-commutative 76.0%

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

   8. Simplified 76.0%

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

   9. Taylor expanded in angle around 0 72.4%

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

   if 1.99999999999999986e-34 < (/.f64 angle 180) < 5.0000000000000003e139

   1. Initial program 80.5%

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

      1. unpow2 80.5%

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

      2. unpow2 80.5%

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

      3. difference-of-squares 80.5%

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

   4. Applied egg-rr 80.5%

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

   5. Taylor expanded in angle around 0 52.7%

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

      1. associate-*r/ 56.2%

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

      2. *-commutative 56.2%

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

      3. clear-num 60.2%

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

      4. *-commutative 60.2%

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

   7. Applied egg-rr 60.2%

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

   if 5.0000000000000003e139 < (/.f64 angle 180) < 6.0000000000000004e197

   1. Initial program 29.5%

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

      1. unpow2 29.5%

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

      2. unpow2 29.5%

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

      3. difference-of-squares 29.5%

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

   4. Applied egg-rr 29.5%

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

   5. Taylor expanded in angle around 0 48.8%

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

      1. clear-num 43.0%

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

      2. un-div-inv 48.0%

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

   7. Applied egg-rr 48.0%

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

      1. associate-/r/ 53.9%

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

   9. Simplified 53.9%

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

   if 6.0000000000000004e197 < (/.f64 angle 180)

   1. Initial program 41.0%

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

      1. unpow2 41.0%

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

      2. unpow2 41.0%

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

      3. difference-of-squares 45.3%

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

   4. Applied egg-rr 45.3%

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

   5. Taylor expanded in angle around 0 17.0%

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

   6. Taylor expanded in angle around 0 36.5%

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

      1. associate-*r* 36.5%

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

      2. sub-neg 36.5%

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

      3. distribute-lft-in 32.1%

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

   8. Applied egg-rr 32.1%

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

      1. distribute-lft-out 36.5%

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

      2. sub-neg 36.5%

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

      3. +-commutative 36.5%

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

   10. Simplified 36.5%

       \[\leadsto \left(0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right)\right) \cdot 1 \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{-34}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(2 \cdot \left(b + a\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+139}:\\ \;\;\;\;\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin \left(\frac{1}{\frac{180}{\pi \cdot angle}}\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 6 \cdot 10^{+197}:\\ \;\;\;\;\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := 2 \cdot \left(\left(b - a\_m\right) \cdot \left(b + a\_m\right)\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 10000000000000:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle\_m}{180}\right) \cdot \left(\left(b - a\_m\right) \cdot \left(\left(2 \cdot \left(b + a\_m\right)\right) \cdot \left(angle\_m \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+139}:\\ \;\;\;\;t\_0 \cdot \sin \left(\frac{1}{\frac{180}{\pi \cdot angle\_m}}\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 6 \cdot 10^{+197}:\\ \;\;\;\;t\_0 \cdot \sin \left(angle\_m \cdot \frac{\pi}{180}\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\left(b - a\_m\right) \cdot \left(\pi \cdot \left(b + a\_m\right)\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (let* ((t_0 (* 2.0 (* (- b a_m) (+ b a_m)))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 10000000000000.0)
      (*
       (cos (* PI (/ angle_m 180.0)))
       (*
        (- b a_m)
        (* (* 2.0 (+ b a_m)) (* angle_m (* PI 0.005555555555555556)))))
      (if (<= (/ angle_m 180.0) 5e+139)
        (* t_0 (sin (/ 1.0 (/ 180.0 (* PI angle_m)))))
        (if (<= (/ angle_m 180.0) 6e+197)
          (* t_0 (sin (* angle_m (/ PI 180.0))))
          (*
           0.011111111111111112
           (* angle_m (* (- b a_m) (* PI (+ b a_m)))))))))))

