Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.4% → 86.7%

Time: 20.1s

Alternatives: 14

Speedup: 3.8×

• Could not converge on a ground truth

  1. reg0 = (bf "2.185745874710962328360030333926872276252e-215")
  2. reg1 = (bf "9.633726260577794647324209028514258095478e-106")
  3. reg2 = (bf "4.532651613588839267051147447129864455613e-290")

• 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: 86.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt[3]{\sin k \cdot \frac{\tan k}{\ell}}\\ t_2 := \frac{\sqrt{2}}{k}\\ t_3 := \frac{\frac{t\_2}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\\ \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-296}:\\ \;\;\;\;\left(t\_2 \cdot \frac{t}{{\left(\frac{t \cdot t\_1}{\sqrt[3]{\ell}}\right)}^{2}}\right) \cdot t\_3\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{-44}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(t \cdot \left(k \cdot k\right)\right) \cdot {\sin k}^{2}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+201}:\\ \;\;\;\;t\_3 \cdot \left(\frac{\sqrt{2}}{k \cdot t} \cdot \sqrt[3]{\frac{{\ell}^{4} \cdot {\cos k}^{2}}{{\sin k}^{4}}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3 \cdot \left(t\_2 \cdot \frac{t}{{\left(t\_1 \cdot \frac{t}{\sqrt[3]{\ell}}\right)}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (cbrt (* (sin k) (/ (tan k) l))))
        (t_2 (/ (sqrt 2.0) k))
        (t_3 (/ (/ t_2 (pow (cbrt l) -2.0)) (cbrt (* (sin k) (tan k))))))
   (if (<= (* l l) 5e-296)
     (* (* t_2 (/ t (pow (/ (* t t_1) (cbrt l)) 2.0))) t_3)
     (if (<= (* l l) 2e-44)
       (/ (* 2.0 (* (pow l 2.0) (cos k))) (* (* t (* k k)) (pow (sin k) 2.0)))
       (if (<= (* l l) 2e+201)
         (*
          t_3
          (*
           (/ (sqrt 2.0) (* k t))
           (cbrt (/ (* (pow l 4.0) (pow (cos k) 2.0)) (pow (sin k) 4.0)))))
         (* t_3 (* t_2 (/ t (pow (* t_1 (/ t (cbrt l))) 2.0)))))))))

C

double code(double t, double l, double k) {
	double t_1 = cbrt((sin(k) * (tan(k) / l)));
	double t_2 = sqrt(2.0) / k;
	double t_3 = (t_2 / pow(cbrt(l), -2.0)) / cbrt((sin(k) * tan(k)));
	double tmp;
	if ((l * l) <= 5e-296) {
		tmp = (t_2 * (t / pow(((t * t_1) / cbrt(l)), 2.0))) * t_3;
	} else if ((l * l) <= 2e-44) {
		tmp = (2.0 * (pow(l, 2.0) * cos(k))) / ((t * (k * k)) * pow(sin(k), 2.0));
	} else if ((l * l) <= 2e+201) {
		tmp = t_3 * ((sqrt(2.0) / (k * t)) * cbrt(((pow(l, 4.0) * pow(cos(k), 2.0)) / pow(sin(k), 4.0))));
	} else {
		tmp = t_3 * (t_2 * (t / pow((t_1 * (t / cbrt(l))), 2.0)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.cbrt((Math.sin(k) * (Math.tan(k) / l)));
	double t_2 = Math.sqrt(2.0) / k;
	double t_3 = (t_2 / Math.pow(Math.cbrt(l), -2.0)) / Math.cbrt((Math.sin(k) * Math.tan(k)));
	double tmp;
	if ((l * l) <= 5e-296) {
		tmp = (t_2 * (t / Math.pow(((t * t_1) / Math.cbrt(l)), 2.0))) * t_3;
	} else if ((l * l) <= 2e-44) {
		tmp = (2.0 * (Math.pow(l, 2.0) * Math.cos(k))) / ((t * (k * k)) * Math.pow(Math.sin(k), 2.0));
	} else if ((l * l) <= 2e+201) {
		tmp = t_3 * ((Math.sqrt(2.0) / (k * t)) * Math.cbrt(((Math.pow(l, 4.0) * Math.pow(Math.cos(k), 2.0)) / Math.pow(Math.sin(k), 4.0))));
	} else {
		tmp = t_3 * (t_2 * (t / Math.pow((t_1 * (t / Math.cbrt(l))), 2.0)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cbrt(Float64(sin(k) * Float64(tan(k) / l)))
	t_2 = Float64(sqrt(2.0) / k)
	t_3 = Float64(Float64(t_2 / (cbrt(l) ^ -2.0)) / cbrt(Float64(sin(k) * tan(k))))
	tmp = 0.0
	if (Float64(l * l) <= 5e-296)
		tmp = Float64(Float64(t_2 * Float64(t / (Float64(Float64(t * t_1) / cbrt(l)) ^ 2.0))) * t_3);
	elseif (Float64(l * l) <= 2e-44)
		tmp = Float64(Float64(2.0 * Float64((l ^ 2.0) * cos(k))) / Float64(Float64(t * Float64(k * k)) * (sin(k) ^ 2.0)));
	elseif (Float64(l * l) <= 2e+201)
		tmp = Float64(t_3 * Float64(Float64(sqrt(2.0) / Float64(k * t)) * cbrt(Float64(Float64((l ^ 4.0) * (cos(k) ^ 2.0)) / (sin(k) ^ 4.0)))));
	else
		tmp = Float64(t_3 * Float64(t_2 * Float64(t / (Float64(t_1 * Float64(t / cbrt(l))) ^ 2.0))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$2 / N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 5e-296], N[(N[(t$95$2 * N[(t / N[Power[N[(N[(t * t$95$1), $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 2e-44], N[(N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 2e+201], N[(t$95$3 * N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[Power[l, 4.0], $MachinePrecision] * N[Power[N[Cos[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 * N[(t$95$2 * N[(t / N[Power[N[(t$95$1 * N[(t / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 l l) < 5.0000000000000003e-296

   1. Initial program 33.4%

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

      1. *-commutative 33.4%

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

      2. associate-/r* 33.4%

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

   3. Simplified 42.1%

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

      1. add-sqr-sqrt 42.1%

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

      2. add-cube-cbrt 42.1%

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

      3. times-frac 42.1%

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

   6. Applied egg-rr 85.6%

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

      1. associate-/r/ 85.6%

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

      2. associate-/r* 85.6%

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

      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-/l* 85.6%

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

      2. div-inv 85.6%

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

      3. pow-flip 85.6%

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

      4. metadata-eval 85.6%

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

   10. Applied egg-rr 85.6%

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

       1. *-commutative 85.6%

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

       2. associate-/r* 85.6%

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

       3. *-inverses 85.6%

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

       4. associate-*l/ 85.6%

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

       5. *-lft-identity 85.6%

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

   12. Simplified 85.6%

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

       1. associate-/l* 90.0%

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

       2. div-inv 90.0%

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

       3. pow-flip 90.0%

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

       4. metadata-eval 90.0%

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

   14. Applied egg-rr 90.0%

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

       1. metadata-eval 90.0%

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

       2. pow-flip 90.0%

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

       3. div-inv 90.0%

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

       4. associate-*l/ 91.6%

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

       5. add-cbrt-cube 68.4%

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

       6. unpow3 68.4%

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

       7. cbrt-prod 67.0%

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

       8. unpow2 67.0%

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

       9. cbrt-prod 53.8%

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

       10. cbrt-div 53.8%

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

       11. associate-/l/ 67.0%

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

       12. cbrt-div 67.0%

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

   16. Applied egg-rr 91.5%

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

       1. associate-/l* 91.4%

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

   18. Simplified 91.4%

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

   if 5.0000000000000003e-296 < (*.f64 l l) < 1.99999999999999991e-44

   1. Initial program 45.7%

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

   3. Simplified 57.2%

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

   5. Taylor expanded in t around 0 97.1%

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

      1. associate-*r/ 97.1%

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

      2. associate-*r* 97.1%

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

   7. Simplified 97.1%

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

      1. unpow2 97.1%

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

   9. Applied egg-rr 97.1%

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

   if 1.99999999999999991e-44 < (*.f64 l l) < 2.00000000000000008e201

   1. Initial program 35.9%

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

      1. *-commutative 35.9%

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

      2. associate-/r* 36.0%

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

   3. Simplified 47.0%

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

      1. add-sqr-sqrt 47.0%

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

      2. add-cube-cbrt 46.9%

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

      3. times-frac 46.9%

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

   6. Applied egg-rr 66.6%

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

      1. associate-/r/ 66.6%

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

      2. associate-/r* 66.5%

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

      3. associate-/r/ 66.6%

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

   8. Simplified 66.6%

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

      1. associate-/l* 66.6%

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

      2. div-inv 66.6%

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

      3. pow-flip 66.6%

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

      4. metadata-eval 66.6%

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

   10. Applied egg-rr 66.6%

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

       1. *-commutative 66.6%

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

       2. associate-/r* 66.6%

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

       3. *-inverses 66.6%

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

       4. associate-*l/ 66.6%

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

       5. *-lft-identity 66.6%

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

   12. Simplified 66.6%

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

   13. Taylor expanded in k around inf 91.0%

       \[\leadsto \color{blue}{\left(\frac{\sqrt{2}}{k \cdot t} \cdot \sqrt[3]{\frac{{\ell}^{4} \cdot {\cos k}^{2}}{{\sin k}^{4}}}\right)} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

   if 2.00000000000000008e201 < (*.f64 l l)

