ab-angle->ABCF B

Percentage Accurate: 54.5% → 57.6%

Time: 31.3s

Alternatives: 12

Speedup: 5.5×

• 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: 54.5% 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: 57.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(b + a\right) \cdot \left(b - a\right)\\ t_1 := \pi \cdot \left(angle_m \cdot 0.005555555555555556\right)\\ t_2 := \frac{angle_m}{180} \cdot \pi\\ t_3 := \cos t_2\\ angle_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle_m}{180} \leq 10^{-241}:\\ \;\;\;\;2 \cdot \left({\left(b \cdot \sqrt{\sin t_1}\right)}^{2} \cdot t_3\right)\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 5 \cdot 10^{+65}:\\ \;\;\;\;2 \cdot \left(t_3 \cdot \left(t_0 \cdot \sin t_2\right)\right)\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 2 \cdot 10^{+78}:\\ \;\;\;\;2 \cdot \left(t_3 \cdot \left(t_0 \cdot \sin \left({\left({\left(0.005555555555555556 \cdot \left(angle_m \cdot \pi\right)\right)}^{3}\right)}^{0.3333333333333333}\right)\right)\right)\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 2 \cdot 10^{+91}:\\ \;\;\;\;2 \cdot \left(\left(t_0 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left({\left({t_1}^{3}\right)}^{0.3333333333333333}\right)\right)\right)\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle_m \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_0 \cdot \left(e^{\mathsf{log1p}\left(\sin \left(\frac{1}{\frac{180}{angle_m \cdot \pi}}\right)\right)} + -1\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* (+ b a) (- b a)))
        (t_1 (* PI (* angle_m 0.005555555555555556)))
        (t_2 (* (/ angle_m 180.0) PI))
        (t_3 (cos t_2)))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 1e-241)
      (* 2.0 (* (pow (* b (sqrt (sin t_1))) 2.0) t_3))
      (if (<= (/ angle_m 180.0) 5e+65)
        (* 2.0 (* t_3 (* t_0 (sin t_2))))
        (if (<= (/ angle_m 180.0) 2e+78)
          (*
           2.0
           (*
            t_3
            (*
             t_0
             (sin
              (pow
               (pow (* 0.005555555555555556 (* angle_m PI)) 3.0)
               0.3333333333333333)))))
          (if (<= (/ angle_m 180.0) 2e+91)
            (*
             2.0
             (*
              (*
               t_0
               (expm1 (log1p (sin (pow (pow t_1 3.0) 0.3333333333333333)))))
              (cos (expm1 (log1p (* angle_m (* PI 0.005555555555555556)))))))
            (*
             2.0
             (*
              t_0
              (+
               (exp (log1p (sin (/ 1.0 (/ 180.0 (* angle_m PI))))))
               -1.0))))))))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (b + a) * (b - a);
	double t_1 = ((double) M_PI) * (angle_m * 0.005555555555555556);
	double t_2 = (angle_m / 180.0) * ((double) M_PI);
	double t_3 = cos(t_2);
	double tmp;
	if ((angle_m / 180.0) <= 1e-241) {
		tmp = 2.0 * (pow((b * sqrt(sin(t_1))), 2.0) * t_3);
	} else if ((angle_m / 180.0) <= 5e+65) {
		tmp = 2.0 * (t_3 * (t_0 * sin(t_2)));
	} else if ((angle_m / 180.0) <= 2e+78) {
		tmp = 2.0 * (t_3 * (t_0 * sin(pow(pow((0.005555555555555556 * (angle_m * ((double) M_PI))), 3.0), 0.3333333333333333))));
	} else if ((angle_m / 180.0) <= 2e+91) {
		tmp = 2.0 * ((t_0 * expm1(log1p(sin(pow(pow(t_1, 3.0), 0.3333333333333333))))) * cos(expm1(log1p((angle_m * (((double) M_PI) * 0.005555555555555556))))));
	} else {
		tmp = 2.0 * (t_0 * (exp(log1p(sin((1.0 / (180.0 / (angle_m * ((double) M_PI))))))) + -1.0));
	}
	return angle_s * tmp;
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (b + a) * (b - a);
	double t_1 = Math.PI * (angle_m * 0.005555555555555556);
	double t_2 = (angle_m / 180.0) * Math.PI;
	double t_3 = Math.cos(t_2);
	double tmp;
	if ((angle_m / 180.0) <= 1e-241) {
		tmp = 2.0 * (Math.pow((b * Math.sqrt(Math.sin(t_1))), 2.0) * t_3);
	} else if ((angle_m / 180.0) <= 5e+65) {
		tmp = 2.0 * (t_3 * (t_0 * Math.sin(t_2)));
	} else if ((angle_m / 180.0) <= 2e+78) {
		tmp = 2.0 * (t_3 * (t_0 * Math.sin(Math.pow(Math.pow((0.005555555555555556 * (angle_m * Math.PI)), 3.0), 0.3333333333333333))));
	} else if ((angle_m / 180.0) <= 2e+91) {
		tmp = 2.0 * ((t_0 * Math.expm1(Math.log1p(Math.sin(Math.pow(Math.pow(t_1, 3.0), 0.3333333333333333))))) * Math.cos(Math.expm1(Math.log1p((angle_m * (Math.PI * 0.005555555555555556))))));
	} else {
		tmp = 2.0 * (t_0 * (Math.exp(Math.log1p(Math.sin((1.0 / (180.0 / (angle_m * Math.PI)))))) + -1.0));
	}
	return angle_s * tmp;
}