C

a_m = fabs(a);
angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = 2.0 * ((b - a_m) * (b + a_m));
	double tmp;
	if ((angle_m / 180.0) <= 10000000000000.0) {
		tmp = cos((((double) M_PI) * (angle_m / 180.0))) * ((b - a_m) * ((2.0 * (b + a_m)) * (angle_m * (((double) M_PI) * 0.005555555555555556))));
	} else if ((angle_m / 180.0) <= 5e+139) {
		tmp = t_0 * sin((1.0 / (180.0 / (((double) M_PI) * angle_m))));
	} else if ((angle_m / 180.0) <= 6e+197) {
		tmp = t_0 * sin((angle_m * (((double) M_PI) / 180.0)));
	} else {
		tmp = 0.011111111111111112 * (angle_m * ((b - a_m) * (((double) M_PI) * (b + a_m))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = 2.0 * ((b - a_m) * (b + a_m));
	double tmp;
	if ((angle_m / 180.0) <= 10000000000000.0) {
		tmp = Math.cos((Math.PI * (angle_m / 180.0))) * ((b - a_m) * ((2.0 * (b + a_m)) * (angle_m * (Math.PI * 0.005555555555555556))));
	} else if ((angle_m / 180.0) <= 5e+139) {
		tmp = t_0 * Math.sin((1.0 / (180.0 / (Math.PI * angle_m))));
	} else if ((angle_m / 180.0) <= 6e+197) {
		tmp = t_0 * Math.sin((angle_m * (Math.PI / 180.0)));
	} else {
		tmp = 0.011111111111111112 * (angle_m * ((b - a_m) * (Math.PI * (b + a_m))));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	t_0 = 2.0 * ((b - a_m) * (b + a_m))
	tmp = 0
	if (angle_m / 180.0) <= 10000000000000.0:
		tmp = math.cos((math.pi * (angle_m / 180.0))) * ((b - a_m) * ((2.0 * (b + a_m)) * (angle_m * (math.pi * 0.005555555555555556))))
	elif (angle_m / 180.0) <= 5e+139:
		tmp = t_0 * math.sin((1.0 / (180.0 / (math.pi * angle_m))))
	elif (angle_m / 180.0) <= 6e+197:
		tmp = t_0 * math.sin((angle_m * (math.pi / 180.0)))
	else:
		tmp = 0.011111111111111112 * (angle_m * ((b - a_m) * (math.pi * (b + a_m))))
	return angle_s * tmp

Julia

a_m = abs(a)
angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	t_0 = Float64(2.0 * Float64(Float64(b - a_m) * Float64(b + a_m)))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 10000000000000.0)
		tmp = Float64(cos(Float64(pi * Float64(angle_m / 180.0))) * Float64(Float64(b - a_m) * Float64(Float64(2.0 * Float64(b + a_m)) * Float64(angle_m * Float64(pi * 0.005555555555555556)))));
	elseif (Float64(angle_m / 180.0) <= 5e+139)
		tmp = Float64(t_0 * sin(Float64(1.0 / Float64(180.0 / Float64(pi * angle_m)))));
	elseif (Float64(angle_m / 180.0) <= 6e+197)
		tmp = Float64(t_0 * sin(Float64(angle_m * Float64(pi / 180.0))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(Float64(b - a_m) * Float64(pi * Float64(b + a_m)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	t_0 = 2.0 * ((b - a_m) * (b + a_m));
	tmp = 0.0;
	if ((angle_m / 180.0) <= 10000000000000.0)
		tmp = cos((pi * (angle_m / 180.0))) * ((b - a_m) * ((2.0 * (b + a_m)) * (angle_m * (pi * 0.005555555555555556))));
	elseif ((angle_m / 180.0) <= 5e+139)
		tmp = t_0 * sin((1.0 / (180.0 / (pi * angle_m))));
	elseif ((angle_m / 180.0) <= 6e+197)
		tmp = t_0 * sin((angle_m * (pi / 180.0)));
	else
		tmp = 0.011111111111111112 * (angle_m * ((b - a_m) * (pi * (b + a_m))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $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_, angle$95$m_] := Block[{t$95$0 = N[(2.0 * N[(N[(b - a$95$m), $MachinePrecision] * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 10000000000000.0], N[(N[Cos[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[(N[(2.0 * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision] * N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+139], N[(t$95$0 * N[Sin[N[(1.0 / N[(180.0 / N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 6e+197], N[(t$95$0 * N[Sin[N[(angle$95$m * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle$95$m * N[(N[(b - a$95$m), $MachinePrecision] * N[(Pi * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle 180) < 1e13

   1. Initial program 63.0%

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

      1. add-cbrt-cube 43.8%

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

      2. pow1/3 30.5%

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

      3. pow3 30.5%

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

      4. *-commutative 30.5%

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

      5. div-inv 30.5%

         \[\leadsto {\left({\left(\sin \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. metadata-eval 30.5%

         \[\leadsto {\left({\left(\sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. Applied egg-rr 30.5%

      \[\leadsto \color{blue}{{\left({\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   5. Step-by-step derivation

