ab-angle->ABCF B

Percentage Accurate: 54.2% → 67.1%

Time: 39.8s

Alternatives: 20

Speedup: 32.2×

• 33.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 67.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle\_m}{180}\\ t_1 := \sin \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right)\\ t_2 := \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin t\_0\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 10^{-6}:\\ \;\;\;\;2 \cdot \left(\left(a \cdot \left(t\_1 \cdot \left(b - b\right) - a \cdot t\_1\right) + {\left(b \cdot \sqrt{t\_1}\right)}^{2}\right) \cdot \cos t\_0\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+172}:\\ \;\;\;\;2 \cdot \left(t\_2 \cdot \cos \left(\frac{angle\_m}{180} \cdot \sqrt[3]{{\pi}^{3}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t\_2 \cdot \sqrt{{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right)}^{2}}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (/ angle_m 180.0)))
        (t_1 (sin (* 0.005555555555555556 (* angle_m PI))))
        (t_2 (* (* (+ b a) (- b a)) (sin t_0))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 1e-6)
      (*
       2.0
       (*
        (+ (* a (- (* t_1 (- b b)) (* a t_1))) (pow (* b (sqrt t_1)) 2.0))
        (cos t_0)))
      (if (<= (/ angle_m 180.0) 2e+172)
        (* 2.0 (* t_2 (cos (* (/ angle_m 180.0) (cbrt (pow PI 3.0))))))
        (*
         2.0
         (*
          t_2
          (sqrt (pow (cos (* PI (* 0.005555555555555556 angle_m))) 2.0)))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m / 180.0);
	double t_1 = sin((0.005555555555555556 * (angle_m * ((double) M_PI))));
	double t_2 = ((b + a) * (b - a)) * sin(t_0);
	double tmp;
	if ((angle_m / 180.0) <= 1e-6) {
		tmp = 2.0 * (((a * ((t_1 * (b - b)) - (a * t_1))) + pow((b * sqrt(t_1)), 2.0)) * cos(t_0));
	} else if ((angle_m / 180.0) <= 2e+172) {
		tmp = 2.0 * (t_2 * cos(((angle_m / 180.0) * cbrt(pow(((double) M_PI), 3.0)))));
	} else {
		tmp = 2.0 * (t_2 * sqrt(pow(cos((((double) M_PI) * (0.005555555555555556 * angle_m))), 2.0)));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.PI * (angle_m / 180.0);
	double t_1 = Math.sin((0.005555555555555556 * (angle_m * Math.PI)));
	double t_2 = ((b + a) * (b - a)) * Math.sin(t_0);
	double tmp;
	if ((angle_m / 180.0) <= 1e-6) {
		tmp = 2.0 * (((a * ((t_1 * (b - b)) - (a * t_1))) + Math.pow((b * Math.sqrt(t_1)), 2.0)) * Math.cos(t_0));
	} else if ((angle_m / 180.0) <= 2e+172) {
		tmp = 2.0 * (t_2 * Math.cos(((angle_m / 180.0) * Math.cbrt(Math.pow(Math.PI, 3.0)))));
	} else {
		tmp = 2.0 * (t_2 * Math.sqrt(Math.pow(Math.cos((Math.PI * (0.005555555555555556 * angle_m))), 2.0)));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(pi * Float64(angle_m / 180.0))
	t_1 = sin(Float64(0.005555555555555556 * Float64(angle_m * pi)))
	t_2 = Float64(Float64(Float64(b + a) * Float64(b - a)) * sin(t_0))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e-6)
		tmp = Float64(2.0 * Float64(Float64(Float64(a * Float64(Float64(t_1 * Float64(b - b)) - Float64(a * t_1))) + (Float64(b * sqrt(t_1)) ^ 2.0)) * cos(t_0)));
	elseif (Float64(angle_m / 180.0) <= 2e+172)
		tmp = Float64(2.0 * Float64(t_2 * cos(Float64(Float64(angle_m / 180.0) * cbrt((pi ^ 3.0))))));
	else
		tmp = Float64(2.0 * Float64(t_2 * sqrt((cos(Float64(pi * Float64(0.005555555555555556 * angle_m))) ^ 2.0))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e-6], N[(2.0 * N[(N[(N[(a * N[(N[(t$95$1 * N[(b - b), $MachinePrecision]), $MachinePrecision] - N[(a * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(b * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+172], N[(2.0 * N[(t$95$2 * N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$2 * N[Sqrt[N[Power[N[Cos[N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 9.99999999999999955e-7

   1. Initial program 55.3%

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

      1. associate-*l* 55.3%

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

      2. associate-*l* 55.3%

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

   3. Simplified 55.3%

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

      1. unpow2 55.3%

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

      2. unpow2 55.3%

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

      3. difference-of-squares 58.1%

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

   6. Applied egg-rr 58.1%

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

   7. Taylor expanded in a around 0 62.2%

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

      1. add-sqr-sqrt 45.2%

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

      2. pow2 45.2%

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

      3. sqrt-prod 34.9%

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

      4. sqrt-pow1 39.4%

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

      5. metadata-eval 39.4%

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

      6. pow1 39.4%

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

   9. Applied egg-rr 39.4%

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

   if 9.99999999999999955e-7 < (/.f64 angle #s(literal 180 binary64)) < 2.0000000000000002e172

   1. Initial program 50.3%

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

      1. associate-*l* 50.3%

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

      2. associate-*l* 50.3%

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

   3. Simplified 50.3%

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

      1. unpow2 50.3%

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

      2. unpow2 50.3%

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

      3. difference-of-squares 50.3%

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

   6. Applied egg-rr 50.3%

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

      1. add-cbrt-cube 50.2%

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

      2. pow3 50.2%

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

   8. Applied egg-rr 50.2%

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

   if 2.0000000000000002e172 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 34.9%

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

      1. associate-*l* 34.9%

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

      2. associate-*l* 34.9%

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

   3. Simplified 34.9%

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

      1. unpow2 34.9%

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

      2. unpow2 34.9%

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

      3. difference-of-squares 38.1%

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

   6. Applied egg-rr 38.1%

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

      1. add-sqr-sqrt 22.2%

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

      2. sqrt-unprod 51.2%

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

      3. pow2 51.2%

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

      4. div-inv 51.1%

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

      5. metadata-eval 51.1%

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

   8. Applied egg-rr 51.1%

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

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{-6}:\\ \;\;\;\;2 \cdot \left(\left(a \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b - b\right) - a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) + {\left(b \cdot \sqrt{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}^{2}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+172}:\\ \;\;\;\;2 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \sqrt[3]{{\pi}^{3}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sqrt{{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 65.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \sin \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -1 \cdot 10^{-250}:\\ \;\;\;\;2 \cdot \left(\left(a \cdot \left(t\_0 \cdot \left(b - b\right) - a \cdot t\_0\right) + {b}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle\_m \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right)\right)\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(b, a\right) \cdot \left(\mathsf{hypot}\left(b, a\right) \cdot \sin \left(2 \cdot \left(angle\_m \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (sin (* 0.005555555555555556 (* angle_m PI)))))
   (*
    angle_s
    (if (<= (- (pow b 2.0) (pow a 2.0)) -1e-250)
      (*
       2.0
       (*
        (+
         (* a (- (* t_0 (- b b)) (* a t_0)))
         (*
          (pow b 2.0)
          (sin
           (*
            0.005555555555555556
            (* angle_m (* (cbrt PI) (pow (cbrt PI) 2.0)))))))
        (expm1 (log1p (cos (* PI (* 0.005555555555555556 angle_m)))))))
      (*
       (hypot b a)
       (*
        (hypot b a)
        (sin (* 2.0 (* angle_m (* 0.005555555555555556 PI))))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = sin((0.005555555555555556 * (angle_m * ((double) M_PI))));
	double tmp;
	if ((pow(b, 2.0) - pow(a, 2.0)) <= -1e-250) {
		tmp = 2.0 * (((a * ((t_0 * (b - b)) - (a * t_0))) + (pow(b, 2.0) * sin((0.005555555555555556 * (angle_m * (cbrt(((double) M_PI)) * pow(cbrt(((double) M_PI)), 2.0))))))) * expm1(log1p(cos((((double) M_PI) * (0.005555555555555556 * angle_m))))));
	} else {
		tmp = hypot(b, a) * (hypot(b, a) * sin((2.0 * (angle_m * (0.005555555555555556 * ((double) M_PI))))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.sin((0.005555555555555556 * (angle_m * Math.PI)));
	double tmp;
	if ((Math.pow(b, 2.0) - Math.pow(a, 2.0)) <= -1e-250) {
		tmp = 2.0 * (((a * ((t_0 * (b - b)) - (a * t_0))) + (Math.pow(b, 2.0) * Math.sin((0.005555555555555556 * (angle_m * (Math.cbrt(Math.PI) * Math.pow(Math.cbrt(Math.PI), 2.0))))))) * Math.expm1(Math.log1p(Math.cos((Math.PI * (0.005555555555555556 * angle_m))))));
	} else {
		tmp = Math.hypot(b, a) * (Math.hypot(b, a) * Math.sin((2.0 * (angle_m * (0.005555555555555556 * Math.PI)))));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = sin(Float64(0.005555555555555556 * Float64(angle_m * pi)))
	tmp = 0.0
	if (Float64((b ^ 2.0) - (a ^ 2.0)) <= -1e-250)
		tmp = Float64(2.0 * Float64(Float64(Float64(a * Float64(Float64(t_0 * Float64(b - b)) - Float64(a * t_0))) + Float64((b ^ 2.0) * sin(Float64(0.005555555555555556 * Float64(angle_m * Float64(cbrt(pi) * (cbrt(pi) ^ 2.0))))))) * expm1(log1p(cos(Float64(pi * Float64(0.005555555555555556 * angle_m)))))));
	else
		tmp = Float64(hypot(b, a) * Float64(hypot(b, a) * sin(Float64(2.0 * Float64(angle_m * Float64(0.005555555555555556 * pi))))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision], -1e-250], N[(2.0 * N[(N[(N[(a * N[(N[(t$95$0 * N[(b - b), $MachinePrecision]), $MachinePrecision] - N[(a * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[b, 2.0], $MachinePrecision] * N[Sin[N[(0.005555555555555556 * N[(angle$95$m * N[(N[Power[Pi, 1/3], $MachinePrecision] * N[Power[N[Power[Pi, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Exp[N[Log[1 + N[Cos[N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[b ^ 2 + a ^ 2], $MachinePrecision] * N[(N[Sqrt[b ^ 2 + a ^ 2], $MachinePrecision] * N[Sin[N[(2.0 * N[(angle$95$m * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -1.0000000000000001e-250

   1. Initial program 50.1%

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

      1. associate-*l* 50.1%

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

      2. associate-*l* 50.1%

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

   3. Simplified 50.1%

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

      1. unpow2 50.1%

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

      2. unpow2 50.1%

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

      3. difference-of-squares 50.1%

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

   6. Applied egg-rr 50.1%

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

   7. Taylor expanded in a around 0 61.3%

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

      1. add-cube-cbrt 61.3%

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

      2. pow2 61.3%

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

   9. Applied egg-rr 61.3%

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

       1. expm1-log1p-u 61.3%

          \[\leadsto 2 \cdot \left(\left(a \cdot \left(-1 \cdot \left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) + \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + -1 \cdot b\right)\right) + {b}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right)\right)\right)\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)}\right) \]

       2. div-inv 63.1%

          \[\leadsto 2 \cdot \left(\left(a \cdot \left(-1 \cdot \left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) + \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + -1 \cdot b\right)\right) + {b}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right)\right)\right)\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right)\right)\right) \]

       3. metadata-eval 63.1%

          \[\leadsto 2 \cdot \left(\left(a \cdot \left(-1 \cdot \left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) + \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + -1 \cdot b\right)\right) + {b}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right)\right)\right)\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 63.1%

       \[\leadsto 2 \cdot \left(\left(a \cdot \left(-1 \cdot \left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) + \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + -1 \cdot b\right)\right) + {b}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right)\right)\right)\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}\right) \]

   if -1.0000000000000001e-250 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 53.7%

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

      1. associate-*l* 53.7%

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

      2. *-commutative 53.7%

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

      3. associate-*l* 53.7%

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

   3. Simplified 53.7%

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

      1. add-cbrt-cube 40.5%

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

      2. pow1/3 30.3%

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

   6. Applied egg-rr 30.3%

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

      1. unpow1/3 41.3%

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

      2. rem-cbrt-cube 54.7%

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

      3. add-sqr-sqrt 54.6%

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

      4. associate-*l* 54.6%

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

   8. Applied egg-rr 72.5%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -1 \cdot 10^{-250}:\\ \;\;\;\;2 \cdot \left(\left(a \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b - b\right) - a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) + {b}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right)\right)\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(b, a\right) \cdot \left(\mathsf{hypot}\left(b, a\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.9% accurate, 0.4× speedup