Python

angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = (b + a) * (b - a)
	t_1 = math.pi * (angle_m * 0.005555555555555556)
	t_2 = (angle_m / 180.0) * math.pi
	t_3 = math.cos(t_2)
	tmp = 0
	if (angle_m / 180.0) <= 1e-241:
		tmp = 2.0 * (math.pow((b * math.sqrt(math.sin(t_1))), 2.0) * t_3)
	elif (angle_m / 180.0) <= 5e+65:
		tmp = 2.0 * (t_3 * (t_0 * math.sin(t_2)))
	elif (angle_m / 180.0) <= 2e+78:
		tmp = 2.0 * (t_3 * (t_0 * math.sin(math.pow(math.pow((0.005555555555555556 * (angle_m * math.pi)), 3.0), 0.3333333333333333))))
	elif (angle_m / 180.0) <= 2e+91:
		tmp = 2.0 * ((t_0 * math.expm1(math.log1p(math.sin(math.pow(math.pow(t_1, 3.0), 0.3333333333333333))))) * math.cos(math.expm1(math.log1p((angle_m * (math.pi * 0.005555555555555556))))))
	else:
		tmp = 2.0 * (t_0 * (math.exp(math.log1p(math.sin((1.0 / (180.0 / (angle_m * math.pi)))))) + -1.0))
	return angle_s * tmp

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(Float64(b + a) * Float64(b - a))
	t_1 = Float64(pi * Float64(angle_m * 0.005555555555555556))
	t_2 = Float64(Float64(angle_m / 180.0) * pi)
	t_3 = cos(t_2)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e-241)
		tmp = Float64(2.0 * Float64((Float64(b * sqrt(sin(t_1))) ^ 2.0) * t_3));
	elseif (Float64(angle_m / 180.0) <= 5e+65)
		tmp = Float64(2.0 * Float64(t_3 * Float64(t_0 * sin(t_2))));
	elseif (Float64(angle_m / 180.0) <= 2e+78)
		tmp = Float64(2.0 * Float64(t_3 * Float64(t_0 * sin(((Float64(0.005555555555555556 * Float64(angle_m * pi)) ^ 3.0) ^ 0.3333333333333333)))));
	elseif (Float64(angle_m / 180.0) <= 2e+91)
		tmp = Float64(2.0 * Float64(Float64(t_0 * expm1(log1p(sin(((t_1 ^ 3.0) ^ 0.3333333333333333))))) * cos(expm1(log1p(Float64(angle_m * Float64(pi * 0.005555555555555556)))))));
	else
		tmp = Float64(2.0 * Float64(t_0 * Float64(exp(log1p(sin(Float64(1.0 / Float64(180.0 / Float64(angle_m * pi)))))) + -1.0)));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$3 = N[Cos[t$95$2], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e-241], N[(2.0 * N[(N[Power[N[(b * N[Sqrt[N[Sin[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+65], N[(2.0 * N[(t$95$3 * N[(t$95$0 * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+78], N[(2.0 * N[(t$95$3 * N[(t$95$0 * N[Sin[N[Power[N[Power[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+91], N[(2.0 * N[(N[(t$95$0 * N[(Exp[N[Log[1 + N[Sin[N[Power[N[Power[t$95$1, 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(Exp[N[Log[1 + N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$0 * N[(N[Exp[N[Log[1 + N[Sin[N[(1.0 / N[(180.0 / N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 angle 180) < 9.9999999999999997e-242

   1. Initial program 48.9%

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

      1. associate-*l* 48.9%

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

      2. associate-*l* 48.9%

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

   3. Simplified 48.9%

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

   4. Taylor expanded in b around inf 37.0%

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

      1. add-sqr-sqrt 20.1%

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

      2. pow2 20.1%

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

      3. div-inv 20.2%

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

      4. metadata-eval 20.2%

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

      5. sqrt-prod 11.2%

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

      6. unpow2 11.2%

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

      7. sqrt-prod 7.5%

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

      8. add-sqr-sqrt 14.3%

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

   6. Applied egg-rr 14.3%

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

   if 9.9999999999999997e-242 < (/.f64 angle 180) < 4.99999999999999973e65

   1. Initial program 67.6%

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

      1. associate-*l* 67.6%

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

      2. associate-*l* 67.6%

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

   3. Simplified 67.6%

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

      1. unpow2 67.6%

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

      2. unpow2 67.6%

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

      3. difference-of-squares 77.1%

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

   5. Applied egg-rr 77.1%

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

   if 4.99999999999999973e65 < (/.f64 angle 180) < 2.00000000000000002e78

   1. Initial program 4.9%

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

      1. associate-*l* 4.9%

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

      2. associate-*l* 4.9%

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

   3. Simplified 4.9%

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

      1. unpow2 4.9%

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

      2. unpow2 4.9%

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

      3. difference-of-squares 29.9%

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

   5. Applied egg-rr 29.9%

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

      1. expm1-log1p-u 29.9%

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

      2. div-inv 29.9%

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

      3. metadata-eval 29.9%

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

   7. Applied egg-rr 29.9%

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

      1. add-cbrt-cube 8.9%

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

      2. pow1/3 54.9%

         \[\leadsto 2 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \color{blue}{\left({\left(\left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{0.3333333333333333}\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. pow3 54.5%

         \[\leadsto 2 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left({\color{blue}{\left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{3}\right)}}^{0.3333333333333333}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   9. Applied egg-rr 54.5%

      \[\leadsto 2 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \color{blue}{\left({\left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{3}\right)}^{0.3333333333333333}\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   10. Taylor expanded in angle around inf 59.2%

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

   if 2.00000000000000002e78 < (/.f64 angle 180) < 2.00000000000000016e91

   1. Initial program 10.8%

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

      1. associate-*l* 10.8%

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

      2. associate-*l* 10.8%

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

   3. Simplified 10.8%

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

      1. unpow2 10.8%

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

      2. unpow2 10.8%

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

      3. difference-of-squares 10.8%

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

   5. Applied egg-rr 10.8%

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

      1. expm1-log1p-u 10.8%

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

      2. div-inv 10.8%

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

      3. metadata-eval 10.8%

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

   7. Applied egg-rr 10.8%

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

      1. add-cbrt-cube 0.7%

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

      2. pow1/3 50.7%

         \[\leadsto 2 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \color{blue}{\left({\left(\left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{0.3333333333333333}\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. pow3 50.7%

         \[\leadsto 2 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left({\color{blue}{\left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{3}\right)}}^{0.3333333333333333}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   9. Applied egg-rr 50.7%

      \[\leadsto 2 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \color{blue}{\left({\left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{3}\right)}^{0.3333333333333333}\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]
   10. Step-by-step derivation

