ab-angle->ABCF B

Percentage Accurate: 53.7% → 64.9%

Time: 35.4s

Alternatives: 14

Speedup: 32.2×

• 32.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.7% 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: 64.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle\_m}{180}\\ t_1 := \cos t\_0\\ t_2 := \pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\\ t_3 := \cos t\_2\\ t_4 := \sin t\_2\\ t_5 := \sqrt{{\left(\left({b\_m}^{2} - {a}^{2}\right) \cdot t\_4\right)}^{2}}\\ t_6 := 0.005555555555555556 \cdot \left(\pi \cdot angle\_m\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 10^{+49}:\\ \;\;\;\;2 \cdot \left(t\_1 \cdot \left(\left(b\_m - a\right) \cdot \left(t\_4 \cdot \left(b\_m + a\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+157}:\\ \;\;\;\;2 \cdot \left(t\_5 \cdot \cos t\_6\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{+262}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot -2, t\_3 \cdot t\_6, 0\right), 2 \cdot \left({b\_m}^{2} \cdot \left(t\_3 \cdot \sin \left(\left(angle\_m \cdot 0.005555555555555556\right) \cdot \sqrt[3]{{\pi}^{3}}\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+276}:\\ \;\;\;\;\left(\left(b\_m - a\right) \cdot \left(b\_m + a\right)\right) \cdot \left(2 \cdot \left(\sin t\_0 \cdot \cos \left(\frac{angle\_m}{180} \cdot {\left(\sqrt{\pi}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t\_1 \cdot t\_5\right)\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (* PI (/ angle_m 180.0)))
        (t_1 (cos t_0))
        (t_2 (* PI (* angle_m 0.005555555555555556)))
        (t_3 (cos t_2))
        (t_4 (sin t_2))
        (t_5 (sqrt (pow (* (- (pow b_m 2.0) (pow a 2.0)) t_4) 2.0)))
        (t_6 (* 0.005555555555555556 (* PI angle_m))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 1e+49)
      (* 2.0 (* t_1 (* (- b_m a) (* t_4 (+ b_m a)))))
      (if (<= (/ angle_m 180.0) 5e+157)
        (* 2.0 (* t_5 (cos t_6)))
        (if (<= (/ angle_m 180.0) 4e+262)
          (fma
           a
           (fma (* a -2.0) (* t_3 t_6) 0.0)
           (*
            2.0
            (*
             (pow b_m 2.0)
             (*
              t_3
              (sin
               (* (* angle_m 0.005555555555555556) (cbrt (pow PI 3.0))))))))
          (if (<= (/ angle_m 180.0) 2e+276)
            (*
             (* (- b_m a) (+ b_m a))
             (*
              2.0
              (* (sin t_0) (cos (* (/ angle_m 180.0) (pow (sqrt PI) 2.0))))))
            (* 2.0 (* t_1 t_5)))))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m / 180.0);
	double t_1 = cos(t_0);
	double t_2 = ((double) M_PI) * (angle_m * 0.005555555555555556);
	double t_3 = cos(t_2);
	double t_4 = sin(t_2);
	double t_5 = sqrt(pow(((pow(b_m, 2.0) - pow(a, 2.0)) * t_4), 2.0));
	double t_6 = 0.005555555555555556 * (((double) M_PI) * angle_m);
	double tmp;
	if ((angle_m / 180.0) <= 1e+49) {
		tmp = 2.0 * (t_1 * ((b_m - a) * (t_4 * (b_m + a))));
	} else if ((angle_m / 180.0) <= 5e+157) {
		tmp = 2.0 * (t_5 * cos(t_6));
	} else if ((angle_m / 180.0) <= 4e+262) {
		tmp = fma(a, fma((a * -2.0), (t_3 * t_6), 0.0), (2.0 * (pow(b_m, 2.0) * (t_3 * sin(((angle_m * 0.005555555555555556) * cbrt(pow(((double) M_PI), 3.0))))))));
	} else if ((angle_m / 180.0) <= 2e+276) {
		tmp = ((b_m - a) * (b_m + a)) * (2.0 * (sin(t_0) * cos(((angle_m / 180.0) * pow(sqrt(((double) M_PI)), 2.0)))));
	} else {
		tmp = 2.0 * (t_1 * t_5);
	}
	return angle_s * tmp;
}

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = Float64(pi * Float64(angle_m / 180.0))
	t_1 = cos(t_0)
	t_2 = Float64(pi * Float64(angle_m * 0.005555555555555556))
	t_3 = cos(t_2)
	t_4 = sin(t_2)
	t_5 = sqrt((Float64(Float64((b_m ^ 2.0) - (a ^ 2.0)) * t_4) ^ 2.0))
	t_6 = Float64(0.005555555555555556 * Float64(pi * angle_m))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e+49)
		tmp = Float64(2.0 * Float64(t_1 * Float64(Float64(b_m - a) * Float64(t_4 * Float64(b_m + a)))));
	elseif (Float64(angle_m / 180.0) <= 5e+157)
		tmp = Float64(2.0 * Float64(t_5 * cos(t_6)));
	elseif (Float64(angle_m / 180.0) <= 4e+262)
		tmp = fma(a, fma(Float64(a * -2.0), Float64(t_3 * t_6), 0.0), Float64(2.0 * Float64((b_m ^ 2.0) * Float64(t_3 * sin(Float64(Float64(angle_m * 0.005555555555555556) * cbrt((pi ^ 3.0))))))));
	elseif (Float64(angle_m / 180.0) <= 2e+276)
		tmp = Float64(Float64(Float64(b_m - a) * Float64(b_m + a)) * Float64(2.0 * Float64(sin(t_0) * cos(Float64(Float64(angle_m / 180.0) * (sqrt(pi) ^ 2.0))))));
	else
		tmp = Float64(2.0 * Float64(t_1 * t_5));
	end
	return Float64(angle_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[Sin[t$95$2], $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[Power[N[(N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+49], N[(2.0 * N[(t$95$1 * N[(N[(b$95$m - a), $MachinePrecision] * N[(t$95$4 * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+157], N[(2.0 * N[(t$95$5 * N[Cos[t$95$6], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+262], N[(a * N[(N[(a * -2.0), $MachinePrecision] * N[(t$95$3 * t$95$6), $MachinePrecision] + 0.0), $MachinePrecision] + N[(2.0 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] * N[(t$95$3 * N[Sin[N[(N[(angle$95$m * 0.005555555555555556), $MachinePrecision] * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+276], N[(N[(N[(b$95$m - a), $MachinePrecision] * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[Sin[t$95$0], $MachinePrecision] * N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$1 * t$95$5), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 9.99999999999999946e48

   1. Initial program 58.7%

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

      1. associate-*l* 59.1%

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

      2. associate-*l* 59.1%

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

   3. Simplified 59.1%

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

      1. add-sqr-sqrt 33.1%

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

      2. sqrt-unprod 36.2%

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

      3. pow2 36.2%

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

      4. *-commutative 36.2%

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

      5. div-inv 36.2%

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

      6. metadata-eval 36.2%

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

   6. Applied egg-rr 36.2%

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

      1. unpow2 36.2%

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

      2. unpow2 36.2%

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

      3. difference-of-squares 37.3%

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

   8. Applied egg-rr 37.3%

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

      1. sqrt-pow1 62.8%

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

      2. metadata-eval 62.8%

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

      3. pow1 62.8%

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

      4. *-commutative 62.8%

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

      5. *-commutative 62.8%

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

      6. associate-*l* 74.6%

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

      7. associate-*r* 72.7%

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

      8. *-commutative 72.7%

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

      9. *-commutative 72.7%

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

      10. associate-*r* 74.6%

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

      11. *-commutative 74.6%

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

   10. Applied egg-rr 74.6%

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

   if 9.99999999999999946e48 < (/.f64 angle #s(literal 180 binary64)) < 4.99999999999999976e157

   1. Initial program 32.6%

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

      1. associate-*l* 32.6%

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

      2. associate-*l* 32.6%

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

   3. Simplified 32.6%

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

      1. add-sqr-sqrt 18.2%

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

      2. sqrt-unprod 35.2%

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

      3. pow2 35.2%

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

      4. *-commutative 35.2%

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

      5. div-inv 35.2%

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

      6. metadata-eval 35.2%

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

   6. Applied egg-rr 35.2%

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

   7. Taylor expanded in angle around inf 40.3%

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

   if 4.99999999999999976e157 < (/.f64 angle #s(literal 180 binary64)) < 4.0000000000000001e262

   1. Initial program 16.8%

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

      1. associate-*l* 16.8%

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

      2. *-commutative 16.8%

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

      3. associate-*l* 16.8%

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

   3. Simplified 16.8%

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

      1. unpow2 16.2%

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

      2. unpow2 16.2%

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

      3. difference-of-squares 16.2%

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

   6. Applied egg-rr 16.8%

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

   7. Taylor expanded in a around 0 17.6%

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

      1. +-commutative 17.6%

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

      2. fma-define 17.6%

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

   9. Simplified 23.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), 0\right), 2 \cdot \left(\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot {b}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. add-cbrt-cube 24.1%

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), 0\right), 2 \cdot \left(\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \sin \left(\color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}} \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot {b}^{2}\right)\right) \]

       2. pow3 24.1%

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), 0\right), 2 \cdot \left(\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \sin \left(\sqrt[3]{\color{blue}{{\pi}^{3}}} \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot {b}^{2}\right)\right) \]

   11. Applied egg-rr 24.1%

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), 0\right), 2 \cdot \left(\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \sin \left(\color{blue}{\sqrt[3]{{\pi}^{3}}} \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot {b}^{2}\right)\right) \]

   12. Taylor expanded in angle around 0 40.6%

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}, 0\right), 2 \cdot \left(\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \sin \left(\sqrt[3]{{\pi}^{3}} \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot {b}^{2}\right)\right) \]

   if 4.0000000000000001e262 < (/.f64 angle #s(literal 180 binary64)) < 2.0000000000000001e276

   1. Initial program 24.4%

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

      1. associate-*l* 24.4%

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

      2. *-commutative 24.4%

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

      3. associate-*l* 24.4%

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

   3. Simplified 24.4%

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

      1. unpow2 19.0%

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

      2. unpow2 19.0%

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

      3. difference-of-squares 35.7%

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

   6. Applied egg-rr 24.4%

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

      1. add-sqr-sqrt 39.4%

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

      2. pow2 39.4%

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

   8. Applied egg-rr 39.4%

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

   if 2.0000000000000001e276 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 19.0%

