Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.4% → 86.0%

Time: 19.0s

Alternatives: 19

Speedup: 3.8×

• 39.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) - 1.0));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) - 1.0d0))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) - 1.0));
}

Python

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) - 1.0))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) - 1.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) - 1.0));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 35.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) - 1.0));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) - 1.0d0))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) - 1.0));
}

Python

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) - 1.0))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) - 1.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) - 1.0));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 86.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t_2 := \sqrt[3]{\sin k \cdot \tan k}\\ t_3 := t\_1 \cdot t\_2\\ t_4 := \frac{\sqrt{2}}{k}\\ \mathbf{if}\;\ell \cdot \ell \leq 10^{-271}:\\ \;\;\;\;\left(\sqrt{2} \cdot \frac{t}{k}\right) \cdot \left({\left(t \cdot t\_3\right)}^{-2} \cdot \frac{t\_4}{t\_3}\right)\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+213}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot t\_4}{{\left(t\_2 \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{\frac{t\_4}{t\_1}}{t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (cbrt l) -2.0))
        (t_2 (cbrt (* (sin k) (tan k))))
        (t_3 (* t_1 t_2))
        (t_4 (/ (sqrt 2.0) k)))
   (if (<= (* l l) 1e-271)
     (* (* (sqrt 2.0) (/ t k)) (* (pow (* t t_3) -2.0) (/ t_4 t_3)))
     (if (<= (* l l) 1e+213)
       (*
        (/ (pow l 2.0) (pow k 2.0))
        (* (/ 2.0 (pow (sin k) 2.0)) (/ (cos k) t)))
       (*
        (/ (* t t_4) (pow (* t_2 (/ t (pow (cbrt l) 2.0))) 2.0))
        (/ (/ t_4 t_1) t_2))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(cbrt(l), -2.0);
	double t_2 = cbrt((sin(k) * tan(k)));
	double t_3 = t_1 * t_2;
	double t_4 = sqrt(2.0) / k;
	double tmp;
	if ((l * l) <= 1e-271) {
		tmp = (sqrt(2.0) * (t / k)) * (pow((t * t_3), -2.0) * (t_4 / t_3));
	} else if ((l * l) <= 1e+213) {
		tmp = (pow(l, 2.0) / pow(k, 2.0)) * ((2.0 / pow(sin(k), 2.0)) * (cos(k) / t));
	} else {
		tmp = ((t * t_4) / pow((t_2 * (t / pow(cbrt(l), 2.0))), 2.0)) * ((t_4 / t_1) / t_2);
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.cbrt(l), -2.0);
	double t_2 = Math.cbrt((Math.sin(k) * Math.tan(k)));
	double t_3 = t_1 * t_2;
	double t_4 = Math.sqrt(2.0) / k;
	double tmp;
	if ((l * l) <= 1e-271) {
		tmp = (Math.sqrt(2.0) * (t / k)) * (Math.pow((t * t_3), -2.0) * (t_4 / t_3));
	} else if ((l * l) <= 1e+213) {
		tmp = (Math.pow(l, 2.0) / Math.pow(k, 2.0)) * ((2.0 / Math.pow(Math.sin(k), 2.0)) * (Math.cos(k) / t));
	} else {
		tmp = ((t * t_4) / Math.pow((t_2 * (t / Math.pow(Math.cbrt(l), 2.0))), 2.0)) * ((t_4 / t_1) / t_2);
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cbrt(l) ^ -2.0
	t_2 = cbrt(Float64(sin(k) * tan(k)))
	t_3 = Float64(t_1 * t_2)
	t_4 = Float64(sqrt(2.0) / k)
	tmp = 0.0
	if (Float64(l * l) <= 1e-271)
		tmp = Float64(Float64(sqrt(2.0) * Float64(t / k)) * Float64((Float64(t * t_3) ^ -2.0) * Float64(t_4 / t_3)));
	elseif (Float64(l * l) <= 1e+213)
		tmp = Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(Float64(2.0 / (sin(k) ^ 2.0)) * Float64(cos(k) / t)));
	else
		tmp = Float64(Float64(Float64(t * t_4) / (Float64(t_2 * Float64(t / (cbrt(l) ^ 2.0))) ^ 2.0)) * Float64(Float64(t_4 / t_1) / t_2));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 1e-271], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / k), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(t * t$95$3), $MachinePrecision], -2.0], $MachinePrecision] * N[(t$95$4 / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 1e+213], N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t * t$95$4), $MachinePrecision] / N[Power[N[(t$95$2 * N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$4 / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 9.99999999999999963e-272

   1. Initial program 27.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 27.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 27.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 39.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 39.9%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 39.9%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 39.9%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 85.2%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 85.2%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}} \]

      2. associate-/r* 85.3%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \color{blue}{\frac{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

      3. associate-/r/ 85.2%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

   8. Simplified 85.2%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 85.2%

         \[\leadsto \color{blue}{\frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

   10. Applied egg-rr 85.3%

       \[\leadsto \color{blue}{\frac{\left(\frac{\sqrt{2} \cdot t}{k} \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}{\sqrt[3]{\sin k \cdot \tan k}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 85.3%

          \[\leadsto \color{blue}{\left(\frac{\sqrt{2} \cdot t}{k} \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

       2. associate-*l* 85.3%

          \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{k} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)} \]

       3. associate-*l/ 85.2%

          \[\leadsto \color{blue}{\left(\frac{\sqrt{2}}{k} \cdot t\right)} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

       4. *-commutative 85.2%

          \[\leadsto \color{blue}{\left(t \cdot \frac{\sqrt{2}}{k}\right)} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

       5. associate-*l* 85.2%

          \[\leadsto \left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left({\color{blue}{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

   12. Simplified 87.7%

       \[\leadsto \color{blue}{\left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)} \]
   13. Step-by-step derivation

       1. pow1 87.7%

          \[\leadsto \color{blue}{{\left(\left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)\right)}^{1}} \]

       2. associate-*r* 87.7%

          \[\leadsto {\color{blue}{\left(\left(\left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2}\right) \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)}}^{1} \]

       3. associate-*r/ 87.7%

          \[\leadsto {\left(\left(\color{blue}{\frac{t \cdot \sqrt{2}}{k}} \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2}\right) \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)}^{1} \]

       4. associate-*r* 85.3%

          \[\leadsto {\left(\left(\frac{t \cdot \sqrt{2}}{k} \cdot {\color{blue}{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}}^{-2}\right) \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)}^{1} \]

   14. Applied egg-rr 85.3%

       \[\leadsto \color{blue}{{\left(\left(\frac{t \cdot \sqrt{2}}{k} \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)}^{1}} \]
   15. Step-by-step derivation

       1. unpow1 85.3%

          \[\leadsto \color{blue}{\left(\frac{t \cdot \sqrt{2}}{k} \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

       2. associate-*l* 85.3%

          \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{k} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)} \]

       3. *-commutative 85.3%

          \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot t}}{k} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

       4. associate-/l* 85.3%

          \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \frac{t}{k}\right)} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

       5. associate-*l* 87.7%

          \[\leadsto \left(\sqrt{2} \cdot \frac{t}{k}\right) \cdot \left({\color{blue}{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

       6. associate-/r* 87.7%

          \[\leadsto \left(\sqrt{2} \cdot \frac{t}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \color{blue}{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}}}\right) \]

   16. Simplified 87.7%

       \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \frac{t}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}}\right)} \]

   if 9.99999999999999963e-272 < (*.f64 l l) < 9.99999999999999984e212

   1. Initial program 41.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 41.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 41.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 49.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 49.0%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 48.9%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 48.9%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 76.3%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 76.3%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}} \]

      2. associate-/r* 76.3%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \color{blue}{\frac{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

      3. associate-/r/ 76.4%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

   8. Simplified 76.4%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

   9. Taylor expanded in k around inf 90.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   10. Step-by-step derivation

       1. times-frac 91.9%

          \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k \cdot {\left(\sqrt{2}\right)}^{2}}{t \cdot {\sin k}^{2}}} \]

       2. *-commutative 91.9%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{{\left(\sqrt{2}\right)}^{2} \cdot \cos k}}{t \cdot {\sin k}^{2}} \]

       3. unpow2 91.9%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)} \cdot \cos k}{t \cdot {\sin k}^{2}} \]

       4. rem-square-sqrt 92.2%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{2} \cdot \cos k}{t \cdot {\sin k}^{2}} \]

       5. *-commutative 92.2%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{2 \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot t}} \]

       6. times-frac 92.2%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \color{blue}{\left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)} \]

   11. Simplified 92.2%

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)} \]

   if 9.99999999999999984e212 < (*.f64 l l)

   1. Initial program 35.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 35.1%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 35.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 35.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 35.1%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 35.1%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 35.1%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 86.2%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 86.2%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}} \]

      2. associate-/r* 86.2%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \color{blue}{\frac{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

      3. associate-/r/ 86.1%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

   8. Simplified 86.1%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]
   9. Step-by-step derivation

      1. div-inv 86.1%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \color{blue}{\left(\frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \frac{1}{\sqrt[3]{\sin k \cdot \tan k}}\right)} \]

      2. associate-/l* 86.2%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \left(\color{blue}{\left(\frac{\sqrt{2}}{k} \cdot \frac{t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\right)} \cdot \frac{1}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

      3. div-inv 86.1%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \left(\left(\frac{\sqrt{2}}{k} \cdot \frac{t}{\color{blue}{t \cdot \frac{1}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}\right) \cdot \frac{1}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

      4. pow-flip 86.1%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \left(\left(\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot \color{blue}{{\left(\sqrt[3]{\ell}\right)}^{\left(-2\right)}}}\right) \cdot \frac{1}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

      5. metadata-eval 86.1%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \left(\left(\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{\color{blue}{-2}}}\right) \cdot \frac{1}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

   10. Applied egg-rr 86.1%

       \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \color{blue}{\left(\left(\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}\right) \cdot \frac{1}{\sqrt[3]{\sin k \cdot \tan k}}\right)} \]
   11. Step-by-step derivation

       1. associate-*r/ 86.2%

          \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \color{blue}{\frac{\left(\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}\right) \cdot 1}{\sqrt[3]{\sin k \cdot \tan k}}} \]

       2. *-rgt-identity 86.2%

          \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

       3. *-commutative 86.2%

          \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\color{blue}{\frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}} \cdot \frac{\sqrt{2}}{k}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

       4. associate-/r* 86.2%

          \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\color{blue}{\frac{\frac{t}{t}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}} \cdot \frac{\sqrt{2}}{k}}{\sqrt[3]{\sin k \cdot \tan k}} \]

       5. *-inverses 86.2%

          \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\color{blue}{1}}{{\left(\sqrt[3]{\ell}\right)}^{-2}} \cdot \frac{\sqrt{2}}{k}}{\sqrt[3]{\sin k \cdot \tan k}} \]

       6. associate-*l/ 86.2%

          \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\color{blue}{\frac{1 \cdot \frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

       7. *-lft-identity 86.2%

          \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\color{blue}{\frac{\sqrt{2}}{k}}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

   12. Simplified 86.2%

       \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \color{blue}{\frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-271}:\\ \;\;\;\;\left(\sqrt{2} \cdot \frac{t}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}}\right)\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+213}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\\ \mathbf{if}\;\ell \cdot \ell \leq 10^{-271} \lor \neg \left(\ell \cdot \ell \leq 10^{+213}\right):\\ \;\;\;\;\left(\sqrt{2} \cdot \frac{t}{k}\right) \cdot \left({\left(t \cdot t\_1\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k}}{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (pow (cbrt l) -2.0) (cbrt (* (sin k) (tan k))))))
   (if (or (<= (* l l) 1e-271) (not (<= (* l l) 1e+213)))
     (*
      (* (sqrt 2.0) (/ t k))
      (* (pow (* t t_1) -2.0) (/ (/ (sqrt 2.0) k) t_1)))
     (*
      (/ (pow l 2.0) (pow k 2.0))
      (* (/ 2.0 (pow (sin k) 2.0)) (/ (cos k) t))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(cbrt(l), -2.0) * cbrt((sin(k) * tan(k)));
	double tmp;
	if (((l * l) <= 1e-271) || !((l * l) <= 1e+213)) {
		tmp = (sqrt(2.0) * (t / k)) * (pow((t * t_1), -2.0) * ((sqrt(2.0) / k) / t_1));
	} else {
		tmp = (pow(l, 2.0) / pow(k, 2.0)) * ((2.0 / pow(sin(k), 2.0)) * (cos(k) / t));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.cbrt(l), -2.0) * Math.cbrt((Math.sin(k) * Math.tan(k)));
	double tmp;
	if (((l * l) <= 1e-271) || !((l * l) <= 1e+213)) {
		tmp = (Math.sqrt(2.0) * (t / k)) * (Math.pow((t * t_1), -2.0) * ((Math.sqrt(2.0) / k) / t_1));
	} else {
		tmp = (Math.pow(l, 2.0) / Math.pow(k, 2.0)) * ((2.0 / Math.pow(Math.sin(k), 2.0)) * (Math.cos(k) / t));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64((cbrt(l) ^ -2.0) * cbrt(Float64(sin(k) * tan(k))))
	tmp = 0.0
	if ((Float64(l * l) <= 1e-271) || !(Float64(l * l) <= 1e+213))
		tmp = Float64(Float64(sqrt(2.0) * Float64(t / k)) * Float64((Float64(t * t_1) ^ -2.0) * Float64(Float64(sqrt(2.0) / k) / t_1)));
	else
		tmp = Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(Float64(2.0 / (sin(k) ^ 2.0)) * Float64(cos(k) / t)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[N[(l * l), $MachinePrecision], 1e-271], N[Not[LessEqual[N[(l * l), $MachinePrecision], 1e+213]], $MachinePrecision]], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / k), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(t * t$95$1), $MachinePrecision], -2.0], $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 9.99999999999999963e-272 or 9.99999999999999984e212 < (*.f64 l l)

   1. Initial program 30.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 30.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 30.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 37.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 37.7%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 37.7%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 37.7%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 85.7%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 85.7%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}} \]

      2. associate-/r* 85.7%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \color{blue}{\frac{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

      3. associate-/r/ 85.6%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

   8. Simplified 85.6%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 85.7%

         \[\leadsto \color{blue}{\frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

   10. Applied egg-rr 85.6%

       \[\leadsto \color{blue}{\frac{\left(\frac{\sqrt{2} \cdot t}{k} \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}{\sqrt[3]{\sin k \cdot \tan k}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 85.6%

          \[\leadsto \color{blue}{\left(\frac{\sqrt{2} \cdot t}{k} \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

       2. associate-*l* 85.1%

          \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{k} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)} \]

       3. associate-*l/ 85.1%

          \[\leadsto \color{blue}{\left(\frac{\sqrt{2}}{k} \cdot t\right)} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

       4. *-commutative 85.1%

          \[\leadsto \color{blue}{\left(t \cdot \frac{\sqrt{2}}{k}\right)} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

       5. associate-*l* 85.1%

          \[\leadsto \left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left({\color{blue}{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

   12. Simplified 86.5%

       \[\leadsto \color{blue}{\left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)} \]
   13. Step-by-step derivation

       1. pow1 86.5%

          \[\leadsto \color{blue}{{\left(\left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)\right)}^{1}} \]

       2. associate-*r* 87.0%

          \[\leadsto {\color{blue}{\left(\left(\left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2}\right) \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)}}^{1} \]

       3. associate-*r/ 87.0%

          \[\leadsto {\left(\left(\color{blue}{\frac{t \cdot \sqrt{2}}{k}} \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2}\right) \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)}^{1} \]

       4. associate-*r* 85.6%

          \[\leadsto {\left(\left(\frac{t \cdot \sqrt{2}}{k} \cdot {\color{blue}{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}}^{-2}\right) \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)}^{1} \]

   14. Applied egg-rr 85.6%

       \[\leadsto \color{blue}{{\left(\left(\frac{t \cdot \sqrt{2}}{k} \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)}^{1}} \]
   15. Step-by-step derivation

       1. unpow1 85.6%

          \[\leadsto \color{blue}{\left(\frac{t \cdot \sqrt{2}}{k} \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

       2. associate-*l* 85.1%

          \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{k} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)} \]

       3. *-commutative 85.1%

          \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot t}}{k} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

       4. associate-/l* 85.1%

          \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \frac{t}{k}\right)} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

       5. associate-*l* 86.4%

          \[\leadsto \left(\sqrt{2} \cdot \frac{t}{k}\right) \cdot \left({\color{blue}{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

       6. associate-/r* 86.5%

          \[\leadsto \left(\sqrt{2} \cdot \frac{t}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \color{blue}{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}}}\right) \]

   16. Simplified 86.5%

       \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \frac{t}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}}\right)} \]

   if 9.99999999999999963e-272 < (*.f64 l l) < 9.99999999999999984e212

   1. Initial program 41.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 41.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 41.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 49.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 49.0%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 48.9%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 48.9%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 76.3%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 76.3%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}} \]

      2. associate-/r* 76.3%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \color{blue}{\frac{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

      3. associate-/r/ 76.4%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

   8. Simplified 76.4%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

   9. Taylor expanded in k around inf 90.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   10. Step-by-step derivation

       1. times-frac 91.9%

          \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k \cdot {\left(\sqrt{2}\right)}^{2}}{t \cdot {\sin k}^{2}}} \]

       2. *-commutative 91.9%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{{\left(\sqrt{2}\right)}^{2} \cdot \cos k}}{t \cdot {\sin k}^{2}} \]

       3. unpow2 91.9%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)} \cdot \cos k}{t \cdot {\sin k}^{2}} \]

       4. rem-square-sqrt 92.2%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{2} \cdot \cos k}{t \cdot {\sin k}^{2}} \]

       5. *-commutative 92.2%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{2 \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot t}} \]

       6. times-frac 92.2%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \color{blue}{\left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)} \]