   1. Initial program 31.0%

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

      1. *-commutative 31.0%

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

      2. associate-/r* 31.0%

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

   3. Simplified 32.1%

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

      1. add-sqr-sqrt 32.1%

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

      2. add-cube-cbrt 32.1%

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

      3. times-frac 32.1%

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

   6. Applied egg-rr 74.0%

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

      1. associate-/r/ 74.0%

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

      2. associate-/r* 74.0%

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

      3. associate-/r/ 76.3%

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

   8. Simplified 76.3%

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

      1. associate-/l* 76.3%

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

      2. div-inv 76.3%

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

      3. pow-flip 76.3%

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

      4. metadata-eval 76.3%

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

   10. Applied egg-rr 76.3%

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

       1. *-commutative 76.3%

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

       2. associate-/r* 76.3%

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

       3. *-inverses 76.3%

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

       4. associate-*l/ 76.3%

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

       5. *-lft-identity 76.3%

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

   12. Simplified 76.3%

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

       1. associate-/l* 84.1%

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

       2. div-inv 84.1%

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

       3. pow-flip 84.1%

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

       4. metadata-eval 84.1%

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

   14. Applied egg-rr 84.1%

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

       1. metadata-eval 84.1%

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

       2. pow-flip 84.1%

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

       3. div-inv 84.1%

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

       4. associate-*l/ 84.1%

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

       5. add-cbrt-cube 60.1%

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

       6. unpow3 60.1%

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

       7. cbrt-prod 60.1%

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

       8. unpow2 60.1%

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

       9. cbrt-prod 54.3%

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

       10. cbrt-div 54.2%

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

       11. associate-/l/ 58.0%

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

       12. cbrt-div 60.1%

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

   16. Applied egg-rr 84.1%

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

       1. *-commutative 84.1%

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

       2. associate-/l* 84.2%

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

       3. associate-/l* 84.2%

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

   18. Simplified 84.2%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-296}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{k} \cdot \frac{t}{{\left(\frac{t \cdot \sqrt[3]{\sin k \cdot \frac{\tan k}{\ell}}}{\sqrt[3]{\ell}}\right)}^{2}}\right) \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{-44}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(t \cdot \left(k \cdot k\right)\right) \cdot {\sin k}^{2}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+201}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}} \cdot \left(\frac{\sqrt{2}}{k \cdot t} \cdot \sqrt[3]{\frac{{\ell}^{4} \cdot {\cos k}^{2}}{{\sin k}^{4}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}} \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{t}{{\left(\sqrt[3]{\sin k \cdot \frac{\tan k}{\ell}} \cdot \frac{t}{\sqrt[3]{\ell}}\right)}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt[3]{\sin k \cdot \frac{\tan k}{\ell}}\\ t_2 := \frac{\sqrt{2}}{k}\\ t_3 := \frac{\frac{t\_2}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\\ \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-296}:\\ \;\;\;\;\left(t\_2 \cdot \frac{t}{{\left(\frac{t \cdot t\_1}{\sqrt[3]{\ell}}\right)}^{2}}\right) \cdot t\_3\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+264}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(t \cdot \left(k \cdot k\right)\right) \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_3 \cdot \left(t\_2 \cdot \frac{t}{{\left(t\_1 \cdot \frac{t}{\sqrt[3]{\ell}}\right)}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (cbrt (* (sin k) (/ (tan k) l))))
        (t_2 (/ (sqrt 2.0) k))
        (t_3 (/ (/ t_2 (pow (cbrt l) -2.0)) (cbrt (* (sin k) (tan k))))))
   (if (<= (* l l) 5e-296)
     (* (* t_2 (/ t (pow (/ (* t t_1) (cbrt l)) 2.0))) t_3)
     (if (<= (* l l) 2e+264)
       (/ (* 2.0 (* (pow l 2.0) (cos k))) (* (* t (* k k)) (pow (sin k) 2.0)))
       (* t_3 (* t_2 (/ t (pow (* t_1 (/ t (cbrt l))) 2.0))))))))

C

double code(double t, double l, double k) {
	double t_1 = cbrt((sin(k) * (tan(k) / l)));
	double t_2 = sqrt(2.0) / k;
	double t_3 = (t_2 / pow(cbrt(l), -2.0)) / cbrt((sin(k) * tan(k)));
	double tmp;
	if ((l * l) <= 5e-296) {
		tmp = (t_2 * (t / pow(((t * t_1) / cbrt(l)), 2.0))) * t_3;
	} else if ((l * l) <= 2e+264) {
		tmp = (2.0 * (pow(l, 2.0) * cos(k))) / ((t * (k * k)) * pow(sin(k), 2.0));
	} else {
		tmp = t_3 * (t_2 * (t / pow((t_1 * (t / cbrt(l))), 2.0)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.cbrt((Math.sin(k) * (Math.tan(k) / l)));
	double t_2 = Math.sqrt(2.0) / k;
	double t_3 = (t_2 / Math.pow(Math.cbrt(l), -2.0)) / Math.cbrt((Math.sin(k) * Math.tan(k)));
	double tmp;
	if ((l * l) <= 5e-296) {
		tmp = (t_2 * (t / Math.pow(((t * t_1) / Math.cbrt(l)), 2.0))) * t_3;
	} else if ((l * l) <= 2e+264) {
		tmp = (2.0 * (Math.pow(l, 2.0) * Math.cos(k))) / ((t * (k * k)) * Math.pow(Math.sin(k), 2.0));
	} else {
		tmp = t_3 * (t_2 * (t / Math.pow((t_1 * (t / Math.cbrt(l))), 2.0)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cbrt(Float64(sin(k) * Float64(tan(k) / l)))
	t_2 = Float64(sqrt(2.0) / k)
	t_3 = Float64(Float64(t_2 / (cbrt(l) ^ -2.0)) / cbrt(Float64(sin(k) * tan(k))))
	tmp = 0.0
	if (Float64(l * l) <= 5e-296)
		tmp = Float64(Float64(t_2 * Float64(t / (Float64(Float64(t * t_1) / cbrt(l)) ^ 2.0))) * t_3);
	elseif (Float64(l * l) <= 2e+264)
		tmp = Float64(Float64(2.0 * Float64((l ^ 2.0) * cos(k))) / Float64(Float64(t * Float64(k * k)) * (sin(k) ^ 2.0)));
	else
		tmp = Float64(t_3 * Float64(t_2 * Float64(t / (Float64(t_1 * Float64(t / cbrt(l))) ^ 2.0))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$2 / N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 5e-296], N[(N[(t$95$2 * N[(t / N[Power[N[(N[(t * t$95$1), $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 2e+264], N[(N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 * N[(t$95$2 * N[(t / N[Power[N[(t$95$1 * N[(t / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 5.0000000000000003e-296

   1. Initial program 33.4%

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

      1. *-commutative 33.4%

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

      2. associate-/r* 33.4%

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

   3. Simplified 42.1%

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

      1. add-sqr-sqrt 42.1%

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

      2. add-cube-cbrt 42.1%

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

      3. times-frac 42.1%

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

   6. Applied egg-rr 85.6%

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

      1. associate-/r/ 85.6%

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

      2. associate-/r* 85.6%

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

      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-/l* 85.6%

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

      2. div-inv 85.6%

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

      3. pow-flip 85.6%

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

      4. metadata-eval 85.6%

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

   10. Applied egg-rr 85.6%

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

       1. *-commutative 85.6%

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

       2. associate-/r* 85.6%

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

       3. *-inverses 85.6%

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

       4. associate-*l/ 85.6%

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

       5. *-lft-identity 85.6%

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

   12. Simplified 85.6%

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

       1. associate-/l* 90.0%

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

       2. div-inv 90.0%

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

       3. pow-flip 90.0%

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

       4. metadata-eval 90.0%

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

   14. Applied egg-rr 90.0%

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

       1. metadata-eval 90.0%

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

       2. pow-flip 90.0%

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

       3. div-inv 90.0%

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

       4. associate-*l/ 91.6%

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

       5. add-cbrt-cube 68.4%

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

       6. unpow3 68.4%

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

       7. cbrt-prod 67.0%

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

       8. unpow2 67.0%

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

       9. cbrt-prod 53.8%

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

       10. cbrt-div 53.8%

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

       11. associate-/l/ 67.0%

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

       12. cbrt-div 67.0%

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

   16. Applied egg-rr 91.5%

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

       1. associate-/l* 91.4%

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

   18. Simplified 91.4%

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

   if 5.0000000000000003e-296 < (*.f64 l l) < 2.00000000000000009e264

   1. Initial program 40.1%

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

   3. Simplified 50.7%

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

   5. Taylor expanded in t around 0 87.7%

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

      1. associate-*r/ 87.7%

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

      2. associate-*r* 87.8%

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

   7. Simplified 87.8%

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

      1. unpow2 87.8%

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

   9. Applied egg-rr 87.8%

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

   if 2.00000000000000009e264 < (*.f64 l l)