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

      1. associate-*l* 19.0%

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

      2. associate-*l* 19.0%

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

   3. Simplified 19.0%

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

      1. add-sqr-sqrt 17.6%

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

      2. sqrt-unprod 50.3%

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

      3. pow2 50.3%

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

      4. *-commutative 50.3%

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

      5. div-inv 50.3%

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

      6. metadata-eval 50.3%

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

   6. Applied egg-rr 50.3%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{+49}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(b + a\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+157}:\\ \;\;\;\;2 \cdot \left(\sqrt{{\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}} \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 4 \cdot 10^{+262}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot -2, \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right), 0\right), 2 \cdot \left({b}^{2} \cdot \left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \sqrt[3]{{\pi}^{3}}\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+276}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\frac{angle}{180} \cdot {\left(\sqrt{\pi}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \sqrt{{\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 65.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\\ t_1 := \pi \cdot \frac{angle\_m}{180}\\ t_2 := \cos t\_1\\ t_3 := \sin t\_0\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b\_m}^{2} - {a}^{2}\right)\right) \cdot \sin t\_1\right) \cdot t\_2 \leq 10^{+107}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot -2, \cos t\_0 \cdot t\_3, 0\right), 2 \cdot \left({b\_m}^{2} \cdot \left(\cos \left(\left(angle\_m \cdot 0.005555555555555556\right) \cdot {\left(\sqrt{\pi}\right)}^{2}\right) \cdot \sin \left(\left(angle\_m \cdot 0.005555555555555556\right) \cdot \sqrt[3]{{\pi}^{3}}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t\_2 \cdot \left(\mathsf{hypot}\left(b\_m, a\right) \cdot \left(t\_3 \cdot \mathsf{hypot}\left(b\_m, a\right)\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m 0.005555555555555556)))
        (t_1 (* PI (/ angle_m 180.0)))
        (t_2 (cos t_1))
        (t_3 (sin t_0)))
   (*
    angle_s
    (if (<= (* (* (* 2.0 (- (pow b_m 2.0) (pow a 2.0))) (sin t_1)) t_2) 1e+107)
      (fma
       a
       (fma (* a -2.0) (* (cos t_0) t_3) 0.0)
       (*
        2.0
        (*
         (pow b_m 2.0)
         (*
          (cos (* (* angle_m 0.005555555555555556) (pow (sqrt PI) 2.0)))
          (sin (* (* angle_m 0.005555555555555556) (cbrt (pow PI 3.0))))))))
      (* 2.0 (* t_2 (* (hypot b_m a) (* t_3 (hypot b_m a)))))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m * 0.005555555555555556);
	double t_1 = ((double) M_PI) * (angle_m / 180.0);
	double t_2 = cos(t_1);
	double t_3 = sin(t_0);
	double tmp;
	if ((((2.0 * (pow(b_m, 2.0) - pow(a, 2.0))) * sin(t_1)) * t_2) <= 1e+107) {
		tmp = fma(a, fma((a * -2.0), (cos(t_0) * t_3), 0.0), (2.0 * (pow(b_m, 2.0) * (cos(((angle_m * 0.005555555555555556) * pow(sqrt(((double) M_PI)), 2.0))) * sin(((angle_m * 0.005555555555555556) * cbrt(pow(((double) M_PI), 3.0))))))));
	} else {
		tmp = 2.0 * (t_2 * (hypot(b_m, a) * (t_3 * hypot(b_m, a))));
	}
	return angle_s * tmp;
}

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = Float64(pi * Float64(angle_m * 0.005555555555555556))
	t_1 = Float64(pi * Float64(angle_m / 180.0))
	t_2 = cos(t_1)
	t_3 = sin(t_0)
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((b_m ^ 2.0) - (a ^ 2.0))) * sin(t_1)) * t_2) <= 1e+107)
		tmp = fma(a, fma(Float64(a * -2.0), Float64(cos(t_0) * t_3), 0.0), Float64(2.0 * Float64((b_m ^ 2.0) * Float64(cos(Float64(Float64(angle_m * 0.005555555555555556) * (sqrt(pi) ^ 2.0))) * sin(Float64(Float64(angle_m * 0.005555555555555556) * cbrt((pi ^ 3.0))))))));
	else
		tmp = Float64(2.0 * Float64(t_2 * Float64(hypot(b_m, a) * Float64(t_3 * hypot(b_m, a)))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Sin[t$95$0], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(N[(N[(2.0 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], 1e+107], N[(a * N[(N[(a * -2.0), $MachinePrecision] * N[(N[Cos[t$95$0], $MachinePrecision] * t$95$3), $MachinePrecision] + 0.0), $MachinePrecision] + N[(2.0 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] * N[(N[Cos[N[(N[(angle$95$m * 0.005555555555555556), $MachinePrecision] * N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(angle$95$m * 0.005555555555555556), $MachinePrecision] * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$2 * N[(N[Sqrt[b$95$m ^ 2 + a ^ 2], $MachinePrecision] * N[(t$95$3 * N[Sqrt[b$95$m ^ 2 + a ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) < 9.9999999999999997e106

   1. Initial program 55.0%

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

      1. associate-*l* 55.0%

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

      2. *-commutative 55.0%

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

      3. associate-*l* 55.0%

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

   3. Simplified 55.0%

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

      1. unpow2 32.5%

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

      2. unpow2 32.5%

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

      3. difference-of-squares 32.5%

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

   6. Applied egg-rr 55.0%

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

   7. Taylor expanded in a around 0 60.2%

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

      1. +-commutative 60.2%

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

      2. fma-define 60.2%

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

   9. Simplified 59.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), 0\right), 2 \cdot \left(\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot {b}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. add-cbrt-cube 59.2%

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), 0\right), 2 \cdot \left(\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \sin \left(\color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}} \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot {b}^{2}\right)\right) \]

       2. pow3 59.2%

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), 0\right), 2 \cdot \left(\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \sin \left(\sqrt[3]{\color{blue}{{\pi}^{3}}} \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot {b}^{2}\right)\right) \]

   11. Applied egg-rr 59.2%

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), 0\right), 2 \cdot \left(\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \sin \left(\color{blue}{\sqrt[3]{{\pi}^{3}}} \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot {b}^{2}\right)\right) \]
   12. Step-by-step derivation

       1. add-sqr-sqrt 57.2%

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

       2. pow2 57.2%

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

   13. Applied egg-rr 59.5%

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), 0\right), 2 \cdot \left(\left(\cos \left(\color{blue}{{\left(\sqrt{\pi}\right)}^{2}} \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \sin \left(\sqrt[3]{{\pi}^{3}} \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot {b}^{2}\right)\right) \]

   if 9.9999999999999997e106 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64)))))

   1. Initial program 40.4%

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

      1. associate-*l* 41.6%

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

      2. associate-*l* 41.6%

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

   3. Simplified 41.6%

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

      1. add-sqr-sqrt 29.5%

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

      2. sqrt-unprod 38.6%

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

      3. pow2 38.6%

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

      4. *-commutative 38.6%

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

      5. div-inv 38.5%

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

      6. metadata-eval 38.5%

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

   6. Applied egg-rr 38.5%

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

      1. sqrt-pow1 44.6%

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

      2. metadata-eval 44.6%

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

      3. pow1 44.6%

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

      4. add-sqr-sqrt 27.2%

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

      5. associate-*r* 27.2%

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

   8. Applied egg-rr 47.2%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \leq 10^{+107}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot -2, \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0\right), 2 \cdot \left({b}^{2} \cdot \left(\cos \left(\left(angle \cdot 0.005555555555555556\right) \cdot {\left(\sqrt{\pi}\right)}^{2}\right) \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \sqrt[3]{{\pi}^{3}}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\mathsf{hypot}\left(b, a\right) \cdot \left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \mathsf{hypot}\left(b, a\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle\_m}{180}\\ t_1 := {b\_m}^{2} - {a}^{2}\\ t_2 := \sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\\ t_3 := \sqrt{{\left(t\_1 \cdot t\_2\right)}^{2}}\\ t_4 := \cos t\_0\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 10^{+49}:\\ \;\;\;\;2 \cdot \left(t\_4 \cdot \left(\left(b\_m - a\right) \cdot \left(t\_2 \cdot \left(b\_m + a\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 10^{+160}:\\ \;\;\;\;2 \cdot \left(t\_3 \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\_m\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+253}:\\ \;\;\;\;\sqrt{{t\_1}^{2}} \cdot \left(2 \cdot \left(\sin t\_0 \cdot t\_4\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 10^{+281}:\\ \;\;\;\;\left(\left(b\_m - a\right) \cdot \left(b\_m + a\right)\right) \cdot \left(2 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(t\_2\right)\right) \cdot \cos \left(\frac{angle\_m}{180} \cdot \sqrt[3]{{\pi}^{3}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t\_4 \cdot t\_3\right)\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (* PI (/ angle_m 180.0)))
        (t_1 (- (pow b_m 2.0) (pow a 2.0)))
        (t_2 (sin (* PI (* angle_m 0.005555555555555556))))
        (t_3 (sqrt (pow (* t_1 t_2) 2.0)))
        (t_4 (cos t_0)))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 1e+49)
      (* 2.0 (* t_4 (* (- b_m a) (* t_2 (+ b_m a)))))
      (if (<= (/ angle_m 180.0) 1e+160)
        (* 2.0 (* t_3 (cos (* 0.005555555555555556 (* PI angle_m)))))
        (if (<= (/ angle_m 180.0) 2e+253)
          (* (sqrt (pow t_1 2.0)) (* 2.0 (* (sin t_0) t_4)))
          (if (<= (/ angle_m 180.0) 1e+281)
            (*
             (* (- b_m a) (+ b_m a))
             (*
              2.0
              (*
               (log1p (expm1 t_2))
               (cos (* (/ angle_m 180.0) (cbrt (pow PI 3.0)))))))
            (* 2.0 (* t_4 t_3)))))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m / 180.0);
	double t_1 = pow(b_m, 2.0) - pow(a, 2.0);
	double t_2 = sin((((double) M_PI) * (angle_m * 0.005555555555555556)));
	double t_3 = sqrt(pow((t_1 * t_2), 2.0));
	double t_4 = cos(t_0);
	double tmp;
	if ((angle_m / 180.0) <= 1e+49) {
		tmp = 2.0 * (t_4 * ((b_m - a) * (t_2 * (b_m + a))));
	} else if ((angle_m / 180.0) <= 1e+160) {
		tmp = 2.0 * (t_3 * cos((0.005555555555555556 * (((double) M_PI) * angle_m))));
	} else if ((angle_m / 180.0) <= 2e+253) {
		tmp = sqrt(pow(t_1, 2.0)) * (2.0 * (sin(t_0) * t_4));
	} else if ((angle_m / 180.0) <= 1e+281) {
		tmp = ((b_m - a) * (b_m + a)) * (2.0 * (log1p(expm1(t_2)) * cos(((angle_m / 180.0) * cbrt(pow(((double) M_PI), 3.0))))));
	} else {
		tmp = 2.0 * (t_4 * t_3);
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = Math.PI * (angle_m / 180.0);
	double t_1 = Math.pow(b_m, 2.0) - Math.pow(a, 2.0);
	double t_2 = Math.sin((Math.PI * (angle_m * 0.005555555555555556)));
	double t_3 = Math.sqrt(Math.pow((t_1 * t_2), 2.0));
	double t_4 = Math.cos(t_0);
	double tmp;
	if ((angle_m / 180.0) <= 1e+49) {
		tmp = 2.0 * (t_4 * ((b_m - a) * (t_2 * (b_m + a))));
	} else if ((angle_m / 180.0) <= 1e+160) {
		tmp = 2.0 * (t_3 * Math.cos((0.005555555555555556 * (Math.PI * angle_m))));
	} else if ((angle_m / 180.0) <= 2e+253) {
		tmp = Math.sqrt(Math.pow(t_1, 2.0)) * (2.0 * (Math.sin(t_0) * t_4));
	} else if ((angle_m / 180.0) <= 1e+281) {
		tmp = ((b_m - a) * (b_m + a)) * (2.0 * (Math.log1p(Math.expm1(t_2)) * Math.cos(((angle_m / 180.0) * Math.cbrt(Math.pow(Math.PI, 3.0))))));
	} else {
		tmp = 2.0 * (t_4 * t_3);
	}
	return angle_s * tmp;
}