   11. Simplified 92.2%

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-271} \lor \neg \left(\ell \cdot \ell \leq 10^{+213}\right):\\ \;\;\;\;\left(\sqrt{2} \cdot \frac{t}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t_2 := \sqrt[3]{\sin k \cdot \tan k}\\ t_3 := \frac{\sqrt{2}}{k}\\ \mathbf{if}\;\ell \cdot \ell \leq 10^{-271} \lor \neg \left(\ell \cdot \ell \leq 10^{+213}\right):\\ \;\;\;\;\left(t \cdot t\_3\right) \cdot \left({\left(t \cdot \left(t\_1 \cdot t\_2\right)\right)}^{-2} \cdot \frac{\frac{t\_3}{t\_1}}{t\_2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (cbrt l) -2.0))
        (t_2 (cbrt (* (sin k) (tan k))))
        (t_3 (/ (sqrt 2.0) k)))
   (if (or (<= (* l l) 1e-271) (not (<= (* l l) 1e+213)))
     (* (* t t_3) (* (pow (* t (* t_1 t_2)) -2.0) (/ (/ t_3 t_1) t_2)))
     (*
      (/ (pow l 2.0) (pow k 2.0))
      (* (/ 2.0 (pow (sin k) 2.0)) (/ (cos k) t))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(cbrt(l), -2.0);
	double t_2 = cbrt((sin(k) * tan(k)));
	double t_3 = sqrt(2.0) / k;
	double tmp;
	if (((l * l) <= 1e-271) || !((l * l) <= 1e+213)) {
		tmp = (t * t_3) * (pow((t * (t_1 * t_2)), -2.0) * ((t_3 / t_1) / t_2));
	} else {
		tmp = (pow(l, 2.0) / pow(k, 2.0)) * ((2.0 / pow(sin(k), 2.0)) * (cos(k) / t));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.cbrt(l), -2.0);
	double t_2 = Math.cbrt((Math.sin(k) * Math.tan(k)));
	double t_3 = Math.sqrt(2.0) / k;
	double tmp;
	if (((l * l) <= 1e-271) || !((l * l) <= 1e+213)) {
		tmp = (t * t_3) * (Math.pow((t * (t_1 * t_2)), -2.0) * ((t_3 / t_1) / t_2));
	} else {
		tmp = (Math.pow(l, 2.0) / Math.pow(k, 2.0)) * ((2.0 / Math.pow(Math.sin(k), 2.0)) * (Math.cos(k) / t));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cbrt(l) ^ -2.0
	t_2 = cbrt(Float64(sin(k) * tan(k)))
	t_3 = Float64(sqrt(2.0) / k)
	tmp = 0.0
	if ((Float64(l * l) <= 1e-271) || !(Float64(l * l) <= 1e+213))
		tmp = Float64(Float64(t * t_3) * Float64((Float64(t * Float64(t_1 * t_2)) ^ -2.0) * Float64(Float64(t_3 / t_1) / t_2)));
	else
		tmp = Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(Float64(2.0 / (sin(k) ^ 2.0)) * Float64(cos(k) / t)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]}, If[Or[LessEqual[N[(l * l), $MachinePrecision], 1e-271], N[Not[LessEqual[N[(l * l), $MachinePrecision], 1e+213]], $MachinePrecision]], N[(N[(t * t$95$3), $MachinePrecision] * N[(N[Power[N[(t * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] * N[(N[(t$95$3 / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 9.99999999999999963e-272 or 9.99999999999999984e212 < (*.f64 l l)

   1. Initial program 30.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 30.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 30.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 37.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 37.7%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 37.7%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 37.7%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 85.7%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 85.7%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}} \]

      2. associate-/r* 85.7%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \color{blue}{\frac{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

      3. associate-/r/ 85.6%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

   8. Simplified 85.6%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 85.7%

         \[\leadsto \color{blue}{\frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

   10. Applied egg-rr 85.6%

       \[\leadsto \color{blue}{\frac{\left(\frac{\sqrt{2} \cdot t}{k} \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}{\sqrt[3]{\sin k \cdot \tan k}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 85.6%

          \[\leadsto \color{blue}{\left(\frac{\sqrt{2} \cdot t}{k} \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

       2. associate-*l* 85.1%

          \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{k} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)} \]

       3. associate-*l/ 85.1%

          \[\leadsto \color{blue}{\left(\frac{\sqrt{2}}{k} \cdot t\right)} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

       4. *-commutative 85.1%

          \[\leadsto \color{blue}{\left(t \cdot \frac{\sqrt{2}}{k}\right)} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

       5. associate-*l* 85.1%

          \[\leadsto \left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left({\color{blue}{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

   12. Simplified 86.5%

       \[\leadsto \color{blue}{\left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)} \]

   if 9.99999999999999963e-272 < (*.f64 l l) < 9.99999999999999984e212

   1. Initial program 41.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 41.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 41.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 49.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 49.0%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 48.9%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 48.9%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 76.3%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 76.3%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}} \]

      2. associate-/r* 76.3%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \color{blue}{\frac{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

      3. associate-/r/ 76.4%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

   8. Simplified 76.4%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

   9. Taylor expanded in k around inf 90.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   10. Step-by-step derivation

       1. times-frac 91.9%

          \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k \cdot {\left(\sqrt{2}\right)}^{2}}{t \cdot {\sin k}^{2}}} \]

       2. *-commutative 91.9%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{{\left(\sqrt{2}\right)}^{2} \cdot \cos k}}{t \cdot {\sin k}^{2}} \]

       3. unpow2 91.9%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)} \cdot \cos k}{t \cdot {\sin k}^{2}} \]

       4. rem-square-sqrt 92.2%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{2} \cdot \cos k}{t \cdot {\sin k}^{2}} \]

       5. *-commutative 92.2%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{2 \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot t}} \]

       6. times-frac 92.2%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \color{blue}{\left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)} \]

   11. Simplified 92.2%

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-271} \lor \neg \left(\ell \cdot \ell \leq 10^{+213}\right):\\ \;\;\;\;\left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t_2 := \sqrt[3]{\sin k \cdot \tan k}\\ t_3 := t\_2 \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ \mathbf{if}\;\ell \leq 1.55 \cdot 10^{-146}:\\ \;\;\;\;\left(t \cdot \frac{\sqrt[3]{{2}^{1.5}}}{k}\right) \cdot \left({\left(t \cdot \left(t\_1 \cdot t\_2\right)\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{t\_1}}{\sqrt[3]{{k}^{2}}}\right)\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+175}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{t\_3}^{2}} \cdot \frac{{\left(\frac{k}{t}\right)}^{-2}}{t\_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (cbrt l) -2.0))
        (t_2 (cbrt (* (sin k) (tan k))))
        (t_3 (* t_2 (/ t (pow (cbrt l) 2.0)))))
   (if (<= l 1.55e-146)
     (*
      (* t (/ (cbrt (pow 2.0 1.5)) k))
      (*
       (pow (* t (* t_1 t_2)) -2.0)
       (/ (/ (/ (sqrt 2.0) k) t_1) (cbrt (pow k 2.0)))))
     (if (<= l 1.2e+175)
       (*
        (/ (pow l 2.0) (pow k 2.0))
        (* (/ 2.0 (pow (sin k) 2.0)) (/ (cos k) t)))
       (* (/ 2.0 (pow t_3 2.0)) (/ (pow (/ k t) -2.0) t_3))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(cbrt(l), -2.0);
	double t_2 = cbrt((sin(k) * tan(k)));
	double t_3 = t_2 * (t / pow(cbrt(l), 2.0));
	double tmp;
	if (l <= 1.55e-146) {
		tmp = (t * (cbrt(pow(2.0, 1.5)) / k)) * (pow((t * (t_1 * t_2)), -2.0) * (((sqrt(2.0) / k) / t_1) / cbrt(pow(k, 2.0))));
	} else if (l <= 1.2e+175) {
		tmp = (pow(l, 2.0) / pow(k, 2.0)) * ((2.0 / pow(sin(k), 2.0)) * (cos(k) / t));
	} else {
		tmp = (2.0 / pow(t_3, 2.0)) * (pow((k / t), -2.0) / t_3);
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.cbrt(l), -2.0);
	double t_2 = Math.cbrt((Math.sin(k) * Math.tan(k)));
	double t_3 = t_2 * (t / Math.pow(Math.cbrt(l), 2.0));
	double tmp;
	if (l <= 1.55e-146) {
		tmp = (t * (Math.cbrt(Math.pow(2.0, 1.5)) / k)) * (Math.pow((t * (t_1 * t_2)), -2.0) * (((Math.sqrt(2.0) / k) / t_1) / Math.cbrt(Math.pow(k, 2.0))));
	} else if (l <= 1.2e+175) {
		tmp = (Math.pow(l, 2.0) / Math.pow(k, 2.0)) * ((2.0 / Math.pow(Math.sin(k), 2.0)) * (Math.cos(k) / t));
	} else {
		tmp = (2.0 / Math.pow(t_3, 2.0)) * (Math.pow((k / t), -2.0) / t_3);
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cbrt(l) ^ -2.0
	t_2 = cbrt(Float64(sin(k) * tan(k)))
	t_3 = Float64(t_2 * Float64(t / (cbrt(l) ^ 2.0)))
	tmp = 0.0
	if (l <= 1.55e-146)
		tmp = Float64(Float64(t * Float64(cbrt((2.0 ^ 1.5)) / k)) * Float64((Float64(t * Float64(t_1 * t_2)) ^ -2.0) * Float64(Float64(Float64(sqrt(2.0) / k) / t_1) / cbrt((k ^ 2.0)))));
	elseif (l <= 1.2e+175)
		tmp = Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(Float64(2.0 / (sin(k) ^ 2.0)) * Float64(cos(k) / t)));
	else
		tmp = Float64(Float64(2.0 / (t_3 ^ 2.0)) * Float64((Float64(k / t) ^ -2.0) / t_3));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 1.55e-146], N[(N[(t * N[(N[Power[N[Power[2.0, 1.5], $MachinePrecision], 1/3], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(t * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] * N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision] / t$95$1), $MachinePrecision] / N[Power[N[Power[k, 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.2e+175], N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(k / t), $MachinePrecision], -2.0], $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.5499999999999999e-146

   1. Initial program 34.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 34.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 34.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 42.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 42.6%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 42.6%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 42.6%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 80.7%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 80.7%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}} \]

      2. associate-/r* 80.7%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \color{blue}{\frac{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

      3. associate-/r/ 80.7%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

   8. Simplified 80.7%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 80.7%

         \[\leadsto \color{blue}{\frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

   10. Applied egg-rr 81.0%

       \[\leadsto \color{blue}{\frac{\left(\frac{\sqrt{2} \cdot t}{k} \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}{\sqrt[3]{\sin k \cdot \tan k}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 81.0%

          \[\leadsto \color{blue}{\left(\frac{\sqrt{2} \cdot t}{k} \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

       2. associate-*l* 81.0%

          \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{k} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)} \]

       3. associate-*l/ 81.0%

          \[\leadsto \color{blue}{\left(\frac{\sqrt{2}}{k} \cdot t\right)} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

       4. *-commutative 81.0%

          \[\leadsto \color{blue}{\left(t \cdot \frac{\sqrt{2}}{k}\right)} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

       5. associate-*l* 81.0%

          \[\leadsto \left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left({\color{blue}{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

   12. Simplified 82.3%

       \[\leadsto \color{blue}{\left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)} \]

   13. Taylor expanded in k around 0 71.9%

       \[\leadsto \left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\color{blue}{\sqrt[3]{{k}^{2}}}}\right) \]
   14. Step-by-step derivation

       1. add-cbrt-cube 71.9%

          \[\leadsto \left(t \cdot \frac{\color{blue}{\sqrt[3]{\left(\sqrt{2} \cdot \sqrt{2}\right) \cdot \sqrt{2}}}}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{{k}^{2}}}\right) \]

       2. pow1/3 71.9%

          \[\leadsto \left(t \cdot \frac{\color{blue}{{\left(\left(\sqrt{2} \cdot \sqrt{2}\right) \cdot \sqrt{2}\right)}^{0.3333333333333333}}}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{{k}^{2}}}\right) \]

       3. pow1/2 71.9%

          \[\leadsto \left(t \cdot \frac{{\left(\left(\color{blue}{{2}^{0.5}} \cdot \sqrt{2}\right) \cdot \sqrt{2}\right)}^{0.3333333333333333}}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{{k}^{2}}}\right) \]

       4. pow1/2 71.9%

          \[\leadsto \left(t \cdot \frac{{\left(\left({2}^{0.5} \cdot \color{blue}{{2}^{0.5}}\right) \cdot \sqrt{2}\right)}^{0.3333333333333333}}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{{k}^{2}}}\right) \]

       5. pow-prod-up 71.9%

          \[\leadsto \left(t \cdot \frac{{\left(\color{blue}{{2}^{\left(0.5 + 0.5\right)}} \cdot \sqrt{2}\right)}^{0.3333333333333333}}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{{k}^{2}}}\right) \]

       6. metadata-eval 71.9%

          \[\leadsto \left(t \cdot \frac{{\left({2}^{\color{blue}{1}} \cdot \sqrt{2}\right)}^{0.3333333333333333}}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{{k}^{2}}}\right) \]

       7. metadata-eval 71.9%

          \[\leadsto \left(t \cdot \frac{{\left(\color{blue}{2} \cdot \sqrt{2}\right)}^{0.3333333333333333}}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{{k}^{2}}}\right) \]

   15. Applied egg-rr 71.9%

       \[\leadsto \left(t \cdot \frac{\color{blue}{{\left(2 \cdot \sqrt{2}\right)}^{0.3333333333333333}}}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{{k}^{2}}}\right) \]
   16. Step-by-step derivation

       1. unpow1/3 71.9%

          \[\leadsto \left(t \cdot \frac{\color{blue}{\sqrt[3]{2 \cdot \sqrt{2}}}}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{{k}^{2}}}\right) \]

       2. *-commutative 71.9%

          \[\leadsto \left(t \cdot \frac{\sqrt[3]{\color{blue}{\sqrt{2} \cdot 2}}}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{{k}^{2}}}\right) \]

       3. unpow1/2 71.9%

          \[\leadsto \left(t \cdot \frac{\sqrt[3]{\color{blue}{{2}^{0.5}} \cdot 2}}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{{k}^{2}}}\right) \]

       4. pow-plus 71.9%

          \[\leadsto \left(t \cdot \frac{\sqrt[3]{\color{blue}{{2}^{\left(0.5 + 1\right)}}}}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{{k}^{2}}}\right) \]

       5. metadata-eval 71.9%

          \[\leadsto \left(t \cdot \frac{\sqrt[3]{{2}^{\color{blue}{1.5}}}}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{{k}^{2}}}\right) \]

   17. Simplified 71.9%

       \[\leadsto \left(t \cdot \frac{\color{blue}{\sqrt[3]{{2}^{1.5}}}}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{{k}^{2}}}\right) \]

   if 1.5499999999999999e-146 < l < 1.2e175

   1. Initial program 39.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 39.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 39.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 46.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 46.8%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 46.8%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 46.7%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 82.1%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 82.2%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}} \]

      2. associate-/r* 82.2%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \color{blue}{\frac{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

      3. associate-/r/ 82.2%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

   8. Simplified 82.2%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

   9. Taylor expanded in k around inf 84.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   10. Step-by-step derivation

       1. times-frac 92.3%

          \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k \cdot {\left(\sqrt{2}\right)}^{2}}{t \cdot {\sin k}^{2}}} \]

       2. *-commutative 92.3%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{{\left(\sqrt{2}\right)}^{2} \cdot \cos k}}{t \cdot {\sin k}^{2}} \]

       3. unpow2 92.3%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)} \cdot \cos k}{t \cdot {\sin k}^{2}} \]

       4. rem-square-sqrt 92.6%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{2} \cdot \cos k}{t \cdot {\sin k}^{2}} \]

       5. *-commutative 92.6%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{2 \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot t}} \]

       6. times-frac 92.6%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \color{blue}{\left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)} \]

   11. Simplified 92.6%

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)} \]

   if 1.2e175 < l

   1. Initial program 26.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 26.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 26.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 26.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cube-cbrt 26.4%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      2. div-inv 26.4%

         \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]

      3. times-frac 26.4%

         \[\leadsto \color{blue}{\frac{2}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\frac{1}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 78.1%

      \[\leadsto \color{blue}{\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{{\left(\frac{k}{t}\right)}^{-2}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.55 \cdot 10^{-146}:\\ \;\;\;\;\left(t \cdot \frac{\sqrt[3]{{2}^{1.5}}}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{{k}^{2}}}\right)\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+175}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{{\left(\frac{k}{t}\right)}^{-2}}{\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t_2 := \sqrt[3]{\sin k \cdot \tan k}\\ t_3 := \frac{\sqrt{2}}{k}\\ t_4 := t\_2 \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ \mathbf{if}\;\ell \leq 5 \cdot 10^{-146}:\\ \;\;\;\;\left(t \cdot t\_3\right) \cdot \left({\left(t \cdot \left(t\_1 \cdot t\_2\right)\right)}^{-2} \cdot \frac{\frac{t\_3}{t\_1}}{\sqrt[3]{{k}^{2}}}\right)\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+175}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{t\_4}^{2}} \cdot \frac{{\left(\frac{k}{t}\right)}^{-2}}{t\_4}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (cbrt l) -2.0))
        (t_2 (cbrt (* (sin k) (tan k))))
        (t_3 (/ (sqrt 2.0) k))
        (t_4 (* t_2 (/ t (pow (cbrt l) 2.0)))))
   (if (<= l 5e-146)
     (*
      (* t t_3)
      (* (pow (* t (* t_1 t_2)) -2.0) (/ (/ t_3 t_1) (cbrt (pow k 2.0)))))
     (if (<= l 1.2e+175)
       (*
        (/ (pow l 2.0) (pow k 2.0))
        (* (/ 2.0 (pow (sin k) 2.0)) (/ (cos k) t)))
       (* (/ 2.0 (pow t_4 2.0)) (/ (pow (/ k t) -2.0) t_4))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(cbrt(l), -2.0);
	double t_2 = cbrt((sin(k) * tan(k)));
	double t_3 = sqrt(2.0) / k;
	double t_4 = t_2 * (t / pow(cbrt(l), 2.0));
	double tmp;
	if (l <= 5e-146) {
		tmp = (t * t_3) * (pow((t * (t_1 * t_2)), -2.0) * ((t_3 / t_1) / cbrt(pow(k, 2.0))));
	} else if (l <= 1.2e+175) {
		tmp = (pow(l, 2.0) / pow(k, 2.0)) * ((2.0 / pow(sin(k), 2.0)) * (cos(k) / t));
	} else {
		tmp = (2.0 / pow(t_4, 2.0)) * (pow((k / t), -2.0) / t_4);
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.cbrt(l), -2.0);
	double t_2 = Math.cbrt((Math.sin(k) * Math.tan(k)));
	double t_3 = Math.sqrt(2.0) / k;
	double t_4 = t_2 * (t / Math.pow(Math.cbrt(l), 2.0));
	double tmp;
	if (l <= 5e-146) {
		tmp = (t * t_3) * (Math.pow((t * (t_1 * t_2)), -2.0) * ((t_3 / t_1) / Math.cbrt(Math.pow(k, 2.0))));
	} else if (l <= 1.2e+175) {
		tmp = (Math.pow(l, 2.0) / Math.pow(k, 2.0)) * ((2.0 / Math.pow(Math.sin(k), 2.0)) * (Math.cos(k) / t));
	} else {
		tmp = (2.0 / Math.pow(t_4, 2.0)) * (Math.pow((k / t), -2.0) / t_4);
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cbrt(l) ^ -2.0
	t_2 = cbrt(Float64(sin(k) * tan(k)))
	t_3 = Float64(sqrt(2.0) / k)
	t_4 = Float64(t_2 * Float64(t / (cbrt(l) ^ 2.0)))
	tmp = 0.0
	if (l <= 5e-146)
		tmp = Float64(Float64(t * t_3) * Float64((Float64(t * Float64(t_1 * t_2)) ^ -2.0) * Float64(Float64(t_3 / t_1) / cbrt((k ^ 2.0)))));
	elseif (l <= 1.2e+175)
		tmp = Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(Float64(2.0 / (sin(k) ^ 2.0)) * Float64(cos(k) / t)));
	else
		tmp = Float64(Float64(2.0 / (t_4 ^ 2.0)) * Float64((Float64(k / t) ^ -2.0) / t_4));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 * N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 5e-146], N[(N[(t * t$95$3), $MachinePrecision] * N[(N[Power[N[(t * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] * N[(N[(t$95$3 / t$95$1), $MachinePrecision] / N[Power[N[Power[k, 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.2e+175], N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Power[t$95$4, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(k / t), $MachinePrecision], -2.0], $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < 4.99999999999999957e-146