Math

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

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (sin (* 0.005555555555555556 (* angle_m PI)))))
   (*
    angle_s
    (if (<= (- (pow b 2.0) (pow a 2.0)) -1e-250)
      (*
       2.0
       (*
        (+ (* a (- (* t_0 (- b b)) (* a t_0))) (* (pow b 2.0) t_0))
        (log1p (expm1 (cos (* PI (* 0.005555555555555556 angle_m)))))))
      (*
       (hypot b a)
       (*
        (hypot b a)
        (sin (* 2.0 (* angle_m (* 0.005555555555555556 PI))))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = sin((0.005555555555555556 * (angle_m * ((double) M_PI))));
	double tmp;
	if ((pow(b, 2.0) - pow(a, 2.0)) <= -1e-250) {
		tmp = 2.0 * (((a * ((t_0 * (b - b)) - (a * t_0))) + (pow(b, 2.0) * t_0)) * log1p(expm1(cos((((double) M_PI) * (0.005555555555555556 * angle_m))))));
	} else {
		tmp = hypot(b, a) * (hypot(b, a) * sin((2.0 * (angle_m * (0.005555555555555556 * ((double) M_PI))))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.sin((0.005555555555555556 * (angle_m * Math.PI)));
	double tmp;
	if ((Math.pow(b, 2.0) - Math.pow(a, 2.0)) <= -1e-250) {
		tmp = 2.0 * (((a * ((t_0 * (b - b)) - (a * t_0))) + (Math.pow(b, 2.0) * t_0)) * Math.log1p(Math.expm1(Math.cos((Math.PI * (0.005555555555555556 * angle_m))))));
	} else {
		tmp = Math.hypot(b, a) * (Math.hypot(b, a) * Math.sin((2.0 * (angle_m * (0.005555555555555556 * Math.PI)))));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = math.sin((0.005555555555555556 * (angle_m * math.pi)))
	tmp = 0
	if (math.pow(b, 2.0) - math.pow(a, 2.0)) <= -1e-250:
		tmp = 2.0 * (((a * ((t_0 * (b - b)) - (a * t_0))) + (math.pow(b, 2.0) * t_0)) * math.log1p(math.expm1(math.cos((math.pi * (0.005555555555555556 * angle_m))))))
	else:
		tmp = math.hypot(b, a) * (math.hypot(b, a) * math.sin((2.0 * (angle_m * (0.005555555555555556 * math.pi)))))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = sin(Float64(0.005555555555555556 * Float64(angle_m * pi)))
	tmp = 0.0
	if (Float64((b ^ 2.0) - (a ^ 2.0)) <= -1e-250)
		tmp = Float64(2.0 * Float64(Float64(Float64(a * Float64(Float64(t_0 * Float64(b - b)) - Float64(a * t_0))) + Float64((b ^ 2.0) * t_0)) * log1p(expm1(cos(Float64(pi * Float64(0.005555555555555556 * angle_m)))))));
	else
		tmp = Float64(hypot(b, a) * Float64(hypot(b, a) * sin(Float64(2.0 * Float64(angle_m * Float64(0.005555555555555556 * pi))))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision], -1e-250], N[(2.0 * N[(N[(N[(a * N[(N[(t$95$0 * N[(b - b), $MachinePrecision]), $MachinePrecision] - N[(a * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[b, 2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * N[Log[1 + N[(Exp[N[Cos[N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[b ^ 2 + a ^ 2], $MachinePrecision] * N[(N[Sqrt[b ^ 2 + a ^ 2], $MachinePrecision] * N[Sin[N[(2.0 * N[(angle$95$m * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -1.0000000000000001e-250

   1. Initial program 50.1%

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

      1. associate-*l* 50.1%

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

      2. associate-*l* 50.1%

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

   3. Simplified 50.1%

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

      1. unpow2 50.1%

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

      2. unpow2 50.1%

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

      3. difference-of-squares 50.1%

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

   6. Applied egg-rr 50.1%

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

   7. Taylor expanded in a around 0 61.3%

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

      1. log1p-expm1-u 61.3%

         \[\leadsto 2 \cdot \left(\left(a \cdot \left(-1 \cdot \left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) + \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + -1 \cdot b\right)\right) + {b}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)}\right) \]

      2. div-inv 63.1%

         \[\leadsto 2 \cdot \left(\left(a \cdot \left(-1 \cdot \left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) + \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + -1 \cdot b\right)\right) + {b}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right)\right)\right) \]

      3. metadata-eval 63.1%

         \[\leadsto 2 \cdot \left(\left(a \cdot \left(-1 \cdot \left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) + \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + -1 \cdot b\right)\right) + {b}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 63.1%

      \[\leadsto 2 \cdot \left(\left(a \cdot \left(-1 \cdot \left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) + \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + -1 \cdot b\right)\right) + {b}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}\right) \]

   if -1.0000000000000001e-250 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 53.7%

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

      1. associate-*l* 53.7%

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

      2. *-commutative 53.7%

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

      3. associate-*l* 53.7%

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

   3. Simplified 53.7%

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

      1. add-cbrt-cube 40.5%

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

      2. pow1/3 30.3%

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

   6. Applied egg-rr 30.3%

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

      1. unpow1/3 41.3%

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

      2. rem-cbrt-cube 54.7%

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

      3. add-sqr-sqrt 54.6%

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

      4. associate-*l* 54.6%

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

   8. Applied egg-rr 72.5%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -1 \cdot 10^{-250}:\\ \;\;\;\;2 \cdot \left(\left(a \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b - b\right) - a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) + {b}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(b, a\right) \cdot \left(\mathsf{hypot}\left(b, a\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.9% accurate, 0.5× speedup

Math

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

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (sin (* 0.005555555555555556 (* angle_m PI)))))
   (*
    angle_s
    (if (<= (- (pow b 2.0) (pow a 2.0)) -1e-250)
      (*
       2.0
       (*
        (cos (* PI (/ angle_m 180.0)))
        (+ (* a (- (* t_0 (- b b)) (* a t_0))) (fabs (* (pow b 2.0) t_0)))))
      (*
       (hypot b a)
       (*
        (hypot b a)
        (sin (* 2.0 (* angle_m (* 0.005555555555555556 PI))))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = sin((0.005555555555555556 * (angle_m * ((double) M_PI))));
	double tmp;
	if ((pow(b, 2.0) - pow(a, 2.0)) <= -1e-250) {
		tmp = 2.0 * (cos((((double) M_PI) * (angle_m / 180.0))) * ((a * ((t_0 * (b - b)) - (a * t_0))) + fabs((pow(b, 2.0) * t_0))));
	} else {
		tmp = hypot(b, a) * (hypot(b, a) * sin((2.0 * (angle_m * (0.005555555555555556 * ((double) M_PI))))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.sin((0.005555555555555556 * (angle_m * Math.PI)));
	double tmp;
	if ((Math.pow(b, 2.0) - Math.pow(a, 2.0)) <= -1e-250) {
		tmp = 2.0 * (Math.cos((Math.PI * (angle_m / 180.0))) * ((a * ((t_0 * (b - b)) - (a * t_0))) + Math.abs((Math.pow(b, 2.0) * t_0))));
	} else {
		tmp = Math.hypot(b, a) * (Math.hypot(b, a) * Math.sin((2.0 * (angle_m * (0.005555555555555556 * Math.PI)))));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = math.sin((0.005555555555555556 * (angle_m * math.pi)))
	tmp = 0
	if (math.pow(b, 2.0) - math.pow(a, 2.0)) <= -1e-250:
		tmp = 2.0 * (math.cos((math.pi * (angle_m / 180.0))) * ((a * ((t_0 * (b - b)) - (a * t_0))) + math.fabs((math.pow(b, 2.0) * t_0))))
	else:
		tmp = math.hypot(b, a) * (math.hypot(b, a) * math.sin((2.0 * (angle_m * (0.005555555555555556 * math.pi)))))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = sin(Float64(0.005555555555555556 * Float64(angle_m * pi)))
	tmp = 0.0
	if (Float64((b ^ 2.0) - (a ^ 2.0)) <= -1e-250)
		tmp = Float64(2.0 * Float64(cos(Float64(pi * Float64(angle_m / 180.0))) * Float64(Float64(a * Float64(Float64(t_0 * Float64(b - b)) - Float64(a * t_0))) + abs(Float64((b ^ 2.0) * t_0)))));
	else
		tmp = Float64(hypot(b, a) * Float64(hypot(b, a) * sin(Float64(2.0 * Float64(angle_m * Float64(0.005555555555555556 * pi))))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = sin((0.005555555555555556 * (angle_m * pi)));
	tmp = 0.0;
	if (((b ^ 2.0) - (a ^ 2.0)) <= -1e-250)
		tmp = 2.0 * (cos((pi * (angle_m / 180.0))) * ((a * ((t_0 * (b - b)) - (a * t_0))) + abs(((b ^ 2.0) * t_0))));
	else
		tmp = hypot(b, a) * (hypot(b, a) * sin((2.0 * (angle_m * (0.005555555555555556 * pi)))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision], -1e-250], N[(2.0 * N[(N[Cos[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(a * N[(N[(t$95$0 * N[(b - b), $MachinePrecision]), $MachinePrecision] - N[(a * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Abs[N[(N[Power[b, 2.0], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[b ^ 2 + a ^ 2], $MachinePrecision] * N[(N[Sqrt[b ^ 2 + a ^ 2], $MachinePrecision] * N[Sin[N[(2.0 * N[(angle$95$m * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -1.0000000000000001e-250

   1. Initial program 50.1%

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

      1. associate-*l* 50.1%

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

      2. associate-*l* 50.1%

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

   3. Simplified 50.1%

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

      1. unpow2 50.1%

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

      2. unpow2 50.1%

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

      3. difference-of-squares 50.1%

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

   6. Applied egg-rr 50.1%

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

   7. Taylor expanded in a around 0 61.3%

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

      1. add-cube-cbrt 61.3%

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

      2. pow2 61.3%

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

   9. Applied egg-rr 61.3%

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

       1. add-sqr-sqrt 52.3%

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

       2. sqrt-unprod 60.3%

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

   11. Applied egg-rr 60.3%

       \[\leadsto 2 \cdot \left(\left(a \cdot \left(-1 \cdot \left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) + \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + -1 \cdot b\right)\right) + \color{blue}{\sqrt{{\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{2} \cdot {b}^{4}}}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]
   12. Step-by-step derivation

       1. *-commutative 60.3%

          \[\leadsto 2 \cdot \left(\left(a \cdot \left(-1 \cdot \left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) + \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + -1 \cdot b\right)\right) + \sqrt{\color{blue}{{b}^{4} \cdot {\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{2}}}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

       2. metadata-eval 60.3%

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

       3. pow-sqr 60.3%

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

       4. unpow2 60.3%

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

       5. unpow2 60.3%

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

       6. *-commutative 60.3%

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

       7. associate-*r* 60.3%

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

       8. *-commutative 60.3%

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

       9. unpow2 60.3%

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

       10. swap-sqr 60.3%

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

       11. rem-sqrt-square 61.3%

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

   13. Simplified 61.3%

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

   if -1.0000000000000001e-250 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 53.7%

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

      1. associate-*l* 53.7%

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

      2. *-commutative 53.7%

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

      3. associate-*l* 53.7%

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

   3. Simplified 53.7%

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

      1. add-cbrt-cube 40.5%

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

      2. pow1/3 30.3%

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

   6. Applied egg-rr 30.3%

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

      1. unpow1/3 41.3%

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

      2. rem-cbrt-cube 54.7%

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

      3. add-sqr-sqrt 54.6%

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

      4. associate-*l* 54.6%

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

   8. Applied egg-rr 72.5%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -1 \cdot 10^{-250}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(a \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b - b\right) - a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) + \left|{b}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right|\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(b, a\right) \cdot \left(\mathsf{hypot}\left(b, a\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.9% accurate, 0.6× speedup

Math

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

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (sin (* 0.005555555555555556 (* angle_m PI)))))
   (*
    angle_s
    (if (<= (- (pow b 2.0) (pow a 2.0)) -1e-250)
      (*
       2.0
       (*
        (+ (* a (- (* t_0 (- b b)) (* a t_0))) (* (pow b 2.0) t_0))
        (cos (* PI (/ angle_m 180.0)))))
      (*
       (hypot b a)
       (*
        (hypot b a)
        (sin (* 2.0 (* angle_m (* 0.005555555555555556 PI))))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = sin((0.005555555555555556 * (angle_m * ((double) M_PI))));
	double tmp;
	if ((pow(b, 2.0) - pow(a, 2.0)) <= -1e-250) {
		tmp = 2.0 * (((a * ((t_0 * (b - b)) - (a * t_0))) + (pow(b, 2.0) * t_0)) * cos((((double) M_PI) * (angle_m / 180.0))));
	} else {
		tmp = hypot(b, a) * (hypot(b, a) * sin((2.0 * (angle_m * (0.005555555555555556 * ((double) M_PI))))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.sin((0.005555555555555556 * (angle_m * Math.PI)));
	double tmp;
	if ((Math.pow(b, 2.0) - Math.pow(a, 2.0)) <= -1e-250) {
		tmp = 2.0 * (((a * ((t_0 * (b - b)) - (a * t_0))) + (Math.pow(b, 2.0) * t_0)) * Math.cos((Math.PI * (angle_m / 180.0))));
	} else {
		tmp = Math.hypot(b, a) * (Math.hypot(b, a) * Math.sin((2.0 * (angle_m * (0.005555555555555556 * Math.PI)))));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = math.sin((0.005555555555555556 * (angle_m * math.pi)))
	tmp = 0
	if (math.pow(b, 2.0) - math.pow(a, 2.0)) <= -1e-250:
		tmp = 2.0 * (((a * ((t_0 * (b - b)) - (a * t_0))) + (math.pow(b, 2.0) * t_0)) * math.cos((math.pi * (angle_m / 180.0))))
	else:
		tmp = math.hypot(b, a) * (math.hypot(b, a) * math.sin((2.0 * (angle_m * (0.005555555555555556 * math.pi)))))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = sin(Float64(0.005555555555555556 * Float64(angle_m * pi)))
	tmp = 0.0
	if (Float64((b ^ 2.0) - (a ^ 2.0)) <= -1e-250)
		tmp = Float64(2.0 * Float64(Float64(Float64(a * Float64(Float64(t_0 * Float64(b - b)) - Float64(a * t_0))) + Float64((b ^ 2.0) * t_0)) * cos(Float64(pi * Float64(angle_m / 180.0)))));
	else
		tmp = Float64(hypot(b, a) * Float64(hypot(b, a) * sin(Float64(2.0 * Float64(angle_m * Float64(0.005555555555555556 * pi))))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = sin((0.005555555555555556 * (angle_m * pi)));
	tmp = 0.0;
	if (((b ^ 2.0) - (a ^ 2.0)) <= -1e-250)
		tmp = 2.0 * (((a * ((t_0 * (b - b)) - (a * t_0))) + ((b ^ 2.0) * t_0)) * cos((pi * (angle_m / 180.0))));
	else
		tmp = hypot(b, a) * (hypot(b, a) * sin((2.0 * (angle_m * (0.005555555555555556 * pi)))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision], -1e-250], N[(2.0 * N[(N[(N[(a * N[(N[(t$95$0 * N[(b - b), $MachinePrecision]), $MachinePrecision] - N[(a * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[b, 2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[b ^ 2 + a ^ 2], $MachinePrecision] * N[(N[Sqrt[b ^ 2 + a ^ 2], $MachinePrecision] * N[Sin[N[(2.0 * N[(angle$95$m * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -1.0000000000000001e-250