       1. div-inv 50.7%

          \[\leadsto 2 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left({\left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{3}\right)}^{0.3333333333333333}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right) \]

       2. metadata-eval 50.7%

          \[\leadsto 2 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left({\left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{3}\right)}^{0.3333333333333333}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right)\right) \]

       3. expm1-log1p-u 58.9%

          \[\leadsto 2 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left({\left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{3}\right)}^{0.3333333333333333}\right)\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}\right) \]

       4. *-commutative 58.9%

          \[\leadsto 2 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left({\left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{3}\right)}^{0.3333333333333333}\right)\right)\right)\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(angle \cdot 0.005555555555555556\right) \cdot \pi}\right)\right)\right)\right) \]

       5. *-commutative 58.9%

          \[\leadsto 2 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left({\left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{3}\right)}^{0.3333333333333333}\right)\right)\right)\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(0.005555555555555556 \cdot angle\right)} \cdot \pi\right)\right)\right)\right) \]

       6. associate-*r* 58.9%

          \[\leadsto 2 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left({\left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{3}\right)}^{0.3333333333333333}\right)\right)\right)\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{0.005555555555555556 \cdot \left(angle \cdot \pi\right)}\right)\right)\right)\right) \]

       7. *-commutative 58.9%

          \[\leadsto 2 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left({\left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{3}\right)}^{0.3333333333333333}\right)\right)\right)\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(angle \cdot \pi\right) \cdot 0.005555555555555556}\right)\right)\right)\right) \]

       8. associate-*l* 58.9%

          \[\leadsto 2 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left({\left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{3}\right)}^{0.3333333333333333}\right)\right)\right)\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{angle \cdot \left(\pi \cdot 0.005555555555555556\right)}\right)\right)\right)\right) \]

   11. Applied egg-rr 58.9%

       \[\leadsto 2 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left({\left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{3}\right)}^{0.3333333333333333}\right)\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}\right) \]

   if 2.00000000000000016e91 < (/.f64 angle 180)

   1. Initial program 21.9%

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

      1. associate-*l* 21.9%

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

      2. associate-*l* 21.9%

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

   3. Simplified 21.9%

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

      1. unpow2 21.9%

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

      2. unpow2 21.9%

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

      3. difference-of-squares 21.9%

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

   5. Applied egg-rr 21.9%

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

   6. Taylor expanded in angle around 0 37.2%

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

      1. expm1-log1p-u 37.2%

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

      2. expm1-udef 37.2%

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

      3. div-inv 36.3%

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

      4. metadata-eval 36.3%

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

   8. Applied egg-rr 36.3%

      \[\leadsto 2 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} - 1\right)}\right) \cdot 1\right) \]
   9. Step-by-step derivation

      1. metadata-eval 36.3%

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

      2. div-inv 37.2%

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

      3. associate-*r/ 40.8%

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

      4. clear-num 43.7%

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

   10. Applied egg-rr 43.7%

       \[\leadsto 2 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(e^{\mathsf{log1p}\left(\sin \color{blue}{\left(\frac{1}{\frac{180}{\pi \cdot angle}}\right)}\right)} - 1\right)\right) \cdot 1\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{-241}:\\ \;\;\;\;2 \cdot \left({\left(b \cdot \sqrt{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)}^{2} \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+65}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+78}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left({\left({\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{3}\right)}^{0.3333333333333333}\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+91}:\\ \;\;\;\;2 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left({\left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{3}\right)}^{0.3333333333333333}\right)\right)\right)\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(e^{\mathsf{log1p}\left(\sin \left(\frac{1}{\frac{180}{angle \cdot \pi}}\right)\right)} + -1\right)\right)\\ \end{array} \]

Alternative 2: 57.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(b + a\right) \cdot \left(b - a\right)\\ t_1 := \frac{angle_m}{180} \cdot \pi\\ t_2 := \cos t_1\\ angle_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle_m}{180} \leq 10^{-241}:\\ \;\;\;\;2 \cdot \left({\left(b \cdot \sqrt{\sin \left(\pi \cdot \left(angle_m \cdot 0.005555555555555556\right)\right)}\right)}^{2} \cdot t_2\right)\\ \mathbf{elif}\;\frac{angle_m}{180} \leq 4 \cdot 10^{+72}:\\ \;\;\;\;2 \cdot \left(t_2 \cdot \left(t_0 \cdot \sin t_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_0 \cdot \left(e^{\mathsf{log1p}\left(\sin \left(\frac{1}{\frac{180}{angle_m \cdot \pi}}\right)\right)} + -1\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* (+ b a) (- b a)))
        (t_1 (* (/ angle_m 180.0) PI))
        (t_2 (cos t_1)))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 1e-241)
      (*
       2.0
       (*
        (pow (* b (sqrt (sin (* PI (* angle_m 0.005555555555555556))))) 2.0)
        t_2))
      (if (<= (/ angle_m 180.0) 4e+72)
        (* 2.0 (* t_2 (* t_0 (sin t_1))))
        (*
         2.0
         (*
          t_0
          (+ (exp (log1p (sin (/ 1.0 (/ 180.0 (* angle_m PI)))))) -1.0))))))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (b + a) * (b - a);
	double t_1 = (angle_m / 180.0) * ((double) M_PI);
	double t_2 = cos(t_1);
	double tmp;
	if ((angle_m / 180.0) <= 1e-241) {
		tmp = 2.0 * (pow((b * sqrt(sin((((double) M_PI) * (angle_m * 0.005555555555555556))))), 2.0) * t_2);
	} else if ((angle_m / 180.0) <= 4e+72) {
		tmp = 2.0 * (t_2 * (t_0 * sin(t_1)));
	} else {
		tmp = 2.0 * (t_0 * (exp(log1p(sin((1.0 / (180.0 / (angle_m * ((double) M_PI))))))) + -1.0));
	}
	return angle_s * tmp;
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (b + a) * (b - a);
	double t_1 = (angle_m / 180.0) * Math.PI;
	double t_2 = Math.cos(t_1);
	double tmp;
	if ((angle_m / 180.0) <= 1e-241) {
		tmp = 2.0 * (Math.pow((b * Math.sqrt(Math.sin((Math.PI * (angle_m * 0.005555555555555556))))), 2.0) * t_2);
	} else if ((angle_m / 180.0) <= 4e+72) {
		tmp = 2.0 * (t_2 * (t_0 * Math.sin(t_1)));
	} else {
		tmp = 2.0 * (t_0 * (Math.exp(Math.log1p(Math.sin((1.0 / (180.0 / (angle_m * Math.PI)))))) + -1.0));
	}
	return angle_s * tmp;
}