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = Float64(pi * Float64(angle_m / 180.0))
	t_1 = Float64((b_m ^ 2.0) - (a ^ 2.0))
	t_2 = sin(Float64(pi * Float64(angle_m * 0.005555555555555556)))
	t_3 = sqrt((Float64(t_1 * t_2) ^ 2.0))
	t_4 = cos(t_0)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e+49)
		tmp = Float64(2.0 * Float64(t_4 * Float64(Float64(b_m - a) * Float64(t_2 * Float64(b_m + a)))));
	elseif (Float64(angle_m / 180.0) <= 1e+160)
		tmp = Float64(2.0 * Float64(t_3 * cos(Float64(0.005555555555555556 * Float64(pi * angle_m)))));
	elseif (Float64(angle_m / 180.0) <= 2e+253)
		tmp = Float64(sqrt((t_1 ^ 2.0)) * Float64(2.0 * Float64(sin(t_0) * t_4)));
	elseif (Float64(angle_m / 180.0) <= 1e+281)
		tmp = Float64(Float64(Float64(b_m - a) * Float64(b_m + a)) * Float64(2.0 * Float64(log1p(expm1(t_2)) * cos(Float64(Float64(angle_m / 180.0) * cbrt((pi ^ 3.0)))))));
	else
		tmp = Float64(2.0 * Float64(t_4 * t_3));
	end
	return Float64(angle_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[Power[N[(t$95$1 * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Cos[t$95$0], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+49], N[(2.0 * N[(t$95$4 * N[(N[(b$95$m - a), $MachinePrecision] * N[(t$95$2 * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+160], N[(2.0 * N[(t$95$3 * N[Cos[N[(0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+253], N[(N[Sqrt[N[Power[t$95$1, 2.0], $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(N[Sin[t$95$0], $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+281], N[(N[(N[(b$95$m - a), $MachinePrecision] * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[Log[1 + N[(Exp[t$95$2] - 1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$4 * t$95$3), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 9.99999999999999946e48

   1. Initial program 58.7%

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

      1. associate-*l* 59.1%

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

      2. associate-*l* 59.1%

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

   3. Simplified 59.1%

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

      1. add-sqr-sqrt 33.1%

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

      2. sqrt-unprod 36.2%

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

      3. pow2 36.2%

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

      4. *-commutative 36.2%

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

      5. div-inv 36.2%

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

      6. metadata-eval 36.2%

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

   6. Applied egg-rr 36.2%

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

      1. unpow2 36.2%

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

      2. unpow2 36.2%

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

      3. difference-of-squares 37.3%

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

   8. Applied egg-rr 37.3%

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

      1. sqrt-pow1 62.8%

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

      2. metadata-eval 62.8%

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

      3. pow1 62.8%

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

      4. *-commutative 62.8%

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

      5. *-commutative 62.8%

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

      6. associate-*l* 74.6%

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

      7. associate-*r* 72.7%

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

      8. *-commutative 72.7%

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

      9. *-commutative 72.7%

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

      10. associate-*r* 74.6%

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

      11. *-commutative 74.6%

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

   10. Applied egg-rr 74.6%

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

   if 9.99999999999999946e48 < (/.f64 angle #s(literal 180 binary64)) < 1.00000000000000001e160

   1. Initial program 31.4%

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

      1. associate-*l* 31.4%

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

      2. associate-*l* 31.4%

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

   3. Simplified 31.4%

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

      1. add-sqr-sqrt 17.6%

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

      2. sqrt-unprod 34.0%

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

      3. pow2 34.0%

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

      4. *-commutative 34.0%

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

      5. div-inv 34.0%

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

      6. metadata-eval 34.0%

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

   6. Applied egg-rr 34.0%

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

   7. Taylor expanded in angle around inf 39.4%

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

   if 1.00000000000000001e160 < (/.f64 angle #s(literal 180 binary64)) < 1.9999999999999999e253

   1. Initial program 14.1%

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

      1. associate-*l* 14.1%

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

      2. *-commutative 14.1%

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

      3. associate-*l* 14.1%

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

   3. Simplified 14.1%

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

      1. add-sqr-sqrt 13.0%

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

      2. sqrt-unprod 46.1%

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

      3. pow2 46.1%

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

   6. Applied egg-rr 46.1%

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

   if 1.9999999999999999e253 < (/.f64 angle #s(literal 180 binary64)) < 1e281

   1. Initial program 30.8%

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

      1. associate-*l* 30.8%

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

      2. *-commutative 30.8%

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

      3. associate-*l* 30.8%

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

   3. Simplified 30.8%

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

      1. unpow2 14.3%

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

      2. unpow2 14.3%

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

      3. difference-of-squares 26.8%

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

   6. Applied egg-rr 30.8%

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

      1. log1p-expm1-u 30.8%

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

      2. div-inv 30.8%

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

      3. metadata-eval 30.8%

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

   8. Applied egg-rr 30.8%

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

      1. add-cbrt-cube 32.8%

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), 0\right), 2 \cdot \left(\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \sin \left(\color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}} \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot {b}^{2}\right)\right) \]

      2. pow3 32.8%

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), 0\right), 2 \cdot \left(\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \sin \left(\sqrt[3]{\color{blue}{{\pi}^{3}}} \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot {b}^{2}\right)\right) \]

   10. Applied egg-rr 42.6%

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

   if 1e281 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 19.0%

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

      1. associate-*l* 19.0%

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

      2. associate-*l* 19.0%

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

   3. Simplified 19.0%

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

      1. add-sqr-sqrt 17.6%

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

      2. sqrt-unprod 50.3%

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

      3. pow2 50.3%

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

      4. *-commutative 50.3%

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

      5. div-inv 50.3%

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

      6. metadata-eval 50.3%

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

   6. Applied egg-rr 50.3%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{+49}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(b + a\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+160}:\\ \;\;\;\;2 \cdot \left(\sqrt{{\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}} \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+253}:\\ \;\;\;\;\sqrt{{\left({b}^{2} - {a}^{2}\right)}^{2}} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+281}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(2 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \sqrt[3]{{\pi}^{3}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \sqrt{{\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle\_m}{180}\\ t_1 := \sin t\_0\\ t_2 := {b\_m}^{2} - {a}^{2}\\ t_3 := \sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\\ t_4 := \cos t\_0\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 10^{+49}:\\ \;\;\;\;2 \cdot \left(t\_4 \cdot \left(\left(b\_m - a\right) \cdot \left(t\_3 \cdot \left(b\_m + a\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 10^{+160}:\\ \;\;\;\;2 \cdot \left(\sqrt{{\left(t\_2 \cdot t\_3\right)}^{2}} \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\_m\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{+253}:\\ \;\;\;\;\sqrt{{t\_2}^{2}} \cdot \left(2 \cdot \left(t\_1 \cdot t\_4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b\_m - a\right) \cdot \left(b\_m + a\right)\right) \cdot \left(2 \cdot \left(t\_1 \cdot \cos \left(0.005555555555555556 \cdot \left(angle\_m \cdot \sqrt[3]{{\pi}^{3}}\right)\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (* PI (/ angle_m 180.0)))
        (t_1 (sin t_0))
        (t_2 (- (pow b_m 2.0) (pow a 2.0)))
        (t_3 (sin (* PI (* angle_m 0.005555555555555556))))
        (t_4 (cos t_0)))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 1e+49)
      (* 2.0 (* t_4 (* (- b_m a) (* t_3 (+ b_m a)))))
      (if (<= (/ angle_m 180.0) 1e+160)
        (*
         2.0
         (*
          (sqrt (pow (* t_2 t_3) 2.0))
          (cos (* 0.005555555555555556 (* PI angle_m)))))
        (if (<= (/ angle_m 180.0) 4e+253)
          (* (sqrt (pow t_2 2.0)) (* 2.0 (* t_1 t_4)))
          (*
           (* (- b_m a) (+ b_m a))
           (*
            2.0
            (*
             t_1
             (cos
              (*
               0.005555555555555556
               (* angle_m (cbrt (pow PI 3.0))))))))))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m / 180.0);
	double t_1 = sin(t_0);
	double t_2 = pow(b_m, 2.0) - pow(a, 2.0);
	double t_3 = sin((((double) M_PI) * (angle_m * 0.005555555555555556)));
	double t_4 = cos(t_0);
	double tmp;
	if ((angle_m / 180.0) <= 1e+49) {
		tmp = 2.0 * (t_4 * ((b_m - a) * (t_3 * (b_m + a))));
	} else if ((angle_m / 180.0) <= 1e+160) {
		tmp = 2.0 * (sqrt(pow((t_2 * t_3), 2.0)) * cos((0.005555555555555556 * (((double) M_PI) * angle_m))));
	} else if ((angle_m / 180.0) <= 4e+253) {
		tmp = sqrt(pow(t_2, 2.0)) * (2.0 * (t_1 * t_4));
	} else {
		tmp = ((b_m - a) * (b_m + a)) * (2.0 * (t_1 * cos((0.005555555555555556 * (angle_m * cbrt(pow(((double) M_PI), 3.0)))))));
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = Math.PI * (angle_m / 180.0);
	double t_1 = Math.sin(t_0);
	double t_2 = Math.pow(b_m, 2.0) - Math.pow(a, 2.0);
	double t_3 = Math.sin((Math.PI * (angle_m * 0.005555555555555556)));
	double t_4 = Math.cos(t_0);
	double tmp;
	if ((angle_m / 180.0) <= 1e+49) {
		tmp = 2.0 * (t_4 * ((b_m - a) * (t_3 * (b_m + a))));
	} else if ((angle_m / 180.0) <= 1e+160) {
		tmp = 2.0 * (Math.sqrt(Math.pow((t_2 * t_3), 2.0)) * Math.cos((0.005555555555555556 * (Math.PI * angle_m))));
	} else if ((angle_m / 180.0) <= 4e+253) {
		tmp = Math.sqrt(Math.pow(t_2, 2.0)) * (2.0 * (t_1 * t_4));
	} else {
		tmp = ((b_m - a) * (b_m + a)) * (2.0 * (t_1 * Math.cos((0.005555555555555556 * (angle_m * Math.cbrt(Math.pow(Math.PI, 3.0)))))));
	}
	return angle_s * tmp;
}

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = Float64(pi * Float64(angle_m / 180.0))
	t_1 = sin(t_0)
	t_2 = Float64((b_m ^ 2.0) - (a ^ 2.0))
	t_3 = sin(Float64(pi * Float64(angle_m * 0.005555555555555556)))
	t_4 = cos(t_0)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e+49)
		tmp = Float64(2.0 * Float64(t_4 * Float64(Float64(b_m - a) * Float64(t_3 * Float64(b_m + a)))));
	elseif (Float64(angle_m / 180.0) <= 1e+160)
		tmp = Float64(2.0 * Float64(sqrt((Float64(t_2 * t_3) ^ 2.0)) * cos(Float64(0.005555555555555556 * Float64(pi * angle_m)))));
	elseif (Float64(angle_m / 180.0) <= 4e+253)
		tmp = Float64(sqrt((t_2 ^ 2.0)) * Float64(2.0 * Float64(t_1 * t_4)));
	else
		tmp = Float64(Float64(Float64(b_m - a) * Float64(b_m + a)) * Float64(2.0 * Float64(t_1 * cos(Float64(0.005555555555555556 * Float64(angle_m * cbrt((pi ^ 3.0))))))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Cos[t$95$0], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+49], N[(2.0 * N[(t$95$4 * N[(N[(b$95$m - a), $MachinePrecision] * N[(t$95$3 * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+160], N[(2.0 * N[(N[Sqrt[N[Power[N[(t$95$2 * t$95$3), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+253], N[(N[Sqrt[N[Power[t$95$2, 2.0], $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(t$95$1 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b$95$m - a), $MachinePrecision] * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(t$95$1 * N[Cos[N[(0.005555555555555556 * N[(angle$95$m * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 9.99999999999999946e48