   1. Initial program 50.1%

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

      1. associate-*l* 50.1%

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

      2. associate-*l* 50.1%

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

   3. Simplified 50.1%

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

      1. unpow2 50.1%

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

      2. unpow2 50.1%

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

      3. difference-of-squares 50.1%

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

   6. Applied egg-rr 50.1%

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

   7. Taylor expanded in a around 0 61.3%

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

   if -1.0000000000000001e-250 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 53.7%

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

      1. associate-*l* 53.7%

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

      2. *-commutative 53.7%

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

      3. associate-*l* 53.7%

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

   3. Simplified 53.7%

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

      1. add-cbrt-cube 40.5%

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

      2. pow1/3 30.3%

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

   6. Applied egg-rr 30.3%

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

      1. unpow1/3 41.3%

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

      2. rem-cbrt-cube 54.7%

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

      3. add-sqr-sqrt 54.6%

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

      4. associate-*l* 54.6%

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

   8. Applied egg-rr 72.5%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -1 \cdot 10^{-250}:\\ \;\;\;\;2 \cdot \left(\left(a \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b - b\right) - a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) + {b}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(b, a\right) \cdot \left(\mathsf{hypot}\left(b, a\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \sin \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq 10^{+60}:\\ \;\;\;\;2 \cdot \left(a \cdot \left(t\_0 \cdot \left(b - b\right) - a \cdot t\_0\right) + {b}^{2} \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(b, a\right) \cdot \left(\mathsf{hypot}\left(b, a\right) \cdot \sin \left(2 \cdot \left(angle\_m \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (sin (* 0.005555555555555556 (* angle_m PI)))))
   (*
    angle_s
    (if (<= (- (pow b 2.0) (pow a 2.0)) 1e+60)
      (* 2.0 (+ (* a (- (* t_0 (- b b)) (* a t_0))) (* (pow b 2.0) t_0)))
      (*
       (hypot b a)
       (*
        (hypot b a)
        (sin (* 2.0 (* angle_m (* 0.005555555555555556 PI))))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = sin((0.005555555555555556 * (angle_m * ((double) M_PI))));
	double tmp;
	if ((pow(b, 2.0) - pow(a, 2.0)) <= 1e+60) {
		tmp = 2.0 * ((a * ((t_0 * (b - b)) - (a * t_0))) + (pow(b, 2.0) * t_0));
	} else {
		tmp = hypot(b, a) * (hypot(b, a) * sin((2.0 * (angle_m * (0.005555555555555556 * ((double) M_PI))))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.sin((0.005555555555555556 * (angle_m * Math.PI)));
	double tmp;
	if ((Math.pow(b, 2.0) - Math.pow(a, 2.0)) <= 1e+60) {
		tmp = 2.0 * ((a * ((t_0 * (b - b)) - (a * t_0))) + (Math.pow(b, 2.0) * t_0));
	} else {
		tmp = Math.hypot(b, a) * (Math.hypot(b, a) * Math.sin((2.0 * (angle_m * (0.005555555555555556 * Math.PI)))));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = math.sin((0.005555555555555556 * (angle_m * math.pi)))
	tmp = 0
	if (math.pow(b, 2.0) - math.pow(a, 2.0)) <= 1e+60:
		tmp = 2.0 * ((a * ((t_0 * (b - b)) - (a * t_0))) + (math.pow(b, 2.0) * t_0))
	else:
		tmp = math.hypot(b, a) * (math.hypot(b, a) * math.sin((2.0 * (angle_m * (0.005555555555555556 * math.pi)))))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = sin(Float64(0.005555555555555556 * Float64(angle_m * pi)))
	tmp = 0.0
	if (Float64((b ^ 2.0) - (a ^ 2.0)) <= 1e+60)
		tmp = Float64(2.0 * Float64(Float64(a * Float64(Float64(t_0 * Float64(b - b)) - Float64(a * t_0))) + Float64((b ^ 2.0) * t_0)));
	else
		tmp = Float64(hypot(b, a) * Float64(hypot(b, a) * sin(Float64(2.0 * Float64(angle_m * Float64(0.005555555555555556 * pi))))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = sin((0.005555555555555556 * (angle_m * pi)));
	tmp = 0.0;
	if (((b ^ 2.0) - (a ^ 2.0)) <= 1e+60)
		tmp = 2.0 * ((a * ((t_0 * (b - b)) - (a * t_0))) + ((b ^ 2.0) * t_0));
	else
		tmp = hypot(b, a) * (hypot(b, a) * sin((2.0 * (angle_m * (0.005555555555555556 * pi)))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision], 1e+60], N[(2.0 * N[(N[(a * N[(N[(t$95$0 * N[(b - b), $MachinePrecision]), $MachinePrecision] - N[(a * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[b, 2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[b ^ 2 + a ^ 2], $MachinePrecision] * N[(N[Sqrt[b ^ 2 + a ^ 2], $MachinePrecision] * N[Sin[N[(2.0 * N[(angle$95$m * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < 9.9999999999999995e59

   1. Initial program 55.2%

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

      1. associate-*l* 55.2%

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

      2. associate-*l* 55.2%

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

   3. Simplified 55.2%

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

      1. unpow2 55.2%

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

      2. unpow2 55.2%

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

      3. difference-of-squares 55.2%

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

   6. Applied egg-rr 55.2%

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

   7. Taylor expanded in a around 0 63.1%

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

   8. Taylor expanded in angle around 0 61.3%

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

   if 9.9999999999999995e59 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 47.3%

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

      1. associate-*l* 47.3%

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

      2. *-commutative 47.3%

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

      3. associate-*l* 47.3%

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

   3. Simplified 47.3%

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

      1. add-cbrt-cube 36.7%

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

      2. pow1/3 23.5%

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

   6. Applied egg-rr 23.5%

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

      1. unpow1/3 37.9%

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

      2. rem-cbrt-cube 48.8%

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

      3. add-sqr-sqrt 48.8%

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

      4. associate-*l* 48.7%

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

   8. Applied egg-rr 73.7%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq 10^{+60}:\\ \;\;\;\;2 \cdot \left(a \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b - b\right) - a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) + {b}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(b, a\right) \cdot \left(\mathsf{hypot}\left(b, a\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := {b}^{2} - {a}^{2}\\ t_1 := \left(b + a\right) \cdot \left(b - a\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right) \cdot t\_1\right)\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(b, \left(angle\_m \cdot 0.011111111111111112\right) \cdot \left(b \cdot \pi\right), \left({a}^{2} \cdot \left(angle\_m \cdot \pi\right)\right) \cdot -0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle\_m \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot t\_1\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (- (pow b 2.0) (pow a 2.0))) (t_1 (* (+ b a) (- b a))))
   (*
    angle_s
    (if (<= t_0 2e+307)
      (*
       2.0
       (*
        (cos (* PI (* 0.005555555555555556 angle_m)))
        (* (sin (* 0.005555555555555556 (* angle_m PI))) t_1)))
      (if (<= t_0 INFINITY)
        (fma
         b
         (* (* angle_m 0.011111111111111112) (* b PI))
         (* (* (pow a 2.0) (* angle_m PI)) -0.011111111111111112))
        (* (* angle_m 0.011111111111111112) (* PI t_1)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = pow(b, 2.0) - pow(a, 2.0);
	double t_1 = (b + a) * (b - a);
	double tmp;
	if (t_0 <= 2e+307) {
		tmp = 2.0 * (cos((((double) M_PI) * (0.005555555555555556 * angle_m))) * (sin((0.005555555555555556 * (angle_m * ((double) M_PI)))) * t_1));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma(b, ((angle_m * 0.011111111111111112) * (b * ((double) M_PI))), ((pow(a, 2.0) * (angle_m * ((double) M_PI))) * -0.011111111111111112));
	} else {
		tmp = (angle_m * 0.011111111111111112) * (((double) M_PI) * t_1);
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64((b ^ 2.0) - (a ^ 2.0))
	t_1 = Float64(Float64(b + a) * Float64(b - a))
	tmp = 0.0
	if (t_0 <= 2e+307)
		tmp = Float64(2.0 * Float64(cos(Float64(pi * Float64(0.005555555555555556 * angle_m))) * Float64(sin(Float64(0.005555555555555556 * Float64(angle_m * pi))) * t_1)));
	elseif (t_0 <= Inf)
		tmp = fma(b, Float64(Float64(angle_m * 0.011111111111111112) * Float64(b * pi)), Float64(Float64((a ^ 2.0) * Float64(angle_m * pi)) * -0.011111111111111112));
	else
		tmp = Float64(Float64(angle_m * 0.011111111111111112) * Float64(pi * t_1));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[t$95$0, 2e+307], N[(2.0 * N[(N[Cos[N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(b * N[(N[(angle$95$m * 0.011111111111111112), $MachinePrecision] * N[(b * Pi), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision] * -0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(N[(angle$95$m * 0.011111111111111112), $MachinePrecision] * N[(Pi * t$95$1), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < 1.99999999999999997e307

   1. Initial program 56.2%

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

      1. associate-*l* 56.2%

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

      2. associate-*l* 56.2%

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

   3. Simplified 56.2%

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

      1. unpow2 56.2%

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

      2. unpow2 56.2%

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

      3. difference-of-squares 56.2%

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

   6. Applied egg-rr 56.2%

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

   7. Taylor expanded in angle around inf 57.5%

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

      1. associate-*r* 57.4%

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

      2. *-commutative 57.4%

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

      3. *-commutative 57.4%

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

      4. *-commutative 57.4%

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

   9. Simplified 57.4%

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

   10. Taylor expanded in angle around inf 58.8%

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

   if 1.99999999999999997e307 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < +inf.0

   1. Initial program 51.9%

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

      1. associate-*l* 51.9%

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

      2. *-commutative 51.9%

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

      3. associate-*l* 51.9%

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

   3. Simplified 51.9%

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

   5. Taylor expanded in angle around 0 47.8%

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

      1. unpow2 51.9%

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

      2. unpow2 51.9%

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

      3. difference-of-squares 51.9%

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

   7. Applied egg-rr 47.8%

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

   8. Taylor expanded in b around 0 68.5%

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

      1. +-commutative 68.5%

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

      2. fma-define 68.5%

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

      3. associate-*r* 68.6%

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

      4. associate-*r* 68.6%

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

      5. distribute-lft-out 68.6%

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

      6. +-commutative 68.6%

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

      7. *-commutative 68.6%

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

      8. distribute-rgt1-in 68.6%

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

      9. metadata-eval 68.6%

         \[\leadsto \mathsf{fma}\left(b, \left(0.011111111111111112 \cdot angle\right) \cdot \left(\left(\color{blue}{0} \cdot a\right) \cdot \pi + b \cdot \pi\right), -0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \]

      10. mul0-lft 68.6%

          \[\leadsto \mathsf{fma}\left(b, \left(0.011111111111111112 \cdot angle\right) \cdot \left(\color{blue}{0} \cdot \pi + b \cdot \pi\right), -0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \]

      11. distribute-rgt-out 68.6%

          \[\leadsto \mathsf{fma}\left(b, \left(0.011111111111111112 \cdot angle\right) \cdot \color{blue}{\left(\pi \cdot \left(0 + b\right)\right)}, -0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \]

      12. *-commutative 68.6%

          \[\leadsto \mathsf{fma}\left(b, \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(0 + b\right)\right), \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right) \cdot -0.011111111111111112}\right) \]

   10. Simplified 68.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(b, \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(0 + b\right)\right), \left(\left(angle \cdot \pi\right) \cdot {a}^{2}\right) \cdot -0.011111111111111112\right)} \]

   if +inf.0 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 0.0%

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

      1. associate-*l* 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-*l* 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in angle around 0 0.0%