      1. unpow1/3 43.3%

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

      2. rem-cbrt-cube 62.5%

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

      3. *-commutative 62.5%

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

      4. *-commutative 62.5%

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

      5. *-commutative 62.5%

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

      6. unpow2 62.5%

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

      7. unpow2 62.5%

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

      8. difference-of-squares 65.5%

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

      9. associate-*r* 65.5%

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

      10. associate-*l* 77.6%

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

   6. Applied egg-rr 77.6%

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

      1. associate-*r* 65.5%

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

      2. *-commutative 65.5%

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

      3. associate-*l* 77.6%

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

      4. *-commutative 77.6%

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

      5. *-commutative 77.6%

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

      6. +-commutative 77.6%

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

   8. Simplified 77.6%

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

   9. Taylor expanded in angle around 0 71.6%

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

       1. associate-*r* 71.6%

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

       2. *-commutative 71.6%

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

       3. associate-*r* 71.6%

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

       4. *-commutative 71.6%

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

       5. *-commutative 71.6%

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

   11. Simplified 71.6%

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

   if 1e13 < (/.f64 angle 180) < 5.0000000000000003e139

   1. Initial program 62.6%

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

      1. unpow2 62.6%

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

      2. unpow2 62.6%

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

      3. difference-of-squares 62.6%

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

   4. Applied egg-rr 62.6%

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

   5. Taylor expanded in angle around 0 30.6%

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

      1. associate-*r/ 37.7%

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

      2. *-commutative 37.7%

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

      3. clear-num 46.0%

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

      4. *-commutative 46.0%

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

   7. Applied egg-rr 46.0%

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

   if 5.0000000000000003e139 < (/.f64 angle 180) < 6.0000000000000004e197

   1. Initial program 29.5%

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

      1. unpow2 29.5%

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

      2. unpow2 29.5%

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

      3. difference-of-squares 29.5%

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

   4. Applied egg-rr 29.5%

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

   5. Taylor expanded in angle around 0 48.8%

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

      1. clear-num 43.0%

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

      2. un-div-inv 48.0%

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

   7. Applied egg-rr 48.0%

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

      1. associate-/r/ 53.9%

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

   9. Simplified 53.9%

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

   if 6.0000000000000004e197 < (/.f64 angle 180)

   1. Initial program 41.0%

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

      1. unpow2 41.0%

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

      2. unpow2 41.0%

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

      3. difference-of-squares 45.3%

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

   4. Applied egg-rr 45.3%

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

   5. Taylor expanded in angle around 0 17.0%

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

   6. Taylor expanded in angle around 0 36.5%

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

      1. associate-*r* 36.5%

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

      2. sub-neg 36.5%

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

      3. distribute-lft-in 32.1%

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

   8. Applied egg-rr 32.1%

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

      1. distribute-lft-out 36.5%

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

      2. sub-neg 36.5%

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

      3. +-commutative 36.5%

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

   10. Simplified 36.5%

       \[\leadsto \left(0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right)\right) \cdot 1 \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10000000000000:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(b - a\right) \cdot \left(\left(2 \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+139}:\\ \;\;\;\;\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin \left(\frac{1}{\frac{180}{\pi \cdot angle}}\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 6 \cdot 10^{+197}:\\ \;\;\;\;\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 55.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 5.5 \cdot 10^{+153}:\\ \;\;\;\;2 \cdot \left(\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\_m\right)\right) \cdot \left(\left(b - a\_m\right) \cdot \left(b + a\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \left(angle\_m \cdot {a\_m}^{2}\right)\right) \cdot -0.011111111111111112\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a_m 5.5e+153)
    (*
     2.0
     (* (sin (* 0.005555555555555556 (* PI angle_m))) (* (- b a_m) (+ b a_m))))
    (* (* PI (* angle_m (pow a_m 2.0))) -0.011111111111111112))))

C

a_m = fabs(a);
angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (a_m <= 5.5e+153) {
		tmp = 2.0 * (sin((0.005555555555555556 * (((double) M_PI) * angle_m))) * ((b - a_m) * (b + a_m)));
	} else {
		tmp = (((double) M_PI) * (angle_m * pow(a_m, 2.0))) * -0.011111111111111112;
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (a_m <= 5.5e+153) {
		tmp = 2.0 * (Math.sin((0.005555555555555556 * (Math.PI * angle_m))) * ((b - a_m) * (b + a_m)));
	} else {
		tmp = (Math.PI * (angle_m * Math.pow(a_m, 2.0))) * -0.011111111111111112;
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	tmp = 0
	if a_m <= 5.5e+153:
		tmp = 2.0 * (math.sin((0.005555555555555556 * (math.pi * angle_m))) * ((b - a_m) * (b + a_m)))
	else:
		tmp = (math.pi * (angle_m * math.pow(a_m, 2.0))) * -0.011111111111111112
	return angle_s * tmp