   1. Initial program 58.7%

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

      1. associate-*l* 59.1%

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

      2. associate-*l* 59.1%

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

   3. Simplified 59.1%

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

      1. add-sqr-sqrt 33.1%

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

      2. sqrt-unprod 36.2%

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

      3. pow2 36.2%

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

      4. *-commutative 36.2%

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

      5. div-inv 36.2%

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

      6. metadata-eval 36.2%

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

   6. Applied egg-rr 36.2%

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

      1. unpow2 36.2%

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

      2. unpow2 36.2%

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

      3. difference-of-squares 37.3%

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

   8. Applied egg-rr 37.3%

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

      1. sqrt-pow1 62.8%

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

      2. metadata-eval 62.8%

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

      3. pow1 62.8%

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

      4. *-commutative 62.8%

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

      5. *-commutative 62.8%

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

      6. associate-*l* 74.6%

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

      7. associate-*r* 72.7%

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

      8. *-commutative 72.7%

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

      9. *-commutative 72.7%

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

      10. associate-*r* 74.6%

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

      11. *-commutative 74.6%

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

   10. Applied egg-rr 74.6%

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

   if 9.99999999999999946e48 < (/.f64 angle #s(literal 180 binary64)) < 1.00000000000000001e160

   1. Initial program 31.4%

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

      1. associate-*l* 31.4%

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

      2. associate-*l* 31.4%

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

   3. Simplified 31.4%

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

      1. add-sqr-sqrt 17.6%

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

      2. sqrt-unprod 34.0%

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

      3. pow2 34.0%

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

      4. *-commutative 34.0%

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

      5. div-inv 34.0%

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

      6. metadata-eval 34.0%

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

   6. Applied egg-rr 34.0%

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

   7. Taylor expanded in angle around inf 39.4%

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

   if 1.00000000000000001e160 < (/.f64 angle #s(literal 180 binary64)) < 3.9999999999999997e253

   1. Initial program 14.1%

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

      1. associate-*l* 14.1%

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

      2. *-commutative 14.1%

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

      3. associate-*l* 14.1%

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

   3. Simplified 14.1%

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

      1. add-sqr-sqrt 13.0%

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

      2. sqrt-unprod 46.1%

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

      3. pow2 46.1%

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

   6. Applied egg-rr 46.1%

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

   if 3.9999999999999997e253 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 25.3%

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

      1. associate-*l* 25.3%

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

      2. *-commutative 25.3%

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

      3. associate-*l* 25.3%

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

   3. Simplified 25.3%

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

      1. unpow2 31.1%

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

      2. unpow2 31.1%

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

      3. difference-of-squares 37.7%

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

   6. Applied egg-rr 25.3%

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

   7. Taylor expanded in angle around inf 31.5%

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

      1. add-cbrt-cube 27.6%

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), 0\right), 2 \cdot \left(\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \sin \left(\color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}} \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot {b}^{2}\right)\right) \]

      2. pow3 27.6%

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), 0\right), 2 \cdot \left(\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \sin \left(\sqrt[3]{\color{blue}{{\pi}^{3}}} \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot {b}^{2}\right)\right) \]

   9. Applied egg-rr 46.7%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{+49}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(b + a\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+160}:\\ \;\;\;\;2 \cdot \left(\sqrt{{\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}} \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 4 \cdot 10^{+253}:\\ \;\;\;\;\sqrt{{\left({b}^{2} - {a}^{2}\right)}^{2}} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \sqrt[3]{{\pi}^{3}}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := {b\_m}^{2} - {a}^{2}\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+273}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left({b\_m}^{2} \cdot \pi\right) - a \cdot \left(a \cdot \left(\pi \cdot angle\_m\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+279}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \frac{angle\_m}{180}\right) \cdot \left(\left(\left(b\_m - a\right) \cdot \left(b\_m + a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\_m\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b\_m, \left(angle\_m \cdot 0.011111111111111112\right) \cdot \left(b\_m \cdot \pi\right), -0.011111111111111112 \cdot \left({a}^{2} \cdot \left(\pi \cdot angle\_m\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (- (pow b_m 2.0) (pow a 2.0))))
   (*
    angle_s
    (if (<= t_0 -5e+273)
      (*
       0.011111111111111112
       (- (* angle_m (* (pow b_m 2.0) PI)) (* a (* a (* PI angle_m)))))
      (if (<= t_0 5e+279)
        (*
         2.0
         (*
          (cos (* PI (/ angle_m 180.0)))
          (*
           (* (- b_m a) (+ b_m a))
           (sin (* 0.005555555555555556 (* PI angle_m))))))
        (fma
         b_m
         (* (* angle_m 0.011111111111111112) (* b_m PI))
         (* -0.011111111111111112 (* (pow a 2.0) (* PI angle_m)))))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = pow(b_m, 2.0) - pow(a, 2.0);
	double tmp;
	if (t_0 <= -5e+273) {
		tmp = 0.011111111111111112 * ((angle_m * (pow(b_m, 2.0) * ((double) M_PI))) - (a * (a * (((double) M_PI) * angle_m))));
	} else if (t_0 <= 5e+279) {
		tmp = 2.0 * (cos((((double) M_PI) * (angle_m / 180.0))) * (((b_m - a) * (b_m + a)) * sin((0.005555555555555556 * (((double) M_PI) * angle_m)))));
	} else {
		tmp = fma(b_m, ((angle_m * 0.011111111111111112) * (b_m * ((double) M_PI))), (-0.011111111111111112 * (pow(a, 2.0) * (((double) M_PI) * angle_m))));
	}
	return angle_s * tmp;
}

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = Float64((b_m ^ 2.0) - (a ^ 2.0))
	tmp = 0.0
	if (t_0 <= -5e+273)
		tmp = Float64(0.011111111111111112 * Float64(Float64(angle_m * Float64((b_m ^ 2.0) * pi)) - Float64(a * Float64(a * Float64(pi * angle_m)))));
	elseif (t_0 <= 5e+279)
		tmp = Float64(2.0 * Float64(cos(Float64(pi * Float64(angle_m / 180.0))) * Float64(Float64(Float64(b_m - a) * Float64(b_m + a)) * sin(Float64(0.005555555555555556 * Float64(pi * angle_m))))));
	else
		tmp = fma(b_m, Float64(Float64(angle_m * 0.011111111111111112) * Float64(b_m * pi)), Float64(-0.011111111111111112 * Float64((a ^ 2.0) * Float64(pi * angle_m))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[t$95$0, -5e+273], N[(0.011111111111111112 * N[(N[(angle$95$m * N[(N[Power[b$95$m, 2.0], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] - N[(a * N[(a * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+279], N[(2.0 * N[(N[Cos[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(b$95$m - a), $MachinePrecision] * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b$95$m * N[(N[(angle$95$m * 0.011111111111111112), $MachinePrecision] * N[(b$95$m * Pi), $MachinePrecision]), $MachinePrecision] + N[(-0.011111111111111112 * N[(N[Power[a, 2.0], $MachinePrecision] * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $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))) < -4.99999999999999961e273

   1. Initial program 45.1%

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

      1. associate-*l* 45.1%

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

      2. *-commutative 45.1%

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

      3. associate-*l* 46.8%

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

   3. Simplified 46.8%

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

   5. Taylor expanded in angle around 0 54.5%

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

      1. unpow2 26.2%

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

      2. unpow2 26.2%

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

      3. difference-of-squares 26.2%

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

   7. Applied egg-rr 54.5%

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

   8. Taylor expanded in a around 0 69.3%

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

      1. +-commutative 69.3%

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

      2. *-commutative 69.3%

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

      3. +-commutative 69.3%

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

      4. associate-*r* 69.3%

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

      5. fma-define 69.3%

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

      6. distribute-rgt1-in 69.3%

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

      7. metadata-eval 69.3%

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

      8. mul0-lft 69.3%

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

      9. mul-1-neg 69.3%

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

      10. fma-neg 69.3%

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

      11. mul0-rgt 69.3%

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

      12. neg-sub0 69.3%

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

      13. mul-1-neg 69.3%

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

   10. Simplified 69.3%

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

   if -4.99999999999999961e273 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < 5.0000000000000002e279

   1. Initial program 56.6%

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

      1. associate-*l* 56.6%

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

      2. associate-*l* 56.6%

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

   3. Simplified 56.6%

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

      1. add-sqr-sqrt 36.3%

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

      2. sqrt-unprod 36.1%

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

      3. pow2 36.1%

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

      4. *-commutative 36.1%

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

      5. div-inv 36.1%

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

      6. metadata-eval 36.1%

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

   6. Applied egg-rr 36.1%

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

      1. unpow2 36.1%

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

      2. unpow2 36.1%

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

      3. difference-of-squares 36.1%

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

   8. Applied egg-rr 36.1%

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

   9. Taylor expanded in angle around inf 56.9%

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

       1. +-commutative 56.9%

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

       2. *-commutative 56.9%

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

   11. Simplified 56.9%

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

   if 5.0000000000000002e279 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 42.8%

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

      1. associate-*l* 42.8%

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

      2. *-commutative 42.8%

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

      3. associate-*l* 42.8%

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

   3. Simplified 42.8%

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

   5. Taylor expanded in angle around 0 45.9%

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

      1. unpow2 37.7%

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

      2. unpow2 37.7%

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

      3. difference-of-squares 42.5%

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

   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 b around 0 62.8%

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

      1. +-commutative 62.8%

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

      2. fma-define 62.8%

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

      3. associate-*r* 62.9%

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

      4. associate-*r* 62.9%

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

      5. distribute-lft-out 62.9%

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

      6. *-commutative 62.9%

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

      7. *-commutative 62.9%

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

      8. distribute-rgt1-in 62.9%

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

      9. metadata-eval 62.9%

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

      10. mul0-lft 62.9%

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

      11. distribute-rgt-out 62.9%

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

      12. +-rgt-identity 62.9%

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

      13. *-commutative 62.9%

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

      14. *-commutative 62.9%

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

   10. Simplified 62.9%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -5 \cdot 10^{+273}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right) - a \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{elif}\;{b}^{2} - {a}^{2} \leq 5 \cdot 10^{+279}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, \left(angle \cdot 0.011111111111111112\right) \cdot \left(b \cdot \pi\right), -0.011111111111111112 \cdot \left({a}^{2} \cdot \left(\pi \cdot angle\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := {b\_m}^{2} - {a}^{2}\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+273}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left({b\_m}^{2} \cdot \pi\right) - a \cdot \left(a \cdot \left(\pi \cdot angle\_m\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\left(\left(b\_m - a\right) \cdot \left(b\_m + a\right)\right) \cdot \sin \left(\left(\pi \cdot angle\_m\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b\_m, \left(angle\_m \cdot 0.011111111111111112\right) \cdot \left(b\_m \cdot \pi\right), -0.011111111111111112 \cdot \left({a}^{2} \cdot \left(\pi \cdot angle\_m\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (- (pow b_m 2.0) (pow a 2.0))))
   (*
    angle_s
    (if (<= t_0 -5e+273)
      (*
       0.011111111111111112
       (- (* angle_m (* (pow b_m 2.0) PI)) (* a (* a (* PI angle_m)))))
      (if (<= t_0 5e+292)
        (*
         (* (- b_m a) (+ b_m a))
         (sin (* (* PI angle_m) 0.011111111111111112)))
        (fma
         b_m
         (* (* angle_m 0.011111111111111112) (* b_m PI))
         (* -0.011111111111111112 (* (pow a 2.0) (* PI angle_m)))))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = pow(b_m, 2.0) - pow(a, 2.0);
	double tmp;
	if (t_0 <= -5e+273) {
		tmp = 0.011111111111111112 * ((angle_m * (pow(b_m, 2.0) * ((double) M_PI))) - (a * (a * (((double) M_PI) * angle_m))));
	} else if (t_0 <= 5e+292) {
		tmp = ((b_m - a) * (b_m + a)) * sin(((((double) M_PI) * angle_m) * 0.011111111111111112));
	} else {
		tmp = fma(b_m, ((angle_m * 0.011111111111111112) * (b_m * ((double) M_PI))), (-0.011111111111111112 * (pow(a, 2.0) * (((double) M_PI) * angle_m))));
	}
	return angle_s * tmp;
}