   1. Initial program 31.5%

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

      1. *-commutative 31.5%

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

      2. associate-/r* 31.5%

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

   3. Simplified 31.5%

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

      1. add-sqr-sqrt 31.5%

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

      2. add-cube-cbrt 31.5%

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

      3. times-frac 31.5%

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

   6. Applied egg-rr 75.2%

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

      1. associate-/r/ 75.2%

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

      2. associate-/r* 75.2%

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

      3. associate-/r/ 77.7%

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

   8. Simplified 77.7%

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

      1. associate-/l* 77.8%

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

      2. div-inv 77.8%

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

      3. pow-flip 77.8%

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

      4. metadata-eval 77.8%

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

   10. Applied egg-rr 77.8%

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

       1. *-commutative 77.8%

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

       2. associate-/r* 77.8%

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

       3. *-inverses 77.8%

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

       4. associate-*l/ 77.8%

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

       5. *-lft-identity 77.8%

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

   12. Simplified 77.8%

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

       1. associate-/l* 86.3%

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

       2. div-inv 86.2%

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

       3. pow-flip 86.3%

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

       4. metadata-eval 86.3%

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

   14. Applied egg-rr 86.3%

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

       1. metadata-eval 86.3%

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

       2. pow-flip 86.2%

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

       3. div-inv 86.3%

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

       4. associate-*l/ 86.2%

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

       5. add-cbrt-cube 62.2%

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

       6. unpow3 62.2%

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

       7. cbrt-prod 62.2%

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

       8. unpow2 62.2%

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

       9. cbrt-prod 55.8%

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

       10. cbrt-div 55.8%

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

       11. associate-/l/ 59.9%

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

       12. cbrt-div 62.2%

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

   16. Applied egg-rr 86.3%

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

       1. *-commutative 86.3%

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

       2. associate-/l* 86.4%

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

       3. associate-/l* 86.3%

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

   18. Simplified 86.3%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-296}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{k} \cdot \frac{t}{{\left(\frac{t \cdot \sqrt[3]{\sin k \cdot \frac{\tan k}{\ell}}}{\sqrt[3]{\ell}}\right)}^{2}}\right) \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+264}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(t \cdot \left(k \cdot k\right)\right) \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}} \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{t}{{\left(\sqrt[3]{\sin k \cdot \frac{\tan k}{\ell}} \cdot \frac{t}{\sqrt[3]{\ell}}\right)}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sqrt{2}}{k}\\ \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-296}:\\ \;\;\;\;e^{2 \cdot \log \ell + \log \left(\frac{2}{t \cdot {k}^{4}}\right)}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+264}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(t \cdot \left(k \cdot k\right)\right) \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}} \cdot \left(t\_1 \cdot \frac{t}{{\left(\sqrt[3]{\sin k \cdot \frac{\tan k}{\ell}} \cdot \frac{t}{\sqrt[3]{\ell}}\right)}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (sqrt 2.0) k)))
   (if (<= (* l l) 5e-296)
     (exp (+ (* 2.0 (log l)) (log (/ 2.0 (* t (pow k 4.0))))))
     (if (<= (* l l) 2e+264)
       (/ (* 2.0 (* (pow l 2.0) (cos k))) (* (* t (* k k)) (pow (sin k) 2.0)))
       (*
        (/ (/ t_1 (pow (cbrt l) -2.0)) (cbrt (* (sin k) (tan k))))
        (*
         t_1
         (/
          t
          (pow (* (cbrt (* (sin k) (/ (tan k) l))) (/ t (cbrt l))) 2.0))))))))

C

double code(double t, double l, double k) {
	double t_1 = sqrt(2.0) / k;
	double tmp;
	if ((l * l) <= 5e-296) {
		tmp = exp(((2.0 * log(l)) + log((2.0 / (t * pow(k, 4.0))))));
	} else if ((l * l) <= 2e+264) {
		tmp = (2.0 * (pow(l, 2.0) * cos(k))) / ((t * (k * k)) * pow(sin(k), 2.0));
	} else {
		tmp = ((t_1 / pow(cbrt(l), -2.0)) / cbrt((sin(k) * tan(k)))) * (t_1 * (t / pow((cbrt((sin(k) * (tan(k) / l))) * (t / cbrt(l))), 2.0)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.sqrt(2.0) / k;
	double tmp;
	if ((l * l) <= 5e-296) {
		tmp = Math.exp(((2.0 * Math.log(l)) + Math.log((2.0 / (t * Math.pow(k, 4.0))))));
	} else if ((l * l) <= 2e+264) {
		tmp = (2.0 * (Math.pow(l, 2.0) * Math.cos(k))) / ((t * (k * k)) * Math.pow(Math.sin(k), 2.0));
	} else {
		tmp = ((t_1 / Math.pow(Math.cbrt(l), -2.0)) / Math.cbrt((Math.sin(k) * Math.tan(k)))) * (t_1 * (t / Math.pow((Math.cbrt((Math.sin(k) * (Math.tan(k) / l))) * (t / Math.cbrt(l))), 2.0)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(sqrt(2.0) / k)
	tmp = 0.0
	if (Float64(l * l) <= 5e-296)
		tmp = exp(Float64(Float64(2.0 * log(l)) + log(Float64(2.0 / Float64(t * (k ^ 4.0))))));
	elseif (Float64(l * l) <= 2e+264)
		tmp = Float64(Float64(2.0 * Float64((l ^ 2.0) * cos(k))) / Float64(Float64(t * Float64(k * k)) * (sin(k) ^ 2.0)));
	else
		tmp = Float64(Float64(Float64(t_1 / (cbrt(l) ^ -2.0)) / cbrt(Float64(sin(k) * tan(k)))) * Float64(t_1 * Float64(t / (Float64(cbrt(Float64(sin(k) * Float64(tan(k) / l))) * Float64(t / cbrt(l))) ^ 2.0))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 5e-296], N[Exp[N[(N[(2.0 * N[Log[l], $MachinePrecision]), $MachinePrecision] + N[Log[N[(2.0 / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 2e+264], N[(N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 / N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(t / N[Power[N[(N[Power[N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(t / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 5.0000000000000003e-296

   1. Initial program 33.4%

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

   3. Simplified 43.5%

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

   5. Taylor expanded in k around 0 74.1%

      \[\leadsto \color{blue}{\frac{2}{{k}^{4} \cdot t}} \cdot \left(\ell \cdot \ell\right) \]
   6. Step-by-step derivation

      1. add-log-exp 74.1%

         \[\leadsto \color{blue}{\log \left(e^{\frac{2}{{k}^{4} \cdot t} \cdot \left(\ell \cdot \ell\right)}\right)} \]

      2. *-commutative 74.1%

         \[\leadsto \log \left(e^{\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{4} \cdot t}}}\right) \]

      3. exp-prod 70.2%

         \[\leadsto \log \color{blue}{\left({\left(e^{\ell \cdot \ell}\right)}^{\left(\frac{2}{{k}^{4} \cdot t}\right)}\right)} \]

      4. pow2 70.2%

         \[\leadsto \log \left({\left(e^{\color{blue}{{\ell}^{2}}}\right)}^{\left(\frac{2}{{k}^{4} \cdot t}\right)}\right) \]

      5. associate-/r* 70.2%

         \[\leadsto \log \left({\left(e^{{\ell}^{2}}\right)}^{\color{blue}{\left(\frac{\frac{2}{{k}^{4}}}{t}\right)}}\right) \]

   7. Applied egg-rr 70.2%

      \[\leadsto \color{blue}{\log \left({\left(e^{{\ell}^{2}}\right)}^{\left(\frac{\frac{2}{{k}^{4}}}{t}\right)}\right)} \]
   8. Step-by-step derivation

      1. pow2 70.2%

         \[\leadsto \log \left({\left(e^{\color{blue}{\ell \cdot \ell}}\right)}^{\left(\frac{\frac{2}{{k}^{4}}}{t}\right)}\right) \]

      2. pow-exp 74.1%

         \[\leadsto \log \color{blue}{\left(e^{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{{k}^{4}}}{t}}\right)} \]

      3. rem-log-exp 74.1%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{{k}^{4}}}{t}} \]

      4. pow2 74.1%

         \[\leadsto \color{blue}{{\ell}^{2}} \cdot \frac{\frac{2}{{k}^{4}}}{t} \]

      5. pow-to-exp 37.8%

         \[\leadsto \color{blue}{e^{\log \ell \cdot 2}} \cdot \frac{\frac{2}{{k}^{4}}}{t} \]

      6. add-exp-log 37.8%

         \[\leadsto e^{\log \ell \cdot 2} \cdot \color{blue}{e^{\log \left(\frac{\frac{2}{{k}^{4}}}{t}\right)}} \]

      7. prod-exp 44.8%

         \[\leadsto \color{blue}{e^{\log \ell \cdot 2 + \log \left(\frac{\frac{2}{{k}^{4}}}{t}\right)}} \]

      8. rem-log-exp 37.8%

         \[\leadsto e^{\color{blue}{\log \left(e^{\log \ell \cdot 2}\right)} + \log \left(\frac{\frac{2}{{k}^{4}}}{t}\right)} \]

      9. pow-to-exp 69.7%

         \[\leadsto e^{\log \color{blue}{\left({\ell}^{2}\right)} + \log \left(\frac{\frac{2}{{k}^{4}}}{t}\right)} \]