Julia

a_m = abs(a)
angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	tmp = 0.0
	if (a_m <= 5.5e+153)
		tmp = Float64(2.0 * Float64(sin(Float64(0.005555555555555556 * Float64(pi * angle_m))) * Float64(Float64(b - a_m) * Float64(b + a_m))));
	else
		tmp = Float64(Float64(pi * Float64(angle_m * (a_m ^ 2.0))) * -0.011111111111111112);
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	tmp = 0.0;
	if (a_m <= 5.5e+153)
		tmp = 2.0 * (sin((0.005555555555555556 * (pi * angle_m))) * ((b - a_m) * (b + a_m)));
	else
		tmp = (pi * (angle_m * (a_m ^ 2.0))) * -0.011111111111111112;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $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_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a$95$m, 5.5e+153], N[(2.0 * N[(N[Sin[N[(0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * N[(angle$95$m * N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.011111111111111112), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 5.5000000000000003e153

   1. Initial program 60.2%

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

      1. unpow2 60.2%

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

      2. unpow2 60.2%

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

      3. difference-of-squares 61.1%

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

   4. Applied egg-rr 61.1%

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

   5. Taylor expanded in angle around 0 54.2%

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

      1. add-cbrt-cube 55.1%

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

      2. pow3 55.1%

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

   7. Applied egg-rr 55.1%

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

   8. Taylor expanded in angle around inf 55.8%

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

   if 5.5000000000000003e153 < a

   1. Initial program 45.1%

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

      1. unpow2 45.1%

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

      2. unpow2 45.1%

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

      3. difference-of-squares 61.8%

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

   4. Applied egg-rr 61.8%

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

   5. Taylor expanded in angle around 0 48.4%

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

   6. Taylor expanded in angle around 0 58.4%

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

   7. Taylor expanded in a around inf 61.8%

      \[\leadsto \color{blue}{\left(-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right)} \cdot 1 \]
   8. Step-by-step derivation

      1. *-commutative 61.8%

         \[\leadsto \color{blue}{\left(\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right) \cdot -0.011111111111111112\right)} \cdot 1 \]

      2. associate-*r* 61.8%

         \[\leadsto \left(\color{blue}{\left(\left({a}^{2} \cdot angle\right) \cdot \pi\right)} \cdot -0.011111111111111112\right) \cdot 1 \]

      3. *-commutative 61.8%

         \[\leadsto \left(\color{blue}{\left(\pi \cdot \left({a}^{2} \cdot angle\right)\right)} \cdot -0.011111111111111112\right) \cdot 1 \]

      4. *-commutative 61.8%

         \[\leadsto \left(\left(\pi \cdot \color{blue}{\left(angle \cdot {a}^{2}\right)}\right) \cdot -0.011111111111111112\right) \cdot 1 \]

   9. Simplified 61.8%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot \left(angle \cdot {a}^{2}\right)\right) \cdot -0.011111111111111112\right)} \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.5 \cdot 10^{+153}:\\ \;\;\;\;2 \cdot \left(\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \left(angle \cdot {a}^{2}\right)\right) \cdot -0.011111111111111112\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 55.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 9.5 \cdot 10^{+153}:\\ \;\;\;\;\left(2 \cdot \left(\left(b - a\_m\right) \cdot \left(b + a\_m\right)\right)\right) \cdot \sin \left(angle\_m \cdot \frac{\pi}{180}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \left(angle\_m \cdot {a\_m}^{2}\right)\right) \cdot -0.011111111111111112\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a_m 9.5e+153)
    (* (* 2.0 (* (- b a_m) (+ b a_m))) (sin (* angle_m (/ PI 180.0))))
    (* (* PI (* angle_m (pow a_m 2.0))) -0.011111111111111112))))

C

a_m = fabs(a);
angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (a_m <= 9.5e+153) {
		tmp = (2.0 * ((b - a_m) * (b + a_m))) * sin((angle_m * (((double) M_PI) / 180.0)));
	} else {
		tmp = (((double) M_PI) * (angle_m * pow(a_m, 2.0))) * -0.011111111111111112;
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (a_m <= 9.5e+153) {
		tmp = (2.0 * ((b - a_m) * (b + a_m))) * Math.sin((angle_m * (Math.PI / 180.0)));
	} else {
		tmp = (Math.PI * (angle_m * Math.pow(a_m, 2.0))) * -0.011111111111111112;
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	tmp = 0
	if a_m <= 9.5e+153:
		tmp = (2.0 * ((b - a_m) * (b + a_m))) * math.sin((angle_m * (math.pi / 180.0)))
	else:
		tmp = (math.pi * (angle_m * math.pow(a_m, 2.0))) * -0.011111111111111112
	return angle_s * tmp