Python

angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = (b + a) * (b - a)
	t_1 = (angle_m / 180.0) * math.pi
	t_2 = math.cos(t_1)
	tmp = 0
	if (angle_m / 180.0) <= 1e-241:
		tmp = 2.0 * (math.pow((b * math.sqrt(math.sin((math.pi * (angle_m * 0.005555555555555556))))), 2.0) * t_2)
	elif (angle_m / 180.0) <= 4e+72:
		tmp = 2.0 * (t_2 * (t_0 * math.sin(t_1)))
	else:
		tmp = 2.0 * (t_0 * (math.exp(math.log1p(math.sin((1.0 / (180.0 / (angle_m * math.pi)))))) + -1.0))
	return angle_s * tmp

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(Float64(b + a) * Float64(b - a))
	t_1 = Float64(Float64(angle_m / 180.0) * pi)
	t_2 = cos(t_1)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e-241)
		tmp = Float64(2.0 * Float64((Float64(b * sqrt(sin(Float64(pi * Float64(angle_m * 0.005555555555555556))))) ^ 2.0) * t_2));
	elseif (Float64(angle_m / 180.0) <= 4e+72)
		tmp = Float64(2.0 * Float64(t_2 * Float64(t_0 * sin(t_1))));
	else
		tmp = Float64(2.0 * Float64(t_0 * Float64(exp(log1p(sin(Float64(1.0 / Float64(180.0 / Float64(angle_m * pi)))))) + -1.0)));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$1], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e-241], N[(2.0 * N[(N[Power[N[(b * N[Sqrt[N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+72], N[(2.0 * N[(t$95$2 * N[(t$95$0 * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$0 * N[(N[Exp[N[Log[1 + N[Sin[N[(1.0 / N[(180.0 / N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < 9.9999999999999997e-242

   1. Initial program 48.9%

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

      1. associate-*l* 48.9%

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

      2. associate-*l* 48.9%

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

   3. Simplified 48.9%

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

   4. Taylor expanded in b around inf 37.0%

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

      1. add-sqr-sqrt 20.1%

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

      2. pow2 20.1%

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

      3. div-inv 20.2%

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

      4. metadata-eval 20.2%

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

      5. sqrt-prod 11.2%

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

      6. unpow2 11.2%

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

      7. sqrt-prod 7.5%

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

      8. add-sqr-sqrt 14.3%

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

   6. Applied egg-rr 14.3%

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

   if 9.9999999999999997e-242 < (/.f64 angle 180) < 3.99999999999999978e72

   1. Initial program 65.5%

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

      1. associate-*l* 65.5%

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

      2. associate-*l* 65.5%

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

   3. Simplified 65.5%

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

      1. unpow2 65.5%

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

      2. unpow2 65.5%

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

      3. difference-of-squares 76.3%

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

   5. Applied egg-rr 76.3%

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

   if 3.99999999999999978e72 < (/.f64 angle 180)

   1. Initial program 20.7%

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

      1. associate-*l* 20.7%

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

      2. associate-*l* 20.7%

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

   3. Simplified 20.7%

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

      1. unpow2 20.7%

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

      2. unpow2 20.7%

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

      3. difference-of-squares 20.8%

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

   5. Applied egg-rr 20.8%

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

   6. Taylor expanded in angle around 0 36.9%

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

      1. expm1-log1p-u 36.9%

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

      2. expm1-udef 36.9%

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

      3. div-inv 36.0%

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

      4. metadata-eval 36.0%

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

   8. Applied egg-rr 36.0%

      \[\leadsto 2 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} - 1\right)}\right) \cdot 1\right) \]
   9. Step-by-step derivation

      1. metadata-eval 36.0%

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

      2. div-inv 36.9%

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

      3. associate-*r/ 39.7%

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

      4. clear-num 42.0%

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

   10. Applied egg-rr 42.0%

       \[\leadsto 2 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(e^{\mathsf{log1p}\left(\sin \color{blue}{\left(\frac{1}{\frac{180}{\pi \cdot angle}}\right)}\right)} - 1\right)\right) \cdot 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{-241}:\\ \;\;\;\;2 \cdot \left({\left(b \cdot \sqrt{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)}^{2} \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 4 \cdot 10^{+72}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(e^{\mathsf{log1p}\left(\sin \left(\frac{1}{\frac{180}{angle \cdot \pi}}\right)\right)} + -1\right)\right)\\ \end{array} \]