   1. Initial program 34.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 34.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 34.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 42.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 42.6%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 42.6%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 42.6%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 80.7%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 80.7%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}} \]

      2. associate-/r* 80.7%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \color{blue}{\frac{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

      3. associate-/r/ 80.7%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

   8. Simplified 80.7%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 80.7%

         \[\leadsto \color{blue}{\frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

   10. Applied egg-rr 81.0%

       \[\leadsto \color{blue}{\frac{\left(\frac{\sqrt{2} \cdot t}{k} \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}{\sqrt[3]{\sin k \cdot \tan k}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 81.0%

          \[\leadsto \color{blue}{\left(\frac{\sqrt{2} \cdot t}{k} \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

       2. associate-*l* 81.0%

          \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{k} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)} \]

       3. associate-*l/ 81.0%

          \[\leadsto \color{blue}{\left(\frac{\sqrt{2}}{k} \cdot t\right)} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

       4. *-commutative 81.0%

          \[\leadsto \color{blue}{\left(t \cdot \frac{\sqrt{2}}{k}\right)} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

       5. associate-*l* 81.0%

          \[\leadsto \left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left({\color{blue}{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

   12. Simplified 82.3%

       \[\leadsto \color{blue}{\left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)} \]

   13. Taylor expanded in k around 0 71.9%

       \[\leadsto \left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\color{blue}{\sqrt[3]{{k}^{2}}}}\right) \]

   if 4.99999999999999957e-146 < l < 1.2e175

   1. Initial program 39.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 39.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 39.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 46.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 46.8%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 46.8%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 46.7%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 82.1%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 82.2%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}} \]

      2. associate-/r* 82.2%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \color{blue}{\frac{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

      3. associate-/r/ 82.2%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

   8. Simplified 82.2%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

   9. Taylor expanded in k around inf 84.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   10. Step-by-step derivation

       1. times-frac 92.3%

          \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k \cdot {\left(\sqrt{2}\right)}^{2}}{t \cdot {\sin k}^{2}}} \]

       2. *-commutative 92.3%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{{\left(\sqrt{2}\right)}^{2} \cdot \cos k}}{t \cdot {\sin k}^{2}} \]

       3. unpow2 92.3%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)} \cdot \cos k}{t \cdot {\sin k}^{2}} \]

       4. rem-square-sqrt 92.6%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{2} \cdot \cos k}{t \cdot {\sin k}^{2}} \]

       5. *-commutative 92.6%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{2 \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot t}} \]

       6. times-frac 92.6%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \color{blue}{\left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)} \]

   11. Simplified 92.6%

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)} \]

   if 1.2e175 < l

   1. Initial program 26.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 26.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 26.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 26.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cube-cbrt 26.4%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      2. div-inv 26.4%

         \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]

      3. times-frac 26.4%

         \[\leadsto \color{blue}{\frac{2}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\frac{1}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 78.1%

      \[\leadsto \color{blue}{\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{{\left(\frac{k}{t}\right)}^{-2}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{-146}:\\ \;\;\;\;\left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{{k}^{2}}}\right)\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+175}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{{\left(\frac{k}{t}\right)}^{-2}}{\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t_2 := \sin k \cdot \tan k\\ t_3 := \frac{\sqrt{2}}{k}\\ \mathbf{if}\;\ell \leq 7.5 \cdot 10^{-147}:\\ \;\;\;\;\left(t \cdot t\_3\right) \cdot \left({\left(t \cdot \left(t\_1 \cdot \sqrt[3]{t\_2}\right)\right)}^{-2} \cdot \frac{\frac{t\_3}{t\_1}}{\sqrt[3]{{k}^{2}}}\right)\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+175}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}{t\_2 \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (cbrt l) -2.0))
        (t_2 (* (sin k) (tan k)))
        (t_3 (/ (sqrt 2.0) k)))
   (if (<= l 7.5e-147)
     (*
      (* t t_3)
      (*
       (pow (* t (* t_1 (cbrt t_2))) -2.0)
       (/ (/ t_3 t_1) (cbrt (pow k 2.0)))))
     (if (<= l 1.2e+175)
       (*
        (/ (pow l 2.0) (pow k 2.0))
        (* (/ 2.0 (pow (sin k) 2.0)) (/ (cos k) t)))
       (/
        (* 2.0 (pow (/ k t) -2.0))
        (* t_2 (pow (/ t (pow (cbrt l) 2.0)) 3.0)))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(cbrt(l), -2.0);
	double t_2 = sin(k) * tan(k);
	double t_3 = sqrt(2.0) / k;
	double tmp;
	if (l <= 7.5e-147) {
		tmp = (t * t_3) * (pow((t * (t_1 * cbrt(t_2))), -2.0) * ((t_3 / t_1) / cbrt(pow(k, 2.0))));
	} else if (l <= 1.2e+175) {
		tmp = (pow(l, 2.0) / pow(k, 2.0)) * ((2.0 / pow(sin(k), 2.0)) * (cos(k) / t));
	} else {
		tmp = (2.0 * pow((k / t), -2.0)) / (t_2 * pow((t / pow(cbrt(l), 2.0)), 3.0));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.cbrt(l), -2.0);
	double t_2 = Math.sin(k) * Math.tan(k);
	double t_3 = Math.sqrt(2.0) / k;
	double tmp;
	if (l <= 7.5e-147) {
		tmp = (t * t_3) * (Math.pow((t * (t_1 * Math.cbrt(t_2))), -2.0) * ((t_3 / t_1) / Math.cbrt(Math.pow(k, 2.0))));
	} else if (l <= 1.2e+175) {
		tmp = (Math.pow(l, 2.0) / Math.pow(k, 2.0)) * ((2.0 / Math.pow(Math.sin(k), 2.0)) * (Math.cos(k) / t));
	} else {
		tmp = (2.0 * Math.pow((k / t), -2.0)) / (t_2 * Math.pow((t / Math.pow(Math.cbrt(l), 2.0)), 3.0));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cbrt(l) ^ -2.0
	t_2 = Float64(sin(k) * tan(k))
	t_3 = Float64(sqrt(2.0) / k)
	tmp = 0.0
	if (l <= 7.5e-147)
		tmp = Float64(Float64(t * t_3) * Float64((Float64(t * Float64(t_1 * cbrt(t_2))) ^ -2.0) * Float64(Float64(t_3 / t_1) / cbrt((k ^ 2.0)))));
	elseif (l <= 1.2e+175)
		tmp = Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(Float64(2.0 / (sin(k) ^ 2.0)) * Float64(cos(k) / t)));
	else
		tmp = Float64(Float64(2.0 * (Float64(k / t) ^ -2.0)) / Float64(t_2 * (Float64(t / (cbrt(l) ^ 2.0)) ^ 3.0)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]}, If[LessEqual[l, 7.5e-147], N[(N[(t * t$95$3), $MachinePrecision] * N[(N[Power[N[(t * N[(t$95$1 * N[Power[t$95$2, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] * N[(N[(t$95$3 / t$95$1), $MachinePrecision] / N[Power[N[Power[k, 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.2e+175], N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Power[N[(k / t), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * N[Power[N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < 7.50000000000000047e-147

   1. Initial program 34.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 34.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 34.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 42.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 42.6%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 42.6%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 42.6%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 80.7%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 80.7%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}} \]

      2. associate-/r* 80.7%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \color{blue}{\frac{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

      3. associate-/r/ 80.7%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

   8. Simplified 80.7%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 80.7%

         \[\leadsto \color{blue}{\frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

   10. Applied egg-rr 81.0%

       \[\leadsto \color{blue}{\frac{\left(\frac{\sqrt{2} \cdot t}{k} \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}{\sqrt[3]{\sin k \cdot \tan k}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 81.0%

          \[\leadsto \color{blue}{\left(\frac{\sqrt{2} \cdot t}{k} \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

       2. associate-*l* 81.0%

          \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{k} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)} \]

       3. associate-*l/ 81.0%

          \[\leadsto \color{blue}{\left(\frac{\sqrt{2}}{k} \cdot t\right)} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

       4. *-commutative 81.0%

          \[\leadsto \color{blue}{\left(t \cdot \frac{\sqrt{2}}{k}\right)} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

       5. associate-*l* 81.0%

          \[\leadsto \left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left({\color{blue}{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

   12. Simplified 82.3%

       \[\leadsto \color{blue}{\left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)} \]

   13. Taylor expanded in k around 0 71.9%

       \[\leadsto \left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\color{blue}{\sqrt[3]{{k}^{2}}}}\right) \]

   if 7.50000000000000047e-147 < l < 1.2e175

   1. Initial program 39.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 39.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 39.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 46.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 46.8%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 46.8%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 46.7%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 82.1%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 82.2%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}} \]

      2. associate-/r* 82.2%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \color{blue}{\frac{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

      3. associate-/r/ 82.2%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

   8. Simplified 82.2%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

   9. Taylor expanded in k around inf 84.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   10. Step-by-step derivation

       1. times-frac 92.3%

          \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k \cdot {\left(\sqrt{2}\right)}^{2}}{t \cdot {\sin k}^{2}}} \]

       2. *-commutative 92.3%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{{\left(\sqrt{2}\right)}^{2} \cdot \cos k}}{t \cdot {\sin k}^{2}} \]

       3. unpow2 92.3%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)} \cdot \cos k}{t \cdot {\sin k}^{2}} \]

       4. rem-square-sqrt 92.6%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{2} \cdot \cos k}{t \cdot {\sin k}^{2}} \]

       5. *-commutative 92.6%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{2 \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot t}} \]

       6. times-frac 92.6%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \color{blue}{\left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)} \]

   11. Simplified 92.6%

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)} \]

   if 1.2e175 < l

   1. Initial program 26.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 26.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 26.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 26.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cube-cbrt 26.4%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      2. div-inv 26.4%

         \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]

      3. times-frac 26.4%

         \[\leadsto \color{blue}{\frac{2}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\frac{1}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 78.1%

      \[\leadsto \color{blue}{\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{{\left(\frac{k}{t}\right)}^{-2}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 78.2%

         \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\left(\frac{k}{t}\right)}^{-2}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. associate-/l* 78.2%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \]

      3. associate-/l/ 70.7%

         \[\leadsto \color{blue}{\frac{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2} \cdot \left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}} \]

      4. unpow2 70.7%

         \[\leadsto \frac{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}{\color{blue}{\left(\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)} \cdot \left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)} \]

      5. unpow3 70.7%

         \[\leadsto \frac{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}{\color{blue}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{3}}} \]

      6. *-commutative 70.7%

         \[\leadsto \frac{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}{{\color{blue}{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}}^{3}} \]

   8. Simplified 67.0%

      \[\leadsto \color{blue}{\frac{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 75.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t_2 := \sqrt[3]{{k}^{2}}\\ t_3 := \frac{\sqrt{2}}{k}\\ \mathbf{if}\;\ell \leq 7.8 \cdot 10^{-147}:\\ \;\;\;\;\left(t \cdot t\_3\right) \cdot \left(\frac{\frac{t\_3}{t\_1}}{t\_2} \cdot {\left(t \cdot \left(t\_1 \cdot t\_2\right)\right)}^{-2}\right)\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+175}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (cbrt l) -2.0))
        (t_2 (cbrt (pow k 2.0)))
        (t_3 (/ (sqrt 2.0) k)))
   (if (<= l 7.8e-147)
     (* (* t t_3) (* (/ (/ t_3 t_1) t_2) (pow (* t (* t_1 t_2)) -2.0)))
     (if (<= l 1.2e+175)
       (*
        (/ (pow l 2.0) (pow k 2.0))
        (* (/ 2.0 (pow (sin k) 2.0)) (/ (cos k) t)))
       (/
        (* 2.0 (pow (/ k t) -2.0))
        (* (* (sin k) (tan k)) (pow (/ t (pow (cbrt l) 2.0)) 3.0)))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(cbrt(l), -2.0);
	double t_2 = cbrt(pow(k, 2.0));
	double t_3 = sqrt(2.0) / k;
	double tmp;
	if (l <= 7.8e-147) {
		tmp = (t * t_3) * (((t_3 / t_1) / t_2) * pow((t * (t_1 * t_2)), -2.0));
	} else if (l <= 1.2e+175) {
		tmp = (pow(l, 2.0) / pow(k, 2.0)) * ((2.0 / pow(sin(k), 2.0)) * (cos(k) / t));
	} else {
		tmp = (2.0 * pow((k / t), -2.0)) / ((sin(k) * tan(k)) * pow((t / pow(cbrt(l), 2.0)), 3.0));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.cbrt(l), -2.0);
	double t_2 = Math.cbrt(Math.pow(k, 2.0));
	double t_3 = Math.sqrt(2.0) / k;
	double tmp;
	if (l <= 7.8e-147) {
		tmp = (t * t_3) * (((t_3 / t_1) / t_2) * Math.pow((t * (t_1 * t_2)), -2.0));
	} else if (l <= 1.2e+175) {
		tmp = (Math.pow(l, 2.0) / Math.pow(k, 2.0)) * ((2.0 / Math.pow(Math.sin(k), 2.0)) * (Math.cos(k) / t));
	} else {
		tmp = (2.0 * Math.pow((k / t), -2.0)) / ((Math.sin(k) * Math.tan(k)) * Math.pow((t / Math.pow(Math.cbrt(l), 2.0)), 3.0));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cbrt(l) ^ -2.0
	t_2 = cbrt((k ^ 2.0))
	t_3 = Float64(sqrt(2.0) / k)
	tmp = 0.0
	if (l <= 7.8e-147)
		tmp = Float64(Float64(t * t_3) * Float64(Float64(Float64(t_3 / t_1) / t_2) * (Float64(t * Float64(t_1 * t_2)) ^ -2.0)));
	elseif (l <= 1.2e+175)
		tmp = Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(Float64(2.0 / (sin(k) ^ 2.0)) * Float64(cos(k) / t)));
	else
		tmp = Float64(Float64(2.0 * (Float64(k / t) ^ -2.0)) / Float64(Float64(sin(k) * tan(k)) * (Float64(t / (cbrt(l) ^ 2.0)) ^ 3.0)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Power[k, 2.0], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]}, If[LessEqual[l, 7.8e-147], N[(N[(t * t$95$3), $MachinePrecision] * N[(N[(N[(t$95$3 / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision] * N[Power[N[(t * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.2e+175], N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Power[N[(k / t), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < 7.7999999999999996e-147

   1. Initial program 34.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 34.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 34.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 42.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 42.6%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 42.6%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 42.6%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 80.7%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 80.7%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}} \]

      2. associate-/r* 80.7%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \color{blue}{\frac{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

      3. associate-/r/ 80.7%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

   8. Simplified 80.7%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 80.7%

         \[\leadsto \color{blue}{\frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

   10. Applied egg-rr 81.0%

       \[\leadsto \color{blue}{\frac{\left(\frac{\sqrt{2} \cdot t}{k} \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}{\sqrt[3]{\sin k \cdot \tan k}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 81.0%

          \[\leadsto \color{blue}{\left(\frac{\sqrt{2} \cdot t}{k} \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

       2. associate-*l* 81.0%

          \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{k} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)} \]

       3. associate-*l/ 81.0%

          \[\leadsto \color{blue}{\left(\frac{\sqrt{2}}{k} \cdot t\right)} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

       4. *-commutative 81.0%

          \[\leadsto \color{blue}{\left(t \cdot \frac{\sqrt{2}}{k}\right)} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

       5. associate-*l* 81.0%

          \[\leadsto \left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left({\color{blue}{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

   12. Simplified 82.3%

       \[\leadsto \color{blue}{\left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)} \]

   13. Taylor expanded in k around 0 71.9%

       \[\leadsto \left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\color{blue}{\sqrt[3]{{k}^{2}}}}\right) \]

   14. Taylor expanded in k around 0 72.0%

       \[\leadsto \left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \color{blue}{\sqrt[3]{{k}^{2}}}\right)\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{{k}^{2}}}\right) \]

   if 7.7999999999999996e-147 < l < 1.2e175

   1. Initial program 39.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 39.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 39.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 46.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 46.8%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 46.8%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 46.7%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 82.1%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 82.2%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}} \]

      2. associate-/r* 82.2%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \color{blue}{\frac{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

      3. associate-/r/ 82.2%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

   8. Simplified 82.2%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

   9. Taylor expanded in k around inf 84.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   10. Step-by-step derivation

       1. times-frac 92.3%

          \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k \cdot {\left(\sqrt{2}\right)}^{2}}{t \cdot {\sin k}^{2}}} \]

       2. *-commutative 92.3%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{{\left(\sqrt{2}\right)}^{2} \cdot \cos k}}{t \cdot {\sin k}^{2}} \]