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

      1. unpow2 0.0%

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

      2. unpow2 0.0%

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

      3. difference-of-squares 42.3%

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

   7. Applied egg-rr 42.3%

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

   8. Taylor expanded in angle around 0 42.3%

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

      1. associate-*r* 42.3%

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

      2. +-commutative 42.3%

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

      3. *-commutative 42.3%

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

   10. Simplified 42.3%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq 2 \cdot 10^{+307}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)\\ \mathbf{elif}\;{b}^{2} - {a}^{2} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(b, \left(angle \cdot 0.011111111111111112\right) \cdot \left(b \cdot \pi\right), \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right) \cdot -0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq 2 \cdot 10^{+191}:\\ \;\;\;\;2 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right)\right) \cdot \cos \left(\frac{angle\_m}{180} \cdot {\left(\sqrt{\pi}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(b, a\right) \cdot \left(\mathsf{hypot}\left(b, a\right) \cdot \sin \left(2 \cdot \left(angle\_m \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (- (pow b 2.0) (pow a 2.0)) 2e+191)
    (*
     2.0
     (*
      (* (* (+ b a) (- b a)) (sin (* PI (* 0.005555555555555556 angle_m))))
      (cos (* (/ angle_m 180.0) (pow (sqrt PI) 2.0)))))
    (*
     (hypot b a)
     (* (hypot b a) (sin (* 2.0 (* angle_m (* 0.005555555555555556 PI)))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((pow(b, 2.0) - pow(a, 2.0)) <= 2e+191) {
		tmp = 2.0 * ((((b + a) * (b - a)) * sin((((double) M_PI) * (0.005555555555555556 * angle_m)))) * cos(((angle_m / 180.0) * pow(sqrt(((double) M_PI)), 2.0))));
	} else {
		tmp = hypot(b, a) * (hypot(b, a) * sin((2.0 * (angle_m * (0.005555555555555556 * ((double) M_PI))))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((Math.pow(b, 2.0) - Math.pow(a, 2.0)) <= 2e+191) {
		tmp = 2.0 * ((((b + a) * (b - a)) * Math.sin((Math.PI * (0.005555555555555556 * angle_m)))) * Math.cos(((angle_m / 180.0) * Math.pow(Math.sqrt(Math.PI), 2.0))));
	} else {
		tmp = Math.hypot(b, a) * (Math.hypot(b, a) * Math.sin((2.0 * (angle_m * (0.005555555555555556 * Math.PI)))));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (math.pow(b, 2.0) - math.pow(a, 2.0)) <= 2e+191:
		tmp = 2.0 * ((((b + a) * (b - a)) * math.sin((math.pi * (0.005555555555555556 * angle_m)))) * math.cos(((angle_m / 180.0) * math.pow(math.sqrt(math.pi), 2.0))))
	else:
		tmp = math.hypot(b, a) * (math.hypot(b, a) * math.sin((2.0 * (angle_m * (0.005555555555555556 * math.pi)))))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64((b ^ 2.0) - (a ^ 2.0)) <= 2e+191)
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(b + a) * Float64(b - a)) * sin(Float64(pi * Float64(0.005555555555555556 * angle_m)))) * cos(Float64(Float64(angle_m / 180.0) * (sqrt(pi) ^ 2.0)))));
	else
		tmp = Float64(hypot(b, a) * Float64(hypot(b, a) * sin(Float64(2.0 * Float64(angle_m * Float64(0.005555555555555556 * pi))))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (((b ^ 2.0) - (a ^ 2.0)) <= 2e+191)
		tmp = 2.0 * ((((b + a) * (b - a)) * sin((pi * (0.005555555555555556 * angle_m)))) * cos(((angle_m / 180.0) * (sqrt(pi) ^ 2.0))));
	else
		tmp = hypot(b, a) * (hypot(b, a) * sin((2.0 * (angle_m * (0.005555555555555556 * pi)))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision], 2e+191], N[(2.0 * N[(N[(N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[b ^ 2 + a ^ 2], $MachinePrecision] * N[(N[Sqrt[b ^ 2 + a ^ 2], $MachinePrecision] * N[Sin[N[(2.0 * N[(angle$95$m * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < 2.00000000000000015e191

   1. Initial program 55.1%

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

      1. associate-*l* 55.1%

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

      2. associate-*l* 55.1%

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

   3. Simplified 55.1%

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

      1. unpow2 55.1%

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

      2. unpow2 55.1%

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

      3. difference-of-squares 55.1%

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

   6. Applied egg-rr 55.1%

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

      1. add-sqr-sqrt 57.9%

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

      2. pow2 57.9%

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

   8. Applied egg-rr 57.9%

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

   9. Taylor expanded in angle around inf 57.0%

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

       1. associate-*r* 58.7%

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

       2. *-commutative 58.7%

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

       3. *-commutative 58.7%

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

       4. *-commutative 58.7%

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

   11. Simplified 58.7%

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

   if 2.00000000000000015e191 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 45.9%

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

      1. associate-*l* 45.9%

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

      2. *-commutative 45.9%

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

      3. associate-*l* 45.9%

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

   3. Simplified 45.9%

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

      1. add-cbrt-cube 38.4%

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

      2. pow1/3 25.4%

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

   6. Applied egg-rr 25.4%

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

      1. unpow1/3 39.7%

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

      2. rem-cbrt-cube 47.6%

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

      3. add-sqr-sqrt 47.6%

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

      4. associate-*l* 47.6%

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

   8. Applied egg-rr 76.9%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq 2 \cdot 10^{+191}:\\ \;\;\;\;2 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot {\left(\sqrt{\pi}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(b, a\right) \cdot \left(\mathsf{hypot}\left(b, a\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -1 \cdot 10^{-250}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(b, a\right) \cdot \left(\mathsf{hypot}\left(b, a\right) \cdot \sin \left(2 \cdot \left(angle\_m \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (- (pow b 2.0) (pow a 2.0)) -1e-250)
    (*
     2.0
     (*
      (cos (* PI (* 0.005555555555555556 angle_m)))
      (* (sin (* 0.005555555555555556 (* angle_m PI))) (* (+ b a) (- b a)))))
    (*
     (hypot b a)
     (* (hypot b a) (sin (* 2.0 (* angle_m (* 0.005555555555555556 PI)))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((pow(b, 2.0) - pow(a, 2.0)) <= -1e-250) {
		tmp = 2.0 * (cos((((double) M_PI) * (0.005555555555555556 * angle_m))) * (sin((0.005555555555555556 * (angle_m * ((double) M_PI)))) * ((b + a) * (b - a))));
	} else {
		tmp = hypot(b, a) * (hypot(b, a) * sin((2.0 * (angle_m * (0.005555555555555556 * ((double) M_PI))))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((Math.pow(b, 2.0) - Math.pow(a, 2.0)) <= -1e-250) {
		tmp = 2.0 * (Math.cos((Math.PI * (0.005555555555555556 * angle_m))) * (Math.sin((0.005555555555555556 * (angle_m * Math.PI))) * ((b + a) * (b - a))));
	} else {
		tmp = Math.hypot(b, a) * (Math.hypot(b, a) * Math.sin((2.0 * (angle_m * (0.005555555555555556 * Math.PI)))));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (math.pow(b, 2.0) - math.pow(a, 2.0)) <= -1e-250:
		tmp = 2.0 * (math.cos((math.pi * (0.005555555555555556 * angle_m))) * (math.sin((0.005555555555555556 * (angle_m * math.pi))) * ((b + a) * (b - a))))
	else:
		tmp = math.hypot(b, a) * (math.hypot(b, a) * math.sin((2.0 * (angle_m * (0.005555555555555556 * math.pi)))))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64((b ^ 2.0) - (a ^ 2.0)) <= -1e-250)
		tmp = Float64(2.0 * Float64(cos(Float64(pi * Float64(0.005555555555555556 * angle_m))) * Float64(sin(Float64(0.005555555555555556 * Float64(angle_m * pi))) * Float64(Float64(b + a) * Float64(b - a)))));
	else
		tmp = Float64(hypot(b, a) * Float64(hypot(b, a) * sin(Float64(2.0 * Float64(angle_m * Float64(0.005555555555555556 * pi))))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (((b ^ 2.0) - (a ^ 2.0)) <= -1e-250)
		tmp = 2.0 * (cos((pi * (0.005555555555555556 * angle_m))) * (sin((0.005555555555555556 * (angle_m * pi))) * ((b + a) * (b - a))));
	else
		tmp = hypot(b, a) * (hypot(b, a) * sin((2.0 * (angle_m * (0.005555555555555556 * pi)))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision], -1e-250], N[(2.0 * N[(N[Cos[N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[b ^ 2 + a ^ 2], $MachinePrecision] * N[(N[Sqrt[b ^ 2 + a ^ 2], $MachinePrecision] * N[Sin[N[(2.0 * N[(angle$95$m * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -1.0000000000000001e-250

   1. Initial program 50.1%

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

      1. associate-*l* 50.1%

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

      2. associate-*l* 50.1%

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

   3. Simplified 50.1%

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

      1. unpow2 50.1%

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

      2. unpow2 50.1%

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

      3. difference-of-squares 50.1%

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

   6. Applied egg-rr 50.1%

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

   7. Taylor expanded in angle around inf 52.0%

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

      1. associate-*r* 51.9%

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

      2. *-commutative 51.9%

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

      3. *-commutative 51.9%

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

      4. *-commutative 51.9%

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

   9. Simplified 51.9%

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

   10. Taylor expanded in angle around inf 54.0%

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

   if -1.0000000000000001e-250 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 53.7%

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

      1. associate-*l* 53.7%

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

      2. *-commutative 53.7%

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

      3. associate-*l* 53.7%

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

   3. Simplified 53.7%

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

      1. add-cbrt-cube 40.5%

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

      2. pow1/3 30.3%

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

   6. Applied egg-rr 30.3%

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

      1. unpow1/3 41.3%

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

      2. rem-cbrt-cube 54.7%

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

      3. add-sqr-sqrt 54.6%

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

      4. associate-*l* 54.6%

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

   8. Applied egg-rr 72.5%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -1 \cdot 10^{-250}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(b, a\right) \cdot \left(\mathsf{hypot}\left(b, a\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq 2 \cdot 10^{+294}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot {\left(\mathsf{hypot}\left(b, a\right) \cdot \sqrt{angle\_m \cdot \pi}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (- (pow b 2.0) (pow a 2.0)) 2e+294)
    (*
     2.0
     (*
      (cos (* PI (* 0.005555555555555556 angle_m)))
      (* (sin (* 0.005555555555555556 (* angle_m PI))) (* (+ b a) (- b a)))))
    (* 0.011111111111111112 (pow (* (hypot b a) (sqrt (* angle_m PI))) 2.0)))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((pow(b, 2.0) - pow(a, 2.0)) <= 2e+294) {
		tmp = 2.0 * (cos((((double) M_PI) * (0.005555555555555556 * angle_m))) * (sin((0.005555555555555556 * (angle_m * ((double) M_PI)))) * ((b + a) * (b - a))));
	} else {
		tmp = 0.011111111111111112 * pow((hypot(b, a) * sqrt((angle_m * ((double) M_PI)))), 2.0);
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((Math.pow(b, 2.0) - Math.pow(a, 2.0)) <= 2e+294) {
		tmp = 2.0 * (Math.cos((Math.PI * (0.005555555555555556 * angle_m))) * (Math.sin((0.005555555555555556 * (angle_m * Math.PI))) * ((b + a) * (b - a))));
	} else {
		tmp = 0.011111111111111112 * Math.pow((Math.hypot(b, a) * Math.sqrt((angle_m * Math.PI))), 2.0);
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (math.pow(b, 2.0) - math.pow(a, 2.0)) <= 2e+294:
		tmp = 2.0 * (math.cos((math.pi * (0.005555555555555556 * angle_m))) * (math.sin((0.005555555555555556 * (angle_m * math.pi))) * ((b + a) * (b - a))))
	else:
		tmp = 0.011111111111111112 * math.pow((math.hypot(b, a) * math.sqrt((angle_m * math.pi))), 2.0)
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64((b ^ 2.0) - (a ^ 2.0)) <= 2e+294)
		tmp = Float64(2.0 * Float64(cos(Float64(pi * Float64(0.005555555555555556 * angle_m))) * Float64(sin(Float64(0.005555555555555556 * Float64(angle_m * pi))) * Float64(Float64(b + a) * Float64(b - a)))));
	else
		tmp = Float64(0.011111111111111112 * (Float64(hypot(b, a) * sqrt(Float64(angle_m * pi))) ^ 2.0));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (((b ^ 2.0) - (a ^ 2.0)) <= 2e+294)
		tmp = 2.0 * (cos((pi * (0.005555555555555556 * angle_m))) * (sin((0.005555555555555556 * (angle_m * pi))) * ((b + a) * (b - a))));
	else
		tmp = 0.011111111111111112 * ((hypot(b, a) * sqrt((angle_m * pi))) ^ 2.0);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision], 2e+294], N[(2.0 * N[(N[Cos[N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[Power[N[(N[Sqrt[b ^ 2 + a ^ 2], $MachinePrecision] * N[Sqrt[N[(angle$95$m * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < 2.00000000000000013e294

   1. Initial program 55.9%

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

      1. associate-*l* 55.9%

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

      2. associate-*l* 55.9%

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

   3. Simplified 55.9%

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

      1. unpow2 55.9%

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

      2. unpow2 55.9%

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

      3. difference-of-squares 55.9%

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

   6. Applied egg-rr 55.9%

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

   7. Taylor expanded in angle around inf 57.3%

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

      1. associate-*r* 57.2%

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

      2. *-commutative 57.2%

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

      3. *-commutative 57.2%

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

      4. *-commutative 57.2%

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

   9. Simplified 57.2%

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

   10. Taylor expanded in angle around inf 58.5%

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

   if 2.00000000000000013e294 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 40.5%