      10. log-pow 44.8%

          \[\leadsto e^{\color{blue}{2 \cdot \log \ell} + \log \left(\frac{\frac{2}{{k}^{4}}}{t}\right)} \]

      11. associate-/l/ 44.8%

          \[\leadsto e^{2 \cdot \log \ell + \log \color{blue}{\left(\frac{2}{t \cdot {k}^{4}}\right)}} \]

   9. Applied egg-rr 44.8%

      \[\leadsto \color{blue}{e^{2 \cdot \log \ell + \log \left(\frac{2}{t \cdot {k}^{4}}\right)}} \]

   if 5.0000000000000003e-296 < (*.f64 l l) < 2.00000000000000009e264

   1. Initial program 40.1%

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

   3. Simplified 50.7%

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

   5. Taylor expanded in t around 0 87.7%

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

      1. associate-*r/ 87.7%

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

      2. associate-*r* 87.8%

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

   7. Simplified 87.8%

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

      1. unpow2 87.8%

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

   9. Applied egg-rr 87.8%

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

   if 2.00000000000000009e264 < (*.f64 l l)

   1. Initial program 31.5%

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

      1. *-commutative 31.5%

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

      2. associate-/r* 31.5%

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

   3. Simplified 31.5%

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

      1. add-sqr-sqrt 31.5%

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

      2. add-cube-cbrt 31.5%

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

      3. times-frac 31.5%

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

   6. Applied egg-rr 75.2%

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

      1. associate-/r/ 75.2%

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

      2. associate-/r* 75.2%

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

      3. associate-/r/ 77.7%

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

   8. Simplified 77.7%

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

      1. associate-/l* 77.8%

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

      2. div-inv 77.8%

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

      3. pow-flip 77.8%

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

      4. metadata-eval 77.8%

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

   10. Applied egg-rr 77.8%

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

       1. *-commutative 77.8%

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

       2. associate-/r* 77.8%

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

       3. *-inverses 77.8%

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

       4. associate-*l/ 77.8%

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

       5. *-lft-identity 77.8%

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

   12. Simplified 77.8%

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

       1. associate-/l* 86.3%

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

       2. div-inv 86.2%

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

       3. pow-flip 86.3%

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

       4. metadata-eval 86.3%

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

   14. Applied egg-rr 86.3%

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

       1. metadata-eval 86.3%

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

       2. pow-flip 86.2%

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

       3. div-inv 86.3%

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

       4. associate-*l/ 86.2%

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

       5. add-cbrt-cube 62.2%

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

       6. unpow3 62.2%

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

       7. cbrt-prod 62.2%

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

       8. unpow2 62.2%

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

       9. cbrt-prod 55.8%

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

       10. cbrt-div 55.8%

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

       11. associate-/l/ 59.9%

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

       12. cbrt-div 62.2%

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

   16. Applied egg-rr 86.3%

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

       1. *-commutative 86.3%

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

       2. associate-/l* 86.4%

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

       3. associate-/l* 86.3%

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

   18. Simplified 86.3%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-296}:\\ \;\;\;\;e^{2 \cdot \log \ell + \log \left(\frac{2}{t \cdot {k}^{4}}\right)}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+264}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(t \cdot \left(k \cdot k\right)\right) \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}} \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{t}{{\left(\sqrt[3]{\sin k \cdot \frac{\tan k}{\ell}} \cdot \frac{t}{\sqrt[3]{\ell}}\right)}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 30.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3700000:\\ \;\;\;\;e^{2 \cdot \log \ell + \log \left(\frac{2}{t \cdot {k}^{4}}\right)}\\ \mathbf{elif}\;k \leq 3.7 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{2}{\frac{k \cdot k}{t \cdot t}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\ell}^{2}}{{\sin k}^{2} \cdot \left(t \cdot {k}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 3700000.0)
   (exp (+ (* 2.0 (log l)) (log (/ 2.0 (* t (pow k 4.0))))))
   (if (<= k 3.7e+62)
     (/
      (/ 2.0 (/ (* k k) (* t t)))
      (* (* (sin k) (tan k)) (* (/ (pow t 2.0) l) (/ t l))))
     (/ (* 2.0 (pow l 2.0)) (* (pow (sin k) 2.0) (* t (pow k 2.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 3700000.0) {
		tmp = exp(((2.0 * log(l)) + log((2.0 / (t * pow(k, 4.0))))));
	} else if (k <= 3.7e+62) {
		tmp = (2.0 / ((k * k) / (t * t))) / ((sin(k) * tan(k)) * ((pow(t, 2.0) / l) * (t / l)));
	} else {
		tmp = (2.0 * pow(l, 2.0)) / (pow(sin(k), 2.0) * (t * pow(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 (k <= 3700000.0d0) then
        tmp = exp(((2.0d0 * log(l)) + log((2.0d0 / (t * (k ** 4.0d0))))))
    else if (k <= 3.7d+62) then
        tmp = (2.0d0 / ((k * k) / (t * t))) / ((sin(k) * tan(k)) * (((t ** 2.0d0) / l) * (t / l)))
    else
        tmp = (2.0d0 * (l ** 2.0d0)) / ((sin(k) ** 2.0d0) * (t * (k ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 3700000.0) {
		tmp = Math.exp(((2.0 * Math.log(l)) + Math.log((2.0 / (t * Math.pow(k, 4.0))))));
	} else if (k <= 3.7e+62) {
		tmp = (2.0 / ((k * k) / (t * t))) / ((Math.sin(k) * Math.tan(k)) * ((Math.pow(t, 2.0) / l) * (t / l)));
	} else {
		tmp = (2.0 * Math.pow(l, 2.0)) / (Math.pow(Math.sin(k), 2.0) * (t * Math.pow(k, 2.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 3700000.0:
		tmp = math.exp(((2.0 * math.log(l)) + math.log((2.0 / (t * math.pow(k, 4.0))))))
	elif k <= 3.7e+62:
		tmp = (2.0 / ((k * k) / (t * t))) / ((math.sin(k) * math.tan(k)) * ((math.pow(t, 2.0) / l) * (t / l)))
	else:
		tmp = (2.0 * math.pow(l, 2.0)) / (math.pow(math.sin(k), 2.0) * (t * math.pow(k, 2.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 3700000.0)
		tmp = exp(Float64(Float64(2.0 * log(l)) + log(Float64(2.0 / Float64(t * (k ^ 4.0))))));
	elseif (k <= 3.7e+62)
		tmp = Float64(Float64(2.0 / Float64(Float64(k * k) / Float64(t * t))) / Float64(Float64(sin(k) * tan(k)) * Float64(Float64((t ^ 2.0) / l) * Float64(t / l))));
	else
		tmp = Float64(Float64(2.0 * (l ^ 2.0)) / Float64((sin(k) ^ 2.0) * Float64(t * (k ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 3700000.0)
		tmp = exp(((2.0 * log(l)) + log((2.0 / (t * (k ^ 4.0))))));
	elseif (k <= 3.7e+62)
		tmp = (2.0 / ((k * k) / (t * t))) / ((sin(k) * tan(k)) * (((t ^ 2.0) / l) * (t / l)));
	else
		tmp = (2.0 * (l ^ 2.0)) / ((sin(k) ^ 2.0) * (t * (k ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 3700000.0], N[Exp[N[(N[(2.0 * N[Log[l], $MachinePrecision]), $MachinePrecision] + N[Log[N[(2.0 / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[k, 3.7e+62], N[(N[(2.0 / N[(N[(k * k), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 3.7e6

   1. Initial program 38.5%

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

   3. Simplified 44.8%

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

   5. Taylor expanded in k around 0 71.1%

      \[\leadsto \color{blue}{\frac{2}{{k}^{4} \cdot t}} \cdot \left(\ell \cdot \ell\right) \]
   6. Step-by-step derivation

      1. add-log-exp 68.1%

         \[\leadsto \color{blue}{\log \left(e^{\frac{2}{{k}^{4} \cdot t} \cdot \left(\ell \cdot \ell\right)}\right)} \]

      2. *-commutative 68.1%

         \[\leadsto \log \left(e^{\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{4} \cdot t}}}\right) \]

      3. exp-prod 56.3%

         \[\leadsto \log \color{blue}{\left({\left(e^{\ell \cdot \ell}\right)}^{\left(\frac{2}{{k}^{4} \cdot t}\right)}\right)} \]

      4. pow2 56.3%

         \[\leadsto \log \left({\left(e^{\color{blue}{{\ell}^{2}}}\right)}^{\left(\frac{2}{{k}^{4} \cdot t}\right)}\right) \]

      5. associate-/r* 56.3%

         \[\leadsto \log \left({\left(e^{{\ell}^{2}}\right)}^{\color{blue}{\left(\frac{\frac{2}{{k}^{4}}}{t}\right)}}\right) \]

   7. Applied egg-rr 56.3%

      \[\leadsto \color{blue}{\log \left({\left(e^{{\ell}^{2}}\right)}^{\left(\frac{\frac{2}{{k}^{4}}}{t}\right)}\right)} \]
   8. Step-by-step derivation

      1. pow2 56.3%

         \[\leadsto \log \left({\left(e^{\color{blue}{\ell \cdot \ell}}\right)}^{\left(\frac{\frac{2}{{k}^{4}}}{t}\right)}\right) \]