       3. unpow2 92.3%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)} \cdot \cos k}{t \cdot {\sin k}^{2}} \]

       4. rem-square-sqrt 92.6%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{2} \cdot \cos k}{t \cdot {\sin k}^{2}} \]

       5. *-commutative 92.6%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{2 \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot t}} \]

       6. times-frac 92.6%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \color{blue}{\left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)} \]

   11. Simplified 92.6%

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)} \]

   if 1.2e175 < l

   1. Initial program 26.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 26.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 26.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 26.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cube-cbrt 26.4%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      2. div-inv 26.4%

         \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]

      3. times-frac 26.4%

         \[\leadsto \color{blue}{\frac{2}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\frac{1}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 78.1%

      \[\leadsto \color{blue}{\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{{\left(\frac{k}{t}\right)}^{-2}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 78.2%

         \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\left(\frac{k}{t}\right)}^{-2}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. associate-/l* 78.2%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \]

      3. associate-/l/ 70.7%

         \[\leadsto \color{blue}{\frac{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2} \cdot \left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}} \]

      4. unpow2 70.7%

         \[\leadsto \frac{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}{\color{blue}{\left(\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)} \cdot \left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)} \]

      5. unpow3 70.7%

         \[\leadsto \frac{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}{\color{blue}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{3}}} \]

      6. *-commutative 70.7%

         \[\leadsto \frac{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}{{\color{blue}{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}}^{3}} \]

   8. Simplified 67.0%

      \[\leadsto \color{blue}{\frac{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7.8 \cdot 10^{-147}:\\ \;\;\;\;\left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left(\frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{{k}^{2}}} \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{{k}^{2}}\right)\right)}^{-2}\right)\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+175}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 48.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6.4 \cdot 10^{-149}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\log \left({\left({\left(e^{{t}^{3}}\right)}^{\left({\ell}^{-2}\right)}\right)}^{\left(\sin k \cdot \tan k\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 6.4e-149)
   (/
    (/ 2.0 (/ (/ k t) (/ t k)))
    (log (pow (pow (exp (pow t 3.0)) (pow l -2.0)) (* (sin k) (tan k)))))
   (*
    (/ (pow l 2.0) (pow k 2.0))
    (* (/ 2.0 (pow (sin k) 2.0)) (/ (cos k) t)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.4e-149) {
		tmp = (2.0 / ((k / t) / (t / k))) / log(pow(pow(exp(pow(t, 3.0)), pow(l, -2.0)), (sin(k) * tan(k))));
	} else {
		tmp = (pow(l, 2.0) / pow(k, 2.0)) * ((2.0 / pow(sin(k), 2.0)) * (cos(k) / t));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 6.4d-149) then
        tmp = (2.0d0 / ((k / t) / (t / k))) / log(((exp((t ** 3.0d0)) ** (l ** (-2.0d0))) ** (sin(k) * tan(k))))
    else
        tmp = ((l ** 2.0d0) / (k ** 2.0d0)) * ((2.0d0 / (sin(k) ** 2.0d0)) * (cos(k) / t))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.4e-149) {
		tmp = (2.0 / ((k / t) / (t / k))) / Math.log(Math.pow(Math.pow(Math.exp(Math.pow(t, 3.0)), Math.pow(l, -2.0)), (Math.sin(k) * Math.tan(k))));
	} else {
		tmp = (Math.pow(l, 2.0) / Math.pow(k, 2.0)) * ((2.0 / Math.pow(Math.sin(k), 2.0)) * (Math.cos(k) / t));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 6.4e-149:
		tmp = (2.0 / ((k / t) / (t / k))) / math.log(math.pow(math.pow(math.exp(math.pow(t, 3.0)), math.pow(l, -2.0)), (math.sin(k) * math.tan(k))))
	else:
		tmp = (math.pow(l, 2.0) / math.pow(k, 2.0)) * ((2.0 / math.pow(math.sin(k), 2.0)) * (math.cos(k) / t))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 6.4e-149)
		tmp = Float64(Float64(2.0 / Float64(Float64(k / t) / Float64(t / k))) / log(((exp((t ^ 3.0)) ^ (l ^ -2.0)) ^ Float64(sin(k) * tan(k)))));
	else
		tmp = Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(Float64(2.0 / (sin(k) ^ 2.0)) * Float64(cos(k) / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 6.4e-149)
		tmp = (2.0 / ((k / t) / (t / k))) / log(((exp((t ^ 3.0)) ^ (l ^ -2.0)) ^ (sin(k) * tan(k))));
	else
		tmp = ((l ^ 2.0) / (k ^ 2.0)) * ((2.0 / (sin(k) ^ 2.0)) * (cos(k) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 6.4e-149], N[(N[(2.0 / N[(N[(k / t), $MachinePrecision] / N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Log[N[Power[N[Power[N[Exp[N[Power[t, 3.0], $MachinePrecision]], $MachinePrecision], N[Power[l, -2.0], $MachinePrecision]], $MachinePrecision], N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.40000000000000004e-149

   1. Initial program 32.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 32.1%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 32.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 36.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 36.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. unpow2 36.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. clear-num 36.5%

         \[\leadsto \frac{\frac{2}{\frac{k}{t} \cdot \color{blue}{\frac{1}{\frac{t}{k}}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      4. un-div-inv 36.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{k}{t}}{\frac{t}{k}}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Applied egg-rr 36.5%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{k}{t}}{\frac{t}{k}}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. add-log-exp 23.9%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\color{blue}{\log \left(e^{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right)}} \]

      2. exp-prod 30.8%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\log \color{blue}{\left({\left(e^{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}^{\left(\sin k \cdot \tan k\right)}\right)}} \]

      3. div-inv 30.8%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\log \left({\left(e^{\color{blue}{{t}^{3} \cdot \frac{1}{\ell \cdot \ell}}}\right)}^{\left(\sin k \cdot \tan k\right)}\right)} \]

      4. exp-prod 27.2%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\log \left({\color{blue}{\left({\left(e^{{t}^{3}}\right)}^{\left(\frac{1}{\ell \cdot \ell}\right)}\right)}}^{\left(\sin k \cdot \tan k\right)}\right)} \]

      5. pow2 27.2%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\log \left({\left({\left(e^{{t}^{3}}\right)}^{\left(\frac{1}{\color{blue}{{\ell}^{2}}}\right)}\right)}^{\left(\sin k \cdot \tan k\right)}\right)} \]

      6. pow-flip 27.2%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\log \left({\left({\left(e^{{t}^{3}}\right)}^{\color{blue}{\left({\ell}^{\left(-2\right)}\right)}}\right)}^{\left(\sin k \cdot \tan k\right)}\right)} \]

      7. metadata-eval 27.2%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\log \left({\left({\left(e^{{t}^{3}}\right)}^{\left({\ell}^{\color{blue}{-2}}\right)}\right)}^{\left(\sin k \cdot \tan k\right)}\right)} \]

   8. Applied egg-rr 27.2%

      \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\color{blue}{\log \left({\left({\left(e^{{t}^{3}}\right)}^{\left({\ell}^{-2}\right)}\right)}^{\left(\sin k \cdot \tan k\right)}\right)}} \]

   if 6.40000000000000004e-149 < k

   1. Initial program 39.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 39.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 39.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 50.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 50.8%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 50.8%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 50.8%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 83.9%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 83.9%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}} \]

      2. associate-/r* 83.9%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \color{blue}{\frac{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

      3. associate-/r/ 83.9%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

   8. Simplified 83.9%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

   9. Taylor expanded in k around inf 81.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   10. Step-by-step derivation

       1. times-frac 80.7%

          \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k \cdot {\left(\sqrt{2}\right)}^{2}}{t \cdot {\sin k}^{2}}} \]

       2. *-commutative 80.7%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{{\left(\sqrt{2}\right)}^{2} \cdot \cos k}}{t \cdot {\sin k}^{2}} \]

       3. unpow2 80.7%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)} \cdot \cos k}{t \cdot {\sin k}^{2}} \]

       4. rem-square-sqrt 80.8%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{2} \cdot \cos k}{t \cdot {\sin k}^{2}} \]

       5. *-commutative 80.8%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{2 \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot t}} \]

       6. times-frac 80.7%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \color{blue}{\left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)} \]

   11. Simplified 80.7%

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 66.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\sin k \cdot \tan k\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\\ \mathbf{if}\;\ell \leq 1.15 \cdot 10^{-171}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{t\_1}\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+175}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (* (sin k) (tan k)) (pow (/ t (pow (cbrt l) 2.0)) 3.0))))
   (if (<= l 1.15e-171)
     (/ (/ 2.0 (/ (/ k t) (/ t k))) t_1)
     (if (<= l 1.2e+175)
       (*
        (/ (pow l 2.0) (pow k 2.0))
        (* (/ 2.0 (pow (sin k) 2.0)) (/ (cos k) t)))
       (/ (* 2.0 (pow (/ k t) -2.0)) t_1)))))

C

double code(double t, double l, double k) {
	double t_1 = (sin(k) * tan(k)) * pow((t / pow(cbrt(l), 2.0)), 3.0);
	double tmp;
	if (l <= 1.15e-171) {
		tmp = (2.0 / ((k / t) / (t / k))) / t_1;
	} else if (l <= 1.2e+175) {
		tmp = (pow(l, 2.0) / pow(k, 2.0)) * ((2.0 / pow(sin(k), 2.0)) * (cos(k) / t));
	} else {
		tmp = (2.0 * pow((k / t), -2.0)) / t_1;
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = (Math.sin(k) * Math.tan(k)) * Math.pow((t / Math.pow(Math.cbrt(l), 2.0)), 3.0);
	double tmp;
	if (l <= 1.15e-171) {
		tmp = (2.0 / ((k / t) / (t / k))) / t_1;
	} else if (l <= 1.2e+175) {
		tmp = (Math.pow(l, 2.0) / Math.pow(k, 2.0)) * ((2.0 / Math.pow(Math.sin(k), 2.0)) * (Math.cos(k) / t));
	} else {
		tmp = (2.0 * Math.pow((k / t), -2.0)) / t_1;
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(Float64(sin(k) * tan(k)) * (Float64(t / (cbrt(l) ^ 2.0)) ^ 3.0))
	tmp = 0.0
	if (l <= 1.15e-171)
		tmp = Float64(Float64(2.0 / Float64(Float64(k / t) / Float64(t / k))) / t_1);
	elseif (l <= 1.2e+175)
		tmp = Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(Float64(2.0 / (sin(k) ^ 2.0)) * Float64(cos(k) / t)));
	else
		tmp = Float64(Float64(2.0 * (Float64(k / t) ^ -2.0)) / t_1);
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 1.15e-171], N[(N[(2.0 / N[(N[(k / t), $MachinePrecision] / N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[l, 1.2e+175], N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Power[N[(k / t), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.14999999999999989e-171

   1. Initial program 34.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 34.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 34.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 42.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 42.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. unpow2 42.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. clear-num 42.8%

         \[\leadsto \frac{\frac{2}{\frac{k}{t} \cdot \color{blue}{\frac{1}{\frac{t}{k}}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      4. un-div-inv 42.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{k}{t}}{\frac{t}{k}}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Applied egg-rr 42.8%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{k}{t}}{\frac{t}{k}}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 42.8%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. associate-*l* 42.8%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. cbrt-div 42.8%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\left(\color{blue}{\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell \cdot \ell}}} \cdot \left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      4. rem-cbrt-cube 42.8%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\left(\frac{\color{blue}{t}}{\sqrt[3]{\ell \cdot \ell}} \cdot \left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. cbrt-prod 42.8%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\left(\frac{t}{\color{blue}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. pow2 42.8%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\left(\frac{t}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      7. pow2 42.8%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}^{2}}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      8. cbrt-div 42.7%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot {\color{blue}{\left(\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell \cdot \ell}}\right)}}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      9. rem-cbrt-cube 53.1%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot {\left(\frac{\color{blue}{t}}{\sqrt[3]{\ell \cdot \ell}}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      10. cbrt-prod 62.8%

          \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot {\left(\frac{t}{\color{blue}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      11. pow2 62.8%

          \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot {\left(\frac{t}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   8. Applied egg-rr 62.8%

      \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\color{blue}{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]
   9. Step-by-step derivation

      1. unpow2 62.8%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      2. cube-mult 62.8%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\color{blue}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \cdot \left(\sin k \cdot \tan k\right)} \]

   10. Simplified 62.8%

       \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\color{blue}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \cdot \left(\sin k \cdot \tan k\right)} \]

   if 1.14999999999999989e-171 < l < 1.2e175

   1. Initial program 37.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 37.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 37.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 45.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 45.8%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 45.7%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 45.7%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 81.3%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 81.3%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}} \]

      2. associate-/r* 81.3%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \color{blue}{\frac{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

      3. associate-/r/ 81.4%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

   8. Simplified 81.4%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

   9. Taylor expanded in k around inf 82.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   10. Step-by-step derivation

       1. times-frac 88.5%

          \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k \cdot {\left(\sqrt{2}\right)}^{2}}{t \cdot {\sin k}^{2}}} \]

       2. *-commutative 88.5%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{{\left(\sqrt{2}\right)}^{2} \cdot \cos k}}{t \cdot {\sin k}^{2}} \]

       3. unpow2 88.5%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)} \cdot \cos k}{t \cdot {\sin k}^{2}} \]

       4. rem-square-sqrt 88.7%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{2} \cdot \cos k}{t \cdot {\sin k}^{2}} \]

       5. *-commutative 88.7%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{2 \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot t}} \]

       6. times-frac 88.7%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \color{blue}{\left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)} \]

   11. Simplified 88.7%

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)} \]

   if 1.2e175 < l

   1. Initial program 26.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 26.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 26.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 26.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cube-cbrt 26.4%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      2. div-inv 26.4%

         \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]

      3. times-frac 26.4%

         \[\leadsto \color{blue}{\frac{2}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\frac{1}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 78.1%

      \[\leadsto \color{blue}{\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{{\left(\frac{k}{t}\right)}^{-2}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 78.2%

         \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\left(\frac{k}{t}\right)}^{-2}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. associate-/l* 78.2%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \]

      3. associate-/l/ 70.7%

         \[\leadsto \color{blue}{\frac{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2} \cdot \left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}} \]

      4. unpow2 70.7%

         \[\leadsto \frac{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}{\color{blue}{\left(\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)} \cdot \left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)} \]

      5. unpow3 70.7%

         \[\leadsto \frac{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}{\color{blue}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{3}}} \]

      6. *-commutative 70.7%

         \[\leadsto \frac{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}{{\color{blue}{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}}^{3}} \]

   8. Simplified 67.0%

      \[\leadsto \color{blue}{\frac{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.15 \cdot 10^{-171}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+175}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-148}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.7e-148)
   (/
    (/ 2.0 (/ (/ k t) (/ t k)))
    (pow (* (cbrt (* (sin k) (tan k))) (/ t (pow (cbrt l) 2.0))) 3.0))
   (*
    (/ (pow l 2.0) (pow k 2.0))
    (* (/ 2.0 (pow (sin k) 2.0)) (/ (cos k) t)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.7e-148) {
		tmp = (2.0 / ((k / t) / (t / k))) / pow((cbrt((sin(k) * tan(k))) * (t / pow(cbrt(l), 2.0))), 3.0);
	} else {
		tmp = (pow(l, 2.0) / pow(k, 2.0)) * ((2.0 / pow(sin(k), 2.0)) * (cos(k) / t));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.7e-148) {
		tmp = (2.0 / ((k / t) / (t / k))) / Math.pow((Math.cbrt((Math.sin(k) * Math.tan(k))) * (t / Math.pow(Math.cbrt(l), 2.0))), 3.0);
	} else {
		tmp = (Math.pow(l, 2.0) / Math.pow(k, 2.0)) * ((2.0 / Math.pow(Math.sin(k), 2.0)) * (Math.cos(k) / t));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.7e-148)
		tmp = Float64(Float64(2.0 / Float64(Float64(k / t) / Float64(t / k))) / (Float64(cbrt(Float64(sin(k) * tan(k))) * Float64(t / (cbrt(l) ^ 2.0))) ^ 3.0));
	else
		tmp = Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(Float64(2.0 / (sin(k) ^ 2.0)) * Float64(cos(k) / t)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.7e-148], N[(N[(2.0 / N[(N[(k / t), $MachinePrecision] / N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.69999999999999988e-148

   1. Initial program 32.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 32.1%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 32.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 36.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 36.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. unpow2 36.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. clear-num 36.5%

         \[\leadsto \frac{\frac{2}{\frac{k}{t} \cdot \color{blue}{\frac{1}{\frac{t}{k}}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      4. un-div-inv 36.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{k}{t}}{\frac{t}{k}}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Applied egg-rr 36.5%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{k}{t}}{\frac{t}{k}}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 36.5%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      2. pow3 36.5%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right)}^{3}}} \]

      3. cbrt-prod 36.4%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}}^{3}} \]

      4. cbrt-div 38.8%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\color{blue}{\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell \cdot \ell}}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{3}} \]

      5. rem-cbrt-cube 56.0%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{\color{blue}{t}}{\sqrt[3]{\ell \cdot \ell}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{3}} \]

      6. cbrt-prod 66.5%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{t}{\color{blue}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{3}} \]

      7. pow2 66.5%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{t}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{3}} \]

   8. Applied egg-rr 66.5%

      \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\color{blue}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{3}}} \]

   if 2.69999999999999988e-148 < k

   1. Initial program 39.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 39.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 39.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 50.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 50.8%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 50.8%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 50.8%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 83.9%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 83.9%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}} \]

      2. associate-/r* 83.9%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \color{blue}{\frac{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

      3. associate-/r/ 83.9%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

   8. Simplified 83.9%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

   9. Taylor expanded in k around inf 81.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   10. Step-by-step derivation

       1. times-frac 80.7%

          \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k \cdot {\left(\sqrt{2}\right)}^{2}}{t \cdot {\sin k}^{2}}} \]

       2. *-commutative 80.7%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{{\left(\sqrt{2}\right)}^{2} \cdot \cos k}}{t \cdot {\sin k}^{2}} \]

       3. unpow2 80.7%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)} \cdot \cos k}{t \cdot {\sin k}^{2}} \]

       4. rem-square-sqrt 80.8%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{2} \cdot \cos k}{t \cdot {\sin k}^{2}} \]

       5. *-commutative 80.8%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{2 \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot t}} \]