Alternative 3: 57.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle_m \cdot \pi\right)\right)\\ angle_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle_m}{180} \leq 10^{+44}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{angle_m}{180} \cdot \pi\right) \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot t_0\\ \end{array} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (+ b a) (- b a))
          (sin (* 0.005555555555555556 (* angle_m PI))))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 1e+44)
      (* 2.0 (* (cos (* (/ angle_m 180.0) PI)) t_0))
      (* 2.0 t_0)))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = ((b + a) * (b - a)) * sin((0.005555555555555556 * (angle_m * ((double) M_PI))));
	double tmp;
	if ((angle_m / 180.0) <= 1e+44) {
		tmp = 2.0 * (cos(((angle_m / 180.0) * ((double) M_PI))) * t_0);
	} else {
		tmp = 2.0 * t_0;
	}
	return angle_s * tmp;
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = ((b + a) * (b - a)) * Math.sin((0.005555555555555556 * (angle_m * Math.PI)));
	double tmp;
	if ((angle_m / 180.0) <= 1e+44) {
		tmp = 2.0 * (Math.cos(((angle_m / 180.0) * Math.PI)) * t_0);
	} else {
		tmp = 2.0 * t_0;
	}
	return angle_s * tmp;
}

Python

angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = ((b + a) * (b - a)) * math.sin((0.005555555555555556 * (angle_m * math.pi)))
	tmp = 0
	if (angle_m / 180.0) <= 1e+44:
		tmp = 2.0 * (math.cos(((angle_m / 180.0) * math.pi)) * t_0)
	else:
		tmp = 2.0 * t_0
	return angle_s * tmp

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(Float64(Float64(b + a) * Float64(b - a)) * sin(Float64(0.005555555555555556 * Float64(angle_m * pi))))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e+44)
		tmp = Float64(2.0 * Float64(cos(Float64(Float64(angle_m / 180.0) * pi)) * t_0));
	else
		tmp = Float64(2.0 * t_0);
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = ((b + a) * (b - a)) * sin((0.005555555555555556 * (angle_m * pi)));
	tmp = 0.0;
	if ((angle_m / 180.0) <= 1e+44)
		tmp = 2.0 * (cos(((angle_m / 180.0) * pi)) * t_0);
	else
		tmp = 2.0 * t_0;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+44], N[(2.0 * N[(N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(2.0 * t$95$0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < 1.0000000000000001e44

   1. Initial program 54.7%

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

      1. associate-*l* 54.7%

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

      2. associate-*l* 54.7%

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

   3. Simplified 54.7%

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

      1. unpow2 54.7%

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

      2. unpow2 54.7%

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

      3. difference-of-squares 61.9%

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

   5. Applied egg-rr 61.9%

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

   6. Taylor expanded in angle around inf 63.7%

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

   if 1.0000000000000001e44 < (/.f64 angle 180)

   1. Initial program 22.3%

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

      1. associate-*l* 22.3%

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

      2. associate-*l* 22.3%

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

   3. Simplified 22.3%

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

      1. unpow2 22.3%

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

      2. unpow2 22.3%

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

      3. difference-of-squares 26.5%

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

   5. Applied egg-rr 26.5%

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

   6. Taylor expanded in angle around 0 38.2%

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

   7. Taylor expanded in angle around inf 40.8%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{+44}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 4: 57.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(b + a\right) \cdot \left(b - a\right)\\ angle_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle_m}{180} \leq 4 \cdot 10^{+72}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{angle_m}{180} \cdot \pi\right) \cdot \left(t_0 \cdot \sin \left(angle_m \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_0 \cdot \sin \left(0.005555555555555556 \cdot \left(angle_m \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* (+ b a) (- b a))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 4e+72)
      (*
       2.0
       (*
        (cos (* (/ angle_m 180.0) PI))
        (* t_0 (sin (* angle_m (* PI 0.005555555555555556))))))
      (* 2.0 (* t_0 (sin (* 0.005555555555555556 (* angle_m PI)))))))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (b + a) * (b - a);
	double tmp;
	if ((angle_m / 180.0) <= 4e+72) {
		tmp = 2.0 * (cos(((angle_m / 180.0) * ((double) M_PI))) * (t_0 * sin((angle_m * (((double) M_PI) * 0.005555555555555556)))));
	} else {
		tmp = 2.0 * (t_0 * sin((0.005555555555555556 * (angle_m * ((double) M_PI)))));
	}
	return angle_s * tmp;
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (b + a) * (b - a);
	double tmp;
	if ((angle_m / 180.0) <= 4e+72) {
		tmp = 2.0 * (Math.cos(((angle_m / 180.0) * Math.PI)) * (t_0 * Math.sin((angle_m * (Math.PI * 0.005555555555555556)))));
	} else {
		tmp = 2.0 * (t_0 * Math.sin((0.005555555555555556 * (angle_m * Math.PI))));
	}
	return angle_s * tmp;
}

Python

angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = (b + a) * (b - a)
	tmp = 0
	if (angle_m / 180.0) <= 4e+72:
		tmp = 2.0 * (math.cos(((angle_m / 180.0) * math.pi)) * (t_0 * math.sin((angle_m * (math.pi * 0.005555555555555556)))))
	else:
		tmp = 2.0 * (t_0 * math.sin((0.005555555555555556 * (angle_m * math.pi))))
	return angle_s * tmp

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(Float64(b + a) * Float64(b - a))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e+72)
		tmp = Float64(2.0 * Float64(cos(Float64(Float64(angle_m / 180.0) * pi)) * Float64(t_0 * sin(Float64(angle_m * Float64(pi * 0.005555555555555556))))));
	else
		tmp = Float64(2.0 * Float64(t_0 * sin(Float64(0.005555555555555556 * Float64(angle_m * pi)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = (b + a) * (b - a);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 4e+72)
		tmp = 2.0 * (cos(((angle_m / 180.0) * pi)) * (t_0 * sin((angle_m * (pi * 0.005555555555555556)))));
	else
		tmp = 2.0 * (t_0 * sin((0.005555555555555556 * (angle_m * pi))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+72], N[(2.0 * N[(N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[Sin[N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$0 * N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < 3.99999999999999978e72