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = Float64((b_m ^ 2.0) - (a ^ 2.0))
	tmp = 0.0
	if (t_0 <= -5e+273)
		tmp = Float64(0.011111111111111112 * Float64(Float64(angle_m * Float64((b_m ^ 2.0) * pi)) - Float64(a * Float64(a * Float64(pi * angle_m)))));
	elseif (t_0 <= 5e+292)
		tmp = Float64(Float64(Float64(b_m - a) * Float64(b_m + a)) * sin(Float64(Float64(pi * angle_m) * 0.011111111111111112)));
	else
		tmp = fma(b_m, Float64(Float64(angle_m * 0.011111111111111112) * Float64(b_m * pi)), Float64(-0.011111111111111112 * Float64((a ^ 2.0) * Float64(pi * angle_m))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[t$95$0, -5e+273], N[(0.011111111111111112 * N[(N[(angle$95$m * N[(N[Power[b$95$m, 2.0], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] - N[(a * N[(a * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+292], N[(N[(N[(b$95$m - a), $MachinePrecision] * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(b$95$m * N[(N[(angle$95$m * 0.011111111111111112), $MachinePrecision] * N[(b$95$m * Pi), $MachinePrecision]), $MachinePrecision] + N[(-0.011111111111111112 * N[(N[Power[a, 2.0], $MachinePrecision] * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $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))) < -4.99999999999999961e273

   1. Initial program 45.1%

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

      1. associate-*l* 45.1%

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

      2. *-commutative 45.1%

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

      3. associate-*l* 46.8%

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

   3. Simplified 46.8%

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

   5. Taylor expanded in angle around 0 54.5%

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

      1. unpow2 26.2%

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

      2. unpow2 26.2%

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

      3. difference-of-squares 26.2%

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

   7. Applied egg-rr 54.5%

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

   8. Taylor expanded in a around 0 69.3%

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

      1. +-commutative 69.3%

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

      2. *-commutative 69.3%

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

      3. +-commutative 69.3%

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

      4. associate-*r* 69.3%

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

      5. fma-define 69.3%

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

      6. distribute-rgt1-in 69.3%

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

      7. metadata-eval 69.3%

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

      8. mul0-lft 69.3%

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

      9. mul-1-neg 69.3%

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

      10. fma-neg 69.3%

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

      11. mul0-rgt 69.3%

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

      12. neg-sub0 69.3%

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

      13. mul-1-neg 69.3%

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

   10. Simplified 69.3%

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

   if -4.99999999999999961e273 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < 4.9999999999999996e292

   1. Initial program 56.5%

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

      1. associate-*l* 56.5%

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

      2. *-commutative 56.5%

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

      3. associate-*l* 56.5%

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

   3. Simplified 56.5%

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

      1. *-commutative 56.5%

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

      2. sub-neg 56.5%

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

      3. distribute-lft-in 56.5%

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

      4. 2-sin 56.5%

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

      5. associate-*r* 56.5%

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

      6. div-inv 56.9%

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

      7. metadata-eval 56.9%

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

   6. Applied egg-rr 57.0%

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

      1. distribute-lft-out 57.0%

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

      2. sub-neg 57.0%

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

      3. *-commutative 57.0%

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

      4. associate-*r* 57.0%

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

      5. associate-*r* 56.7%

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

      6. *-commutative 56.7%

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

      7. *-commutative 56.7%

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

      8. associate-*r* 56.7%

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

      9. metadata-eval 56.7%

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

   8. Simplified 56.7%

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

      1. unpow2 35.5%

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

      2. unpow2 35.5%

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

      3. difference-of-squares 35.5%

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

   10. Applied egg-rr 56.7%

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

   if 4.9999999999999996e292 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 42.6%

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

      1. associate-*l* 42.6%

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

      2. *-commutative 42.6%

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

      3. associate-*l* 42.6%

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

   3. Simplified 42.6%

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

   5. Taylor expanded in angle around 0 45.6%

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

      1. unpow2 38.8%

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

      2. unpow2 38.8%

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

      3. difference-of-squares 43.7%

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

   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 b around 0 63.1%

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

      1. +-commutative 63.1%

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

      2. fma-define 63.1%

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

      3. associate-*r* 63.1%

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

      4. associate-*r* 63.1%

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

      5. distribute-lft-out 63.1%

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

      6. *-commutative 63.1%

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

      7. *-commutative 63.1%

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

      8. distribute-rgt1-in 63.1%

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

      9. metadata-eval 63.1%

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

      10. mul0-lft 63.1%

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

      11. distribute-rgt-out 63.1%

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

      12. +-rgt-identity 63.1%

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

      13. *-commutative 63.1%

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

      14. *-commutative 63.1%

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

   10. Simplified 63.1%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -5 \cdot 10^{+273}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right) - a \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{elif}\;{b}^{2} - {a}^{2} \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, \left(angle \cdot 0.011111111111111112\right) \cdot \left(b \cdot \pi\right), -0.011111111111111112 \cdot \left({a}^{2} \cdot \left(\pi \cdot angle\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := {b\_m}^{2} - {a}^{2}\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+273}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left({b\_m}^{2} \cdot \pi\right) - a \cdot \left(a \cdot \left(\pi \cdot angle\_m\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\left(\left(b\_m - a\right) \cdot \left(b\_m + a\right)\right) \cdot \sin \left(\left(\pi \cdot angle\_m\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(\pi \cdot angle\_m\right)\right) + b\_m \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(b\_m \cdot \pi\right)\right) + 0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(a - a\right)\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (- (pow b_m 2.0) (pow a 2.0))))
   (*
    angle_s
    (if (<= t_0 -5e+273)
      (*
       0.011111111111111112
       (- (* angle_m (* (pow b_m 2.0) PI)) (* a (* a (* PI angle_m)))))
      (if (<= t_0 5e+292)
        (*
         (* (- b_m a) (+ b_m a))
         (sin (* (* PI angle_m) 0.011111111111111112)))
        (+
         (* -0.011111111111111112 (* (pow a 2.0) (* PI angle_m)))
         (*
          b_m
          (+
           (* 0.011111111111111112 (* angle_m (* b_m PI)))
           (* 0.011111111111111112 (* angle_m (* PI (- a a))))))))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = pow(b_m, 2.0) - pow(a, 2.0);
	double tmp;
	if (t_0 <= -5e+273) {
		tmp = 0.011111111111111112 * ((angle_m * (pow(b_m, 2.0) * ((double) M_PI))) - (a * (a * (((double) M_PI) * angle_m))));
	} else if (t_0 <= 5e+292) {
		tmp = ((b_m - a) * (b_m + a)) * sin(((((double) M_PI) * angle_m) * 0.011111111111111112));
	} else {
		tmp = (-0.011111111111111112 * (pow(a, 2.0) * (((double) M_PI) * angle_m))) + (b_m * ((0.011111111111111112 * (angle_m * (b_m * ((double) M_PI)))) + (0.011111111111111112 * (angle_m * (((double) M_PI) * (a - a))))));
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = Math.pow(b_m, 2.0) - Math.pow(a, 2.0);
	double tmp;
	if (t_0 <= -5e+273) {
		tmp = 0.011111111111111112 * ((angle_m * (Math.pow(b_m, 2.0) * Math.PI)) - (a * (a * (Math.PI * angle_m))));
	} else if (t_0 <= 5e+292) {
		tmp = ((b_m - a) * (b_m + a)) * Math.sin(((Math.PI * angle_m) * 0.011111111111111112));
	} else {
		tmp = (-0.011111111111111112 * (Math.pow(a, 2.0) * (Math.PI * angle_m))) + (b_m * ((0.011111111111111112 * (angle_m * (b_m * Math.PI))) + (0.011111111111111112 * (angle_m * (Math.PI * (a - a))))));
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	t_0 = math.pow(b_m, 2.0) - math.pow(a, 2.0)
	tmp = 0
	if t_0 <= -5e+273:
		tmp = 0.011111111111111112 * ((angle_m * (math.pow(b_m, 2.0) * math.pi)) - (a * (a * (math.pi * angle_m))))
	elif t_0 <= 5e+292:
		tmp = ((b_m - a) * (b_m + a)) * math.sin(((math.pi * angle_m) * 0.011111111111111112))
	else:
		tmp = (-0.011111111111111112 * (math.pow(a, 2.0) * (math.pi * angle_m))) + (b_m * ((0.011111111111111112 * (angle_m * (b_m * math.pi))) + (0.011111111111111112 * (angle_m * (math.pi * (a - a))))))
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = Float64((b_m ^ 2.0) - (a ^ 2.0))
	tmp = 0.0
	if (t_0 <= -5e+273)
		tmp = Float64(0.011111111111111112 * Float64(Float64(angle_m * Float64((b_m ^ 2.0) * pi)) - Float64(a * Float64(a * Float64(pi * angle_m)))));
	elseif (t_0 <= 5e+292)
		tmp = Float64(Float64(Float64(b_m - a) * Float64(b_m + a)) * sin(Float64(Float64(pi * angle_m) * 0.011111111111111112)));
	else
		tmp = Float64(Float64(-0.011111111111111112 * Float64((a ^ 2.0) * Float64(pi * angle_m))) + Float64(b_m * Float64(Float64(0.011111111111111112 * Float64(angle_m * Float64(b_m * pi))) + Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(a - a)))))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b_m, angle_m)
	t_0 = (b_m ^ 2.0) - (a ^ 2.0);
	tmp = 0.0;
	if (t_0 <= -5e+273)
		tmp = 0.011111111111111112 * ((angle_m * ((b_m ^ 2.0) * pi)) - (a * (a * (pi * angle_m))));
	elseif (t_0 <= 5e+292)
		tmp = ((b_m - a) * (b_m + a)) * sin(((pi * angle_m) * 0.011111111111111112));
	else
		tmp = (-0.011111111111111112 * ((a ^ 2.0) * (pi * angle_m))) + (b_m * ((0.011111111111111112 * (angle_m * (b_m * pi))) + (0.011111111111111112 * (angle_m * (pi * (a - a))))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[t$95$0, -5e+273], N[(0.011111111111111112 * N[(N[(angle$95$m * N[(N[Power[b$95$m, 2.0], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] - N[(a * N[(a * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+292], N[(N[(N[(b$95$m - a), $MachinePrecision] * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(-0.011111111111111112 * N[(N[Power[a, 2.0], $MachinePrecision] * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b$95$m * N[(N[(0.011111111111111112 * N[(angle$95$m * N[(b$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(a - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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))) < -4.99999999999999961e273