Julia

a_m = abs(a)
angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	tmp = 0.0
	if (a_m <= 9.5e+153)
		tmp = Float64(Float64(2.0 * Float64(Float64(b - a_m) * Float64(b + a_m))) * sin(Float64(angle_m * Float64(pi / 180.0))));
	else
		tmp = Float64(Float64(pi * Float64(angle_m * (a_m ^ 2.0))) * -0.011111111111111112);
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	tmp = 0.0;
	if (a_m <= 9.5e+153)
		tmp = (2.0 * ((b - a_m) * (b + a_m))) * sin((angle_m * (pi / 180.0)));
	else
		tmp = (pi * (angle_m * (a_m ^ 2.0))) * -0.011111111111111112;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $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_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a$95$m, 9.5e+153], N[(N[(2.0 * N[(N[(b - a$95$m), $MachinePrecision] * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(angle$95$m * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(Pi * N[(angle$95$m * N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.011111111111111112), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 9.4999999999999995e153

   1. Initial program 60.2%

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

      1. unpow2 60.2%

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

      2. unpow2 60.2%

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

      3. difference-of-squares 61.1%

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

   4. Applied egg-rr 61.1%

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

   5. Taylor expanded in angle around 0 54.2%

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

      1. clear-num 54.3%

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

      2. un-div-inv 55.4%

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

   7. Applied egg-rr 55.4%

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

      1. associate-/r/ 56.2%

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

   9. Simplified 56.2%

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

   if 9.4999999999999995e153 < a

   1. Initial program 45.1%

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

      1. unpow2 45.1%

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

      2. unpow2 45.1%

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

      3. difference-of-squares 61.8%

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

   4. Applied egg-rr 61.8%

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

   5. Taylor expanded in angle around 0 48.4%

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

   6. Taylor expanded in angle around 0 58.4%

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

   7. Taylor expanded in a around inf 61.8%

      \[\leadsto \color{blue}{\left(-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right)} \cdot 1 \]
   8. Step-by-step derivation

      1. *-commutative 61.8%

         \[\leadsto \color{blue}{\left(\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right) \cdot -0.011111111111111112\right)} \cdot 1 \]

      2. associate-*r* 61.8%

         \[\leadsto \left(\color{blue}{\left(\left({a}^{2} \cdot angle\right) \cdot \pi\right)} \cdot -0.011111111111111112\right) \cdot 1 \]

      3. *-commutative 61.8%

         \[\leadsto \left(\color{blue}{\left(\pi \cdot \left({a}^{2} \cdot angle\right)\right)} \cdot -0.011111111111111112\right) \cdot 1 \]

      4. *-commutative 61.8%

         \[\leadsto \left(\left(\pi \cdot \color{blue}{\left(angle \cdot {a}^{2}\right)}\right) \cdot -0.011111111111111112\right) \cdot 1 \]

   9. Simplified 61.8%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot \left(angle \cdot {a}^{2}\right)\right) \cdot -0.011111111111111112\right)} \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 9.5 \cdot 10^{+153}:\\ \;\;\;\;\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \left(angle \cdot {a}^{2}\right)\right) \cdot -0.011111111111111112\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 54.9% accurate, 32.2× speedup

Math

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

FPCore

a_m = (fabs.f64 a)
angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (* 0.011111111111111112 (* angle_m (* PI (* (- b a_m) (+ b a_m)))))))

C

a_m = fabs(a);
angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * (angle_m * (((double) M_PI) * ((b - a_m) * (b + a_m)))));
}

Java

a_m = Math.abs(a);
angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * (angle_m * (Math.PI * ((b - a_m) * (b + a_m)))));
}

Python

a_m = math.fabs(a)
angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	return angle_s * (0.011111111111111112 * (angle_m * (math.pi * ((b - a_m) * (b + a_m)))))

Julia

a_m = abs(a)
angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	return Float64(angle_s * Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(Float64(b - a_m) * Float64(b + a_m))))))
end