       6. times-frac 80.7%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \color{blue}{\left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)} \]

   11. Simplified 80.7%

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-148}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 72.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.95 \cdot 10^{-98}:\\ \;\;\;\;\cos k \cdot \left(2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{\left(k \cdot \sin k\right)}^{2}}\right)\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+85}:\\ \;\;\;\;\frac{\frac{t}{k} \cdot \frac{2}{\frac{k}{t}}}{\frac{\left(\sin k \cdot \tan k\right) \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 2.95e-98)
   (* (cos k) (* 2.0 (/ (/ (pow l 2.0) t) (pow (* k (sin k)) 2.0))))
   (if (<= t 2.8e+85)
     (/
      (* (/ t k) (/ 2.0 (/ k t)))
      (/ (* (* (sin k) (tan k)) (/ (pow t 3.0) l)) l))
     (*
      (/ (pow l 2.0) (pow k 2.0))
      (* (/ 2.0 (pow (sin k) 2.0)) (/ (cos k) t))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 2.95e-98) {
		tmp = cos(k) * (2.0 * ((pow(l, 2.0) / t) / pow((k * sin(k)), 2.0)));
	} else if (t <= 2.8e+85) {
		tmp = ((t / k) * (2.0 / (k / t))) / (((sin(k) * tan(k)) * (pow(t, 3.0) / l)) / l);
	} else {
		tmp = (pow(l, 2.0) / pow(k, 2.0)) * ((2.0 / pow(sin(k), 2.0)) * (cos(k) / t));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 2.95d-98) then
        tmp = cos(k) * (2.0d0 * (((l ** 2.0d0) / t) / ((k * sin(k)) ** 2.0d0)))
    else if (t <= 2.8d+85) then
        tmp = ((t / k) * (2.0d0 / (k / t))) / (((sin(k) * tan(k)) * ((t ** 3.0d0) / l)) / l)
    else
        tmp = ((l ** 2.0d0) / (k ** 2.0d0)) * ((2.0d0 / (sin(k) ** 2.0d0)) * (cos(k) / t))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 2.95e-98) {
		tmp = Math.cos(k) * (2.0 * ((Math.pow(l, 2.0) / t) / Math.pow((k * Math.sin(k)), 2.0)));
	} else if (t <= 2.8e+85) {
		tmp = ((t / k) * (2.0 / (k / t))) / (((Math.sin(k) * Math.tan(k)) * (Math.pow(t, 3.0) / l)) / l);
	} else {
		tmp = (Math.pow(l, 2.0) / Math.pow(k, 2.0)) * ((2.0 / Math.pow(Math.sin(k), 2.0)) * (Math.cos(k) / t));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 2.95e-98:
		tmp = math.cos(k) * (2.0 * ((math.pow(l, 2.0) / t) / math.pow((k * math.sin(k)), 2.0)))
	elif t <= 2.8e+85:
		tmp = ((t / k) * (2.0 / (k / t))) / (((math.sin(k) * math.tan(k)) * (math.pow(t, 3.0) / l)) / l)
	else:
		tmp = (math.pow(l, 2.0) / math.pow(k, 2.0)) * ((2.0 / math.pow(math.sin(k), 2.0)) * (math.cos(k) / t))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 2.95e-98)
		tmp = Float64(cos(k) * Float64(2.0 * Float64(Float64((l ^ 2.0) / t) / (Float64(k * sin(k)) ^ 2.0))));
	elseif (t <= 2.8e+85)
		tmp = Float64(Float64(Float64(t / k) * Float64(2.0 / Float64(k / t))) / Float64(Float64(Float64(sin(k) * tan(k)) * Float64((t ^ 3.0) / l)) / l));
	else
		tmp = Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(Float64(2.0 / (sin(k) ^ 2.0)) * Float64(cos(k) / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 2.95e-98)
		tmp = cos(k) * (2.0 * (((l ^ 2.0) / t) / ((k * sin(k)) ^ 2.0)));
	elseif (t <= 2.8e+85)
		tmp = ((t / k) * (2.0 / (k / t))) / (((sin(k) * tan(k)) * ((t ^ 3.0) / l)) / l);
	else
		tmp = ((l ^ 2.0) / (k ^ 2.0)) * ((2.0 / (sin(k) ^ 2.0)) * (cos(k) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 2.95e-98], N[(N[Cos[k], $MachinePrecision] * N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / t), $MachinePrecision] / N[Power[N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.8e+85], N[(N[(N[(t / k), $MachinePrecision] * N[(2.0 / N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.94999999999999996e-98

   1. Initial program 32.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 32.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 32.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 38.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 38.4%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 38.4%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 38.4%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 81.4%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 81.4%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}} \]

      2. associate-/r* 81.4%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \color{blue}{\frac{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

      3. associate-/r/ 81.4%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

   8. Simplified 81.4%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 81.4%

         \[\leadsto \color{blue}{\frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

   10. Applied egg-rr 81.7%

       \[\leadsto \color{blue}{\frac{\left(\frac{\sqrt{2} \cdot t}{k} \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}{\sqrt[3]{\sin k \cdot \tan k}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 81.7%

          \[\leadsto \color{blue}{\left(\frac{\sqrt{2} \cdot t}{k} \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

       2. associate-*l* 81.5%

          \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{k} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)} \]

       3. associate-*l/ 81.5%

          \[\leadsto \color{blue}{\left(\frac{\sqrt{2}}{k} \cdot t\right)} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

       4. *-commutative 81.5%

          \[\leadsto \color{blue}{\left(t \cdot \frac{\sqrt{2}}{k}\right)} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

       5. associate-*l* 81.5%

          \[\leadsto \left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left({\color{blue}{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

   12. Simplified 82.7%

       \[\leadsto \color{blue}{\left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)} \]

   13. Taylor expanded in t around 0 75.9%

       \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   14. Step-by-step derivation

       1. associate-*r* 75.9%

          \[\leadsto \frac{\color{blue}{\left({\ell}^{2} \cdot \cos k\right) \cdot {\left(\sqrt{2}\right)}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

       2. unpow2 75.9%

          \[\leadsto \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

       3. rem-square-sqrt 76.0%

          \[\leadsto \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot \color{blue}{2}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

       4. associate-*l/ 76.0%

          \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot 2} \]

       5. *-commutative 76.0%

          \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

       6. *-commutative 76.0%

          \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

       7. associate-/l* 76.0%

          \[\leadsto 2 \cdot \color{blue}{\left(\cos k \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)} \]

       8. associate-*r* 76.0%

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}}\right) \]

       9. *-commutative 76.0%

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{\color{blue}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}}\right) \]

   15. Simplified 76.0%

       \[\leadsto \color{blue}{2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}\right)} \]

   16. Taylor expanded in k around inf 76.0%

       \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   17. Step-by-step derivation

       1. *-commutative 76.0%

          \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot 2} \]

       2. *-commutative 76.0%

          \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{\left(t \cdot {\sin k}^{2}\right) \cdot {k}^{2}}} \cdot 2 \]

       3. associate-*l* 75.4%

          \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{t \cdot \left({\sin k}^{2} \cdot {k}^{2}\right)}} \cdot 2 \]

       4. unpow2 75.4%

          \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{t \cdot \left(\color{blue}{\left(\sin k \cdot \sin k\right)} \cdot {k}^{2}\right)} \cdot 2 \]

       5. unpow2 75.4%

          \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{t \cdot \left(\left(\sin k \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot 2 \]

       6. swap-sqr 75.4%

          \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{t \cdot \color{blue}{\left(\left(\sin k \cdot k\right) \cdot \left(\sin k \cdot k\right)\right)}} \cdot 2 \]

       7. unpow2 75.4%

          \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{t \cdot \color{blue}{{\left(\sin k \cdot k\right)}^{2}}} \cdot 2 \]

       8. *-commutative 75.4%

          \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\left(\sin k \cdot k\right)}^{2} \cdot t}} \cdot 2 \]

       9. associate-*l/ 75.5%

          \[\leadsto \color{blue}{\left(\frac{{\ell}^{2}}{{\left(\sin k \cdot k\right)}^{2} \cdot t} \cdot \cos k\right)} \cdot 2 \]

       10. *-commutative 75.5%

           \[\leadsto \color{blue}{\left(\cos k \cdot \frac{{\ell}^{2}}{{\left(\sin k \cdot k\right)}^{2} \cdot t}\right)} \cdot 2 \]

       11. associate-*l* 75.5%

           \[\leadsto \color{blue}{\cos k \cdot \left(\frac{{\ell}^{2}}{{\left(\sin k \cdot k\right)}^{2} \cdot t} \cdot 2\right)} \]

   18. Simplified 73.7%

       \[\leadsto \color{blue}{\cos k \cdot \left(\frac{\frac{{\ell}^{2}}{t}}{{\left(k \cdot \sin k\right)}^{2}} \cdot 2\right)} \]

   if 2.94999999999999996e-98 < t < 2.7999999999999999e85

   1. Initial program 69.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 69.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 69.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 69.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 69.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. unpow2 69.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. clear-num 69.8%

         \[\leadsto \frac{\frac{2}{\frac{k}{t} \cdot \color{blue}{\frac{1}{\frac{t}{k}}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      4. un-div-inv 69.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{k}{t}}{\frac{t}{k}}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Applied egg-rr 69.8%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{k}{t}}{\frac{t}{k}}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. associate-/r/ 69.9%

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{k}{t}} \cdot \frac{t}{k}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

   8. Applied egg-rr 69.9%

      \[\leadsto \frac{\color{blue}{\frac{2}{\frac{k}{t}} \cdot \frac{t}{k}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]
   9. Step-by-step derivation

      1. associate-/r* 77.4%

         \[\leadsto \frac{\frac{2}{\frac{k}{t}} \cdot \frac{t}{k}}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. associate-*l/ 80.0%

         \[\leadsto \frac{\frac{2}{\frac{k}{t}} \cdot \frac{t}{k}}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}{\ell}}} \]

   10. Applied egg-rr 80.0%

       \[\leadsto \frac{\frac{2}{\frac{k}{t}} \cdot \frac{t}{k}}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}{\ell}}} \]

   if 2.7999999999999999e85 < t

   1. Initial program 10.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 10.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 10.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 28.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 28.9%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 28.9%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 28.9%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 76.6%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 76.6%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}} \]

      2. associate-/r* 76.6%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \color{blue}{\frac{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

      3. associate-/r/ 76.7%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

   8. Simplified 76.7%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

   9. Taylor expanded in k around inf 69.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   10. Step-by-step derivation

       1. times-frac 68.0%

          \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k \cdot {\left(\sqrt{2}\right)}^{2}}{t \cdot {\sin k}^{2}}} \]

       2. *-commutative 68.0%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{{\left(\sqrt{2}\right)}^{2} \cdot \cos k}}{t \cdot {\sin k}^{2}} \]

       3. unpow2 68.0%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)} \cdot \cos k}{t \cdot {\sin k}^{2}} \]

       4. rem-square-sqrt 68.1%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{2} \cdot \cos k}{t \cdot {\sin k}^{2}} \]

       5. *-commutative 68.1%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{2 \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot t}} \]

       6. times-frac 68.0%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2}} \cdot \color{blue}{\left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)} \]

   11. Simplified 68.0%

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.95 \cdot 10^{-98}:\\ \;\;\;\;\cos k \cdot \left(2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{\left(k \cdot \sin k\right)}^{2}}\right)\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+85}:\\ \;\;\;\;\frac{\frac{t}{k} \cdot \frac{2}{\frac{k}{t}}}{\frac{\left(\sin k \cdot \tan k\right) \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{2}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{-98}:\\ \;\;\;\;\cos k \cdot \left(2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{\left(k \cdot \sin k\right)}^{2}}\right)\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+86}:\\ \;\;\;\;\frac{\frac{t}{k} \cdot \frac{2}{\frac{k}{t}}}{\frac{\left(\sin k \cdot \tan k\right) \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 2.8e-98)
   (* (cos k) (* 2.0 (/ (/ (pow l 2.0) t) (pow (* k (sin k)) 2.0))))
   (if (<= t 8e+86)
     (/
      (* (/ t k) (/ 2.0 (/ k t)))
      (/ (* (* (sin k) (tan k)) (/ (pow t 3.0) l)) l))
     (* (* l l) (* 2.0 (/ (/ (cos k) (pow k 2.0)) (* t (pow (sin k) 2.0))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 2.8e-98) {
		tmp = cos(k) * (2.0 * ((pow(l, 2.0) / t) / pow((k * sin(k)), 2.0)));
	} else if (t <= 8e+86) {
		tmp = ((t / k) * (2.0 / (k / t))) / (((sin(k) * tan(k)) * (pow(t, 3.0) / l)) / l);
	} else {
		tmp = (l * l) * (2.0 * ((cos(k) / pow(k, 2.0)) / (t * pow(sin(k), 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 2.8d-98) then
        tmp = cos(k) * (2.0d0 * (((l ** 2.0d0) / t) / ((k * sin(k)) ** 2.0d0)))
    else if (t <= 8d+86) then
        tmp = ((t / k) * (2.0d0 / (k / t))) / (((sin(k) * tan(k)) * ((t ** 3.0d0) / l)) / l)
    else
        tmp = (l * l) * (2.0d0 * ((cos(k) / (k ** 2.0d0)) / (t * (sin(k) ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 2.8e-98) {
		tmp = Math.cos(k) * (2.0 * ((Math.pow(l, 2.0) / t) / Math.pow((k * Math.sin(k)), 2.0)));
	} else if (t <= 8e+86) {
		tmp = ((t / k) * (2.0 / (k / t))) / (((Math.sin(k) * Math.tan(k)) * (Math.pow(t, 3.0) / l)) / l);
	} else {
		tmp = (l * l) * (2.0 * ((Math.cos(k) / Math.pow(k, 2.0)) / (t * Math.pow(Math.sin(k), 2.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 2.8e-98:
		tmp = math.cos(k) * (2.0 * ((math.pow(l, 2.0) / t) / math.pow((k * math.sin(k)), 2.0)))
	elif t <= 8e+86:
		tmp = ((t / k) * (2.0 / (k / t))) / (((math.sin(k) * math.tan(k)) * (math.pow(t, 3.0) / l)) / l)
	else:
		tmp = (l * l) * (2.0 * ((math.cos(k) / math.pow(k, 2.0)) / (t * math.pow(math.sin(k), 2.0))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 2.8e-98)
		tmp = Float64(cos(k) * Float64(2.0 * Float64(Float64((l ^ 2.0) / t) / (Float64(k * sin(k)) ^ 2.0))));
	elseif (t <= 8e+86)
		tmp = Float64(Float64(Float64(t / k) * Float64(2.0 / Float64(k / t))) / Float64(Float64(Float64(sin(k) * tan(k)) * Float64((t ^ 3.0) / l)) / l));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(Float64(cos(k) / (k ^ 2.0)) / Float64(t * (sin(k) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 2.8e-98)
		tmp = cos(k) * (2.0 * (((l ^ 2.0) / t) / ((k * sin(k)) ^ 2.0)));
	elseif (t <= 8e+86)
		tmp = ((t / k) * (2.0 / (k / t))) / (((sin(k) * tan(k)) * ((t ^ 3.0) / l)) / l);
	else
		tmp = (l * l) * (2.0 * ((cos(k) / (k ^ 2.0)) / (t * (sin(k) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 2.8e-98], N[(N[Cos[k], $MachinePrecision] * N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / t), $MachinePrecision] / N[Power[N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8e+86], N[(N[(N[(t / k), $MachinePrecision] * N[(2.0 / N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.7999999999999999e-98

   1. Initial program 32.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 32.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 32.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 38.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 38.4%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 38.4%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 38.4%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 81.4%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 81.4%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}} \]

      2. associate-/r* 81.4%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \color{blue}{\frac{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

      3. associate-/r/ 81.4%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

   8. Simplified 81.4%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 81.4%

         \[\leadsto \color{blue}{\frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

   10. Applied egg-rr 81.7%

       \[\leadsto \color{blue}{\frac{\left(\frac{\sqrt{2} \cdot t}{k} \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}{\sqrt[3]{\sin k \cdot \tan k}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 81.7%

          \[\leadsto \color{blue}{\left(\frac{\sqrt{2} \cdot t}{k} \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

       2. associate-*l* 81.5%

          \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{k} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)} \]

       3. associate-*l/ 81.5%

          \[\leadsto \color{blue}{\left(\frac{\sqrt{2}}{k} \cdot t\right)} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

       4. *-commutative 81.5%

          \[\leadsto \color{blue}{\left(t \cdot \frac{\sqrt{2}}{k}\right)} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

       5. associate-*l* 81.5%

          \[\leadsto \left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left({\color{blue}{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

   12. Simplified 82.7%

       \[\leadsto \color{blue}{\left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)} \]

   13. Taylor expanded in t around 0 75.9%

       \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   14. Step-by-step derivation

       1. associate-*r* 75.9%

          \[\leadsto \frac{\color{blue}{\left({\ell}^{2} \cdot \cos k\right) \cdot {\left(\sqrt{2}\right)}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

       2. unpow2 75.9%

          \[\leadsto \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

       3. rem-square-sqrt 76.0%

          \[\leadsto \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot \color{blue}{2}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

       4. associate-*l/ 76.0%

          \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot 2} \]

       5. *-commutative 76.0%

          \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

       6. *-commutative 76.0%

          \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

       7. associate-/l* 76.0%

          \[\leadsto 2 \cdot \color{blue}{\left(\cos k \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)} \]

       8. associate-*r* 76.0%

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}}\right) \]

       9. *-commutative 76.0%

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{\color{blue}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}}\right) \]

   15. Simplified 76.0%

       \[\leadsto \color{blue}{2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}\right)} \]

   16. Taylor expanded in k around inf 76.0%

       \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   17. Step-by-step derivation

       1. *-commutative 76.0%

          \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot 2} \]

       2. *-commutative 76.0%

          \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{\left(t \cdot {\sin k}^{2}\right) \cdot {k}^{2}}} \cdot 2 \]

       3. associate-*l* 75.4%

          \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{t \cdot \left({\sin k}^{2} \cdot {k}^{2}\right)}} \cdot 2 \]

       4. unpow2 75.4%

          \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{t \cdot \left(\color{blue}{\left(\sin k \cdot \sin k\right)} \cdot {k}^{2}\right)} \cdot 2 \]

       5. unpow2 75.4%

          \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{t \cdot \left(\left(\sin k \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot 2 \]