   1. Initial program 45.1%

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

      1. associate-*l* 45.1%

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

      2. *-commutative 45.1%

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

      3. associate-*l* 46.8%

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

   3. Simplified 46.8%

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

   5. Taylor expanded in angle around 0 54.5%

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

      1. unpow2 26.2%

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

      2. unpow2 26.2%

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

      3. difference-of-squares 26.2%

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

   7. Applied egg-rr 54.5%

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

   8. Taylor expanded in a around 0 69.3%

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

      1. +-commutative 69.3%

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

      2. *-commutative 69.3%

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

      3. +-commutative 69.3%

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

      4. associate-*r* 69.3%

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

      5. fma-define 69.3%

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

      6. distribute-rgt1-in 69.3%

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

      7. metadata-eval 69.3%

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

      8. mul0-lft 69.3%

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

      9. mul-1-neg 69.3%

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

      10. fma-neg 69.3%

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

      11. mul0-rgt 69.3%

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

      12. neg-sub0 69.3%

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

      13. mul-1-neg 69.3%

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

   10. Simplified 69.3%

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

   if -4.99999999999999961e273 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < 4.9999999999999996e292

   1. Initial program 56.5%

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

      1. associate-*l* 56.5%

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

      2. *-commutative 56.5%

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

      3. associate-*l* 56.5%

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

   3. Simplified 56.5%

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

      1. *-commutative 56.5%

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

      2. sub-neg 56.5%

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

      3. distribute-lft-in 56.5%

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

      4. 2-sin 56.5%

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

      5. associate-*r* 56.5%

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

      6. div-inv 56.9%

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

      7. metadata-eval 56.9%

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

   6. Applied egg-rr 57.0%

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

      1. distribute-lft-out 57.0%

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

      2. sub-neg 57.0%

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

      3. *-commutative 57.0%

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

      4. associate-*r* 57.0%

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

      5. associate-*r* 56.7%

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

      6. *-commutative 56.7%

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

      7. *-commutative 56.7%

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

      8. associate-*r* 56.7%

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

      9. metadata-eval 56.7%

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

   8. Simplified 56.7%

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

      1. unpow2 35.5%

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

      2. unpow2 35.5%

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

      3. difference-of-squares 35.5%

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

   10. Applied egg-rr 56.7%

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

   if 4.9999999999999996e292 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 42.6%

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

      1. associate-*l* 42.6%

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

      2. *-commutative 42.6%

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

      3. associate-*l* 42.6%

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

   3. Simplified 42.6%

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

   5. Taylor expanded in angle around 0 45.6%

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

      1. unpow2 38.8%

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

      2. unpow2 38.8%

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

      3. difference-of-squares 43.7%

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

   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 b around 0 63.1%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -5 \cdot 10^{+273}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right) - a \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{elif}\;{b}^{2} - {a}^{2} \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(\pi \cdot angle\right)\right) + b \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right) + 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(a - a\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.9% accurate, 1.9× speedup

Math

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

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (*
  angle_s
  (*
   2.0
   (*
    (cos (* PI (/ angle_m 180.0)))
    (*
     (- b_m a)
     (* (sin (* PI (* angle_m 0.005555555555555556))) (+ b_m a)))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.5%

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

   1. associate-*l* 50.9%

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

   2. associate-*l* 50.9%

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

3. Simplified 50.9%

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

   1. add-sqr-sqrt 28.5%

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

   2. sqrt-unprod 34.4%

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

   3. pow2 34.4%

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

   4. *-commutative 34.4%

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

   5. div-inv 34.4%

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

   6. metadata-eval 34.4%

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

6. Applied egg-rr 34.4%

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

   1. unpow2 34.4%

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

   2. unpow2 34.4%

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

   3. difference-of-squares 35.6%

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

8. Applied egg-rr 35.6%

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

   1. sqrt-pow1 54.6%

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

   2. metadata-eval 54.6%

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

   3. pow1 54.6%

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

   4. *-commutative 54.6%

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

   5. *-commutative 54.6%

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

   6. associate-*l* 63.7%

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

   7. associate-*r* 61.4%

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

   8. *-commutative 61.4%

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

   9. *-commutative 61.4%

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

   10. associate-*r* 63.7%

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

   11. *-commutative 63.7%

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

10. Applied egg-rr 63.7%

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

11. Final simplification 63.7%

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

Alternative 9: 60.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(b\_m - a\right) \cdot \left(b\_m + a\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{-272}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left({b\_m}^{2} \cdot \pi\right) - a \cdot \left(a \cdot \left(\pi \cdot angle\_m\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{-129}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b\_m \cdot \left(angle\_m \cdot \left(b\_m \cdot \pi\right) + angle\_m \cdot \left(\pi \cdot \left(a - a\right)\right)\right) - {a}^{2} \cdot \left(\pi \cdot angle\_m\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 10^{+73}:\\ \;\;\;\;t\_0 \cdot \sin \left(\left(\pi \cdot angle\_m\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle\_m}{180}\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (* (- b_m a) (+ b_m a))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 2e-272)
      (*
       0.011111111111111112
       (- (* angle_m (* (pow b_m 2.0) PI)) (* a (* a (* PI angle_m)))))
      (if (<= (/ angle_m 180.0) 5e-129)
        (*
         0.011111111111111112
         (-
          (* b_m (+ (* angle_m (* b_m PI)) (* angle_m (* PI (- a a)))))
          (* (pow a 2.0) (* PI angle_m))))
        (if (<= (/ angle_m 180.0) 1e+73)
          (* t_0 (sin (* (* PI angle_m) 0.011111111111111112)))
          (* t_0 (* 2.0 (sin (* PI (/ angle_m 180.0)))))))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = (b_m - a) * (b_m + a);
	double tmp;
	if ((angle_m / 180.0) <= 2e-272) {
		tmp = 0.011111111111111112 * ((angle_m * (pow(b_m, 2.0) * ((double) M_PI))) - (a * (a * (((double) M_PI) * angle_m))));
	} else if ((angle_m / 180.0) <= 5e-129) {
		tmp = 0.011111111111111112 * ((b_m * ((angle_m * (b_m * ((double) M_PI))) + (angle_m * (((double) M_PI) * (a - a))))) - (pow(a, 2.0) * (((double) M_PI) * angle_m)));
	} else if ((angle_m / 180.0) <= 1e+73) {
		tmp = t_0 * sin(((((double) M_PI) * angle_m) * 0.011111111111111112));
	} else {
		tmp = t_0 * (2.0 * sin((((double) M_PI) * (angle_m / 180.0))));
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = (b_m - a) * (b_m + a);
	double tmp;
	if ((angle_m / 180.0) <= 2e-272) {
		tmp = 0.011111111111111112 * ((angle_m * (Math.pow(b_m, 2.0) * Math.PI)) - (a * (a * (Math.PI * angle_m))));
	} else if ((angle_m / 180.0) <= 5e-129) {
		tmp = 0.011111111111111112 * ((b_m * ((angle_m * (b_m * Math.PI)) + (angle_m * (Math.PI * (a - a))))) - (Math.pow(a, 2.0) * (Math.PI * angle_m)));
	} else if ((angle_m / 180.0) <= 1e+73) {
		tmp = t_0 * Math.sin(((Math.PI * angle_m) * 0.011111111111111112));
	} else {
		tmp = t_0 * (2.0 * Math.sin((Math.PI * (angle_m / 180.0))));
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	t_0 = (b_m - a) * (b_m + a)
	tmp = 0
	if (angle_m / 180.0) <= 2e-272:
		tmp = 0.011111111111111112 * ((angle_m * (math.pow(b_m, 2.0) * math.pi)) - (a * (a * (math.pi * angle_m))))
	elif (angle_m / 180.0) <= 5e-129:
		tmp = 0.011111111111111112 * ((b_m * ((angle_m * (b_m * math.pi)) + (angle_m * (math.pi * (a - a))))) - (math.pow(a, 2.0) * (math.pi * angle_m)))
	elif (angle_m / 180.0) <= 1e+73:
		tmp = t_0 * math.sin(((math.pi * angle_m) * 0.011111111111111112))
	else:
		tmp = t_0 * (2.0 * math.sin((math.pi * (angle_m / 180.0))))
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = Float64(Float64(b_m - a) * Float64(b_m + a))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e-272)
		tmp = Float64(0.011111111111111112 * Float64(Float64(angle_m * Float64((b_m ^ 2.0) * pi)) - Float64(a * Float64(a * Float64(pi * angle_m)))));
	elseif (Float64(angle_m / 180.0) <= 5e-129)
		tmp = Float64(0.011111111111111112 * Float64(Float64(b_m * Float64(Float64(angle_m * Float64(b_m * pi)) + Float64(angle_m * Float64(pi * Float64(a - a))))) - Float64((a ^ 2.0) * Float64(pi * angle_m))));
	elseif (Float64(angle_m / 180.0) <= 1e+73)
		tmp = Float64(t_0 * sin(Float64(Float64(pi * angle_m) * 0.011111111111111112)));
	else
		tmp = Float64(t_0 * Float64(2.0 * sin(Float64(pi * Float64(angle_m / 180.0)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b_m, angle_m)
	t_0 = (b_m - a) * (b_m + a);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 2e-272)
		tmp = 0.011111111111111112 * ((angle_m * ((b_m ^ 2.0) * pi)) - (a * (a * (pi * angle_m))));
	elseif ((angle_m / 180.0) <= 5e-129)
		tmp = 0.011111111111111112 * ((b_m * ((angle_m * (b_m * pi)) + (angle_m * (pi * (a - a))))) - ((a ^ 2.0) * (pi * angle_m)));
	elseif ((angle_m / 180.0) <= 1e+73)
		tmp = t_0 * sin(((pi * angle_m) * 0.011111111111111112));
	else
		tmp = t_0 * (2.0 * sin((pi * (angle_m / 180.0))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(N[(b$95$m - a), $MachinePrecision] * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e-272], N[(0.011111111111111112 * N[(N[(angle$95$m * N[(N[Power[b$95$m, 2.0], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] - N[(a * N[(a * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e-129], N[(0.011111111111111112 * N[(N[(b$95$m * N[(N[(angle$95$m * N[(b$95$m * Pi), $MachinePrecision]), $MachinePrecision] + N[(angle$95$m * N[(Pi * N[(a - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[a, 2.0], $MachinePrecision] * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+73], N[(t$95$0 * N[Sin[N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(2.0 * 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)) < 1.99999999999999986e-272