      2. pow-exp 68.1%

         \[\leadsto \log \color{blue}{\left(e^{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{{k}^{4}}}{t}}\right)} \]

      3. rem-log-exp 71.1%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{{k}^{4}}}{t}} \]

      4. pow2 71.1%

         \[\leadsto \color{blue}{{\ell}^{2}} \cdot \frac{\frac{2}{{k}^{4}}}{t} \]

      5. pow-to-exp 36.8%

         \[\leadsto \color{blue}{e^{\log \ell \cdot 2}} \cdot \frac{\frac{2}{{k}^{4}}}{t} \]

      6. add-exp-log 28.5%

         \[\leadsto e^{\log \ell \cdot 2} \cdot \color{blue}{e^{\log \left(\frac{\frac{2}{{k}^{4}}}{t}\right)}} \]

      7. prod-exp 31.1%

         \[\leadsto \color{blue}{e^{\log \ell \cdot 2 + \log \left(\frac{\frac{2}{{k}^{4}}}{t}\right)}} \]

      8. rem-log-exp 28.6%

         \[\leadsto e^{\color{blue}{\log \left(e^{\log \ell \cdot 2}\right)} + \log \left(\frac{\frac{2}{{k}^{4}}}{t}\right)} \]

      9. pow-to-exp 50.8%

         \[\leadsto e^{\log \color{blue}{\left({\ell}^{2}\right)} + \log \left(\frac{\frac{2}{{k}^{4}}}{t}\right)} \]

      10. log-pow 31.1%

          \[\leadsto e^{\color{blue}{2 \cdot \log \ell} + \log \left(\frac{\frac{2}{{k}^{4}}}{t}\right)} \]

      11. associate-/l/ 31.1%

          \[\leadsto e^{2 \cdot \log \ell + \log \color{blue}{\left(\frac{2}{t \cdot {k}^{4}}\right)}} \]

   9. Applied egg-rr 31.1%

      \[\leadsto \color{blue}{e^{2 \cdot \log \ell + \log \left(\frac{2}{t \cdot {k}^{4}}\right)}} \]

   if 3.7e6 < k < 3.70000000000000014e62

   1. Initial program 20.2%

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

      1. *-commutative 20.2%

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

      2. associate-/r* 20.0%

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

   3. Simplified 40.1%

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

      1. +-rgt-identity 40.1%

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

      2. unpow2 40.1%

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

      3. frac-2neg 40.1%

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

      4. frac-times 40.1%

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

   6. Applied egg-rr 40.1%

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

      1. unpow3 40.1%

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

      2. times-frac 59.7%

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

      3. pow2 59.7%

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

   8. Applied egg-rr 59.7%

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

   if 3.70000000000000014e62 < k

   1. Initial program 27.6%

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

   3. Simplified 35.4%

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

   5. Taylor expanded in t around 0 59.5%

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

      1. associate-*r/ 59.5%

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

      2. associate-*r* 59.5%

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

   7. Simplified 59.5%

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

   8. Taylor expanded in k around 0 48.7%

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

4. Final simplification 35.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3700000:\\ \;\;\;\;e^{2 \cdot \log \ell + \log \left(\frac{2}{t \cdot {k}^{4}}\right)}\\ \mathbf{elif}\;k \leq 3.7 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{2}{\frac{k \cdot k}{t \cdot t}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\ell}^{2}}{{\sin k}^{2} \cdot \left(t \cdot {k}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-296}:\\ \;\;\;\;e^{2 \cdot \log \ell + \log \left(\frac{2}{t \cdot {k}^{4}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(t \cdot \left(k \cdot k\right)\right) \cdot {\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 5e-296)
   (exp (+ (* 2.0 (log l)) (log (/ 2.0 (* t (pow k 4.0))))))
   (/ (* 2.0 (* (pow l 2.0) (cos k))) (* (* t (* k k)) (pow (sin k) 2.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 5e-296) {
		tmp = exp(((2.0 * log(l)) + log((2.0 / (t * pow(k, 4.0))))));
	} else {
		tmp = (2.0 * (pow(l, 2.0) * cos(k))) / ((t * (k * k)) * 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 ((l * l) <= 5d-296) then
        tmp = exp(((2.0d0 * log(l)) + log((2.0d0 / (t * (k ** 4.0d0))))))
    else
        tmp = (2.0d0 * ((l ** 2.0d0) * cos(k))) / ((t * (k * k)) * (sin(k) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 5e-296) {
		tmp = Math.exp(((2.0 * Math.log(l)) + Math.log((2.0 / (t * Math.pow(k, 4.0))))));
	} else {
		tmp = (2.0 * (Math.pow(l, 2.0) * Math.cos(k))) / ((t * (k * k)) * Math.pow(Math.sin(k), 2.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 5e-296:
		tmp = math.exp(((2.0 * math.log(l)) + math.log((2.0 / (t * math.pow(k, 4.0))))))
	else:
		tmp = (2.0 * (math.pow(l, 2.0) * math.cos(k))) / ((t * (k * k)) * math.pow(math.sin(k), 2.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 5e-296)
		tmp = exp(Float64(Float64(2.0 * log(l)) + log(Float64(2.0 / Float64(t * (k ^ 4.0))))));
	else
		tmp = Float64(Float64(2.0 * Float64((l ^ 2.0) * cos(k))) / Float64(Float64(t * Float64(k * k)) * (sin(k) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 5e-296)
		tmp = exp(((2.0 * log(l)) + log((2.0 / (t * (k ^ 4.0))))));
	else
		tmp = (2.0 * ((l ^ 2.0) * cos(k))) / ((t * (k * k)) * (sin(k) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 5e-296], N[Exp[N[(N[(2.0 * N[Log[l], $MachinePrecision]), $MachinePrecision] + N[Log[N[(2.0 / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 5.0000000000000003e-296

   1. Initial program 33.4%

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

   3. Simplified 43.5%

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

   5. Taylor expanded in k around 0 74.1%

      \[\leadsto \color{blue}{\frac{2}{{k}^{4} \cdot t}} \cdot \left(\ell \cdot \ell\right) \]
   6. Step-by-step derivation

      1. add-log-exp 74.1%

         \[\leadsto \color{blue}{\log \left(e^{\frac{2}{{k}^{4} \cdot t} \cdot \left(\ell \cdot \ell\right)}\right)} \]

      2. *-commutative 74.1%

         \[\leadsto \log \left(e^{\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{4} \cdot t}}}\right) \]

      3. exp-prod 70.2%

         \[\leadsto \log \color{blue}{\left({\left(e^{\ell \cdot \ell}\right)}^{\left(\frac{2}{{k}^{4} \cdot t}\right)}\right)} \]

      4. pow2 70.2%

         \[\leadsto \log \left({\left(e^{\color{blue}{{\ell}^{2}}}\right)}^{\left(\frac{2}{{k}^{4} \cdot t}\right)}\right) \]

      5. associate-/r* 70.2%

         \[\leadsto \log \left({\left(e^{{\ell}^{2}}\right)}^{\color{blue}{\left(\frac{\frac{2}{{k}^{4}}}{t}\right)}}\right) \]

   7. Applied egg-rr 70.2%

      \[\leadsto \color{blue}{\log \left({\left(e^{{\ell}^{2}}\right)}^{\left(\frac{\frac{2}{{k}^{4}}}{t}\right)}\right)} \]
   8. Step-by-step derivation

      1. pow2 70.2%

         \[\leadsto \log \left({\left(e^{\color{blue}{\ell \cdot \ell}}\right)}^{\left(\frac{\frac{2}{{k}^{4}}}{t}\right)}\right) \]

      2. pow-exp 74.1%

         \[\leadsto \log \color{blue}{\left(e^{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{{k}^{4}}}{t}}\right)} \]

      3. rem-log-exp 74.1%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{{k}^{4}}}{t}} \]

      4. pow2 74.1%

         \[\leadsto \color{blue}{{\ell}^{2}} \cdot \frac{\frac{2}{{k}^{4}}}{t} \]

      5. pow-to-exp 37.8%

         \[\leadsto \color{blue}{e^{\log \ell \cdot 2}} \cdot \frac{\frac{2}{{k}^{4}}}{t} \]

      6. add-exp-log 37.8%

         \[\leadsto e^{\log \ell \cdot 2} \cdot \color{blue}{e^{\log \left(\frac{\frac{2}{{k}^{4}}}{t}\right)}} \]

      7. prod-exp 44.8%

         \[\leadsto \color{blue}{e^{\log \ell \cdot 2 + \log \left(\frac{\frac{2}{{k}^{4}}}{t}\right)}} \]

      8. rem-log-exp 37.8%

         \[\leadsto e^{\color{blue}{\log \left(e^{\log \ell \cdot 2}\right)} + \log \left(\frac{\frac{2}{{k}^{4}}}{t}\right)} \]

      9. pow-to-exp 69.7%

         \[\leadsto e^{\log \color{blue}{\left({\ell}^{2}\right)} + \log \left(\frac{\frac{2}{{k}^{4}}}{t}\right)} \]

      10. log-pow 44.8%

          \[\leadsto e^{\color{blue}{2 \cdot \log \ell} + \log \left(\frac{\frac{2}{{k}^{4}}}{t}\right)} \]