   1. Initial program 53.9%

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

      1. associate-*l* 53.9%

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

      2. associate-*l* 53.9%

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

   3. Simplified 53.9%

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

      1. unpow2 53.9%

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

      2. unpow2 53.9%

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

      3. difference-of-squares 61.9%

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

   5. Applied egg-rr 61.9%

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

   6. Taylor expanded in angle around inf 62.2%

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

      1. *-commutative 62.2%

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

      2. associate-*l* 63.3%

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

   8. Simplified 63.3%

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

   if 3.99999999999999978e72 < (/.f64 angle 180)

   1. Initial program 20.7%

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

      1. associate-*l* 20.7%

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

      2. associate-*l* 20.7%

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

   3. Simplified 20.7%

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

      1. unpow2 20.7%

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

      2. unpow2 20.7%

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

      3. difference-of-squares 20.8%

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

   5. Applied egg-rr 20.8%

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

   6. Taylor expanded in angle around 0 36.9%

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

   7. Taylor expanded in angle around inf 42.3%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{+72}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 5: 57.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(b + a\right) \cdot \left(b - a\right)\\ angle_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle_m}{180} \leq 4 \cdot 10^{+72}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{angle_m}{180} \cdot \pi\right) \cdot \left(\sin \left(\pi \cdot \left(angle_m \cdot 0.005555555555555556\right)\right) \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_0 \cdot \sin \left(0.005555555555555556 \cdot \left(angle_m \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* (+ b a) (- b a))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 4e+72)
      (*
       2.0
       (*
        (cos (* (/ angle_m 180.0) PI))
        (* (sin (* PI (* angle_m 0.005555555555555556))) t_0)))
      (* 2.0 (* t_0 (sin (* 0.005555555555555556 (* angle_m PI)))))))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (b + a) * (b - a);
	double tmp;
	if ((angle_m / 180.0) <= 4e+72) {
		tmp = 2.0 * (cos(((angle_m / 180.0) * ((double) M_PI))) * (sin((((double) M_PI) * (angle_m * 0.005555555555555556))) * t_0));
	} else {
		tmp = 2.0 * (t_0 * sin((0.005555555555555556 * (angle_m * ((double) M_PI)))));
	}
	return angle_s * tmp;
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (b + a) * (b - a);
	double tmp;
	if ((angle_m / 180.0) <= 4e+72) {
		tmp = 2.0 * (Math.cos(((angle_m / 180.0) * Math.PI)) * (Math.sin((Math.PI * (angle_m * 0.005555555555555556))) * t_0));
	} else {
		tmp = 2.0 * (t_0 * Math.sin((0.005555555555555556 * (angle_m * Math.PI))));
	}
	return angle_s * tmp;
}

Python

angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = (b + a) * (b - a)
	tmp = 0
	if (angle_m / 180.0) <= 4e+72:
		tmp = 2.0 * (math.cos(((angle_m / 180.0) * math.pi)) * (math.sin((math.pi * (angle_m * 0.005555555555555556))) * t_0))
	else:
		tmp = 2.0 * (t_0 * math.sin((0.005555555555555556 * (angle_m * math.pi))))
	return angle_s * tmp

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(Float64(b + a) * Float64(b - a))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e+72)
		tmp = Float64(2.0 * Float64(cos(Float64(Float64(angle_m / 180.0) * pi)) * Float64(sin(Float64(pi * Float64(angle_m * 0.005555555555555556))) * t_0)));
	else
		tmp = Float64(2.0 * Float64(t_0 * sin(Float64(0.005555555555555556 * Float64(angle_m * pi)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = (b + a) * (b - a);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 4e+72)
		tmp = 2.0 * (cos(((angle_m / 180.0) * pi)) * (sin((pi * (angle_m * 0.005555555555555556))) * t_0));
	else
		tmp = 2.0 * (t_0 * sin((0.005555555555555556 * (angle_m * pi))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+72], N[(2.0 * N[(N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$0 * N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < 3.99999999999999978e72

   1. Initial program 53.9%

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

      1. associate-*l* 53.9%

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

      2. associate-*l* 53.9%

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

   3. Simplified 53.9%

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

      1. unpow2 53.9%

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

      2. unpow2 53.9%

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

      3. difference-of-squares 61.9%

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

   5. Applied egg-rr 61.9%

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

   6. Taylor expanded in angle around inf 62.2%

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

      1. *-commutative 62.2%

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

      2. *-commutative 62.2%

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

      3. associate-*r* 61.9%

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

   8. Simplified 61.9%

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

   if 3.99999999999999978e72 < (/.f64 angle 180)

   1. Initial program 20.7%

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

      1. associate-*l* 20.7%

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

      2. associate-*l* 20.7%

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

   3. Simplified 20.7%

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

      1. unpow2 20.7%

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

      2. unpow2 20.7%

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

      3. difference-of-squares 20.8%

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

   5. Applied egg-rr 20.8%

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

   6. Taylor expanded in angle around 0 36.9%

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

   7. Taylor expanded in angle around inf 42.3%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{+72}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 6: 57.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \frac{angle_m}{180} \cdot \pi\\ t_1 := \left(b + a\right) \cdot \left(b - a\right)\\ angle_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle_m}{180} \leq 4 \cdot 10^{+72}:\\ \;\;\;\;2 \cdot \left(\cos t_0 \cdot \left(t_1 \cdot \sin t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \sin \left(0.005555555555555556 \cdot \left(angle_m \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* (/ angle_m 180.0) PI)) (t_1 (* (+ b a) (- b a))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 4e+72)
      (* 2.0 (* (cos t_0) (* t_1 (sin t_0))))
      (* 2.0 (* t_1 (sin (* 0.005555555555555556 (* angle_m PI)))))))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (angle_m / 180.0) * ((double) M_PI);
	double t_1 = (b + a) * (b - a);
	double tmp;
	if ((angle_m / 180.0) <= 4e+72) {
		tmp = 2.0 * (cos(t_0) * (t_1 * sin(t_0)));
	} else {
		tmp = 2.0 * (t_1 * sin((0.005555555555555556 * (angle_m * ((double) M_PI)))));
	}
	return angle_s * tmp;
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (angle_m / 180.0) * Math.PI;
	double t_1 = (b + a) * (b - a);
	double tmp;
	if ((angle_m / 180.0) <= 4e+72) {
		tmp = 2.0 * (Math.cos(t_0) * (t_1 * Math.sin(t_0)));
	} else {
		tmp = 2.0 * (t_1 * Math.sin((0.005555555555555556 * (angle_m * Math.PI))));
	}
	return angle_s * tmp;
}