       6. swap-sqr 75.4%

          \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{t \cdot \color{blue}{\left(\left(\sin k \cdot k\right) \cdot \left(\sin k \cdot k\right)\right)}} \cdot 2 \]

       7. unpow2 75.4%

          \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{t \cdot \color{blue}{{\left(\sin k \cdot k\right)}^{2}}} \cdot 2 \]

       8. *-commutative 75.4%

          \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\left(\sin k \cdot k\right)}^{2} \cdot t}} \cdot 2 \]

       9. associate-*l/ 75.5%

          \[\leadsto \color{blue}{\left(\frac{{\ell}^{2}}{{\left(\sin k \cdot k\right)}^{2} \cdot t} \cdot \cos k\right)} \cdot 2 \]

       10. *-commutative 75.5%

           \[\leadsto \color{blue}{\left(\cos k \cdot \frac{{\ell}^{2}}{{\left(\sin k \cdot k\right)}^{2} \cdot t}\right)} \cdot 2 \]

       11. associate-*l* 75.5%

           \[\leadsto \color{blue}{\cos k \cdot \left(\frac{{\ell}^{2}}{{\left(\sin k \cdot k\right)}^{2} \cdot t} \cdot 2\right)} \]

   18. Simplified 73.7%

       \[\leadsto \color{blue}{\cos k \cdot \left(\frac{\frac{{\ell}^{2}}{t}}{{\left(k \cdot \sin k\right)}^{2}} \cdot 2\right)} \]

   if 2.7999999999999999e-98 < t < 8.0000000000000001e86

   1. Initial program 69.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 69.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 69.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 69.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 69.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. unpow2 69.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. clear-num 69.8%

         \[\leadsto \frac{\frac{2}{\frac{k}{t} \cdot \color{blue}{\frac{1}{\frac{t}{k}}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      4. un-div-inv 69.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{k}{t}}{\frac{t}{k}}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Applied egg-rr 69.8%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{k}{t}}{\frac{t}{k}}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. associate-/r/ 69.9%

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{k}{t}} \cdot \frac{t}{k}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

   8. Applied egg-rr 69.9%

      \[\leadsto \frac{\color{blue}{\frac{2}{\frac{k}{t}} \cdot \frac{t}{k}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]
   9. Step-by-step derivation

      1. associate-/r* 77.4%

         \[\leadsto \frac{\frac{2}{\frac{k}{t}} \cdot \frac{t}{k}}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. associate-*l/ 80.0%

         \[\leadsto \frac{\frac{2}{\frac{k}{t}} \cdot \frac{t}{k}}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}{\ell}}} \]

   10. Applied egg-rr 80.0%

       \[\leadsto \frac{\frac{2}{\frac{k}{t}} \cdot \frac{t}{k}}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}{\ell}}} \]

   if 8.0000000000000001e86 < t

   1. Initial program 10.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 31.6%

      \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 69.9%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)} \cdot \left(\ell \cdot \ell\right) \]
   6. Step-by-step derivation

      1. associate-/r* 70.0%

         \[\leadsto \left(2 \cdot \color{blue}{\frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}}\right) \cdot \left(\ell \cdot \ell\right) \]

   7. Simplified 70.0%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)} \cdot \left(\ell \cdot \ell\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{-98}:\\ \;\;\;\;\cos k \cdot \left(2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{\left(k \cdot \sin k\right)}^{2}}\right)\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+86}:\\ \;\;\;\;\frac{\frac{t}{k} \cdot \frac{2}{\frac{k}{t}}}{\frac{\left(\sin k \cdot \tan k\right) \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 72.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.35 \cdot 10^{-97} \lor \neg \left(t \leq 2.9 \cdot 10^{+84}\right):\\ \;\;\;\;2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{t \cdot {\left(k \cdot \sin k\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t}{k} \cdot \frac{2}{\frac{k}{t}}}{\frac{\left(\sin k \cdot \tan k\right) \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t 2.35e-97) (not (<= t 2.9e+84)))
   (* 2.0 (* (cos k) (/ (pow l 2.0) (* t (pow (* k (sin k)) 2.0)))))
   (/
    (* (/ t k) (/ 2.0 (/ k t)))
    (/ (* (* (sin k) (tan k)) (/ (pow t 3.0) l)) l))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= 2.35e-97) || !(t <= 2.9e+84)) {
		tmp = 2.0 * (cos(k) * (pow(l, 2.0) / (t * pow((k * sin(k)), 2.0))));
	} else {
		tmp = ((t / k) * (2.0 / (k / t))) / (((sin(k) * tan(k)) * (pow(t, 3.0) / l)) / l);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= 2.35d-97) .or. (.not. (t <= 2.9d+84))) then
        tmp = 2.0d0 * (cos(k) * ((l ** 2.0d0) / (t * ((k * sin(k)) ** 2.0d0))))
    else
        tmp = ((t / k) * (2.0d0 / (k / t))) / (((sin(k) * tan(k)) * ((t ** 3.0d0) / l)) / l)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= 2.35e-97) || !(t <= 2.9e+84)) {
		tmp = 2.0 * (Math.cos(k) * (Math.pow(l, 2.0) / (t * Math.pow((k * Math.sin(k)), 2.0))));
	} else {
		tmp = ((t / k) * (2.0 / (k / t))) / (((Math.sin(k) * Math.tan(k)) * (Math.pow(t, 3.0) / l)) / l);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= 2.35e-97) or not (t <= 2.9e+84):
		tmp = 2.0 * (math.cos(k) * (math.pow(l, 2.0) / (t * math.pow((k * math.sin(k)), 2.0))))
	else:
		tmp = ((t / k) * (2.0 / (k / t))) / (((math.sin(k) * math.tan(k)) * (math.pow(t, 3.0) / l)) / l)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= 2.35e-97) || !(t <= 2.9e+84))
		tmp = Float64(2.0 * Float64(cos(k) * Float64((l ^ 2.0) / Float64(t * (Float64(k * sin(k)) ^ 2.0)))));
	else
		tmp = Float64(Float64(Float64(t / k) * Float64(2.0 / Float64(k / t))) / Float64(Float64(Float64(sin(k) * tan(k)) * Float64((t ^ 3.0) / l)) / l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= 2.35e-97) || ~((t <= 2.9e+84)))
		tmp = 2.0 * (cos(k) * ((l ^ 2.0) / (t * ((k * sin(k)) ^ 2.0))));
	else
		tmp = ((t / k) * (2.0 / (k / t))) / (((sin(k) * tan(k)) * ((t ^ 3.0) / l)) / l);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, 2.35e-97], N[Not[LessEqual[t, 2.9e+84]], $MachinePrecision]], N[(2.0 * N[(N[Cos[k], $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t * N[Power[N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t / k), $MachinePrecision] * N[(2.0 / N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.3500000000000001e-97 or 2.89999999999999989e84 < t

   1. Initial program 28.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 28.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 28.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 36.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 36.8%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 36.7%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 36.7%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 80.5%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 80.5%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}} \]

      2. associate-/r* 80.6%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \color{blue}{\frac{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

      3. associate-/r/ 80.6%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

   8. Simplified 80.6%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 80.6%

         \[\leadsto \color{blue}{\frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

   10. Applied egg-rr 80.8%

       \[\leadsto \color{blue}{\frac{\left(\frac{\sqrt{2} \cdot t}{k} \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}{\sqrt[3]{\sin k \cdot \tan k}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 80.8%

          \[\leadsto \color{blue}{\left(\frac{\sqrt{2} \cdot t}{k} \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

       2. associate-*l* 80.6%

          \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{k} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)} \]

       3. associate-*l/ 80.6%

          \[\leadsto \color{blue}{\left(\frac{\sqrt{2}}{k} \cdot t\right)} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

       4. *-commutative 80.6%

          \[\leadsto \color{blue}{\left(t \cdot \frac{\sqrt{2}}{k}\right)} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

       5. associate-*l* 80.6%

          \[\leadsto \left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left({\color{blue}{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

   12. Simplified 81.7%

       \[\leadsto \color{blue}{\left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)} \]

   13. Taylor expanded in t around 0 74.8%

       \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   14. Step-by-step derivation

       1. associate-*r* 74.9%

          \[\leadsto \frac{\color{blue}{\left({\ell}^{2} \cdot \cos k\right) \cdot {\left(\sqrt{2}\right)}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

       2. unpow2 74.9%

          \[\leadsto \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

       3. rem-square-sqrt 75.0%

          \[\leadsto \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot \color{blue}{2}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

       4. associate-*l/ 75.0%

          \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot 2} \]

       5. *-commutative 75.0%

          \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

       6. *-commutative 75.0%

          \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

       7. associate-/l* 75.0%

          \[\leadsto 2 \cdot \color{blue}{\left(\cos k \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)} \]

       8. associate-*r* 75.0%

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}}\right) \]

       9. *-commutative 75.0%

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{\color{blue}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}}\right) \]

   15. Simplified 75.0%

       \[\leadsto \color{blue}{2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}\right)} \]

   16. Taylor expanded in k around inf 75.0%

       \[\leadsto 2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}}\right) \]
   17. Step-by-step derivation

       1. *-commutative 75.0%

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{\color{blue}{\left(t \cdot {\sin k}^{2}\right) \cdot {k}^{2}}}\right) \]

       2. associate-*l* 73.6%

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{\color{blue}{t \cdot \left({\sin k}^{2} \cdot {k}^{2}\right)}}\right) \]

       3. unpow2 73.6%

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{t \cdot \left(\color{blue}{\left(\sin k \cdot \sin k\right)} \cdot {k}^{2}\right)}\right) \]

       4. unpow2 73.6%

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{t \cdot \left(\left(\sin k \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)}\right) \]

       5. swap-sqr 73.6%

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{t \cdot \color{blue}{\left(\left(\sin k \cdot k\right) \cdot \left(\sin k \cdot k\right)\right)}}\right) \]

       6. unpow2 73.6%

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{t \cdot \color{blue}{{\left(\sin k \cdot k\right)}^{2}}}\right) \]

       7. *-commutative 73.6%

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{t \cdot {\color{blue}{\left(k \cdot \sin k\right)}}^{2}}\right) \]

   18. Simplified 73.6%

       \[\leadsto 2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{\color{blue}{t \cdot {\left(k \cdot \sin k\right)}^{2}}}\right) \]

   if 2.3500000000000001e-97 < t < 2.89999999999999989e84

   1. Initial program 69.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 69.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 69.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 69.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 69.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. unpow2 69.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. clear-num 69.8%

         \[\leadsto \frac{\frac{2}{\frac{k}{t} \cdot \color{blue}{\frac{1}{\frac{t}{k}}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      4. un-div-inv 69.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{k}{t}}{\frac{t}{k}}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Applied egg-rr 69.8%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{k}{t}}{\frac{t}{k}}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. associate-/r/ 69.9%

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{k}{t}} \cdot \frac{t}{k}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

   8. Applied egg-rr 69.9%

      \[\leadsto \frac{\color{blue}{\frac{2}{\frac{k}{t}} \cdot \frac{t}{k}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]
   9. Step-by-step derivation

      1. associate-/r* 77.4%

         \[\leadsto \frac{\frac{2}{\frac{k}{t}} \cdot \frac{t}{k}}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. associate-*l/ 80.0%

         \[\leadsto \frac{\frac{2}{\frac{k}{t}} \cdot \frac{t}{k}}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}{\ell}}} \]

   10. Applied egg-rr 80.0%

       \[\leadsto \frac{\frac{2}{\frac{k}{t}} \cdot \frac{t}{k}}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.35 \cdot 10^{-97} \lor \neg \left(t \leq 2.9 \cdot 10^{+84}\right):\\ \;\;\;\;2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{t \cdot {\left(k \cdot \sin k\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t}{k} \cdot \frac{2}{\frac{k}{t}}}{\frac{\left(\sin k \cdot \tan k\right) \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 71.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(k \cdot \sin k\right)}^{2}\\ \mathbf{if}\;t \leq 2.35 \cdot 10^{-97}:\\ \;\;\;\;\cos k \cdot \left(2 \cdot \frac{\frac{{\ell}^{2}}{t}}{t\_1}\right)\\ \mathbf{elif}\;t \leq 10^{+88}:\\ \;\;\;\;\frac{\frac{t}{k} \cdot \frac{2}{\frac{k}{t}}}{\frac{\left(\sin k \cdot \tan k\right) \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{t \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (* k (sin k)) 2.0)))
   (if (<= t 2.35e-97)
     (* (cos k) (* 2.0 (/ (/ (pow l 2.0) t) t_1)))
     (if (<= t 1e+88)
       (/
        (* (/ t k) (/ 2.0 (/ k t)))
        (/ (* (* (sin k) (tan k)) (/ (pow t 3.0) l)) l))
       (* 2.0 (/ (* (pow l 2.0) (cos k)) (* t t_1)))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k * sin(k)), 2.0);
	double tmp;
	if (t <= 2.35e-97) {
		tmp = cos(k) * (2.0 * ((pow(l, 2.0) / t) / t_1));
	} else if (t <= 1e+88) {
		tmp = ((t / k) * (2.0 / (k / t))) / (((sin(k) * tan(k)) * (pow(t, 3.0) / l)) / l);
	} else {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k)) / (t * t_1));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (k * sin(k)) ** 2.0d0
    if (t <= 2.35d-97) then
        tmp = cos(k) * (2.0d0 * (((l ** 2.0d0) / t) / t_1))
    else if (t <= 1d+88) then
        tmp = ((t / k) * (2.0d0 / (k / t))) / (((sin(k) * tan(k)) * ((t ** 3.0d0) / l)) / l)
    else
        tmp = 2.0d0 * (((l ** 2.0d0) * cos(k)) / (t * t_1))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k * Math.sin(k)), 2.0);
	double tmp;
	if (t <= 2.35e-97) {
		tmp = Math.cos(k) * (2.0 * ((Math.pow(l, 2.0) / t) / t_1));
	} else if (t <= 1e+88) {
		tmp = ((t / k) * (2.0 / (k / t))) / (((Math.sin(k) * Math.tan(k)) * (Math.pow(t, 3.0) / l)) / l);
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / (t * t_1));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow((k * math.sin(k)), 2.0)
	tmp = 0
	if t <= 2.35e-97:
		tmp = math.cos(k) * (2.0 * ((math.pow(l, 2.0) / t) / t_1))
	elif t <= 1e+88:
		tmp = ((t / k) * (2.0 / (k / t))) / (((math.sin(k) * math.tan(k)) * (math.pow(t, 3.0) / l)) / l)
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) * math.cos(k)) / (t * t_1))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(k * sin(k)) ^ 2.0
	tmp = 0.0
	if (t <= 2.35e-97)
		tmp = Float64(cos(k) * Float64(2.0 * Float64(Float64((l ^ 2.0) / t) / t_1)));
	elseif (t <= 1e+88)
		tmp = Float64(Float64(Float64(t / k) * Float64(2.0 / Float64(k / t))) / Float64(Float64(Float64(sin(k) * tan(k)) * Float64((t ^ 3.0) / l)) / l));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / Float64(t * t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k * sin(k)) ^ 2.0;
	tmp = 0.0;
	if (t <= 2.35e-97)
		tmp = cos(k) * (2.0 * (((l ^ 2.0) / t) / t_1));
	elseif (t <= 1e+88)
		tmp = ((t / k) * (2.0 / (k / t))) / (((sin(k) * tan(k)) * ((t ^ 3.0) / l)) / l);
	else
		tmp = 2.0 * (((l ^ 2.0) * cos(k)) / (t * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t, 2.35e-97], N[(N[Cos[k], $MachinePrecision] * N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / t), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1e+88], N[(N[(N[(t / k), $MachinePrecision] * N[(2.0 / N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.3500000000000001e-97

   1. Initial program 32.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 32.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 32.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 38.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 38.4%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 38.4%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 38.4%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 81.4%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 81.4%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}} \]

      2. associate-/r* 81.4%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \color{blue}{\frac{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

      3. associate-/r/ 81.4%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

   8. Simplified 81.4%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 81.4%

         \[\leadsto \color{blue}{\frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

   10. Applied egg-rr 81.7%

       \[\leadsto \color{blue}{\frac{\left(\frac{\sqrt{2} \cdot t}{k} \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}{\sqrt[3]{\sin k \cdot \tan k}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 81.7%

          \[\leadsto \color{blue}{\left(\frac{\sqrt{2} \cdot t}{k} \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

       2. associate-*l* 81.5%

          \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{k} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)} \]

       3. associate-*l/ 81.5%

          \[\leadsto \color{blue}{\left(\frac{\sqrt{2}}{k} \cdot t\right)} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

       4. *-commutative 81.5%

          \[\leadsto \color{blue}{\left(t \cdot \frac{\sqrt{2}}{k}\right)} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

       5. associate-*l* 81.5%

          \[\leadsto \left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left({\color{blue}{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

   12. Simplified 82.7%

       \[\leadsto \color{blue}{\left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)} \]

   13. Taylor expanded in t around 0 75.9%

       \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   14. Step-by-step derivation

       1. associate-*r* 75.9%

          \[\leadsto \frac{\color{blue}{\left({\ell}^{2} \cdot \cos k\right) \cdot {\left(\sqrt{2}\right)}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

       2. unpow2 75.9%

          \[\leadsto \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

       3. rem-square-sqrt 76.0%

          \[\leadsto \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot \color{blue}{2}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

       4. associate-*l/ 76.0%

          \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot 2} \]

       5. *-commutative 76.0%

          \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

       6. *-commutative 76.0%

          \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

       7. associate-/l* 76.0%

          \[\leadsto 2 \cdot \color{blue}{\left(\cos k \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)} \]

       8. associate-*r* 76.0%

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}}\right) \]

       9. *-commutative 76.0%

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{\color{blue}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}}\right) \]

   15. Simplified 76.0%

       \[\leadsto \color{blue}{2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}\right)} \]

   16. Taylor expanded in k around inf 76.0%

       \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   17. Step-by-step derivation

       1. *-commutative 76.0%

          \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot 2} \]

       2. *-commutative 76.0%

          \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{\left(t \cdot {\sin k}^{2}\right) \cdot {k}^{2}}} \cdot 2 \]

       3. associate-*l* 75.4%

          \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{t \cdot \left({\sin k}^{2} \cdot {k}^{2}\right)}} \cdot 2 \]