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

      1. associate-*l* 40.5%

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

      2. *-commutative 40.5%

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

      3. associate-*l* 40.5%

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

   3. Simplified 40.5%

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

   5. Taylor expanded in angle around 0 37.4%

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

      1. add-sqr-sqrt 20.8%

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

      2. pow2 20.8%

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

   7. Applied egg-rr 35.8%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq 2 \cdot 10^{+294}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot {\left(\mathsf{hypot}\left(b, a\right) \cdot \sqrt{angle \cdot \pi}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{-230}:\\ \;\;\;\;\mathsf{fma}\left(b, \left(angle\_m \cdot 0.011111111111111112\right) \cdot \left(b \cdot \pi\right), \left({a}^{2} \cdot \left(angle\_m \cdot \pi\right)\right) \cdot -0.011111111111111112\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{-155}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left({b}^{2} \cdot \pi\right)\right) + a \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(angle\_m \cdot \pi\right)\right) + 0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(b - b\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 10^{+76}:\\ \;\;\;\;\left(angle\_m \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\pi \cdot \frac{angle\_m}{180}\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 2e-230)
    (fma
     b
     (* (* angle_m 0.011111111111111112) (* b PI))
     (* (* (pow a 2.0) (* angle_m PI)) -0.011111111111111112))
    (if (<= (/ angle_m 180.0) 4e-155)
      (+
       (* 0.011111111111111112 (* angle_m (* (pow b 2.0) PI)))
       (*
        a
        (+
         (* -0.011111111111111112 (* a (* angle_m PI)))
         (* 0.011111111111111112 (* angle_m (* PI (- b b)))))))
      (if (<= (/ angle_m 180.0) 1e+76)
        (* (* angle_m 0.011111111111111112) (* (- b a) (* PI (+ b a))))
        (* 2.0 (* (* (+ b a) (- b a)) (sin (* PI (/ angle_m 180.0))))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 2e-230) {
		tmp = fma(b, ((angle_m * 0.011111111111111112) * (b * ((double) M_PI))), ((pow(a, 2.0) * (angle_m * ((double) M_PI))) * -0.011111111111111112));
	} else if ((angle_m / 180.0) <= 4e-155) {
		tmp = (0.011111111111111112 * (angle_m * (pow(b, 2.0) * ((double) M_PI)))) + (a * ((-0.011111111111111112 * (a * (angle_m * ((double) M_PI)))) + (0.011111111111111112 * (angle_m * (((double) M_PI) * (b - b))))));
	} else if ((angle_m / 180.0) <= 1e+76) {
		tmp = (angle_m * 0.011111111111111112) * ((b - a) * (((double) M_PI) * (b + a)));
	} else {
		tmp = 2.0 * (((b + a) * (b - a)) * sin((((double) M_PI) * (angle_m / 180.0))));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e-230)
		tmp = fma(b, Float64(Float64(angle_m * 0.011111111111111112) * Float64(b * pi)), Float64(Float64((a ^ 2.0) * Float64(angle_m * pi)) * -0.011111111111111112));
	elseif (Float64(angle_m / 180.0) <= 4e-155)
		tmp = Float64(Float64(0.011111111111111112 * Float64(angle_m * Float64((b ^ 2.0) * pi))) + Float64(a * Float64(Float64(-0.011111111111111112 * Float64(a * Float64(angle_m * pi))) + Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(b - b)))))));
	elseif (Float64(angle_m / 180.0) <= 1e+76)
		tmp = Float64(Float64(angle_m * 0.011111111111111112) * Float64(Float64(b - a) * Float64(pi * Float64(b + a))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(b + a) * Float64(b - a)) * sin(Float64(pi * Float64(angle_m / 180.0)))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e-230], N[(b * N[(N[(angle$95$m * 0.011111111111111112), $MachinePrecision] * N[(b * Pi), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision] * -0.011111111111111112), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e-155], N[(N[(0.011111111111111112 * N[(angle$95$m * N[(N[Power[b, 2.0], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.011111111111111112 * N[(a * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(b - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+76], N[(N[(angle$95$m * 0.011111111111111112), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2.00000000000000009e-230

   1. Initial program 44.5%

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

      1. associate-*l* 44.5%

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

      2. *-commutative 44.5%

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

      3. associate-*l* 44.5%

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

   3. Simplified 44.5%

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

   5. Taylor expanded in angle around 0 44.7%

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

      1. unpow2 44.5%

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

      2. unpow2 44.5%

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

      3. difference-of-squares 46.9%

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

   7. Applied egg-rr 46.3%

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

   8. Taylor expanded in b around 0 47.8%

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

      1. +-commutative 47.8%

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

      2. fma-define 47.8%

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

      3. associate-*r* 47.8%

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

      4. associate-*r* 47.8%

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

      5. distribute-lft-out 47.8%

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

      6. +-commutative 47.8%

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

      7. *-commutative 47.8%

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

      8. distribute-rgt1-in 47.8%

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

      9. metadata-eval 47.8%

         \[\leadsto \mathsf{fma}\left(b, \left(0.011111111111111112 \cdot angle\right) \cdot \left(\left(\color{blue}{0} \cdot a\right) \cdot \pi + b \cdot \pi\right), -0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \]

      10. mul0-lft 47.8%

          \[\leadsto \mathsf{fma}\left(b, \left(0.011111111111111112 \cdot angle\right) \cdot \left(\color{blue}{0} \cdot \pi + b \cdot \pi\right), -0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \]

      11. distribute-rgt-out 47.8%

          \[\leadsto \mathsf{fma}\left(b, \left(0.011111111111111112 \cdot angle\right) \cdot \color{blue}{\left(\pi \cdot \left(0 + b\right)\right)}, -0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \]

      12. *-commutative 47.8%

          \[\leadsto \mathsf{fma}\left(b, \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(0 + b\right)\right), \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right) \cdot -0.011111111111111112}\right) \]

   10. Simplified 47.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(b, \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(0 + b\right)\right), \left(\left(angle \cdot \pi\right) \cdot {a}^{2}\right) \cdot -0.011111111111111112\right)} \]

   if 2.00000000000000009e-230 < (/.f64 angle #s(literal 180 binary64)) < 4.00000000000000006e-155

   1. Initial program 82.0%

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

      1. associate-*l* 82.0%

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

      2. *-commutative 82.0%

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

      3. associate-*l* 82.0%

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

   3. Simplified 82.0%

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

   5. Taylor expanded in angle around 0 81.8%

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

      1. unpow2 82.0%

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

      2. unpow2 82.0%

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

      3. difference-of-squares 82.0%

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

   7. Applied egg-rr 81.8%

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

   8. Taylor expanded in a around 0 99.4%

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

   if 4.00000000000000006e-155 < (/.f64 angle #s(literal 180 binary64)) < 1e76

   1. Initial program 74.5%

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

      1. associate-*l* 74.5%

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

      2. *-commutative 74.5%

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

      3. associate-*l* 74.5%

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

   3. Simplified 74.5%

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

   5. Taylor expanded in angle around 0 71.0%

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

      1. unpow2 74.5%

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

      2. unpow2 74.5%

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

      3. difference-of-squares 78.6%

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

   7. Applied egg-rr 75.1%

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

   8. Taylor expanded in angle around 0 75.1%

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

      1. associate-*r* 75.2%

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

      2. +-commutative 75.2%

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

      3. *-commutative 75.2%

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

   10. Simplified 75.2%

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

   11. Taylor expanded in angle around 0 75.1%

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

       1. associate-*r* 75.2%

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

       2. *-commutative 75.2%

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

       3. associate-*r* 75.2%

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

       4. +-commutative 75.2%

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

   13. Simplified 75.2%

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

   if 1e76 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 41.4%

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

      1. associate-*l* 41.4%

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

      2. associate-*l* 41.4%

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

   3. Simplified 41.4%

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

      1. unpow2 41.4%

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

      2. unpow2 41.4%

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

      3. difference-of-squares 43.3%

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

   6. Applied egg-rr 43.3%

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

   7. Taylor expanded in angle around 0 50.1%

      \[\leadsto 2 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{1}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{-230}:\\ \;\;\;\;\mathsf{fma}\left(b, \left(angle \cdot 0.011111111111111112\right) \cdot \left(b \cdot \pi\right), \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right) \cdot -0.011111111111111112\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 4 \cdot 10^{-155}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right)\right) + a \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right) + 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - b\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+76}:\\ \;\;\;\;\left(angle \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{-230}:\\ \;\;\;\;\left({a}^{2} \cdot \left(angle\_m \cdot \pi\right)\right) \cdot -0.011111111111111112 + b \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(b \cdot \pi\right)\right) + 0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(a - a\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{-155}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left({b}^{2} \cdot \pi\right)\right) + a \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(angle\_m \cdot \pi\right)\right) + 0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(b - b\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 10^{+76}:\\ \;\;\;\;\left(angle\_m \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\pi \cdot \frac{angle\_m}{180}\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 2e-230)
    (+
     (* (* (pow a 2.0) (* angle_m PI)) -0.011111111111111112)
     (*
      b
      (+
       (* 0.011111111111111112 (* angle_m (* b PI)))
       (* 0.011111111111111112 (* angle_m (* PI (- a a)))))))
    (if (<= (/ angle_m 180.0) 4e-155)
      (+
       (* 0.011111111111111112 (* angle_m (* (pow b 2.0) PI)))
       (*
        a
        (+
         (* -0.011111111111111112 (* a (* angle_m PI)))
         (* 0.011111111111111112 (* angle_m (* PI (- b b)))))))
      (if (<= (/ angle_m 180.0) 1e+76)
        (* (* angle_m 0.011111111111111112) (* (- b a) (* PI (+ b a))))
        (* 2.0 (* (* (+ b a) (- b a)) (sin (* PI (/ angle_m 180.0))))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 2e-230) {
		tmp = ((pow(a, 2.0) * (angle_m * ((double) M_PI))) * -0.011111111111111112) + (b * ((0.011111111111111112 * (angle_m * (b * ((double) M_PI)))) + (0.011111111111111112 * (angle_m * (((double) M_PI) * (a - a))))));
	} else if ((angle_m / 180.0) <= 4e-155) {
		tmp = (0.011111111111111112 * (angle_m * (pow(b, 2.0) * ((double) M_PI)))) + (a * ((-0.011111111111111112 * (a * (angle_m * ((double) M_PI)))) + (0.011111111111111112 * (angle_m * (((double) M_PI) * (b - b))))));
	} else if ((angle_m / 180.0) <= 1e+76) {
		tmp = (angle_m * 0.011111111111111112) * ((b - a) * (((double) M_PI) * (b + a)));
	} else {
		tmp = 2.0 * (((b + a) * (b - a)) * sin((((double) M_PI) * (angle_m / 180.0))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 2e-230) {
		tmp = ((Math.pow(a, 2.0) * (angle_m * Math.PI)) * -0.011111111111111112) + (b * ((0.011111111111111112 * (angle_m * (b * Math.PI))) + (0.011111111111111112 * (angle_m * (Math.PI * (a - a))))));
	} else if ((angle_m / 180.0) <= 4e-155) {
		tmp = (0.011111111111111112 * (angle_m * (Math.pow(b, 2.0) * Math.PI))) + (a * ((-0.011111111111111112 * (a * (angle_m * Math.PI))) + (0.011111111111111112 * (angle_m * (Math.PI * (b - b))))));
	} else if ((angle_m / 180.0) <= 1e+76) {
		tmp = (angle_m * 0.011111111111111112) * ((b - a) * (Math.PI * (b + a)));
	} else {
		tmp = 2.0 * (((b + a) * (b - a)) * Math.sin((Math.PI * (angle_m / 180.0))));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 2e-230:
		tmp = ((math.pow(a, 2.0) * (angle_m * math.pi)) * -0.011111111111111112) + (b * ((0.011111111111111112 * (angle_m * (b * math.pi))) + (0.011111111111111112 * (angle_m * (math.pi * (a - a))))))
	elif (angle_m / 180.0) <= 4e-155:
		tmp = (0.011111111111111112 * (angle_m * (math.pow(b, 2.0) * math.pi))) + (a * ((-0.011111111111111112 * (a * (angle_m * math.pi))) + (0.011111111111111112 * (angle_m * (math.pi * (b - b))))))
	elif (angle_m / 180.0) <= 1e+76:
		tmp = (angle_m * 0.011111111111111112) * ((b - a) * (math.pi * (b + a)))
	else:
		tmp = 2.0 * (((b + a) * (b - a)) * math.sin((math.pi * (angle_m / 180.0))))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e-230)
		tmp = Float64(Float64(Float64((a ^ 2.0) * Float64(angle_m * pi)) * -0.011111111111111112) + Float64(b * Float64(Float64(0.011111111111111112 * Float64(angle_m * Float64(b * pi))) + Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(a - a)))))));
	elseif (Float64(angle_m / 180.0) <= 4e-155)
		tmp = Float64(Float64(0.011111111111111112 * Float64(angle_m * Float64((b ^ 2.0) * pi))) + Float64(a * Float64(Float64(-0.011111111111111112 * Float64(a * Float64(angle_m * pi))) + Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(b - b)))))));
	elseif (Float64(angle_m / 180.0) <= 1e+76)
		tmp = Float64(Float64(angle_m * 0.011111111111111112) * Float64(Float64(b - a) * Float64(pi * Float64(b + a))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(b + a) * Float64(b - a)) * sin(Float64(pi * Float64(angle_m / 180.0)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if ((angle_m / 180.0) <= 2e-230)
		tmp = (((a ^ 2.0) * (angle_m * pi)) * -0.011111111111111112) + (b * ((0.011111111111111112 * (angle_m * (b * pi))) + (0.011111111111111112 * (angle_m * (pi * (a - a))))));
	elseif ((angle_m / 180.0) <= 4e-155)
		tmp = (0.011111111111111112 * (angle_m * ((b ^ 2.0) * pi))) + (a * ((-0.011111111111111112 * (a * (angle_m * pi))) + (0.011111111111111112 * (angle_m * (pi * (b - b))))));
	elseif ((angle_m / 180.0) <= 1e+76)
		tmp = (angle_m * 0.011111111111111112) * ((b - a) * (pi * (b + a)));
	else
		tmp = 2.0 * (((b + a) * (b - a)) * sin((pi * (angle_m / 180.0))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e-230], N[(N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision] * -0.011111111111111112), $MachinePrecision] + N[(b * N[(N[(0.011111111111111112 * N[(angle$95$m * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(a - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e-155], N[(N[(0.011111111111111112 * N[(angle$95$m * N[(N[Power[b, 2.0], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.011111111111111112 * N[(a * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(b - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+76], N[(N[(angle$95$m * 0.011111111111111112), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2.00000000000000009e-230

   1. Initial program 44.5%

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

      1. associate-*l* 44.5%

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

      2. *-commutative 44.5%

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

      3. associate-*l* 44.5%

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

   3. Simplified 44.5%

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

   5. Taylor expanded in angle around 0 44.7%

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

      1. unpow2 44.5%

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

      2. unpow2 44.5%

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

      3. difference-of-squares 46.9%

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

   7. Applied egg-rr 46.3%

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

   8. Taylor expanded in b around 0 47.8%

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

   if 2.00000000000000009e-230 < (/.f64 angle #s(literal 180 binary64)) < 4.00000000000000006e-155