Python

angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = (angle_m / 180.0) * math.pi
	t_1 = (b + a) * (b - a)
	tmp = 0
	if (angle_m / 180.0) <= 4e+72:
		tmp = 2.0 * (math.cos(t_0) * (t_1 * math.sin(t_0)))
	else:
		tmp = 2.0 * (t_1 * math.sin((0.005555555555555556 * (angle_m * math.pi))))
	return angle_s * tmp

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(Float64(angle_m / 180.0) * pi)
	t_1 = Float64(Float64(b + a) * Float64(b - a))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e+72)
		tmp = Float64(2.0 * Float64(cos(t_0) * Float64(t_1 * sin(t_0))));
	else
		tmp = Float64(2.0 * Float64(t_1 * sin(Float64(0.005555555555555556 * Float64(angle_m * pi)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = (angle_m / 180.0) * pi;
	t_1 = (b + a) * (b - a);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 4e+72)
		tmp = 2.0 * (cos(t_0) * (t_1 * sin(t_0)));
	else
		tmp = 2.0 * (t_1 * sin((0.005555555555555556 * (angle_m * pi))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+72], N[(2.0 * N[(N[Cos[t$95$0], $MachinePrecision] * N[(t$95$1 * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$1 * N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < 3.99999999999999978e72

   1. Initial program 53.9%

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

      1. associate-*l* 53.9%

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

      2. associate-*l* 53.9%

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

   3. Simplified 53.9%

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

      1. unpow2 53.9%

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

      2. unpow2 53.9%

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

      3. difference-of-squares 61.9%

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

   5. Applied egg-rr 61.9%

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

   if 3.99999999999999978e72 < (/.f64 angle 180)

   1. Initial program 20.7%

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

      1. associate-*l* 20.7%

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

      2. associate-*l* 20.7%

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

   3. Simplified 20.7%

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

      1. unpow2 20.7%

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

      2. unpow2 20.7%

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

      3. difference-of-squares 20.8%

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

   5. Applied egg-rr 20.8%

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

   6. Taylor expanded in angle around 0 36.9%

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

   7. Taylor expanded in angle around inf 42.3%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{+72}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 7: 58.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \frac{angle_m}{180} \cdot \pi\\ t_1 := \left(b + a\right) \cdot \left(b - a\right)\\ angle_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle_m}{180} \leq 4 \cdot 10^{+72}:\\ \;\;\;\;2 \cdot \left(\cos t_0 \cdot \left(t_1 \cdot \sin t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \left(e^{\mathsf{log1p}\left(\sin \left(\frac{1}{\frac{180}{angle_m \cdot \pi}}\right)\right)} + -1\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* (/ angle_m 180.0) PI)) (t_1 (* (+ b a) (- b a))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 4e+72)
      (* 2.0 (* (cos t_0) (* t_1 (sin t_0))))
      (*
       2.0
       (*
        t_1
        (+ (exp (log1p (sin (/ 1.0 (/ 180.0 (* angle_m PI)))))) -1.0)))))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (angle_m / 180.0) * ((double) M_PI);
	double t_1 = (b + a) * (b - a);
	double tmp;
	if ((angle_m / 180.0) <= 4e+72) {
		tmp = 2.0 * (cos(t_0) * (t_1 * sin(t_0)));
	} else {
		tmp = 2.0 * (t_1 * (exp(log1p(sin((1.0 / (180.0 / (angle_m * ((double) M_PI))))))) + -1.0));
	}
	return angle_s * tmp;
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (angle_m / 180.0) * Math.PI;
	double t_1 = (b + a) * (b - a);
	double tmp;
	if ((angle_m / 180.0) <= 4e+72) {
		tmp = 2.0 * (Math.cos(t_0) * (t_1 * Math.sin(t_0)));
	} else {
		tmp = 2.0 * (t_1 * (Math.exp(Math.log1p(Math.sin((1.0 / (180.0 / (angle_m * Math.PI)))))) + -1.0));
	}
	return angle_s * tmp;
}

Python

angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = (angle_m / 180.0) * math.pi
	t_1 = (b + a) * (b - a)
	tmp = 0
	if (angle_m / 180.0) <= 4e+72:
		tmp = 2.0 * (math.cos(t_0) * (t_1 * math.sin(t_0)))
	else:
		tmp = 2.0 * (t_1 * (math.exp(math.log1p(math.sin((1.0 / (180.0 / (angle_m * math.pi)))))) + -1.0))
	return angle_s * tmp

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(Float64(angle_m / 180.0) * pi)
	t_1 = Float64(Float64(b + a) * Float64(b - a))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e+72)
		tmp = Float64(2.0 * Float64(cos(t_0) * Float64(t_1 * sin(t_0))));
	else
		tmp = Float64(2.0 * Float64(t_1 * Float64(exp(log1p(sin(Float64(1.0 / Float64(180.0 / Float64(angle_m * pi)))))) + -1.0)));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+72], N[(2.0 * N[(N[Cos[t$95$0], $MachinePrecision] * N[(t$95$1 * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$1 * N[(N[Exp[N[Log[1 + N[Sin[N[(1.0 / N[(180.0 / N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < 3.99999999999999978e72