      11. associate-/l/ 44.8%

          \[\leadsto e^{2 \cdot \log \ell + \log \color{blue}{\left(\frac{2}{t \cdot {k}^{4}}\right)}} \]

   9. Applied egg-rr 44.8%

      \[\leadsto \color{blue}{e^{2 \cdot \log \ell + \log \left(\frac{2}{t \cdot {k}^{4}}\right)}} \]

   if 5.0000000000000003e-296 < (*.f64 l l)

   1. Initial program 36.4%

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

   3. Simplified 42.4%

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

   5. Taylor expanded in t around 0 76.2%

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

      1. associate-*r/ 76.2%

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

      2. associate-*r* 76.2%

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

   7. Simplified 76.2%

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

      1. unpow2 76.2%

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

   9. Applied egg-rr 76.2%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-296}:\\ \;\;\;\;e^{2 \cdot \log \ell + \log \left(\frac{2}{t \cdot {k}^{4}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(t \cdot \left(k \cdot k\right)\right) \cdot {\sin k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-296}:\\ \;\;\;\;e^{2 \cdot \log \ell + \log \left(\frac{2}{t \cdot {k}^{4}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 5e-296)
   (exp (+ (* 2.0 (log l)) (log (/ 2.0 (* t (pow k 4.0))))))
   (* (* 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 ((l * l) <= 5e-296) {
		tmp = exp(((2.0 * log(l)) + log((2.0 / (t * pow(k, 4.0))))));
	} 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 ((l * l) <= 5d-296) then
        tmp = exp(((2.0d0 * log(l)) + log((2.0d0 / (t * (k ** 4.0d0))))))
    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 ((l * l) <= 5e-296) {
		tmp = Math.exp(((2.0 * Math.log(l)) + Math.log((2.0 / (t * Math.pow(k, 4.0))))));
	} 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 (l * l) <= 5e-296:
		tmp = math.exp(((2.0 * math.log(l)) + math.log((2.0 / (t * math.pow(k, 4.0))))))
	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 (Float64(l * l) <= 5e-296)
		tmp = exp(Float64(Float64(2.0 * log(l)) + log(Float64(2.0 / Float64(t * (k ^ 4.0))))));
	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 ((l * l) <= 5e-296)
		tmp = exp(((2.0 * log(l)) + log((2.0 / (t * (k ^ 4.0))))));
	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[N[(l * l), $MachinePrecision], 5e-296], N[Exp[N[(N[(2.0 * N[Log[l], $MachinePrecision]), $MachinePrecision] + N[Log[N[(2.0 / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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 2 regimes

2. if (*.f64 l l) < 5.0000000000000003e-296

   1. Initial program 33.4%

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

   3. Simplified 43.5%

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

   5. Taylor expanded in k around 0 74.1%

      \[\leadsto \color{blue}{\frac{2}{{k}^{4} \cdot t}} \cdot \left(\ell \cdot \ell\right) \]
   6. Step-by-step derivation

      1. add-log-exp 74.1%

         \[\leadsto \color{blue}{\log \left(e^{\frac{2}{{k}^{4} \cdot t} \cdot \left(\ell \cdot \ell\right)}\right)} \]

      2. *-commutative 74.1%

         \[\leadsto \log \left(e^{\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{4} \cdot t}}}\right) \]

      3. exp-prod 70.2%

         \[\leadsto \log \color{blue}{\left({\left(e^{\ell \cdot \ell}\right)}^{\left(\frac{2}{{k}^{4} \cdot t}\right)}\right)} \]

      4. pow2 70.2%

         \[\leadsto \log \left({\left(e^{\color{blue}{{\ell}^{2}}}\right)}^{\left(\frac{2}{{k}^{4} \cdot t}\right)}\right) \]

      5. associate-/r* 70.2%

         \[\leadsto \log \left({\left(e^{{\ell}^{2}}\right)}^{\color{blue}{\left(\frac{\frac{2}{{k}^{4}}}{t}\right)}}\right) \]

   7. Applied egg-rr 70.2%

      \[\leadsto \color{blue}{\log \left({\left(e^{{\ell}^{2}}\right)}^{\left(\frac{\frac{2}{{k}^{4}}}{t}\right)}\right)} \]
   8. Step-by-step derivation

      1. pow2 70.2%

         \[\leadsto \log \left({\left(e^{\color{blue}{\ell \cdot \ell}}\right)}^{\left(\frac{\frac{2}{{k}^{4}}}{t}\right)}\right) \]

      2. pow-exp 74.1%

         \[\leadsto \log \color{blue}{\left(e^{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{{k}^{4}}}{t}}\right)} \]

      3. rem-log-exp 74.1%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{{k}^{4}}}{t}} \]

      4. pow2 74.1%

         \[\leadsto \color{blue}{{\ell}^{2}} \cdot \frac{\frac{2}{{k}^{4}}}{t} \]

      5. pow-to-exp 37.8%

         \[\leadsto \color{blue}{e^{\log \ell \cdot 2}} \cdot \frac{\frac{2}{{k}^{4}}}{t} \]

      6. add-exp-log 37.8%

         \[\leadsto e^{\log \ell \cdot 2} \cdot \color{blue}{e^{\log \left(\frac{\frac{2}{{k}^{4}}}{t}\right)}} \]

      7. prod-exp 44.8%

         \[\leadsto \color{blue}{e^{\log \ell \cdot 2 + \log \left(\frac{\frac{2}{{k}^{4}}}{t}\right)}} \]

      8. rem-log-exp 37.8%

         \[\leadsto e^{\color{blue}{\log \left(e^{\log \ell \cdot 2}\right)} + \log \left(\frac{\frac{2}{{k}^{4}}}{t}\right)} \]

      9. pow-to-exp 69.7%

         \[\leadsto e^{\log \color{blue}{\left({\ell}^{2}\right)} + \log \left(\frac{\frac{2}{{k}^{4}}}{t}\right)} \]

      10. log-pow 44.8%

          \[\leadsto e^{\color{blue}{2 \cdot \log \ell} + \log \left(\frac{\frac{2}{{k}^{4}}}{t}\right)} \]

      11. associate-/l/ 44.8%

          \[\leadsto e^{2 \cdot \log \ell + \log \color{blue}{\left(\frac{2}{t \cdot {k}^{4}}\right)}} \]

   9. Applied egg-rr 44.8%

      \[\leadsto \color{blue}{e^{2 \cdot \log \ell + \log \left(\frac{2}{t \cdot {k}^{4}}\right)}} \]

   if 5.0000000000000003e-296 < (*.f64 l l)

   1. Initial program 36.4%

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

   3. Simplified 42.4%

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

   5. Taylor expanded in t around 0 75.9%

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

      1. associate-/r* 76.0%

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

   7. Simplified 76.0%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-296}:\\ \;\;\;\;e^{2 \cdot \log \ell + \log \left(\frac{2}{t \cdot {k}^{4}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 22.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{2 \cdot \log \ell + \log \left(\frac{2}{t \cdot {k}^{4}}\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (exp (+ (* 2.0 (log l)) (log (/ 2.0 (* t (pow k 4.0)))))))

C

double code(double t, double l, double k) {
	return exp(((2.0 * log(l)) + log((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 = exp(((2.0d0 * log(l)) + log((2.0d0 / (t * (k ** 4.0d0))))))
end function

Java

public static double code(double t, double l, double k) {
	return Math.exp(((2.0 * Math.log(l)) + Math.log((2.0 / (t * Math.pow(k, 4.0))))));
}

Python

def code(t, l, k):
	return math.exp(((2.0 * math.log(l)) + math.log((2.0 / (t * math.pow(k, 4.0))))))

Julia

function code(t, l, k)
	return exp(Float64(Float64(2.0 * log(l)) + log(Float64(2.0 / Float64(t * (k ^ 4.0))))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = exp(((2.0 * log(l)) + log((2.0 / (t * (k ^ 4.0))))));
end

Wolfram

code[t_, l_, k_] := N[Exp[N[(N[(2.0 * N[Log[l], $MachinePrecision]), $MachinePrecision] + N[Log[N[(2.0 / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 35.6%

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

3. Simplified 42.7%

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

5. Taylor expanded in k around 0 64.6%

   \[\leadsto \color{blue}{\frac{2}{{k}^{4} \cdot t}} \cdot \left(\ell \cdot \ell\right) \]
6. Step-by-step derivation

   1. add-log-exp 62.3%

      \[\leadsto \color{blue}{\log \left(e^{\frac{2}{{k}^{4} \cdot t} \cdot \left(\ell \cdot \ell\right)}\right)} \]

   2. *-commutative 62.3%

      \[\leadsto \log \left(e^{\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{4} \cdot t}}}\right) \]

   3. exp-prod 53.9%

      \[\leadsto \log \color{blue}{\left({\left(e^{\ell \cdot \ell}\right)}^{\left(\frac{2}{{k}^{4} \cdot t}\right)}\right)} \]

   4. pow2 53.9%

      \[\leadsto \log \left({\left(e^{\color{blue}{{\ell}^{2}}}\right)}^{\left(\frac{2}{{k}^{4} \cdot t}\right)}\right) \]

   5. associate-/r* 53.9%

      \[\leadsto \log \left({\left(e^{{\ell}^{2}}\right)}^{\color{blue}{\left(\frac{\frac{2}{{k}^{4}}}{t}\right)}}\right) \]