   1. Initial program 82.0%

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

      1. associate-*l* 82.0%

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

      2. *-commutative 82.0%

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

      3. associate-*l* 82.0%

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

   3. Simplified 82.0%

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

   5. Taylor expanded in angle around 0 81.8%

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

      1. unpow2 82.0%

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

      2. unpow2 82.0%

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

      3. difference-of-squares 82.0%

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

   7. Applied egg-rr 81.8%

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

   8. Taylor expanded in a around 0 99.4%

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

   if 4.00000000000000006e-155 < (/.f64 angle #s(literal 180 binary64)) < 1e76

   1. Initial program 74.5%

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

      1. associate-*l* 74.5%

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

      2. *-commutative 74.5%

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

      3. associate-*l* 74.5%

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

   3. Simplified 74.5%

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

   5. Taylor expanded in angle around 0 71.0%

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

      1. unpow2 74.5%

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

      2. unpow2 74.5%

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

      3. difference-of-squares 78.6%

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

   7. Applied egg-rr 75.1%

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

   8. Taylor expanded in angle around 0 75.1%

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

      1. associate-*r* 75.2%

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

      2. +-commutative 75.2%

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

      3. *-commutative 75.2%

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

   10. Simplified 75.2%

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

   11. Taylor expanded in angle around 0 75.1%

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

       1. associate-*r* 75.2%

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

       2. *-commutative 75.2%

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

       3. associate-*r* 75.2%

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

       4. +-commutative 75.2%

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

   13. Simplified 75.2%

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

   if 1e76 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 41.4%

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

      1. associate-*l* 41.4%

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

      2. associate-*l* 41.4%

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

   3. Simplified 41.4%

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

      1. unpow2 41.4%

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

      2. unpow2 41.4%

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

      3. difference-of-squares 43.3%

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

   6. Applied egg-rr 43.3%

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

   7. Taylor expanded in angle around 0 50.1%

      \[\leadsto 2 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{1}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{-230}:\\ \;\;\;\;\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right) \cdot -0.011111111111111112 + b \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right) + 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(a - a\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 4 \cdot 10^{-155}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right)\right) + a \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right) + 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - b\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+76}:\\ \;\;\;\;\left(angle \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 58.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{-230}:\\ \;\;\;\;\left({a}^{2} \cdot \left(angle\_m \cdot \pi\right)\right) \cdot -0.011111111111111112 + b \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(b \cdot \pi\right)\right) + 0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(a - a\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{-155}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left({b}^{2} \cdot \pi\right) - a \cdot \left(a \cdot \left(angle\_m \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 6 \cdot 10^{+69}:\\ \;\;\;\;\left(angle\_m \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\pi \cdot \frac{angle\_m}{180}\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 2e-230)
    (+
     (* (* (pow a 2.0) (* angle_m PI)) -0.011111111111111112)
     (*
      b
      (+
       (* 0.011111111111111112 (* angle_m (* b PI)))
       (* 0.011111111111111112 (* angle_m (* PI (- a a)))))))
    (if (<= (/ angle_m 180.0) 4e-155)
      (*
       0.011111111111111112
       (- (* angle_m (* (pow b 2.0) PI)) (* a (* a (* angle_m PI)))))
      (if (<= (/ angle_m 180.0) 6e+69)
        (* (* angle_m 0.011111111111111112) (* (- b a) (* PI (+ b a))))
        (* 2.0 (* (* (+ b a) (- b a)) (sin (* PI (/ angle_m 180.0))))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 2e-230) {
		tmp = ((pow(a, 2.0) * (angle_m * ((double) M_PI))) * -0.011111111111111112) + (b * ((0.011111111111111112 * (angle_m * (b * ((double) M_PI)))) + (0.011111111111111112 * (angle_m * (((double) M_PI) * (a - a))))));
	} else if ((angle_m / 180.0) <= 4e-155) {
		tmp = 0.011111111111111112 * ((angle_m * (pow(b, 2.0) * ((double) M_PI))) - (a * (a * (angle_m * ((double) M_PI)))));
	} else if ((angle_m / 180.0) <= 6e+69) {
		tmp = (angle_m * 0.011111111111111112) * ((b - a) * (((double) M_PI) * (b + a)));
	} else {
		tmp = 2.0 * (((b + a) * (b - a)) * sin((((double) M_PI) * (angle_m / 180.0))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 2e-230) {
		tmp = ((Math.pow(a, 2.0) * (angle_m * Math.PI)) * -0.011111111111111112) + (b * ((0.011111111111111112 * (angle_m * (b * Math.PI))) + (0.011111111111111112 * (angle_m * (Math.PI * (a - a))))));
	} else if ((angle_m / 180.0) <= 4e-155) {
		tmp = 0.011111111111111112 * ((angle_m * (Math.pow(b, 2.0) * Math.PI)) - (a * (a * (angle_m * Math.PI))));
	} else if ((angle_m / 180.0) <= 6e+69) {
		tmp = (angle_m * 0.011111111111111112) * ((b - a) * (Math.PI * (b + a)));
	} else {
		tmp = 2.0 * (((b + a) * (b - a)) * Math.sin((Math.PI * (angle_m / 180.0))));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 2e-230:
		tmp = ((math.pow(a, 2.0) * (angle_m * math.pi)) * -0.011111111111111112) + (b * ((0.011111111111111112 * (angle_m * (b * math.pi))) + (0.011111111111111112 * (angle_m * (math.pi * (a - a))))))
	elif (angle_m / 180.0) <= 4e-155:
		tmp = 0.011111111111111112 * ((angle_m * (math.pow(b, 2.0) * math.pi)) - (a * (a * (angle_m * math.pi))))
	elif (angle_m / 180.0) <= 6e+69:
		tmp = (angle_m * 0.011111111111111112) * ((b - a) * (math.pi * (b + a)))
	else:
		tmp = 2.0 * (((b + a) * (b - a)) * math.sin((math.pi * (angle_m / 180.0))))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e-230)
		tmp = Float64(Float64(Float64((a ^ 2.0) * Float64(angle_m * pi)) * -0.011111111111111112) + Float64(b * Float64(Float64(0.011111111111111112 * Float64(angle_m * Float64(b * pi))) + Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(a - a)))))));
	elseif (Float64(angle_m / 180.0) <= 4e-155)
		tmp = Float64(0.011111111111111112 * Float64(Float64(angle_m * Float64((b ^ 2.0) * pi)) - Float64(a * Float64(a * Float64(angle_m * pi)))));
	elseif (Float64(angle_m / 180.0) <= 6e+69)
		tmp = Float64(Float64(angle_m * 0.011111111111111112) * Float64(Float64(b - a) * Float64(pi * Float64(b + a))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(b + a) * Float64(b - a)) * sin(Float64(pi * Float64(angle_m / 180.0)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if ((angle_m / 180.0) <= 2e-230)
		tmp = (((a ^ 2.0) * (angle_m * pi)) * -0.011111111111111112) + (b * ((0.011111111111111112 * (angle_m * (b * pi))) + (0.011111111111111112 * (angle_m * (pi * (a - a))))));
	elseif ((angle_m / 180.0) <= 4e-155)
		tmp = 0.011111111111111112 * ((angle_m * ((b ^ 2.0) * pi)) - (a * (a * (angle_m * pi))));
	elseif ((angle_m / 180.0) <= 6e+69)
		tmp = (angle_m * 0.011111111111111112) * ((b - a) * (pi * (b + a)));
	else
		tmp = 2.0 * (((b + a) * (b - a)) * sin((pi * (angle_m / 180.0))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e-230], N[(N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision] * -0.011111111111111112), $MachinePrecision] + N[(b * N[(N[(0.011111111111111112 * N[(angle$95$m * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(a - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e-155], N[(0.011111111111111112 * N[(N[(angle$95$m * N[(N[Power[b, 2.0], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] - N[(a * N[(a * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 6e+69], N[(N[(angle$95$m * 0.011111111111111112), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2.00000000000000009e-230

   1. Initial program 44.5%

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

      1. associate-*l* 44.5%

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

      2. *-commutative 44.5%

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

      3. associate-*l* 44.5%

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

   3. Simplified 44.5%

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

   5. Taylor expanded in angle around 0 44.7%

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

      1. unpow2 44.5%

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

      2. unpow2 44.5%

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

      3. difference-of-squares 46.9%

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

   7. Applied egg-rr 46.3%

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

   8. Taylor expanded in b around 0 47.8%

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

   if 2.00000000000000009e-230 < (/.f64 angle #s(literal 180 binary64)) < 4.00000000000000006e-155

   1. Initial program 82.0%

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

      1. associate-*l* 82.0%

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

      2. *-commutative 82.0%

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

      3. associate-*l* 82.0%

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

   3. Simplified 82.0%

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

   5. Taylor expanded in angle around 0 81.8%

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

      1. unpow2 82.0%

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

      2. unpow2 82.0%

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

      3. difference-of-squares 82.0%

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

   7. Applied egg-rr 81.8%

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

   8. Taylor expanded in a around 0 99.2%

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

      1. +-commutative 99.2%

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

      2. *-commutative 99.2%

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

      3. +-commutative 99.2%

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

      4. associate-*r* 99.2%

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

      5. fma-define 99.2%

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

      6. distribute-rgt1-in 99.2%

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

      7. metadata-eval 99.2%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right) + \mathsf{fma}\left(angle \cdot \pi, \color{blue}{0} \cdot b, -1 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right) \cdot a\right) \]

      8. mul0-lft 99.2%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right) + \mathsf{fma}\left(angle \cdot \pi, \color{blue}{0}, -1 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right) \cdot a\right) \]

      9. mul-1-neg 99.2%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right) + \mathsf{fma}\left(angle \cdot \pi, 0, \color{blue}{-a \cdot \left(angle \cdot \pi\right)}\right) \cdot a\right) \]

      10. fma-neg 99.2%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right) + \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0 - a \cdot \left(angle \cdot \pi\right)\right)} \cdot a\right) \]

      11. mul0-rgt 99.2%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right) + \left(\color{blue}{0} - a \cdot \left(angle \cdot \pi\right)\right) \cdot a\right) \]

      12. neg-sub0 99.2%

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

      13. mul-1-neg 99.2%

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

   10. Simplified 99.2%

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

   if 4.00000000000000006e-155 < (/.f64 angle #s(literal 180 binary64)) < 5.99999999999999967e69

   1. Initial program 76.0%

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

      1. associate-*l* 76.0%

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

      2. *-commutative 76.0%

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

      3. associate-*l* 76.0%

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

   3. Simplified 76.0%

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

   5. Taylor expanded in angle around 0 70.4%

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

      1. unpow2 76.0%

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

      2. unpow2 76.0%

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

      3. difference-of-squares 80.2%

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

   7. Applied egg-rr 74.6%

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

   8. Taylor expanded in angle around 0 74.6%

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

      1. associate-*r* 74.7%

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

      2. +-commutative 74.7%

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

      3. *-commutative 74.7%

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

   10. Simplified 74.7%

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

   11. Taylor expanded in angle around 0 74.6%

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

       1. associate-*r* 74.7%

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

       2. *-commutative 74.7%

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

       3. associate-*r* 74.7%

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

       4. +-commutative 74.7%

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

   13. Simplified 74.7%

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

   if 5.99999999999999967e69 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 40.6%