MATLAB

a_m = abs(a);
angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a_m, b, angle_m)
	tmp = angle_s * (0.011111111111111112 * (angle_m * (pi * ((b - a_m) * (b + a_m)))));
end

Wolfram

a_m = N[Abs[a], $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_, angle$95$m_] := N[(angle$95$s * N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(N[(b - a$95$m), $MachinePrecision] * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.4%

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

   1. unpow2 58.4%

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

   2. unpow2 58.4%

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

   3. difference-of-squares 61.2%

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

4. Applied egg-rr 61.2%

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

5. Taylor expanded in angle around 0 53.5%

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

6. Taylor expanded in angle around 0 54.7%

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

7. Final simplification 54.7%

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

Alternative 15: 54.9% accurate, 32.2× speedup

Math

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

FPCore

a_m = (fabs.f64 a)
angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (* 0.011111111111111112 (* angle_m (* (- b a_m) (* PI (+ b a_m)))))))

C

a_m = fabs(a);
angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * (angle_m * ((b - a_m) * (((double) M_PI) * (b + a_m)))));
}

Java

a_m = Math.abs(a);
angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * (angle_m * ((b - a_m) * (Math.PI * (b + a_m)))));
}

Python

a_m = math.fabs(a)
angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	return angle_s * (0.011111111111111112 * (angle_m * ((b - a_m) * (math.pi * (b + a_m)))))

Julia

a_m = abs(a)
angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	return Float64(angle_s * Float64(0.011111111111111112 * Float64(angle_m * Float64(Float64(b - a_m) * Float64(pi * Float64(b + a_m))))))
end

MATLAB

a_m = abs(a);
angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a_m, b, angle_m)
	tmp = angle_s * (0.011111111111111112 * (angle_m * ((b - a_m) * (pi * (b + a_m)))));
end

Wolfram

a_m = N[Abs[a], $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_, angle$95$m_] := N[(angle$95$s * N[(0.011111111111111112 * N[(angle$95$m * N[(N[(b - a$95$m), $MachinePrecision] * N[(Pi * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.4%

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

   1. unpow2 58.4%

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

   2. unpow2 58.4%

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

   3. difference-of-squares 61.2%

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

4. Applied egg-rr 61.2%

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

5. Taylor expanded in angle around 0 53.5%

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

6. Taylor expanded in angle around 0 54.7%

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

   1. associate-*r* 54.7%

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

   2. sub-neg 54.7%

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

   3. distribute-lft-in 52.3%

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

8. Applied egg-rr 52.3%

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

   1. distribute-lft-out 54.7%

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

   2. sub-neg 54.7%

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

   3. +-commutative 54.7%

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

10. Simplified 54.7%

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

11. Final simplification 54.7%

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

Alternative 16: 54.9% accurate, 32.2× speedup

Math

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

FPCore

a_m = (fabs.f64 a)
angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (* (* PI (* (- b a_m) (+ b a_m))) (* angle_m 0.011111111111111112))))

C

a_m = fabs(a);
angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	return angle_s * ((((double) M_PI) * ((b - a_m) * (b + a_m))) * (angle_m * 0.011111111111111112));
}

Java

a_m = Math.abs(a);
angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	return angle_s * ((Math.PI * ((b - a_m) * (b + a_m))) * (angle_m * 0.011111111111111112));
}

Python

a_m = math.fabs(a)
angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	return angle_s * ((math.pi * ((b - a_m) * (b + a_m))) * (angle_m * 0.011111111111111112))

Julia

a_m = abs(a)
angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	return Float64(angle_s * Float64(Float64(pi * Float64(Float64(b - a_m) * Float64(b + a_m))) * Float64(angle_m * 0.011111111111111112)))
end

MATLAB

a_m = abs(a);
angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a_m, b, angle_m)
	tmp = angle_s * ((pi * ((b - a_m) * (b + a_m))) * (angle_m * 0.011111111111111112));
end

Wolfram

a_m = N[Abs[a], $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_, angle$95$m_] := N[(angle$95$s * N[(N[(Pi * N[(N[(b - a$95$m), $MachinePrecision] * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.4%

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

   1. unpow2 58.4%

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

   2. unpow2 58.4%

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

   3. difference-of-squares 61.2%

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

4. Applied egg-rr 61.2%

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

5. Taylor expanded in angle around 0 53.5%

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

6. Taylor expanded in angle around 0 54.7%

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

   1. associate-*r* 54.7%

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

8. Simplified 54.7%

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

9. Final simplification 54.7%

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

Reproduce

herbie shell --seed 2024031 
(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)))))