   1. Initial program 47.7%

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

      1. associate-*l* 47.7%

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

      2. *-commutative 47.7%

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

      3. associate-*l* 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 50.5%

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

      1. unpow2 32.7%

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

      2. unpow2 32.7%

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

      3. difference-of-squares 34.3%

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

   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 a around 0 54.2%

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

      1. +-commutative 54.2%

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

      2. *-commutative 54.2%

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

      3. +-commutative 54.2%

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

      4. associate-*r* 54.2%

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

      5. fma-define 54.2%

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

      6. distribute-rgt1-in 54.2%

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

      7. metadata-eval 54.2%

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

      8. mul0-lft 54.2%

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

      9. mul-1-neg 54.2%

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

      10. fma-neg 54.2%

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

      11. mul0-rgt 54.2%

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

      12. neg-sub0 54.2%

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

      13. mul-1-neg 54.2%

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

   10. Simplified 54.2%

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

   if 1.99999999999999986e-272 < (/.f64 angle #s(literal 180 binary64)) < 5.00000000000000027e-129

   1. Initial program 71.1%

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

      1. associate-*l* 71.1%

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

      2. *-commutative 71.1%

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

      3. associate-*l* 71.1%

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

   3. Simplified 71.1%

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

   5. Taylor expanded in angle around 0 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 51.5%

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

      2. unpow2 51.5%

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

      3. difference-of-squares 51.9%

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

   7. Applied egg-rr 75.0%

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

   8. Taylor expanded in b around 0 83.9%

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

   if 5.00000000000000027e-129 < (/.f64 angle #s(literal 180 binary64)) < 9.99999999999999983e72

   1. Initial program 77.6%

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

      1. associate-*l* 77.6%

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

      2. *-commutative 77.6%

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

      3. associate-*l* 77.6%

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

   3. Simplified 77.6%

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

      1. *-commutative 77.6%

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

      2. sub-neg 77.6%

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

      3. distribute-lft-in 77.6%

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

      4. 2-sin 77.6%

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

      5. associate-*r* 77.6%

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

      6. div-inv 77.6%

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

      7. metadata-eval 77.6%

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

   6. Applied egg-rr 77.7%

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

      1. distribute-lft-out 77.7%

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

      2. sub-neg 77.7%

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

      3. *-commutative 77.7%

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

      4. associate-*r* 77.7%

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

      5. associate-*r* 77.2%

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

      6. *-commutative 77.2%

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

      7. *-commutative 77.2%

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

      8. associate-*r* 77.2%

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

      9. metadata-eval 77.2%

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

   8. Simplified 77.2%

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

      1. unpow2 34.1%

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

      2. unpow2 34.1%

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

      3. difference-of-squares 34.3%

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

   10. Applied egg-rr 77.4%

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

   if 9.99999999999999983e72 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 25.1%

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

      1. associate-*l* 25.1%

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

      2. *-commutative 25.1%

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

      3. associate-*l* 25.1%

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

   3. Simplified 25.1%

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

      1. unpow2 29.3%

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

      2. unpow2 29.3%

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

      3. difference-of-squares 31.0%

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

   6. Applied egg-rr 25.1%

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

   7. Taylor expanded in angle around 0 30.4%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{-272}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right) - a \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{-129}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \left(b \cdot \pi\right) + angle \cdot \left(\pi \cdot \left(a - a\right)\right)\right) - {a}^{2} \cdot \left(\pi \cdot angle\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+73}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(b\_m - a\right) \cdot \left(b\_m + a\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{-56}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left({b\_m}^{2} \cdot \pi\right) - a \cdot \left(a \cdot \left(\pi \cdot angle\_m\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 10^{+73}:\\ \;\;\;\;t\_0 \cdot \sin \left(\left(\pi \cdot angle\_m\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle\_m}{180}\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (* (- b_m a) (+ b_m a))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 5e-56)
      (*
       0.011111111111111112
       (- (* angle_m (* (pow b_m 2.0) PI)) (* a (* a (* PI angle_m)))))
      (if (<= (/ angle_m 180.0) 1e+73)
        (* t_0 (sin (* (* PI angle_m) 0.011111111111111112)))
        (* t_0 (* 2.0 (sin (* PI (/ angle_m 180.0))))))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = (b_m - a) * (b_m + a);
	double tmp;
	if ((angle_m / 180.0) <= 5e-56) {
		tmp = 0.011111111111111112 * ((angle_m * (pow(b_m, 2.0) * ((double) M_PI))) - (a * (a * (((double) M_PI) * angle_m))));
	} else if ((angle_m / 180.0) <= 1e+73) {
		tmp = t_0 * sin(((((double) M_PI) * angle_m) * 0.011111111111111112));
	} else {
		tmp = t_0 * (2.0 * sin((((double) M_PI) * (angle_m / 180.0))));
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = (b_m - a) * (b_m + a);
	double tmp;
	if ((angle_m / 180.0) <= 5e-56) {
		tmp = 0.011111111111111112 * ((angle_m * (Math.pow(b_m, 2.0) * Math.PI)) - (a * (a * (Math.PI * angle_m))));
	} else if ((angle_m / 180.0) <= 1e+73) {
		tmp = t_0 * Math.sin(((Math.PI * angle_m) * 0.011111111111111112));
	} else {
		tmp = t_0 * (2.0 * Math.sin((Math.PI * (angle_m / 180.0))));
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	t_0 = (b_m - a) * (b_m + a)
	tmp = 0
	if (angle_m / 180.0) <= 5e-56:
		tmp = 0.011111111111111112 * ((angle_m * (math.pow(b_m, 2.0) * math.pi)) - (a * (a * (math.pi * angle_m))))
	elif (angle_m / 180.0) <= 1e+73:
		tmp = t_0 * math.sin(((math.pi * angle_m) * 0.011111111111111112))
	else:
		tmp = t_0 * (2.0 * math.sin((math.pi * (angle_m / 180.0))))
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = Float64(Float64(b_m - a) * Float64(b_m + a))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e-56)
		tmp = Float64(0.011111111111111112 * Float64(Float64(angle_m * Float64((b_m ^ 2.0) * pi)) - Float64(a * Float64(a * Float64(pi * angle_m)))));
	elseif (Float64(angle_m / 180.0) <= 1e+73)
		tmp = Float64(t_0 * sin(Float64(Float64(pi * angle_m) * 0.011111111111111112)));
	else
		tmp = Float64(t_0 * Float64(2.0 * sin(Float64(pi * Float64(angle_m / 180.0)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b_m, angle_m)
	t_0 = (b_m - a) * (b_m + a);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 5e-56)
		tmp = 0.011111111111111112 * ((angle_m * ((b_m ^ 2.0) * pi)) - (a * (a * (pi * angle_m))));
	elseif ((angle_m / 180.0) <= 1e+73)
		tmp = t_0 * sin(((pi * angle_m) * 0.011111111111111112));
	else
		tmp = t_0 * (2.0 * sin((pi * (angle_m / 180.0))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(N[(b$95$m - a), $MachinePrecision] * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e-56], N[(0.011111111111111112 * N[(N[(angle$95$m * N[(N[Power[b$95$m, 2.0], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] - N[(a * N[(a * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+73], N[(t$95$0 * N[Sin[N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(2.0 * N[Sin[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.99999999999999997e-56

   1. Initial program 57.3%

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

      1. associate-*l* 57.3%

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

      2. *-commutative 57.3%

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

      3. associate-*l* 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 59.3%

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

      1. unpow2 37.1%

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

      2. unpow2 37.1%

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

      3. difference-of-squares 38.4%

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

   7. Applied egg-rr 62.3%

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

   8. Taylor expanded in a around 0 63.6%

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

      1. +-commutative 63.6%

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

      2. *-commutative 63.6%

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

      3. +-commutative 63.6%

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

      4. associate-*r* 63.6%

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

      5. fma-define 63.6%

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

      6. distribute-rgt1-in 63.6%

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

      7. metadata-eval 63.6%

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

      8. mul0-lft 63.6%

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

      9. mul-1-neg 63.6%

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

      10. fma-neg 63.6%

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

      11. mul0-rgt 63.6%

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

      12. neg-sub0 63.6%

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

      13. mul-1-neg 63.6%

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

   10. Simplified 63.6%

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

   if 4.99999999999999997e-56 < (/.f64 angle #s(literal 180 binary64)) < 9.99999999999999983e72

   1. Initial program 61.6%

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

      1. associate-*l* 61.7%

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

      2. *-commutative 61.7%

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

      3. associate-*l* 61.7%

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

   3. Simplified 61.7%

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

      1. *-commutative 61.7%

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

      2. sub-neg 61.7%

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

      3. distribute-lft-in 61.7%

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

      4. 2-sin 61.7%

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

      5. associate-*r* 61.7%

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

      6. div-inv 61.8%

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

      7. metadata-eval 61.8%

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

   6. Applied egg-rr 61.8%

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

      1. distribute-lft-out 61.8%

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

      2. sub-neg 61.8%

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

      3. *-commutative 61.8%

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

      4. associate-*r* 61.8%

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

      5. associate-*r* 61.1%

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

      6. *-commutative 61.1%

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

      7. *-commutative 61.1%

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

      8. associate-*r* 61.1%

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

      9. metadata-eval 61.1%

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

   8. Simplified 61.1%

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

      1. unpow2 25.7%

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

      2. unpow2 25.7%

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

      3. difference-of-squares 25.7%

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

   10. Applied egg-rr 61.1%

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

   if 9.99999999999999983e72 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 25.1%

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

      1. associate-*l* 25.1%

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

      2. *-commutative 25.1%

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

      3. associate-*l* 25.1%

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

   3. Simplified 25.1%

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

      1. unpow2 29.3%

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

      2. unpow2 29.3%

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

      3. difference-of-squares 31.0%

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

   6. Applied egg-rr 25.1%

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

   7. Taylor expanded in angle around 0 30.4%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-56}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right) - a \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+73}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(b\_m - a\right) \cdot \left(b\_m + a\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 8 \cdot 10^{-51}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left({b\_m}^{2} \cdot \pi\right) - a \cdot \left(a \cdot \left(\pi \cdot angle\_m\right)\right)\right)\\ \mathbf{elif}\;angle\_m \leq 5.2 \cdot 10^{+163}:\\ \;\;\;\;t\_0 \cdot \sin \left(\left(\pi \cdot angle\_m\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot t\_0\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (* (- b_m a) (+ b_m a))))
   (*
    angle_s
    (if (<= angle_m 8e-51)
      (*
       0.011111111111111112
       (- (* angle_m (* (pow b_m 2.0) PI)) (* a (* a (* PI angle_m)))))
      (if (<= angle_m 5.2e+163)
        (* t_0 (sin (* (* PI angle_m) 0.011111111111111112)))
        (* 0.011111111111111112 (* angle_m (* PI t_0))))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = (b_m - a) * (b_m + a);
	double tmp;
	if (angle_m <= 8e-51) {
		tmp = 0.011111111111111112 * ((angle_m * (pow(b_m, 2.0) * ((double) M_PI))) - (a * (a * (((double) M_PI) * angle_m))));
	} else if (angle_m <= 5.2e+163) {
		tmp = t_0 * sin(((((double) M_PI) * angle_m) * 0.011111111111111112));
	} else {
		tmp = 0.011111111111111112 * (angle_m * (((double) M_PI) * t_0));
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = (b_m - a) * (b_m + a);
	double tmp;
	if (angle_m <= 8e-51) {
		tmp = 0.011111111111111112 * ((angle_m * (Math.pow(b_m, 2.0) * Math.PI)) - (a * (a * (Math.PI * angle_m))));
	} else if (angle_m <= 5.2e+163) {
		tmp = t_0 * Math.sin(((Math.PI * angle_m) * 0.011111111111111112));
	} else {
		tmp = 0.011111111111111112 * (angle_m * (Math.PI * t_0));
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	t_0 = (b_m - a) * (b_m + a)
	tmp = 0
	if angle_m <= 8e-51:
		tmp = 0.011111111111111112 * ((angle_m * (math.pow(b_m, 2.0) * math.pi)) - (a * (a * (math.pi * angle_m))))
	elif angle_m <= 5.2e+163:
		tmp = t_0 * math.sin(((math.pi * angle_m) * 0.011111111111111112))
	else:
		tmp = 0.011111111111111112 * (angle_m * (math.pi * t_0))
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = Float64(Float64(b_m - a) * Float64(b_m + a))
	tmp = 0.0
	if (angle_m <= 8e-51)
		tmp = Float64(0.011111111111111112 * Float64(Float64(angle_m * Float64((b_m ^ 2.0) * pi)) - Float64(a * Float64(a * Float64(pi * angle_m)))));
	elseif (angle_m <= 5.2e+163)
		tmp = Float64(t_0 * sin(Float64(Float64(pi * angle_m) * 0.011111111111111112)));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * t_0)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b_m, angle_m)
	t_0 = (b_m - a) * (b_m + a);
	tmp = 0.0;
	if (angle_m <= 8e-51)
		tmp = 0.011111111111111112 * ((angle_m * ((b_m ^ 2.0) * pi)) - (a * (a * (pi * angle_m))));
	elseif (angle_m <= 5.2e+163)
		tmp = t_0 * sin(((pi * angle_m) * 0.011111111111111112));
	else
		tmp = 0.011111111111111112 * (angle_m * (pi * t_0));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(N[(b$95$m - a), $MachinePrecision] * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[angle$95$m, 8e-51], N[(0.011111111111111112 * N[(N[(angle$95$m * N[(N[Power[b$95$m, 2.0], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] - N[(a * N[(a * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle$95$m, 5.2e+163], N[(t$95$0 * N[Sin[N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if angle < 8.0000000000000001e-51