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

      1. associate-*l* 40.6%

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

      2. associate-*l* 40.6%

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

   3. Simplified 40.6%

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

      1. unpow2 40.6%

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

      2. unpow2 40.6%

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

      3. difference-of-squares 42.5%

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

   6. Applied egg-rr 42.5%

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

   7. Taylor expanded in angle around 0 51.0%

      \[\leadsto 2 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{1}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{-230}:\\ \;\;\;\;\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right) \cdot -0.011111111111111112 + b \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right) + 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(a - a\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 4 \cdot 10^{-155}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right) - a \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 6 \cdot 10^{+69}:\\ \;\;\;\;\left(angle \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{-230}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b \cdot \left(\pi \cdot \left(b \cdot angle\_m\right)\right) - {a}^{2} \cdot \left(angle\_m \cdot \pi\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{-155}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left({b}^{2} \cdot \pi\right) - a \cdot \left(a \cdot \left(angle\_m \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 6 \cdot 10^{+69}:\\ \;\;\;\;\left(angle\_m \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\pi \cdot \frac{angle\_m}{180}\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 2e-230)
    (*
     0.011111111111111112
     (- (* b (* PI (* b angle_m))) (* (pow a 2.0) (* angle_m PI))))
    (if (<= (/ angle_m 180.0) 4e-155)
      (*
       0.011111111111111112
       (- (* angle_m (* (pow b 2.0) PI)) (* a (* a (* angle_m PI)))))
      (if (<= (/ angle_m 180.0) 6e+69)
        (* (* angle_m 0.011111111111111112) (* (- b a) (* PI (+ b a))))
        (* 2.0 (* (* (+ b a) (- b a)) (sin (* PI (/ angle_m 180.0))))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 2e-230) {
		tmp = 0.011111111111111112 * ((b * (((double) M_PI) * (b * angle_m))) - (pow(a, 2.0) * (angle_m * ((double) M_PI))));
	} else if ((angle_m / 180.0) <= 4e-155) {
		tmp = 0.011111111111111112 * ((angle_m * (pow(b, 2.0) * ((double) M_PI))) - (a * (a * (angle_m * ((double) M_PI)))));
	} else if ((angle_m / 180.0) <= 6e+69) {
		tmp = (angle_m * 0.011111111111111112) * ((b - a) * (((double) M_PI) * (b + a)));
	} else {
		tmp = 2.0 * (((b + a) * (b - a)) * sin((((double) M_PI) * (angle_m / 180.0))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 2e-230) {
		tmp = 0.011111111111111112 * ((b * (Math.PI * (b * angle_m))) - (Math.pow(a, 2.0) * (angle_m * Math.PI)));
	} else if ((angle_m / 180.0) <= 4e-155) {
		tmp = 0.011111111111111112 * ((angle_m * (Math.pow(b, 2.0) * Math.PI)) - (a * (a * (angle_m * Math.PI))));
	} else if ((angle_m / 180.0) <= 6e+69) {
		tmp = (angle_m * 0.011111111111111112) * ((b - a) * (Math.PI * (b + a)));
	} else {
		tmp = 2.0 * (((b + a) * (b - a)) * Math.sin((Math.PI * (angle_m / 180.0))));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 2e-230:
		tmp = 0.011111111111111112 * ((b * (math.pi * (b * angle_m))) - (math.pow(a, 2.0) * (angle_m * math.pi)))
	elif (angle_m / 180.0) <= 4e-155:
		tmp = 0.011111111111111112 * ((angle_m * (math.pow(b, 2.0) * math.pi)) - (a * (a * (angle_m * math.pi))))
	elif (angle_m / 180.0) <= 6e+69:
		tmp = (angle_m * 0.011111111111111112) * ((b - a) * (math.pi * (b + a)))
	else:
		tmp = 2.0 * (((b + a) * (b - a)) * math.sin((math.pi * (angle_m / 180.0))))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e-230)
		tmp = Float64(0.011111111111111112 * Float64(Float64(b * Float64(pi * Float64(b * angle_m))) - Float64((a ^ 2.0) * Float64(angle_m * pi))));
	elseif (Float64(angle_m / 180.0) <= 4e-155)
		tmp = Float64(0.011111111111111112 * Float64(Float64(angle_m * Float64((b ^ 2.0) * pi)) - Float64(a * Float64(a * Float64(angle_m * pi)))));
	elseif (Float64(angle_m / 180.0) <= 6e+69)
		tmp = Float64(Float64(angle_m * 0.011111111111111112) * Float64(Float64(b - a) * Float64(pi * Float64(b + a))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(b + a) * Float64(b - a)) * sin(Float64(pi * Float64(angle_m / 180.0)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if ((angle_m / 180.0) <= 2e-230)
		tmp = 0.011111111111111112 * ((b * (pi * (b * angle_m))) - ((a ^ 2.0) * (angle_m * pi)));
	elseif ((angle_m / 180.0) <= 4e-155)
		tmp = 0.011111111111111112 * ((angle_m * ((b ^ 2.0) * pi)) - (a * (a * (angle_m * pi))));
	elseif ((angle_m / 180.0) <= 6e+69)
		tmp = (angle_m * 0.011111111111111112) * ((b - a) * (pi * (b + a)));
	else
		tmp = 2.0 * (((b + a) * (b - a)) * sin((pi * (angle_m / 180.0))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e-230], N[(0.011111111111111112 * N[(N[(b * N[(Pi * N[(b * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[a, 2.0], $MachinePrecision] * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e-155], N[(0.011111111111111112 * N[(N[(angle$95$m * N[(N[Power[b, 2.0], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] - N[(a * N[(a * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 6e+69], N[(N[(angle$95$m * 0.011111111111111112), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2.00000000000000009e-230

   1. Initial program 44.5%

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

      1. associate-*l* 44.5%

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

      2. *-commutative 44.5%

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

      3. associate-*l* 44.5%

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

   3. Simplified 44.5%

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

   5. Taylor expanded in angle around 0 44.7%

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

      1. unpow2 44.5%

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

      2. unpow2 44.5%

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

      3. difference-of-squares 46.9%

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

   7. Applied egg-rr 46.3%

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

   8. Taylor expanded in b around 0 47.2%

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

      1. +-commutative 47.2%

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

      2. mul-1-neg 47.2%

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

      3. unsub-neg 47.2%

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

      4. *-commutative 47.2%

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

   10. Simplified 47.2%

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

   if 2.00000000000000009e-230 < (/.f64 angle #s(literal 180 binary64)) < 4.00000000000000006e-155

   1. Initial program 82.0%

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

      1. associate-*l* 82.0%

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

      2. *-commutative 82.0%

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

      3. associate-*l* 82.0%

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

   3. Simplified 82.0%

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

   5. Taylor expanded in angle around 0 81.8%

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

      1. unpow2 82.0%

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

      2. unpow2 82.0%

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

      3. difference-of-squares 82.0%

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

   7. Applied egg-rr 81.8%

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

   8. Taylor expanded in a around 0 99.2%

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

      1. +-commutative 99.2%

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

      2. *-commutative 99.2%

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

      3. +-commutative 99.2%

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

      4. associate-*r* 99.2%

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

      5. fma-define 99.2%

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

      6. distribute-rgt1-in 99.2%

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

      7. metadata-eval 99.2%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right) + \mathsf{fma}\left(angle \cdot \pi, \color{blue}{0} \cdot b, -1 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right) \cdot a\right) \]

      8. mul0-lft 99.2%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right) + \mathsf{fma}\left(angle \cdot \pi, \color{blue}{0}, -1 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right) \cdot a\right) \]

      9. mul-1-neg 99.2%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right) + \mathsf{fma}\left(angle \cdot \pi, 0, \color{blue}{-a \cdot \left(angle \cdot \pi\right)}\right) \cdot a\right) \]

      10. fma-neg 99.2%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right) + \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0 - a \cdot \left(angle \cdot \pi\right)\right)} \cdot a\right) \]

      11. mul0-rgt 99.2%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right) + \left(\color{blue}{0} - a \cdot \left(angle \cdot \pi\right)\right) \cdot a\right) \]

      12. neg-sub0 99.2%

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

      13. mul-1-neg 99.2%

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

   10. Simplified 99.2%

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

   if 4.00000000000000006e-155 < (/.f64 angle #s(literal 180 binary64)) < 5.99999999999999967e69

   1. Initial program 76.0%

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

      1. associate-*l* 76.0%

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

      2. *-commutative 76.0%

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

      3. associate-*l* 76.0%

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

   3. Simplified 76.0%

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

   5. Taylor expanded in angle around 0 70.4%

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

      1. unpow2 76.0%

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

      2. unpow2 76.0%

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

      3. difference-of-squares 80.2%

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

   7. Applied egg-rr 74.6%

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

   8. Taylor expanded in angle around 0 74.6%

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

      1. associate-*r* 74.7%

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

      2. +-commutative 74.7%

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

      3. *-commutative 74.7%

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

   10. Simplified 74.7%

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

   11. Taylor expanded in angle around 0 74.6%

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

       1. associate-*r* 74.7%

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

       2. *-commutative 74.7%

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

       3. associate-*r* 74.7%

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

       4. +-commutative 74.7%

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

   13. Simplified 74.7%

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

   if 5.99999999999999967e69 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 40.6%

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

      1. associate-*l* 40.6%

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

      2. associate-*l* 40.6%

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

   3. Simplified 40.6%

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

      1. unpow2 40.6%

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

      2. unpow2 40.6%

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

      3. difference-of-squares 42.5%

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

   6. Applied egg-rr 42.5%

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

   7. Taylor expanded in angle around 0 51.0%

      \[\leadsto 2 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{1}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{-230}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b \cdot \left(\pi \cdot \left(b \cdot angle\right)\right) - {a}^{2} \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 4 \cdot 10^{-155}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right) - a \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 6 \cdot 10^{+69}:\\ \;\;\;\;\left(angle \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 53.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := angle\_m \cdot \left(\pi \cdot 0.011111111111111112\right)\\ t_1 := 0.011111111111111112 \cdot \left(b \cdot \left(\pi \cdot \left(b \cdot angle\_m\right)\right) - {a}^{2} \cdot \left(angle\_m \cdot \pi\right)\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 3.1 \cdot 10^{-227}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;angle\_m \leq 2.6 \cdot 10^{-151}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left({b}^{2} \cdot \pi\right) - a \cdot \left(a \cdot \left(angle\_m \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;angle\_m \leq 9.2 \cdot 10^{-114}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot \left(b - a\right)\right) \cdot t\_0 + t\_0 \cdot \left(a \cdot \left(b - a\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* angle_m (* PI 0.011111111111111112)))
        (t_1
         (*
          0.011111111111111112
          (- (* b (* PI (* b angle_m))) (* (pow a 2.0) (* angle_m PI))))))
   (*
    angle_s
    (if (<= angle_m 3.1e-227)
      t_1
      (if (<= angle_m 2.6e-151)
        (*
         0.011111111111111112
         (- (* angle_m (* (pow b 2.0) PI)) (* a (* a (* angle_m PI)))))
        (if (<= angle_m 9.2e-114)
          t_1
          (+ (* (* b (- b a)) t_0) (* t_0 (* a (- b a))))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = angle_m * (((double) M_PI) * 0.011111111111111112);
	double t_1 = 0.011111111111111112 * ((b * (((double) M_PI) * (b * angle_m))) - (pow(a, 2.0) * (angle_m * ((double) M_PI))));
	double tmp;
	if (angle_m <= 3.1e-227) {
		tmp = t_1;
	} else if (angle_m <= 2.6e-151) {
		tmp = 0.011111111111111112 * ((angle_m * (pow(b, 2.0) * ((double) M_PI))) - (a * (a * (angle_m * ((double) M_PI)))));
	} else if (angle_m <= 9.2e-114) {
		tmp = t_1;
	} else {
		tmp = ((b * (b - a)) * t_0) + (t_0 * (a * (b - a)));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = angle_m * (Math.PI * 0.011111111111111112);
	double t_1 = 0.011111111111111112 * ((b * (Math.PI * (b * angle_m))) - (Math.pow(a, 2.0) * (angle_m * Math.PI)));
	double tmp;
	if (angle_m <= 3.1e-227) {
		tmp = t_1;
	} else if (angle_m <= 2.6e-151) {
		tmp = 0.011111111111111112 * ((angle_m * (Math.pow(b, 2.0) * Math.PI)) - (a * (a * (angle_m * Math.PI))));
	} else if (angle_m <= 9.2e-114) {
		tmp = t_1;
	} else {
		tmp = ((b * (b - a)) * t_0) + (t_0 * (a * (b - a)));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = angle_m * (math.pi * 0.011111111111111112)
	t_1 = 0.011111111111111112 * ((b * (math.pi * (b * angle_m))) - (math.pow(a, 2.0) * (angle_m * math.pi)))
	tmp = 0
	if angle_m <= 3.1e-227:
		tmp = t_1
	elif angle_m <= 2.6e-151:
		tmp = 0.011111111111111112 * ((angle_m * (math.pow(b, 2.0) * math.pi)) - (a * (a * (angle_m * math.pi))))
	elif angle_m <= 9.2e-114:
		tmp = t_1
	else:
		tmp = ((b * (b - a)) * t_0) + (t_0 * (a * (b - a)))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(angle_m * Float64(pi * 0.011111111111111112))
	t_1 = Float64(0.011111111111111112 * Float64(Float64(b * Float64(pi * Float64(b * angle_m))) - Float64((a ^ 2.0) * Float64(angle_m * pi))))
	tmp = 0.0
	if (angle_m <= 3.1e-227)
		tmp = t_1;
	elseif (angle_m <= 2.6e-151)
		tmp = Float64(0.011111111111111112 * Float64(Float64(angle_m * Float64((b ^ 2.0) * pi)) - Float64(a * Float64(a * Float64(angle_m * pi)))));
	elseif (angle_m <= 9.2e-114)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(b * Float64(b - a)) * t_0) + Float64(t_0 * Float64(a * Float64(b - a))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = angle_m * (pi * 0.011111111111111112);
	t_1 = 0.011111111111111112 * ((b * (pi * (b * angle_m))) - ((a ^ 2.0) * (angle_m * pi)));
	tmp = 0.0;
	if (angle_m <= 3.1e-227)
		tmp = t_1;
	elseif (angle_m <= 2.6e-151)
		tmp = 0.011111111111111112 * ((angle_m * ((b ^ 2.0) * pi)) - (a * (a * (angle_m * pi))));
	elseif (angle_m <= 9.2e-114)
		tmp = t_1;
	else
		tmp = ((b * (b - a)) * t_0) + (t_0 * (a * (b - a)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(angle$95$m * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.011111111111111112 * N[(N[(b * N[(Pi * N[(b * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[a, 2.0], $MachinePrecision] * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[angle$95$m, 3.1e-227], t$95$1, If[LessEqual[angle$95$m, 2.6e-151], N[(0.011111111111111112 * N[(N[(angle$95$m * N[(N[Power[b, 2.0], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] - N[(a * N[(a * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle$95$m, 9.2e-114], t$95$1, N[(N[(N[(b * N[(b - a), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(t$95$0 * N[(a * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if angle < 3.09999999999999979e-227 or 2.6e-151 < angle < 9.1999999999999997e-114

   1. Initial program 44.4%

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

      1. associate-*l* 44.4%

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

      2. *-commutative 44.4%

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

      3. associate-*l* 44.4%

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

   3. Simplified 44.4%

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

   5. Taylor expanded in angle around 0 44.5%

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

      1. unpow2 44.4%

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

      2. unpow2 44.4%

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

      3. difference-of-squares 47.4%

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

   7. Applied egg-rr 46.9%

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

   8. Taylor expanded in b around 0 47.7%

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

      1. +-commutative 47.7%

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

      2. mul-1-neg 47.7%

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

      3. unsub-neg 47.7%

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

      4. *-commutative 47.7%

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

   10. Simplified 47.7%

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

   if 3.09999999999999979e-227 < angle < 2.6e-151

   1. Initial program 82.0%

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

      1. associate-*l* 82.0%

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

      2. *-commutative 82.0%

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

      3. associate-*l* 82.0%

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

   3. Simplified 82.0%

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

   5. Taylor expanded in angle around 0 81.8%

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

      1. unpow2 82.0%

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

      2. unpow2 82.0%

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

      3. difference-of-squares 82.0%

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

   7. Applied egg-rr 81.8%

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

   8. Taylor expanded in a around 0 99.2%

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

      1. +-commutative 99.2%

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

      2. *-commutative 99.2%

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

      3. +-commutative 99.2%

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

      4. associate-*r* 99.2%

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

      5. fma-define 99.2%

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

      6. distribute-rgt1-in 99.2%

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

      7. metadata-eval 99.2%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right) + \mathsf{fma}\left(angle \cdot \pi, \color{blue}{0} \cdot b, -1 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right) \cdot a\right) \]