       4. unpow2 75.4%

          \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{t \cdot \left(\color{blue}{\left(\sin k \cdot \sin k\right)} \cdot {k}^{2}\right)} \cdot 2 \]

       5. unpow2 75.4%

          \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{t \cdot \left(\left(\sin k \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot 2 \]

       6. swap-sqr 75.4%

          \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{t \cdot \color{blue}{\left(\left(\sin k \cdot k\right) \cdot \left(\sin k \cdot k\right)\right)}} \cdot 2 \]

       7. unpow2 75.4%

          \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{t \cdot \color{blue}{{\left(\sin k \cdot k\right)}^{2}}} \cdot 2 \]

       8. *-commutative 75.4%

          \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\left(\sin k \cdot k\right)}^{2} \cdot t}} \cdot 2 \]

       9. associate-*l/ 75.5%

          \[\leadsto \color{blue}{\left(\frac{{\ell}^{2}}{{\left(\sin k \cdot k\right)}^{2} \cdot t} \cdot \cos k\right)} \cdot 2 \]

       10. *-commutative 75.5%

           \[\leadsto \color{blue}{\left(\cos k \cdot \frac{{\ell}^{2}}{{\left(\sin k \cdot k\right)}^{2} \cdot t}\right)} \cdot 2 \]

       11. associate-*l* 75.5%

           \[\leadsto \color{blue}{\cos k \cdot \left(\frac{{\ell}^{2}}{{\left(\sin k \cdot k\right)}^{2} \cdot t} \cdot 2\right)} \]

   18. Simplified 73.7%

       \[\leadsto \color{blue}{\cos k \cdot \left(\frac{\frac{{\ell}^{2}}{t}}{{\left(k \cdot \sin k\right)}^{2}} \cdot 2\right)} \]

   if 2.3500000000000001e-97 < t < 9.99999999999999959e87

   1. Initial program 69.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 69.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 69.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 69.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 69.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. unpow2 69.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. clear-num 69.8%

         \[\leadsto \frac{\frac{2}{\frac{k}{t} \cdot \color{blue}{\frac{1}{\frac{t}{k}}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      4. un-div-inv 69.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{k}{t}}{\frac{t}{k}}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Applied egg-rr 69.8%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{k}{t}}{\frac{t}{k}}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. associate-/r/ 69.9%

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{k}{t}} \cdot \frac{t}{k}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

   8. Applied egg-rr 69.9%

      \[\leadsto \frac{\color{blue}{\frac{2}{\frac{k}{t}} \cdot \frac{t}{k}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]
   9. Step-by-step derivation

      1. associate-/r* 77.4%

         \[\leadsto \frac{\frac{2}{\frac{k}{t}} \cdot \frac{t}{k}}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. associate-*l/ 80.0%

         \[\leadsto \frac{\frac{2}{\frac{k}{t}} \cdot \frac{t}{k}}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}{\ell}}} \]

   10. Applied egg-rr 80.0%

       \[\leadsto \frac{\frac{2}{\frac{k}{t}} \cdot \frac{t}{k}}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}{\ell}}} \]

   if 9.99999999999999959e87 < t

   1. Initial program 10.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 10.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 10.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 28.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 28.9%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 28.9%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 28.9%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 76.6%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 76.6%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}} \]

      2. associate-/r* 76.6%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \color{blue}{\frac{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

      3. associate-/r/ 76.7%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

   8. Simplified 76.7%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 76.7%

         \[\leadsto \color{blue}{\frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

   10. Applied egg-rr 76.6%

       \[\leadsto \color{blue}{\frac{\left(\frac{\sqrt{2} \cdot t}{k} \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}{\sqrt[3]{\sin k \cdot \tan k}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 76.6%

          \[\leadsto \color{blue}{\left(\frac{\sqrt{2} \cdot t}{k} \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

       2. associate-*l* 76.6%

          \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{k} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)} \]

       3. associate-*l/ 76.7%

          \[\leadsto \color{blue}{\left(\frac{\sqrt{2}}{k} \cdot t\right)} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

       4. *-commutative 76.7%

          \[\leadsto \color{blue}{\left(t \cdot \frac{\sqrt{2}}{k}\right)} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

       5. associate-*l* 76.8%

          \[\leadsto \left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left({\color{blue}{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

   12. Simplified 76.9%

       \[\leadsto \color{blue}{\left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)} \]

   13. Taylor expanded in t around 0 69.8%

       \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   14. Step-by-step derivation

       1. associate-*r* 69.8%

          \[\leadsto \frac{\color{blue}{\left({\ell}^{2} \cdot \cos k\right) \cdot {\left(\sqrt{2}\right)}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

       2. unpow2 69.8%

          \[\leadsto \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

       3. rem-square-sqrt 70.0%

          \[\leadsto \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot \color{blue}{2}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

       4. associate-*l/ 70.0%

          \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot 2} \]

       5. *-commutative 70.0%

          \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

       6. *-commutative 70.0%

          \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

       7. associate-/l* 70.0%

          \[\leadsto 2 \cdot \color{blue}{\left(\cos k \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)} \]

       8. associate-*r* 69.9%

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}}\right) \]

       9. *-commutative 69.9%

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{\color{blue}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}}\right) \]

   15. Simplified 69.9%

       \[\leadsto \color{blue}{2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}\right)} \]
   16. Step-by-step derivation

       1. pow2 69.9%

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}\right) \]

       2. associate-*r/ 70.0%

          \[\leadsto 2 \cdot \color{blue}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}} \]

       3. pow2 70.0%

          \[\leadsto 2 \cdot \frac{\cos k \cdot \color{blue}{{\ell}^{2}}}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)} \]

       4. associate-*r* 65.1%

          \[\leadsto 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{\color{blue}{\left({\sin k}^{2} \cdot {k}^{2}\right) \cdot t}} \]

       5. pow-prod-down 65.1%

          \[\leadsto 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{\color{blue}{{\left(\sin k \cdot k\right)}^{2}} \cdot t} \]

   17. Applied egg-rr 65.1%

       \[\leadsto 2 \cdot \color{blue}{\frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k \cdot k\right)}^{2} \cdot t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.35 \cdot 10^{-97}:\\ \;\;\;\;\cos k \cdot \left(2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{\left(k \cdot \sin k\right)}^{2}}\right)\\ \mathbf{elif}\;t \leq 10^{+88}:\\ \;\;\;\;\frac{\frac{t}{k} \cdot \frac{2}{\frac{k}{t}}}{\frac{\left(\sin k \cdot \tan k\right) \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{t \cdot {\left(k \cdot \sin k\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 72.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot {\left(k \cdot \sin k\right)}^{2}\\ \mathbf{if}\;t \leq 6.2 \cdot 10^{-99}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{t\_1}\right)\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+90}:\\ \;\;\;\;\frac{\frac{t}{k} \cdot \frac{2}{\frac{k}{t}}}{\frac{\left(\sin k \cdot \tan k\right) \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* t (pow (* k (sin k)) 2.0))))
   (if (<= t 6.2e-99)
     (* 2.0 (* (cos k) (/ (pow l 2.0) t_1)))
     (if (<= t 7e+90)
       (/
        (* (/ t k) (/ 2.0 (/ k t)))
        (/ (* (* (sin k) (tan k)) (/ (pow t 3.0) l)) l))
       (* 2.0 (/ (* (pow l 2.0) (cos k)) t_1))))))

C

double code(double t, double l, double k) {
	double t_1 = t * pow((k * sin(k)), 2.0);
	double tmp;
	if (t <= 6.2e-99) {
		tmp = 2.0 * (cos(k) * (pow(l, 2.0) / t_1));
	} else if (t <= 7e+90) {
		tmp = ((t / k) * (2.0 / (k / t))) / (((sin(k) * tan(k)) * (pow(t, 3.0) / l)) / l);
	} else {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k)) / t_1);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * ((k * sin(k)) ** 2.0d0)
    if (t <= 6.2d-99) then
        tmp = 2.0d0 * (cos(k) * ((l ** 2.0d0) / t_1))
    else if (t <= 7d+90) then
        tmp = ((t / k) * (2.0d0 / (k / t))) / (((sin(k) * tan(k)) * ((t ** 3.0d0) / l)) / l)
    else
        tmp = 2.0d0 * (((l ** 2.0d0) * cos(k)) / t_1)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = t * Math.pow((k * Math.sin(k)), 2.0);
	double tmp;
	if (t <= 6.2e-99) {
		tmp = 2.0 * (Math.cos(k) * (Math.pow(l, 2.0) / t_1));
	} else if (t <= 7e+90) {
		tmp = ((t / k) * (2.0 / (k / t))) / (((Math.sin(k) * Math.tan(k)) * (Math.pow(t, 3.0) / l)) / l);
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / t_1);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = t * math.pow((k * math.sin(k)), 2.0)
	tmp = 0
	if t <= 6.2e-99:
		tmp = 2.0 * (math.cos(k) * (math.pow(l, 2.0) / t_1))
	elif t <= 7e+90:
		tmp = ((t / k) * (2.0 / (k / t))) / (((math.sin(k) * math.tan(k)) * (math.pow(t, 3.0) / l)) / l)
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) * math.cos(k)) / t_1)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(t * (Float64(k * sin(k)) ^ 2.0))
	tmp = 0.0
	if (t <= 6.2e-99)
		tmp = Float64(2.0 * Float64(cos(k) * Float64((l ^ 2.0) / t_1)));
	elseif (t <= 7e+90)
		tmp = Float64(Float64(Float64(t / k) * Float64(2.0 / Float64(k / t))) / Float64(Float64(Float64(sin(k) * tan(k)) * Float64((t ^ 3.0) / l)) / l));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = t * ((k * sin(k)) ^ 2.0);
	tmp = 0.0;
	if (t <= 6.2e-99)
		tmp = 2.0 * (cos(k) * ((l ^ 2.0) / t_1));
	elseif (t <= 7e+90)
		tmp = ((t / k) * (2.0 / (k / t))) / (((sin(k) * tan(k)) * ((t ^ 3.0) / l)) / l);
	else
		tmp = 2.0 * (((l ^ 2.0) * cos(k)) / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(t * N[Power[N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 6.2e-99], N[(2.0 * N[(N[Cos[k], $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7e+90], N[(N[(N[(t / k), $MachinePrecision] * N[(2.0 / N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 6.1999999999999997e-99

   1. Initial program 32.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 32.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 32.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 38.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 38.4%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 38.4%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 38.4%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 81.4%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 81.4%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}} \]

      2. associate-/r* 81.4%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \color{blue}{\frac{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

      3. associate-/r/ 81.4%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

   8. Simplified 81.4%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 81.4%

         \[\leadsto \color{blue}{\frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

   10. Applied egg-rr 81.7%

       \[\leadsto \color{blue}{\frac{\left(\frac{\sqrt{2} \cdot t}{k} \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}{\sqrt[3]{\sin k \cdot \tan k}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 81.7%

          \[\leadsto \color{blue}{\left(\frac{\sqrt{2} \cdot t}{k} \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

       2. associate-*l* 81.5%

          \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{k} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)} \]

       3. associate-*l/ 81.5%

          \[\leadsto \color{blue}{\left(\frac{\sqrt{2}}{k} \cdot t\right)} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

       4. *-commutative 81.5%

          \[\leadsto \color{blue}{\left(t \cdot \frac{\sqrt{2}}{k}\right)} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

       5. associate-*l* 81.5%

          \[\leadsto \left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left({\color{blue}{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

   12. Simplified 82.7%

       \[\leadsto \color{blue}{\left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)} \]

   13. Taylor expanded in t around 0 75.9%

       \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   14. Step-by-step derivation

       1. associate-*r* 75.9%

          \[\leadsto \frac{\color{blue}{\left({\ell}^{2} \cdot \cos k\right) \cdot {\left(\sqrt{2}\right)}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

       2. unpow2 75.9%

          \[\leadsto \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

       3. rem-square-sqrt 76.0%

          \[\leadsto \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot \color{blue}{2}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

       4. associate-*l/ 76.0%

          \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot 2} \]

       5. *-commutative 76.0%

          \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

       6. *-commutative 76.0%

          \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

       7. associate-/l* 76.0%

          \[\leadsto 2 \cdot \color{blue}{\left(\cos k \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)} \]

       8. associate-*r* 76.0%

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}}\right) \]

       9. *-commutative 76.0%

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{\color{blue}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}}\right) \]

   15. Simplified 76.0%

       \[\leadsto \color{blue}{2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}\right)} \]

   16. Taylor expanded in k around inf 76.0%

       \[\leadsto 2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}}\right) \]
   17. Step-by-step derivation

       1. *-commutative 76.0%

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{\color{blue}{\left(t \cdot {\sin k}^{2}\right) \cdot {k}^{2}}}\right) \]

       2. associate-*l* 75.5%

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{\color{blue}{t \cdot \left({\sin k}^{2} \cdot {k}^{2}\right)}}\right) \]

       3. unpow2 75.5%

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{t \cdot \left(\color{blue}{\left(\sin k \cdot \sin k\right)} \cdot {k}^{2}\right)}\right) \]

       4. unpow2 75.5%

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{t \cdot \left(\left(\sin k \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)}\right) \]

       5. swap-sqr 75.5%

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{t \cdot \color{blue}{\left(\left(\sin k \cdot k\right) \cdot \left(\sin k \cdot k\right)\right)}}\right) \]

       6. unpow2 75.5%

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{t \cdot \color{blue}{{\left(\sin k \cdot k\right)}^{2}}}\right) \]

       7. *-commutative 75.5%

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{t \cdot {\color{blue}{\left(k \cdot \sin k\right)}}^{2}}\right) \]

   18. Simplified 75.5%

       \[\leadsto 2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{\color{blue}{t \cdot {\left(k \cdot \sin k\right)}^{2}}}\right) \]

   if 6.1999999999999997e-99 < t < 6.9999999999999997e90

   1. Initial program 69.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 69.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 69.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 69.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 69.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. unpow2 69.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. clear-num 69.8%

         \[\leadsto \frac{\frac{2}{\frac{k}{t} \cdot \color{blue}{\frac{1}{\frac{t}{k}}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      4. un-div-inv 69.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{k}{t}}{\frac{t}{k}}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Applied egg-rr 69.8%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{k}{t}}{\frac{t}{k}}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. associate-/r/ 69.9%

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{k}{t}} \cdot \frac{t}{k}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

   8. Applied egg-rr 69.9%

      \[\leadsto \frac{\color{blue}{\frac{2}{\frac{k}{t}} \cdot \frac{t}{k}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]
   9. Step-by-step derivation

      1. associate-/r* 77.4%

         \[\leadsto \frac{\frac{2}{\frac{k}{t}} \cdot \frac{t}{k}}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. associate-*l/ 80.0%

         \[\leadsto \frac{\frac{2}{\frac{k}{t}} \cdot \frac{t}{k}}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}{\ell}}} \]

   10. Applied egg-rr 80.0%

       \[\leadsto \frac{\frac{2}{\frac{k}{t}} \cdot \frac{t}{k}}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}{\ell}}} \]

   if 6.9999999999999997e90 < t

   1. Initial program 10.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 10.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 10.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 28.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 28.9%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 28.9%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 28.9%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 76.6%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 76.6%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}} \]

      2. associate-/r* 76.6%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \color{blue}{\frac{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

      3. associate-/r/ 76.7%

         \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

   8. Simplified 76.7%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 76.7%

         \[\leadsto \color{blue}{\frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

   10. Applied egg-rr 76.6%

       \[\leadsto \color{blue}{\frac{\left(\frac{\sqrt{2} \cdot t}{k} \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}{\sqrt[3]{\sin k \cdot \tan k}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 76.6%

          \[\leadsto \color{blue}{\left(\frac{\sqrt{2} \cdot t}{k} \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

       2. associate-*l* 76.6%

          \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{k} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)} \]

       3. associate-*l/ 76.7%

          \[\leadsto \color{blue}{\left(\frac{\sqrt{2}}{k} \cdot t\right)} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

       4. *-commutative 76.7%

          \[\leadsto \color{blue}{\left(t \cdot \frac{\sqrt{2}}{k}\right)} \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

       5. associate-*l* 76.8%

          \[\leadsto \left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left({\color{blue}{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}}^{-2} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \]

   12. Simplified 76.9%

       \[\leadsto \color{blue}{\left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)} \]

   13. Taylor expanded in t around 0 69.8%

       \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   14. Step-by-step derivation

       1. associate-*r* 69.8%

          \[\leadsto \frac{\color{blue}{\left({\ell}^{2} \cdot \cos k\right) \cdot {\left(\sqrt{2}\right)}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

       2. unpow2 69.8%

          \[\leadsto \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

       3. rem-square-sqrt 70.0%

          \[\leadsto \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot \color{blue}{2}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

       4. associate-*l/ 70.0%

          \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot 2} \]

       5. *-commutative 70.0%

          \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

       6. *-commutative 70.0%

          \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

       7. associate-/l* 70.0%

          \[\leadsto 2 \cdot \color{blue}{\left(\cos k \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)} \]

       8. associate-*r* 69.9%

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}}\right) \]

       9. *-commutative 69.9%

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{\color{blue}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}}\right) \]

   15. Simplified 69.9%

       \[\leadsto \color{blue}{2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}\right)} \]
   16. Step-by-step derivation

       1. pow2 69.9%

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}\right) \]

       2. associate-*r/ 70.0%

          \[\leadsto 2 \cdot \color{blue}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}} \]

       3. pow2 70.0%

          \[\leadsto 2 \cdot \frac{\cos k \cdot \color{blue}{{\ell}^{2}}}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)} \]

       4. associate-*r* 65.1%

          \[\leadsto 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{\color{blue}{\left({\sin k}^{2} \cdot {k}^{2}\right) \cdot t}} \]

       5. pow-prod-down 65.1%

          \[\leadsto 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{\color{blue}{{\left(\sin k \cdot k\right)}^{2}} \cdot t} \]