   1. Initial program 53.9%

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

      1. associate-*l* 53.9%

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

      2. associate-*l* 53.9%

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

   3. Simplified 53.9%

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

      1. unpow2 53.9%

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

      2. unpow2 53.9%

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

      3. difference-of-squares 61.9%

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

   5. Applied egg-rr 61.9%

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

   if 3.99999999999999978e72 < (/.f64 angle 180)

   1. Initial program 20.7%

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

      1. associate-*l* 20.7%

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

      2. associate-*l* 20.7%

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

   3. Simplified 20.7%

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

      1. unpow2 20.7%

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

      2. unpow2 20.7%

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

      3. difference-of-squares 20.8%

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

   5. Applied egg-rr 20.8%

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

   6. Taylor expanded in angle around 0 36.9%

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

      1. expm1-log1p-u 36.9%

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

      2. expm1-udef 36.9%

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

      3. div-inv 36.0%

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

      4. metadata-eval 36.0%

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

   8. Applied egg-rr 36.0%

      \[\leadsto 2 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} - 1\right)}\right) \cdot 1\right) \]
   9. Step-by-step derivation

      1. metadata-eval 36.0%

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

      2. div-inv 36.9%

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

      3. associate-*r/ 39.7%

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

      4. clear-num 42.0%

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

   10. Applied egg-rr 42.0%

       \[\leadsto 2 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(e^{\mathsf{log1p}\left(\sin \color{blue}{\left(\frac{1}{\frac{180}{\pi \cdot angle}}\right)}\right)} - 1\right)\right) \cdot 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{+72}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(e^{\mathsf{log1p}\left(\sin \left(\frac{1}{\frac{180}{angle \cdot \pi}}\right)\right)} + -1\right)\right)\\ \end{array} \]

Alternative 8: 56.1% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(2.0 * N[(N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.6%

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

   1. associate-*l* 48.6%

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

   2. associate-*l* 48.6%

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

3. Simplified 48.6%

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

   1. unpow2 48.6%

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

   2. unpow2 48.6%

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

   3. difference-of-squares 55.3%

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

5. Applied egg-rr 55.3%

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

6. Taylor expanded in angle around 0 56.9%

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

7. Taylor expanded in angle around inf 59.0%

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

8. Final simplification 59.0%

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

Alternative 9: 54.6% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(2.0 * N[(N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.6%

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

   1. associate-*l* 48.6%

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

   2. associate-*l* 48.6%

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

3. Simplified 48.6%

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

   1. unpow2 48.6%

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

   2. unpow2 48.6%

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

   3. difference-of-squares 55.3%

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

5. Applied egg-rr 55.3%

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

6. Taylor expanded in angle around 0 56.9%

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

7. Taylor expanded in angle around 0 52.2%

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

8. Final simplification 52.2%

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

Alternative 10: 54.6% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(2.0 * N[(N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.6%

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

   1. associate-*l* 48.6%

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

   2. associate-*l* 48.6%

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

3. Simplified 48.6%

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

   1. unpow2 48.6%

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

   2. unpow2 48.6%

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

   3. difference-of-squares 55.3%

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

5. Applied egg-rr 55.3%

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

6. Taylor expanded in angle around 0 56.9%

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

7. Taylor expanded in angle around 0 52.2%

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

   1. *-commutative 52.2%

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

   2. associate-*l* 52.2%

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

9. Simplified 52.2%

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

10. Final simplification 52.2%

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

Alternative 11: 54.6% accurate, 5.5× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (* angle_s (* 0.011111111111111112 (* angle_m (* PI (* (+ b a) (- b a)))))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * (angle_m * (((double) M_PI) * ((b + a) * (b - a)))));
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * (angle_m * (Math.PI * ((b + a) * (b - a)))));
}

Python

angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * (0.011111111111111112 * (angle_m * (math.pi * ((b + a) * (b - a)))))

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(Float64(b + a) * Float64(b - a))))))
end

MATLAB

angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * (0.011111111111111112 * (angle_m * (pi * ((b + a) * (b - a)))));
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.6%

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

   1. associate-*l* 48.6%

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

   2. associate-*l* 48.6%

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

3. Simplified 48.6%

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

   1. unpow2 48.6%

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

   2. unpow2 48.6%

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

   3. difference-of-squares 55.3%

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

5. Applied egg-rr 55.3%

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

6. Taylor expanded in angle around 0 56.9%

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

7. Taylor expanded in angle around 0 52.1%

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

8. Final simplification 52.1%

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

Alternative 12: 54.6% accurate, 5.5× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (* angle_s (* 0.011111111111111112 (* (* (+ b a) (- b a)) (* angle_m PI)))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * (((b + a) * (b - a)) * (angle_m * ((double) M_PI))));
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * (((b + a) * (b - a)) * (angle_m * Math.PI)));
}

Python

angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * (0.011111111111111112 * (((b + a) * (b - a)) * (angle_m * math.pi)))

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(0.011111111111111112 * Float64(Float64(Float64(b + a) * Float64(b - a)) * Float64(angle_m * pi))))
end

MATLAB

angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * (0.011111111111111112 * (((b + a) * (b - a)) * (angle_m * pi)));
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(0.011111111111111112 * N[(N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.6%

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

   1. associate-*l* 48.6%

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

   2. associate-*l* 48.6%

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

3. Simplified 48.6%

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

   1. unpow2 48.6%

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

   2. unpow2 48.6%

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

   3. difference-of-squares 55.3%

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

5. Applied egg-rr 55.3%

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

6. Taylor expanded in angle around 0 56.9%

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

7. Taylor expanded in angle around 0 52.1%

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

   1. associate-*r* 52.1%

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

9. Simplified 52.1%

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

10. Final simplification 52.1%

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

Reproduce

herbie shell --seed 2023333 
(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)))))