      8. mul0-lft 99.2%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right) + \mathsf{fma}\left(angle \cdot \pi, \color{blue}{0}, -1 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right) \cdot a\right) \]

      9. mul-1-neg 99.2%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right) + \mathsf{fma}\left(angle \cdot \pi, 0, \color{blue}{-a \cdot \left(angle \cdot \pi\right)}\right) \cdot a\right) \]

      10. fma-neg 99.2%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right) + \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0 - a \cdot \left(angle \cdot \pi\right)\right)} \cdot a\right) \]

      11. mul0-rgt 99.2%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right) + \left(\color{blue}{0} - a \cdot \left(angle \cdot \pi\right)\right) \cdot a\right) \]

      12. neg-sub0 99.2%

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

      13. mul-1-neg 99.2%

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

   10. Simplified 99.2%

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

   if 9.1999999999999997e-114 < angle

   1. Initial program 58.6%

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

      1. associate-*l* 58.6%

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

      2. *-commutative 58.6%

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

      3. associate-*l* 58.6%

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

   3. Simplified 58.6%

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

   5. Taylor expanded in angle around 0 50.2%

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

      1. unpow2 58.6%

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

      2. unpow2 58.6%

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

      3. difference-of-squares 60.7%

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

   7. Applied egg-rr 53.3%

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

   8. Taylor expanded in angle around 0 53.3%

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

      1. associate-*r* 53.4%

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

      2. +-commutative 53.4%

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

      3. *-commutative 53.4%

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

   10. Simplified 53.4%

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

       1. associate-*r* 53.4%

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

       2. distribute-lft-in 50.3%

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

       3. distribute-rgt-in 44.0%

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

       4. *-commutative 44.0%

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

       5. *-commutative 44.0%

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

       6. associate-*l* 44.0%

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

       7. *-commutative 44.0%

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

       8. *-commutative 44.0%

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

       9. associate-*l* 44.0%

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

   12. Applied egg-rr 44.0%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 3.1 \cdot 10^{-227}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b \cdot \left(\pi \cdot \left(b \cdot angle\right)\right) - {a}^{2} \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{elif}\;angle \leq 2.6 \cdot 10^{-151}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right) - a \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;angle \leq 9.2 \cdot 10^{-114}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b \cdot \left(\pi \cdot \left(b \cdot angle\right)\right) - {a}^{2} \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot \left(b - a\right)\right) \cdot \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) + \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(a \cdot \left(b - a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 57.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 1.25 \cdot 10^{-152}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left({b}^{2} \cdot \pi\right) - a \cdot \left(a \cdot \left(angle\_m \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle\_m \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 1.25e-152)
    (*
     0.011111111111111112
     (- (* angle_m (* (pow b 2.0) PI)) (* a (* a (* angle_m PI)))))
    (* (* angle_m 0.011111111111111112) (* (- b a) (* PI (+ b a)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 1.25e-152) {
		tmp = 0.011111111111111112 * ((angle_m * (pow(b, 2.0) * ((double) M_PI))) - (a * (a * (angle_m * ((double) M_PI)))));
	} else {
		tmp = (angle_m * 0.011111111111111112) * ((b - a) * (((double) M_PI) * (b + a)));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 1.25e-152) {
		tmp = 0.011111111111111112 * ((angle_m * (Math.pow(b, 2.0) * Math.PI)) - (a * (a * (angle_m * Math.PI))));
	} else {
		tmp = (angle_m * 0.011111111111111112) * ((b - a) * (Math.PI * (b + a)));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if angle_m <= 1.25e-152:
		tmp = 0.011111111111111112 * ((angle_m * (math.pow(b, 2.0) * math.pi)) - (a * (a * (angle_m * math.pi))))
	else:
		tmp = (angle_m * 0.011111111111111112) * ((b - a) * (math.pi * (b + a)))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (angle_m <= 1.25e-152)
		tmp = Float64(0.011111111111111112 * Float64(Float64(angle_m * Float64((b ^ 2.0) * pi)) - Float64(a * Float64(a * Float64(angle_m * pi)))));
	else
		tmp = Float64(Float64(angle_m * 0.011111111111111112) * Float64(Float64(b - a) * Float64(pi * Float64(b + a))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (angle_m <= 1.25e-152)
		tmp = 0.011111111111111112 * ((angle_m * ((b ^ 2.0) * pi)) - (a * (a * (angle_m * pi))));
	else
		tmp = (angle_m * 0.011111111111111112) * ((b - a) * (pi * (b + a)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 1.25e-152], N[(0.011111111111111112 * N[(N[(angle$95$m * N[(N[Power[b, 2.0], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] - N[(a * N[(a * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(angle$95$m * 0.011111111111111112), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 1.2499999999999999e-152

   1. Initial program 48.4%

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

      1. associate-*l* 48.4%

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

      2. *-commutative 48.4%

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

      3. associate-*l* 48.4%

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

   3. Simplified 48.4%

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

   5. Taylor expanded in angle around 0 48.5%

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

      1. unpow2 48.4%

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

      2. unpow2 48.4%

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

      3. difference-of-squares 50.5%

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

   7. Applied egg-rr 50.0%

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

   8. Taylor expanded in a around 0 49.6%

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

      1. +-commutative 49.6%

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

      2. *-commutative 49.6%

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

      3. +-commutative 49.6%

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

      4. associate-*r* 49.6%

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

      5. fma-define 49.6%

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

      6. distribute-rgt1-in 49.6%

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

      7. metadata-eval 49.6%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right) + \mathsf{fma}\left(angle \cdot \pi, \color{blue}{0} \cdot b, -1 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right) \cdot a\right) \]

      8. mul0-lft 49.6%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right) + \mathsf{fma}\left(angle \cdot \pi, \color{blue}{0}, -1 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right) \cdot a\right) \]

      9. mul-1-neg 49.6%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right) + \mathsf{fma}\left(angle \cdot \pi, 0, \color{blue}{-a \cdot \left(angle \cdot \pi\right)}\right) \cdot a\right) \]

      10. fma-neg 49.6%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right) + \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0 - a \cdot \left(angle \cdot \pi\right)\right)} \cdot a\right) \]

      11. mul0-rgt 49.6%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right) + \left(\color{blue}{0} - a \cdot \left(angle \cdot \pi\right)\right) \cdot a\right) \]

      12. neg-sub0 49.6%

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

      13. mul-1-neg 49.6%

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

   10. Simplified 49.6%

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

   if 1.2499999999999999e-152 < angle

   1. Initial program 57.8%

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

      1. associate-*l* 57.8%

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

      2. *-commutative 57.8%

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

      3. associate-*l* 57.8%

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

   3. Simplified 57.8%

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

   5. Taylor expanded in angle around 0 49.7%

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

      1. unpow2 57.8%

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

      2. unpow2 57.8%

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

      3. difference-of-squares 60.8%

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

   7. Applied egg-rr 53.7%

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

   8. Taylor expanded in angle around 0 53.7%

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

      1. associate-*r* 53.8%

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

      2. +-commutative 53.8%

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

      3. *-commutative 53.8%

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

   10. Simplified 53.8%

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

   11. Taylor expanded in angle around 0 53.7%

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

       1. associate-*r* 53.8%

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

       2. *-commutative 53.8%

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

       3. associate-*r* 53.8%

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

       4. +-commutative 53.8%

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

   13. Simplified 53.8%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 1.25 \cdot 10^{-152}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right) - a \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 49.3% accurate, 18.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := angle\_m \cdot \left(\pi \cdot 0.011111111111111112\right)\\ angle\_s \cdot \left(\left(b \cdot \left(b - a\right)\right) \cdot t\_0 + t\_0 \cdot \left(a \cdot \left(b - a\right)\right)\right) \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* angle_m (* PI 0.011111111111111112))))
   (* angle_s (+ (* (* b (- b a)) t_0) (* t_0 (* a (- b a)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = angle_m * (((double) M_PI) * 0.011111111111111112);
	return angle_s * (((b * (b - a)) * t_0) + (t_0 * (a * (b - a))));
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = angle_m * (Math.PI * 0.011111111111111112);
	return angle_s * (((b * (b - a)) * t_0) + (t_0 * (a * (b - a))));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = angle_m * (math.pi * 0.011111111111111112)
	return angle_s * (((b * (b - a)) * t_0) + (t_0 * (a * (b - a))))

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(angle_m * Float64(pi * 0.011111111111111112))
	return Float64(angle_s * Float64(Float64(Float64(b * Float64(b - a)) * t_0) + Float64(t_0 * Float64(a * Float64(b - a)))))
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	t_0 = angle_m * (pi * 0.011111111111111112);
	tmp = angle_s * (((b * (b - a)) * t_0) + (t_0 * (a * (b - a))));
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(angle$95$m * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * N[(N[(N[(b * N[(b - a), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(t$95$0 * N[(a * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 52.1%

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

   1. associate-*l* 52.1%

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

   2. *-commutative 52.1%

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

   3. associate-*l* 52.1%

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

3. Simplified 52.1%

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

5. Taylor expanded in angle around 0 49.0%

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

   1. unpow2 52.1%

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

   2. unpow2 52.1%

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

   3. difference-of-squares 54.6%

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

7. Applied egg-rr 51.5%

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

8. Taylor expanded in angle around 0 51.5%

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

   1. associate-*r* 51.5%

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

   2. +-commutative 51.5%

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

   3. *-commutative 51.5%

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

10. Simplified 51.5%

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

    1. associate-*r* 51.5%

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

    2. distribute-lft-in 47.5%

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

    3. distribute-rgt-in 45.1%

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

    4. *-commutative 45.1%

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

    5. *-commutative 45.1%

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

    6. associate-*l* 45.2%

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

    7. *-commutative 45.2%

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

    8. *-commutative 45.2%

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

    9. associate-*l* 45.2%

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

12. Applied egg-rr 45.2%

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

13. Final simplification 45.2%

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

Alternative 18: 54.5% accurate, 32.2× speedup

Math

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

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (* angle_s (* (* angle_m 0.011111111111111112) (* (- b a) (* PI (+ b a))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * ((angle_m * 0.011111111111111112) * ((b - a) * (((double) M_PI) * (b + a))));
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * ((angle_m * 0.011111111111111112) * ((b - a) * (Math.PI * (b + a))));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * ((angle_m * 0.011111111111111112) * ((b - a) * (math.pi * (b + a))))

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(Float64(angle_m * 0.011111111111111112) * Float64(Float64(b - a) * Float64(pi * Float64(b + a)))))
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * ((angle_m * 0.011111111111111112) * ((b - a) * (pi * (b + a))));
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(N[(angle$95$m * 0.011111111111111112), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.1%

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

   1. associate-*l* 52.1%

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

   2. *-commutative 52.1%

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

   3. associate-*l* 52.1%

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

3. Simplified 52.1%

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

5. Taylor expanded in angle around 0 49.0%

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

   1. unpow2 52.1%

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

   2. unpow2 52.1%

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

   3. difference-of-squares 54.6%

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

7. Applied egg-rr 51.5%

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

8. Taylor expanded in angle around 0 51.5%

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

   1. associate-*r* 51.5%

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

   2. +-commutative 51.5%

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

   3. *-commutative 51.5%

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

10. Simplified 51.5%

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

11. Taylor expanded in angle around 0 51.5%

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

    1. associate-*r* 51.5%

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

    2. *-commutative 51.5%

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

    3. associate-*r* 51.5%

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

    4. +-commutative 51.5%

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

13. Simplified 51.5%

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

14. Final simplification 51.5%

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

Alternative 19: 54.5% accurate, 32.2× speedup

Math

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

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (* angle_s (* (* angle_m 0.011111111111111112) (* PI (* (+ b a) (- b a))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * ((angle_m * 0.011111111111111112) * (((double) M_PI) * ((b + a) * (b - a))));
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * ((angle_m * 0.011111111111111112) * (Math.PI * ((b + a) * (b - a))));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * ((angle_m * 0.011111111111111112) * (math.pi * ((b + a) * (b - a))))

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(Float64(angle_m * 0.011111111111111112) * Float64(pi * Float64(Float64(b + a) * Float64(b - a)))))
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * ((angle_m * 0.011111111111111112) * (pi * ((b + a) * (b - a))));
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(N[(angle$95$m * 0.011111111111111112), $MachinePrecision] * N[(Pi * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.1%

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

   1. associate-*l* 52.1%

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

   2. *-commutative 52.1%

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

   3. associate-*l* 52.1%

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

3. Simplified 52.1%

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

5. Taylor expanded in angle around 0 49.0%

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

   1. unpow2 52.1%

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

   2. unpow2 52.1%

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

   3. difference-of-squares 54.6%

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

7. Applied egg-rr 51.5%

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

8. Taylor expanded in angle around 0 51.5%

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

   1. associate-*r* 51.5%

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

   2. +-commutative 51.5%

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

   3. *-commutative 51.5%

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

10. Simplified 51.5%

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

11. Final simplification 51.5%

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

Alternative 20: 54.5% accurate, 32.2× speedup

Math

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

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (* angle_s (* 0.011111111111111112 (* angle_m (* PI (* (+ b a) (- b a)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * (angle_m * (((double) M_PI) * ((b + a) * (b - a)))));
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * (angle_m * (Math.PI * ((b + a) * (b - a)))));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * (0.011111111111111112 * (angle_m * (math.pi * ((b + a) * (b - a)))))

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(Float64(b + a) * Float64(b - a))))))
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * (0.011111111111111112 * (angle_m * (pi * ((b + a) * (b - a)))));
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.1%

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

   1. associate-*l* 52.1%

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

   2. *-commutative 52.1%

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

   3. associate-*l* 52.1%

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

3. Simplified 52.1%

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

5. Taylor expanded in angle around 0 49.0%

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

   1. unpow2 52.1%

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

   2. unpow2 52.1%

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

   3. difference-of-squares 54.6%

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

7. Applied egg-rr 51.5%

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

Reproduce

herbie shell --seed 2024096 
(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)))))