7. Applied egg-rr 53.9%

   \[\leadsto \color{blue}{\log \left({\left(e^{{\ell}^{2}}\right)}^{\left(\frac{\frac{2}{{k}^{4}}}{t}\right)}\right)} \]
8. Step-by-step derivation

   1. pow2 53.9%

      \[\leadsto \log \left({\left(e^{\color{blue}{\ell \cdot \ell}}\right)}^{\left(\frac{\frac{2}{{k}^{4}}}{t}\right)}\right) \]

   2. pow-exp 62.3%

      \[\leadsto \log \color{blue}{\left(e^{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{{k}^{4}}}{t}}\right)} \]

   3. rem-log-exp 64.6%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{{k}^{4}}}{t}} \]

   4. pow2 64.6%

      \[\leadsto \color{blue}{{\ell}^{2}} \cdot \frac{\frac{2}{{k}^{4}}}{t} \]

   5. pow-to-exp 32.9%

      \[\leadsto \color{blue}{e^{\log \ell \cdot 2}} \cdot \frac{\frac{2}{{k}^{4}}}{t} \]

   6. add-exp-log 26.6%

      \[\leadsto e^{\log \ell \cdot 2} \cdot \color{blue}{e^{\log \left(\frac{\frac{2}{{k}^{4}}}{t}\right)}} \]

   7. prod-exp 28.7%

      \[\leadsto \color{blue}{e^{\log \ell \cdot 2 + \log \left(\frac{\frac{2}{{k}^{4}}}{t}\right)}} \]

   8. rem-log-exp 26.6%

      \[\leadsto e^{\color{blue}{\log \left(e^{\log \ell \cdot 2}\right)} + \log \left(\frac{\frac{2}{{k}^{4}}}{t}\right)} \]

   9. pow-to-exp 48.7%

      \[\leadsto e^{\log \color{blue}{\left({\ell}^{2}\right)} + \log \left(\frac{\frac{2}{{k}^{4}}}{t}\right)} \]

   10. log-pow 28.7%

       \[\leadsto e^{\color{blue}{2 \cdot \log \ell} + \log \left(\frac{\frac{2}{{k}^{4}}}{t}\right)} \]

   11. associate-/l/ 28.7%

       \[\leadsto e^{2 \cdot \log \ell + \log \color{blue}{\left(\frac{2}{t \cdot {k}^{4}}\right)}} \]

9. Applied egg-rr 28.7%

   \[\leadsto \color{blue}{e^{2 \cdot \log \ell + \log \left(\frac{2}{t \cdot {k}^{4}}\right)}} \]
10. Add Preprocessing

Alternative 8: 64.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{-129}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{+79}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{t \cdot \frac{t}{k}}}}{\frac{\frac{\left(\sin k \cdot \tan k\right) \cdot {t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{4} \cdot \left(t + {k}^{2} \cdot \left(\left(t \cdot {k}^{2}\right) \cdot 0.08611111111111111 + t \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 7.2e-129)
   (* 2.0 (/ (/ (pow l 2.0) t) (pow k 4.0)))
   (if (<= t 3.6e+79)
     (/
      (/ 2.0 (/ k (* t (/ t k))))
      (/ (/ (* (* (sin k) (tan k)) (pow t 3.0)) l) l))
     (*
      (* l l)
      (/
       2.0
       (*
        (pow k 4.0)
        (+
         t
         (*
          (pow k 2.0)
          (+
           (* (* t (pow k 2.0)) 0.08611111111111111)
           (* t 0.16666666666666666))))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 7.2e-129) {
		tmp = 2.0 * ((pow(l, 2.0) / t) / pow(k, 4.0));
	} else if (t <= 3.6e+79) {
		tmp = (2.0 / (k / (t * (t / k)))) / ((((sin(k) * tan(k)) * pow(t, 3.0)) / l) / l);
	} else {
		tmp = (l * l) * (2.0 / (pow(k, 4.0) * (t + (pow(k, 2.0) * (((t * pow(k, 2.0)) * 0.08611111111111111) + (t * 0.16666666666666666))))));
	}
	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 <= 7.2d-129) then
        tmp = 2.0d0 * (((l ** 2.0d0) / t) / (k ** 4.0d0))
    else if (t <= 3.6d+79) then
        tmp = (2.0d0 / (k / (t * (t / k)))) / ((((sin(k) * tan(k)) * (t ** 3.0d0)) / l) / l)
    else
        tmp = (l * l) * (2.0d0 / ((k ** 4.0d0) * (t + ((k ** 2.0d0) * (((t * (k ** 2.0d0)) * 0.08611111111111111d0) + (t * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 7.2e-129) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / t) / Math.pow(k, 4.0));
	} else if (t <= 3.6e+79) {
		tmp = (2.0 / (k / (t * (t / k)))) / ((((Math.sin(k) * Math.tan(k)) * Math.pow(t, 3.0)) / l) / l);
	} else {
		tmp = (l * l) * (2.0 / (Math.pow(k, 4.0) * (t + (Math.pow(k, 2.0) * (((t * Math.pow(k, 2.0)) * 0.08611111111111111) + (t * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 7.2e-129:
		tmp = 2.0 * ((math.pow(l, 2.0) / t) / math.pow(k, 4.0))
	elif t <= 3.6e+79:
		tmp = (2.0 / (k / (t * (t / k)))) / ((((math.sin(k) * math.tan(k)) * math.pow(t, 3.0)) / l) / l)
	else:
		tmp = (l * l) * (2.0 / (math.pow(k, 4.0) * (t + (math.pow(k, 2.0) * (((t * math.pow(k, 2.0)) * 0.08611111111111111) + (t * 0.16666666666666666))))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 7.2e-129)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / t) / (k ^ 4.0)));
	elseif (t <= 3.6e+79)
		tmp = Float64(Float64(2.0 / Float64(k / Float64(t * Float64(t / k)))) / Float64(Float64(Float64(Float64(sin(k) * tan(k)) * (t ^ 3.0)) / l) / l));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64((k ^ 4.0) * Float64(t + Float64((k ^ 2.0) * Float64(Float64(Float64(t * (k ^ 2.0)) * 0.08611111111111111) + Float64(t * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 7.2e-129)
		tmp = 2.0 * (((l ^ 2.0) / t) / (k ^ 4.0));
	elseif (t <= 3.6e+79)
		tmp = (2.0 / (k / (t * (t / k)))) / ((((sin(k) * tan(k)) * (t ^ 3.0)) / l) / l);
	else
		tmp = (l * l) * (2.0 / ((k ^ 4.0) * (t + ((k ^ 2.0) * (((t * (k ^ 2.0)) * 0.08611111111111111) + (t * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 7.2e-129], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / t), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.6e+79], N[(N[(2.0 / N[(k / N[(t * N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Power[k, 4.0], $MachinePrecision] * N[(t + N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[(N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * 0.08611111111111111), $MachinePrecision] + N[(t * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 7.2e-129

   1. Initial program 33.4%

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

   3. Simplified 37.1%

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

   5. Taylor expanded in k around 0 61.1%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   6. Step-by-step derivation

      1. *-commutative 61.1%

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}} \]

      2. associate-/r* 60.7%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]

   7. Simplified 60.7%

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]