   1. Initial program 57.3%

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

      1. associate-*l* 57.3%

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

      2. *-commutative 57.3%

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

      3. associate-*l* 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 59.3%

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

      1. unpow2 37.1%

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

      2. unpow2 37.1%

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

      3. difference-of-squares 38.4%

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

   7. Applied egg-rr 62.3%

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

   8. Taylor expanded in a around 0 63.6%

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

      1. +-commutative 63.6%

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

      2. *-commutative 63.6%

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

      3. +-commutative 63.6%

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

      4. associate-*r* 63.6%

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

      5. fma-define 63.6%

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

      6. distribute-rgt1-in 63.6%

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

      7. metadata-eval 63.6%

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

      8. mul0-lft 63.6%

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

      9. mul-1-neg 63.6%

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

      10. fma-neg 63.6%

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

      11. mul0-rgt 63.6%

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

      12. neg-sub0 63.6%

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

      13. mul-1-neg 63.6%

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

   10. Simplified 63.6%

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

   if 8.0000000000000001e-51 < angle < 5.2000000000000003e163

   1. Initial program 47.8%

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

      1. associate-*l* 47.8%

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

      2. *-commutative 47.8%

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

      3. associate-*l* 47.8%

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

   3. Simplified 47.8%

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

      1. *-commutative 47.8%

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

      2. sub-neg 47.8%

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

      3. distribute-lft-in 47.8%

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

      4. 2-sin 47.8%

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

      5. associate-*r* 47.8%

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

      6. div-inv 48.3%

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

      7. metadata-eval 48.3%

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

   6. Applied egg-rr 48.2%

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

      1. distribute-lft-out 48.2%

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

      2. sub-neg 48.2%

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

      3. *-commutative 48.2%

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

      4. associate-*r* 48.2%

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

      5. associate-*r* 48.8%

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

      6. *-commutative 48.8%

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

      7. *-commutative 48.8%

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

      8. associate-*r* 48.8%

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

      9. metadata-eval 48.8%

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

   8. Simplified 48.8%

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

      1. unpow2 31.3%

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

      2. unpow2 31.3%

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

      3. difference-of-squares 31.3%

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

   10. Applied egg-rr 48.8%

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

   if 5.2000000000000003e163 < angle

   1. Initial program 19.0%

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

      1. associate-*l* 19.0%

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

      2. *-commutative 19.0%

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

      3. associate-*l* 19.0%

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

   3. Simplified 19.0%

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

   5. Taylor expanded in angle around 0 25.2%

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

      1. unpow2 24.2%

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

      2. unpow2 24.2%

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

      3. difference-of-squares 27.1%

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

   7. Applied egg-rr 25.2%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 8 \cdot 10^{-51}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right) - a \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{elif}\;angle \leq 5.2 \cdot 10^{+163}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(b\_m - a\right) \cdot \left(b\_m + a\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 5.2 \cdot 10^{+163}:\\ \;\;\;\;t\_0 \cdot \sin \left(\left(\pi \cdot angle\_m\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot t\_0\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (* (- b_m a) (+ b_m a))))
   (*
    angle_s
    (if (<= angle_m 5.2e+163)
      (* t_0 (sin (* (* PI angle_m) 0.011111111111111112)))
      (* 0.011111111111111112 (* angle_m (* PI t_0)))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = (b_m - a) * (b_m + a);
	double tmp;
	if (angle_m <= 5.2e+163) {
		tmp = t_0 * sin(((((double) M_PI) * angle_m) * 0.011111111111111112));
	} else {
		tmp = 0.011111111111111112 * (angle_m * (((double) M_PI) * t_0));
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = (b_m - a) * (b_m + a);
	double tmp;
	if (angle_m <= 5.2e+163) {
		tmp = t_0 * Math.sin(((Math.PI * angle_m) * 0.011111111111111112));
	} else {
		tmp = 0.011111111111111112 * (angle_m * (Math.PI * t_0));
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	t_0 = (b_m - a) * (b_m + a)
	tmp = 0
	if angle_m <= 5.2e+163:
		tmp = t_0 * math.sin(((math.pi * angle_m) * 0.011111111111111112))
	else:
		tmp = 0.011111111111111112 * (angle_m * (math.pi * t_0))
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = Float64(Float64(b_m - a) * Float64(b_m + a))
	tmp = 0.0
	if (angle_m <= 5.2e+163)
		tmp = Float64(t_0 * sin(Float64(Float64(pi * angle_m) * 0.011111111111111112)));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * t_0)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b_m, angle_m)
	t_0 = (b_m - a) * (b_m + a);
	tmp = 0.0;
	if (angle_m <= 5.2e+163)
		tmp = t_0 * sin(((pi * angle_m) * 0.011111111111111112));
	else
		tmp = 0.011111111111111112 * (angle_m * (pi * t_0));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(N[(b$95$m - a), $MachinePrecision] * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[angle$95$m, 5.2e+163], N[(t$95$0 * N[Sin[N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 5.2000000000000003e163

   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* 55.4%

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

      2. *-commutative 55.4%

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

      3. associate-*l* 55.8%

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

   3. Simplified 55.8%

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

      1. *-commutative 55.8%

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

      2. sub-neg 55.8%

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

      3. distribute-lft-in 55.8%

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

      4. 2-sin 55.8%

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

      5. associate-*r* 55.8%

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

      6. div-inv 56.3%

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

      7. metadata-eval 56.3%

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

   6. Applied egg-rr 56.3%

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

      1. distribute-lft-out 56.3%

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

      2. sub-neg 56.3%

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

      3. *-commutative 56.3%

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

      4. associate-*r* 56.3%

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

      5. associate-*r* 56.4%

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

      6. *-commutative 56.4%

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

      7. *-commutative 56.4%

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

      8. associate-*r* 56.4%

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

      9. metadata-eval 56.4%

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

   8. Simplified 56.4%

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

      1. unpow2 35.9%

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

      2. unpow2 35.9%

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

      3. difference-of-squares 36.9%

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

   10. Applied egg-rr 58.3%

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

   if 5.2000000000000003e163 < angle

   1. Initial program 19.0%

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

      1. associate-*l* 19.0%

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

      2. *-commutative 19.0%

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

      3. associate-*l* 19.0%

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

   3. Simplified 19.0%

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

   5. Taylor expanded in angle around 0 25.2%

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

      1. unpow2 24.2%

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

      2. unpow2 24.2%

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

      3. difference-of-squares 27.1%

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

   7. Applied egg-rr 25.2%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 5.2 \cdot 10^{+163}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 54.1% accurate, 32.2× speedup

Math

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

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (*
  angle_s
  (* 0.011111111111111112 (* angle_m (* PI (* (- b_m a) (+ b_m a)))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	return angle_s * (0.011111111111111112 * (angle_m * (((double) M_PI) * ((b_m - a) * (b_m + a)))));
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	return angle_s * (0.011111111111111112 * (angle_m * (Math.PI * ((b_m - a) * (b_m + a)))));
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	return angle_s * (0.011111111111111112 * (angle_m * (math.pi * ((b_m - a) * (b_m + a)))))

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	return Float64(angle_s * Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(Float64(b_m - a) * Float64(b_m + a))))))
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b_m, angle_m)
	tmp = angle_s * (0.011111111111111112 * (angle_m * (pi * ((b_m - a) * (b_m + a)))));
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := N[(angle$95$s * N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(N[(b$95$m - a), $MachinePrecision] * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.5%

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

   1. associate-*l* 50.5%

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

   2. *-commutative 50.5%

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

   3. associate-*l* 50.9%

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

3. Simplified 50.9%

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

5. Taylor expanded in angle around 0 50.0%

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

   1. unpow2 34.4%

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

   2. unpow2 34.4%

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

   3. difference-of-squares 35.6%

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

7. Applied egg-rr 52.1%

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

8. Final simplification 52.1%

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

Alternative 14: 54.1% accurate, 32.2× speedup

Math

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

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (*
  angle_s
  (* (* angle_m 0.011111111111111112) (* PI (* (- b_m a) (+ b_m a))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	return angle_s * ((angle_m * 0.011111111111111112) * (((double) M_PI) * ((b_m - a) * (b_m + a))));
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	return angle_s * ((angle_m * 0.011111111111111112) * (Math.PI * ((b_m - a) * (b_m + a))));
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	return angle_s * ((angle_m * 0.011111111111111112) * (math.pi * ((b_m - a) * (b_m + a))))

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	return Float64(angle_s * Float64(Float64(angle_m * 0.011111111111111112) * Float64(pi * Float64(Float64(b_m - a) * Float64(b_m + a)))))
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b_m, angle_m)
	tmp = angle_s * ((angle_m * 0.011111111111111112) * (pi * ((b_m - a) * (b_m + a))));
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := N[(angle$95$s * N[(N[(angle$95$m * 0.011111111111111112), $MachinePrecision] * N[(Pi * N[(N[(b$95$m - a), $MachinePrecision] * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.5%

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

   1. associate-*l* 50.5%

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

   2. *-commutative 50.5%

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

   3. associate-*l* 50.9%

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

3. Simplified 50.9%

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

5. Taylor expanded in angle around 0 50.0%

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

   1. unpow2 34.4%

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

   2. unpow2 34.4%

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

   3. difference-of-squares 35.6%

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

7. Applied egg-rr 52.1%

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

8. Taylor expanded in angle around 0 52.1%

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

   1. associate-*r* 52.1%

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

   2. *-commutative 52.1%

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

   3. +-commutative 52.1%

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

10. Simplified 52.1%

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

11. Final simplification 52.1%

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

Reproduce

herbie shell --seed 2024080 
(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)))))