   17. Applied egg-rr 65.1%

       \[\leadsto 2 \cdot \color{blue}{\frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k \cdot k\right)}^{2} \cdot t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.2 \cdot 10^{-99}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{t \cdot {\left(k \cdot \sin k\right)}^{2}}\right)\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+90}:\\ \;\;\;\;\frac{\frac{t}{k} \cdot \frac{2}{\frac{k}{t}}}{\frac{\left(\sin k \cdot \tan k\right) \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{t \cdot {\left(k \cdot \sin k\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 64.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin k \cdot \tan k\\ \mathbf{if}\;t \leq 1.35 \cdot 10^{-167}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-90}:\\ \;\;\;\;\frac{\frac{t}{k} \cdot \frac{2}{\frac{k}{t}}}{t\_1 \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+95}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\frac{\frac{t\_1 \cdot {t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (sin k) (tan k))))
   (if (<= t 1.35e-167)
     (* 2.0 (/ (/ (pow l 2.0) t) (pow k 4.0)))
     (if (<= t 1.3e-90)
       (/ (* (/ t k) (/ 2.0 (/ k t))) (* t_1 (* (/ (pow t 2.0) l) (/ t l))))
       (if (<= t 2e+95)
         (/ (/ 2.0 (/ (/ k t) (/ t k))) (/ (/ (* t_1 (pow t 3.0)) l) l))
         (* 2.0 (/ (pow l 2.0) (* t (pow k 4.0)))))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(k) * tan(k);
	double tmp;
	if (t <= 1.35e-167) {
		tmp = 2.0 * ((pow(l, 2.0) / t) / pow(k, 4.0));
	} else if (t <= 1.3e-90) {
		tmp = ((t / k) * (2.0 / (k / t))) / (t_1 * ((pow(t, 2.0) / l) * (t / l)));
	} else if (t <= 2e+95) {
		tmp = (2.0 / ((k / t) / (t / k))) / (((t_1 * pow(t, 3.0)) / l) / l);
	} else {
		tmp = 2.0 * (pow(l, 2.0) / (t * pow(k, 4.0)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(k) * tan(k)
    if (t <= 1.35d-167) then
        tmp = 2.0d0 * (((l ** 2.0d0) / t) / (k ** 4.0d0))
    else if (t <= 1.3d-90) then
        tmp = ((t / k) * (2.0d0 / (k / t))) / (t_1 * (((t ** 2.0d0) / l) * (t / l)))
    else if (t <= 2d+95) then
        tmp = (2.0d0 / ((k / t) / (t / k))) / (((t_1 * (t ** 3.0d0)) / l) / l)
    else
        tmp = 2.0d0 * ((l ** 2.0d0) / (t * (k ** 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.sin(k) * Math.tan(k);
	double tmp;
	if (t <= 1.35e-167) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / t) / Math.pow(k, 4.0));
	} else if (t <= 1.3e-90) {
		tmp = ((t / k) * (2.0 / (k / t))) / (t_1 * ((Math.pow(t, 2.0) / l) * (t / l)));
	} else if (t <= 2e+95) {
		tmp = (2.0 / ((k / t) / (t / k))) / (((t_1 * Math.pow(t, 3.0)) / l) / l);
	} else {
		tmp = 2.0 * (Math.pow(l, 2.0) / (t * Math.pow(k, 4.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.sin(k) * math.tan(k)
	tmp = 0
	if t <= 1.35e-167:
		tmp = 2.0 * ((math.pow(l, 2.0) / t) / math.pow(k, 4.0))
	elif t <= 1.3e-90:
		tmp = ((t / k) * (2.0 / (k / t))) / (t_1 * ((math.pow(t, 2.0) / l) * (t / l)))
	elif t <= 2e+95:
		tmp = (2.0 / ((k / t) / (t / k))) / (((t_1 * math.pow(t, 3.0)) / l) / l)
	else:
		tmp = 2.0 * (math.pow(l, 2.0) / (t * math.pow(k, 4.0)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(sin(k) * tan(k))
	tmp = 0.0
	if (t <= 1.35e-167)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / t) / (k ^ 4.0)));
	elseif (t <= 1.3e-90)
		tmp = Float64(Float64(Float64(t / k) * Float64(2.0 / Float64(k / t))) / Float64(t_1 * Float64(Float64((t ^ 2.0) / l) * Float64(t / l))));
	elseif (t <= 2e+95)
		tmp = Float64(Float64(2.0 / Float64(Float64(k / t) / Float64(t / k))) / Float64(Float64(Float64(t_1 * (t ^ 3.0)) / l) / l));
	else
		tmp = Float64(2.0 * Float64((l ^ 2.0) / Float64(t * (k ^ 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(k) * tan(k);
	tmp = 0.0;
	if (t <= 1.35e-167)
		tmp = 2.0 * (((l ^ 2.0) / t) / (k ^ 4.0));
	elseif (t <= 1.3e-90)
		tmp = ((t / k) * (2.0 / (k / t))) / (t_1 * (((t ^ 2.0) / l) * (t / l)));
	elseif (t <= 2e+95)
		tmp = (2.0 / ((k / t) / (t / k))) / (((t_1 * (t ^ 3.0)) / l) / l);
	else
		tmp = 2.0 * ((l ^ 2.0) / (t * (k ^ 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 1.35e-167], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / t), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.3e-90], N[(N[(N[(t / k), $MachinePrecision] * N[(2.0 / N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2e+95], N[(N[(2.0 / N[(N[(k / t), $MachinePrecision] / N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t$95$1 * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 1.35e-167

   1. Initial program 32.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 39.1%

      \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 64.5%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   6. Step-by-step derivation

      1. *-commutative 64.5%

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}} \]

      2. associate-/r* 63.0%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]

   7. Simplified 63.0%

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]

   if 1.35e-167 < t < 1.3e-90

   1. Initial program 29.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 29.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 29.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 29.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 29.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. unpow2 29.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. clear-num 29.0%

         \[\leadsto \frac{\frac{2}{\frac{k}{t} \cdot \color{blue}{\frac{1}{\frac{t}{k}}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      4. un-div-inv 29.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{k}{t}}{\frac{t}{k}}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Applied egg-rr 29.0%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{k}{t}}{\frac{t}{k}}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. associate-/r/ 29.0%

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{k}{t}} \cdot \frac{t}{k}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

   8. Applied egg-rr 29.0%

      \[\leadsto \frac{\color{blue}{\frac{2}{\frac{k}{t}} \cdot \frac{t}{k}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]
   9. Step-by-step derivation

      1. unpow3 29.0%

         \[\leadsto \frac{\frac{2}{\frac{k}{t}} \cdot \frac{t}{k}}{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. times-frac 64.0%

         \[\leadsto \frac{\frac{2}{\frac{k}{t}} \cdot \frac{t}{k}}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. pow2 64.0%

         \[\leadsto \frac{\frac{2}{\frac{k}{t}} \cdot \frac{t}{k}}{\left(\frac{\color{blue}{{t}^{2}}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   10. Applied egg-rr 64.0%

       \[\leadsto \frac{\frac{2}{\frac{k}{t}} \cdot \frac{t}{k}}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   if 1.3e-90 < t < 2.00000000000000004e95

   1. Initial program 70.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 70.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 70.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 70.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 70.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. unpow2 70.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. clear-num 70.9%

         \[\leadsto \frac{\frac{2}{\frac{k}{t} \cdot \color{blue}{\frac{1}{\frac{t}{k}}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      4. un-div-inv 70.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{k}{t}}{\frac{t}{k}}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Applied egg-rr 70.9%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{k}{t}}{\frac{t}{k}}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 73.6%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\color{blue}{\frac{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}{\ell \cdot \ell}}} \]

      2. associate-/r* 84.3%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\color{blue}{\frac{\frac{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}{\ell}}{\ell}}} \]

   8. Applied egg-rr 84.3%

      \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\color{blue}{\frac{\frac{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}{\ell}}{\ell}}} \]

   if 2.00000000000000004e95 < t

   1. Initial program 10.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 29.7%

      \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 56.5%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.35 \cdot 10^{-167}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-90}:\\ \;\;\;\;\frac{\frac{t}{k} \cdot \frac{2}{\frac{k}{t}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+95}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\frac{\frac{\left(\sin k \cdot \tan k\right) \cdot {t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 63.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.05 \cdot 10^{-111}:\\ \;\;\;\;\frac{{\ell}^{2} \cdot \frac{2}{t}}{{k}^{4}}\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{+95}:\\ \;\;\;\;\frac{\frac{t}{k} \cdot \frac{2}{\frac{k}{t}}}{\frac{\left(\sin k \cdot \tan k\right) \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 3.05e-111)
   (/ (* (pow l 2.0) (/ 2.0 t)) (pow k 4.0))
   (if (<= t 1.95e+95)
     (/
      (* (/ t k) (/ 2.0 (/ k t)))
      (/ (* (* (sin k) (tan k)) (/ (pow t 3.0) l)) l))
     (* 2.0 (/ (pow l 2.0) (* t (pow k 4.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 3.05e-111) {
		tmp = (pow(l, 2.0) * (2.0 / t)) / pow(k, 4.0);
	} else if (t <= 1.95e+95) {
		tmp = ((t / k) * (2.0 / (k / t))) / (((sin(k) * tan(k)) * (pow(t, 3.0) / l)) / l);
	} else {
		tmp = 2.0 * (pow(l, 2.0) / (t * pow(k, 4.0)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 3.05d-111) then
        tmp = ((l ** 2.0d0) * (2.0d0 / t)) / (k ** 4.0d0)
    else if (t <= 1.95d+95) then
        tmp = ((t / k) * (2.0d0 / (k / t))) / (((sin(k) * tan(k)) * ((t ** 3.0d0) / l)) / l)
    else
        tmp = 2.0d0 * ((l ** 2.0d0) / (t * (k ** 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 3.05e-111) {
		tmp = (Math.pow(l, 2.0) * (2.0 / t)) / Math.pow(k, 4.0);
	} else if (t <= 1.95e+95) {
		tmp = ((t / k) * (2.0 / (k / t))) / (((Math.sin(k) * Math.tan(k)) * (Math.pow(t, 3.0) / l)) / l);
	} else {
		tmp = 2.0 * (Math.pow(l, 2.0) / (t * Math.pow(k, 4.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 3.05e-111:
		tmp = (math.pow(l, 2.0) * (2.0 / t)) / math.pow(k, 4.0)
	elif t <= 1.95e+95:
		tmp = ((t / k) * (2.0 / (k / t))) / (((math.sin(k) * math.tan(k)) * (math.pow(t, 3.0) / l)) / l)
	else:
		tmp = 2.0 * (math.pow(l, 2.0) / (t * math.pow(k, 4.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 3.05e-111)
		tmp = Float64(Float64((l ^ 2.0) * Float64(2.0 / t)) / (k ^ 4.0));
	elseif (t <= 1.95e+95)
		tmp = Float64(Float64(Float64(t / k) * Float64(2.0 / Float64(k / t))) / Float64(Float64(Float64(sin(k) * tan(k)) * Float64((t ^ 3.0) / l)) / l));
	else
		tmp = Float64(2.0 * Float64((l ^ 2.0) / Float64(t * (k ^ 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 3.05e-111)
		tmp = ((l ^ 2.0) * (2.0 / t)) / (k ^ 4.0);
	elseif (t <= 1.95e+95)
		tmp = ((t / k) * (2.0 / (k / t))) / (((sin(k) * tan(k)) * ((t ^ 3.0) / l)) / l);
	else
		tmp = 2.0 * ((l ^ 2.0) / (t * (k ^ 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 3.05e-111], N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(2.0 / t), $MachinePrecision]), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.95e+95], N[(N[(N[(t / k), $MachinePrecision] * N[(2.0 / N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 3.0500000000000001e-111

   1. Initial program 31.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 37.8%

      \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 64.1%

      \[\leadsto \color{blue}{\frac{2}{{k}^{4} \cdot t}} \cdot \left(\ell \cdot \ell\right) \]
   6. Step-by-step derivation

      1. *-commutative 64.1%

         \[\leadsto \frac{2}{\color{blue}{t \cdot {k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

      2. associate-/r* 64.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

   7. Simplified 64.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]
   8. Step-by-step derivation

      1. associate-*l/ 62.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{t} \cdot \left(\ell \cdot \ell\right)}{{k}^{4}}} \]

      2. pow2 62.8%

         \[\leadsto \frac{\frac{2}{t} \cdot \color{blue}{{\ell}^{2}}}{{k}^{4}} \]

   9. Applied egg-rr 62.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{t} \cdot {\ell}^{2}}{{k}^{4}}} \]

   if 3.0500000000000001e-111 < t < 1.9499999999999999e95

   1. Initial program 67.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 67.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 67.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 67.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 67.2%

         \[\leadsto \frac{\frac{2}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. unpow2 67.2%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. clear-num 67.2%

         \[\leadsto \frac{\frac{2}{\frac{k}{t} \cdot \color{blue}{\frac{1}{\frac{t}{k}}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      4. un-div-inv 67.2%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{k}{t}}{\frac{t}{k}}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Applied egg-rr 67.2%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{k}{t}}{\frac{t}{k}}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. associate-/r/ 67.2%

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{k}{t}} \cdot \frac{t}{k}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

   8. Applied egg-rr 67.2%

      \[\leadsto \frac{\color{blue}{\frac{2}{\frac{k}{t}} \cdot \frac{t}{k}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]
   9. Step-by-step derivation

      1. associate-/r* 76.0%

         \[\leadsto \frac{\frac{2}{\frac{k}{t}} \cdot \frac{t}{k}}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. associate-*l/ 80.5%

         \[\leadsto \frac{\frac{2}{\frac{k}{t}} \cdot \frac{t}{k}}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}{\ell}}} \]

   10. Applied egg-rr 80.5%

       \[\leadsto \frac{\frac{2}{\frac{k}{t}} \cdot \frac{t}{k}}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}{\ell}}} \]

   if 1.9499999999999999e95 < t

   1. Initial program 10.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 29.7%

      \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 56.5%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.05 \cdot 10^{-111}:\\ \;\;\;\;\frac{{\ell}^{2} \cdot \frac{2}{t}}{{k}^{4}}\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{+95}:\\ \;\;\;\;\frac{\frac{t}{k} \cdot \frac{2}{\frac{k}{t}}}{\frac{\left(\sin k \cdot \tan k\right) \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 61.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{{\ell}^{2} \cdot \left(2 \cdot {k}^{-4}\right)}{t} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (/ (* (pow l 2.0) (* 2.0 (pow k -4.0))) t))

C

double code(double t, double l, double k) {
	return (pow(l, 2.0) * (2.0 * pow(k, -4.0))) / t;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = ((l ** 2.0d0) * (2.0d0 * (k ** (-4.0d0)))) / t
end function

Java

public static double code(double t, double l, double k) {
	return (Math.pow(l, 2.0) * (2.0 * Math.pow(k, -4.0))) / t;
}

Python

def code(t, l, k):
	return (math.pow(l, 2.0) * (2.0 * math.pow(k, -4.0))) / t

Julia

function code(t, l, k)
	return Float64(Float64((l ^ 2.0) * Float64(2.0 * (k ^ -4.0))) / t)
end

MATLAB

function tmp = code(t, l, k)
	tmp = ((l ^ 2.0) * (2.0 * (k ^ -4.0))) / t;
end

Wolfram

code[t_, l_, k_] := N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(2.0 * N[Power[k, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]

Derivation

1. Initial program 34.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

3. Simplified 42.2%

   \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 62.9%

   \[\leadsto \color{blue}{\frac{2}{{k}^{4} \cdot t}} \cdot \left(\ell \cdot \ell\right) \]
6. Step-by-step derivation

   1. *-commutative 62.9%

      \[\leadsto \frac{2}{\color{blue}{t \cdot {k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

   2. associate-/r* 62.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

7. Simplified 62.9%

   \[\leadsto \color{blue}{\frac{\frac{2}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]
8. Step-by-step derivation

   1. div-inv 62.6%

      \[\leadsto \color{blue}{\left(\frac{2}{t} \cdot \frac{1}{{k}^{4}}\right)} \cdot \left(\ell \cdot \ell\right) \]

   2. pow-flip 62.6%

      \[\leadsto \left(\frac{2}{t} \cdot \color{blue}{{k}^{\left(-4\right)}}\right) \cdot \left(\ell \cdot \ell\right) \]

   3. metadata-eval 62.6%

      \[\leadsto \left(\frac{2}{t} \cdot {k}^{\color{blue}{-4}}\right) \cdot \left(\ell \cdot \ell\right) \]

9. Applied egg-rr 62.6%

   \[\leadsto \color{blue}{\left(\frac{2}{t} \cdot {k}^{-4}\right)} \cdot \left(\ell \cdot \ell\right) \]
10. Step-by-step derivation

    1. associate-*l/ 62.6%

       \[\leadsto \color{blue}{\frac{2 \cdot {k}^{-4}}{t}} \cdot \left(\ell \cdot \ell\right) \]

11. Simplified 62.6%

    \[\leadsto \color{blue}{\frac{2 \cdot {k}^{-4}}{t}} \cdot \left(\ell \cdot \ell\right) \]
12. Step-by-step derivation

    1. associate-*l/ 63.5%

       \[\leadsto \color{blue}{\frac{\left(2 \cdot {k}^{-4}\right) \cdot \left(\ell \cdot \ell\right)}{t}} \]

    2. pow2 63.5%

       \[\leadsto \frac{\left(2 \cdot {k}^{-4}\right) \cdot \color{blue}{{\ell}^{2}}}{t} \]

13. Applied egg-rr 63.5%

    \[\leadsto \color{blue}{\frac{\left(2 \cdot {k}^{-4}\right) \cdot {\ell}^{2}}{t}} \]

14. Final simplification 63.5%

    \[\leadsto \frac{{\ell}^{2} \cdot \left(2 \cdot {k}^{-4}\right)}{t} \]
15. Add Preprocessing

Alternative 19: 61.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot {k}^{4}} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* (* l l) (/ 2.0 (* t (pow k 4.0)))))

C

double code(double t, double l, double k) {
	return (l * l) * (2.0 / (t * pow(k, 4.0)));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (l * l) * (2.0d0 / (t * (k ** 4.0d0)))
end function

Java

public static double code(double t, double l, double k) {
	return (l * l) * (2.0 / (t * Math.pow(k, 4.0)));
}

Python

def code(t, l, k):
	return (l * l) * (2.0 / (t * math.pow(k, 4.0)))

Julia

function code(t, l, k)
	return Float64(Float64(l * l) * Float64(2.0 / Float64(t * (k ^ 4.0))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l * l) * (2.0 / (t * (k ^ 4.0)));
end

Wolfram

code[t_, l_, k_] := N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

3. Simplified 42.2%

   \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 62.9%

   \[\leadsto \color{blue}{\frac{2}{{k}^{4} \cdot t}} \cdot \left(\ell \cdot \ell\right) \]

6. Final simplification 62.9%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot {k}^{4}} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024181 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  :precision binary64
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))