   if 7.2e-129 < t < 3.5999999999999999e79

   1. Initial program 64.1%

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

      1. *-commutative 64.1%

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

      2. associate-/r* 64.1%

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

   3. Simplified 70.4%

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

      1. associate-*l/ 74.7%

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

      2. associate-/r* 83.5%

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

   6. Applied egg-rr 83.5%

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

      1. +-rgt-identity 83.5%

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

      2. unpow2 83.5%

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

      3. clear-num 83.5%

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

      4. frac-times 83.6%

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

      5. *-un-lft-identity 83.6%

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

   8. Applied egg-rr 83.6%

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

   if 3.5999999999999999e79 < t

   1. Initial program 15.5%

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

   3. Simplified 35.4%

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

   5. Taylor expanded in k around 0 77.8%

      \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \left(t + {k}^{2} \cdot \left(0.08611111111111111 \cdot \left({k}^{2} \cdot t\right) + 0.16666666666666666 \cdot t\right)\right)}} \cdot \left(\ell \cdot \ell\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{-129}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{+79}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{t \cdot \frac{t}{k}}}}{\frac{\frac{\left(\sin k \cdot \tan k\right) \cdot {t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{4} \cdot \left(t + {k}^{2} \cdot \left(\left(t \cdot {k}^{2}\right) \cdot 0.08611111111111111 + t \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 4.15 \cdot 10^{-128}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{+79}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{t \cdot \frac{t}{k}}}}{\frac{\frac{\left(\sin k \cdot \tan k\right) \cdot {t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{4} \cdot \left(t + \left(t \cdot {k}^{2}\right) \cdot 0.16666666666666666\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 4.15e-128)
   (* 2.0 (/ (/ (pow l 2.0) t) (pow k 4.0)))
   (if (<= t 3.6e+79)
     (/
      (/ 2.0 (/ k (* t (/ t k))))
      (/ (/ (* (* (sin k) (tan k)) (pow t 3.0)) l) l))
     (*
      (* l l)
      (/
       2.0
       (* (pow k 4.0) (+ t (* (* t (pow k 2.0)) 0.16666666666666666))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 4.15e-128) {
		tmp = 2.0 * ((pow(l, 2.0) / t) / pow(k, 4.0));
	} else if (t <= 3.6e+79) {
		tmp = (2.0 / (k / (t * (t / k)))) / ((((sin(k) * tan(k)) * pow(t, 3.0)) / l) / l);
	} else {
		tmp = (l * l) * (2.0 / (pow(k, 4.0) * (t + ((t * pow(k, 2.0)) * 0.16666666666666666))));
	}
	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 <= 4.15d-128) then
        tmp = 2.0d0 * (((l ** 2.0d0) / t) / (k ** 4.0d0))
    else if (t <= 3.6d+79) then
        tmp = (2.0d0 / (k / (t * (t / k)))) / ((((sin(k) * tan(k)) * (t ** 3.0d0)) / l) / l)
    else
        tmp = (l * l) * (2.0d0 / ((k ** 4.0d0) * (t + ((t * (k ** 2.0d0)) * 0.16666666666666666d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 4.15e-128) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / t) / Math.pow(k, 4.0));
	} else if (t <= 3.6e+79) {
		tmp = (2.0 / (k / (t * (t / k)))) / ((((Math.sin(k) * Math.tan(k)) * Math.pow(t, 3.0)) / l) / l);
	} else {
		tmp = (l * l) * (2.0 / (Math.pow(k, 4.0) * (t + ((t * Math.pow(k, 2.0)) * 0.16666666666666666))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 4.15e-128:
		tmp = 2.0 * ((math.pow(l, 2.0) / t) / math.pow(k, 4.0))
	elif t <= 3.6e+79:
		tmp = (2.0 / (k / (t * (t / k)))) / ((((math.sin(k) * math.tan(k)) * math.pow(t, 3.0)) / l) / l)
	else:
		tmp = (l * l) * (2.0 / (math.pow(k, 4.0) * (t + ((t * math.pow(k, 2.0)) * 0.16666666666666666))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 4.15e-128)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / t) / (k ^ 4.0)));
	elseif (t <= 3.6e+79)
		tmp = Float64(Float64(2.0 / Float64(k / Float64(t * Float64(t / k)))) / Float64(Float64(Float64(Float64(sin(k) * tan(k)) * (t ^ 3.0)) / l) / l));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64((k ^ 4.0) * Float64(t + Float64(Float64(t * (k ^ 2.0)) * 0.16666666666666666)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 4.15e-128)
		tmp = 2.0 * (((l ^ 2.0) / t) / (k ^ 4.0));
	elseif (t <= 3.6e+79)
		tmp = (2.0 / (k / (t * (t / k)))) / ((((sin(k) * tan(k)) * (t ^ 3.0)) / l) / l);
	else
		tmp = (l * l) * (2.0 / ((k ^ 4.0) * (t + ((t * (k ^ 2.0)) * 0.16666666666666666))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 4.15e-128], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / t), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.6e+79], N[(N[(2.0 / N[(k / N[(t * N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Power[k, 4.0], $MachinePrecision] * N[(t + N[(N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 4.15000000000000008e-128

   1. Initial program 33.4%

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

   3. Simplified 37.1%

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

   5. Taylor expanded in k around 0 61.1%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   6. Step-by-step derivation

      1. *-commutative 61.1%

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}} \]

      2. associate-/r* 60.7%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]

   7. Simplified 60.7%

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]

   if 4.15000000000000008e-128 < t < 3.5999999999999999e79

   1. Initial program 64.1%

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

      1. *-commutative 64.1%

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

      2. associate-/r* 64.1%

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

   3. Simplified 70.4%

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

      1. associate-*l/ 74.7%

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

      2. associate-/r* 83.5%

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

   6. Applied egg-rr 83.5%

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

      1. +-rgt-identity 83.5%

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

      2. unpow2 83.5%

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

      3. clear-num 83.5%

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

      4. frac-times 83.6%

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

      5. *-un-lft-identity 83.6%

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

   8. Applied egg-rr 83.6%

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

   if 3.5999999999999999e79 < t

   1. Initial program 15.5%

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

   3. Simplified 35.4%

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

   5. Taylor expanded in k around 0 77.8%

      \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \left(t + 0.16666666666666666 \cdot \left({k}^{2} \cdot t\right)\right)}} \cdot \left(\ell \cdot \ell\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.15 \cdot 10^{-128}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{+79}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{t \cdot \frac{t}{k}}}}{\frac{\frac{\left(\sin k \cdot \tan k\right) \cdot {t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{4} \cdot \left(t + \left(t \cdot {k}^{2}\right) \cdot 0.16666666666666666\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.8 \cdot 10^{+76}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333 \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 4.8e+76)
   (*
    (* l l)
    (/
     (+ (* -0.3333333333333333 (/ (pow k 2.0) t)) (* 2.0 (/ 1.0 t)))
     (pow k 4.0)))
   0.0))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.8e+76) {
		tmp = (l * l) * (((-0.3333333333333333 * (pow(k, 2.0) / t)) + (2.0 * (1.0 / t))) / pow(k, 4.0));
	} else {
		tmp = 0.0;
	}
	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 (k <= 4.8d+76) then
        tmp = (l * l) * ((((-0.3333333333333333d0) * ((k ** 2.0d0) / t)) + (2.0d0 * (1.0d0 / t))) / (k ** 4.0d0))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.8e+76) {
		tmp = (l * l) * (((-0.3333333333333333 * (Math.pow(k, 2.0) / t)) + (2.0 * (1.0 / t))) / Math.pow(k, 4.0));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 4.8e+76:
		tmp = (l * l) * (((-0.3333333333333333 * (math.pow(k, 2.0) / t)) + (2.0 * (1.0 / t))) / math.pow(k, 4.0))
	else:
		tmp = 0.0
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 4.8e+76)
		tmp = Float64(Float64(l * l) * Float64(Float64(Float64(-0.3333333333333333 * Float64((k ^ 2.0) / t)) + Float64(2.0 * Float64(1.0 / t))) / (k ^ 4.0)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 4.8e+76)
		tmp = (l * l) * (((-0.3333333333333333 * ((k ^ 2.0) / t)) + (2.0 * (1.0 / t))) / (k ^ 4.0));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 4.8e+76], N[(N[(l * l), $MachinePrecision] * N[(N[(N[(-0.3333333333333333 * N[(N[Power[k, 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if k < 4.8e76

   1. Initial program 37.7%

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

   3. Simplified 44.6%

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

   5. Taylor expanded in k around 0 54.0%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

   if 4.8e76 < k

   1. Initial program 26.6%

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

   3. Simplified 34.7%

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

   5. Taylor expanded in k around 0 47.7%

      \[\leadsto \color{blue}{\frac{2}{{k}^{4} \cdot t}} \cdot \left(\ell \cdot \ell\right) \]
   6. Step-by-step derivation

      1. add-log-exp 47.7%

         \[\leadsto \color{blue}{\log \left(e^{\frac{2}{{k}^{4} \cdot t} \cdot \left(\ell \cdot \ell\right)}\right)} \]

      2. *-commutative 47.7%

         \[\leadsto \log \left(e^{\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{4} \cdot t}}}\right) \]

      3. exp-prod 50.8%

         \[\leadsto \log \color{blue}{\left({\left(e^{\ell \cdot \ell}\right)}^{\left(\frac{2}{{k}^{4} \cdot t}\right)}\right)} \]

      4. pow2 50.8%

         \[\leadsto \log \left({\left(e^{\color{blue}{{\ell}^{2}}}\right)}^{\left(\frac{2}{{k}^{4} \cdot t}\right)}\right) \]

      5. associate-/r* 50.8%

         \[\leadsto \log \left({\left(e^{{\ell}^{2}}\right)}^{\color{blue}{\left(\frac{\frac{2}{{k}^{4}}}{t}\right)}}\right) \]

   7. Applied egg-rr 50.8%

      \[\leadsto \color{blue}{\log \left({\left(e^{{\ell}^{2}}\right)}^{\left(\frac{\frac{2}{{k}^{4}}}{t}\right)}\right)} \]

   8. Taylor expanded in l around 0 50.8%

      \[\leadsto \log \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.8 \cdot 10^{+76}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333 \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{4} \cdot \left(t + \left(t \cdot {k}^{2}\right) \cdot 0.16666666666666666\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (*
  (* l l)
  (/ 2.0 (* (pow k 4.0) (+ t (* (* t (pow k 2.0)) 0.16666666666666666))))))

C

double code(double t, double l, double k) {
	return (l * l) * (2.0 / (pow(k, 4.0) * (t + ((t * pow(k, 2.0)) * 0.16666666666666666))));
}

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 / ((k ** 4.0d0) * (t + ((t * (k ** 2.0d0)) * 0.16666666666666666d0))))
end function

Java

public static double code(double t, double l, double k) {
	return (l * l) * (2.0 / (Math.pow(k, 4.0) * (t + ((t * Math.pow(k, 2.0)) * 0.16666666666666666))));
}

Python

def code(t, l, k):
	return (l * l) * (2.0 / (math.pow(k, 4.0) * (t + ((t * math.pow(k, 2.0)) * 0.16666666666666666))))

Julia

function code(t, l, k)
	return Float64(Float64(l * l) * Float64(2.0 / Float64((k ^ 4.0) * Float64(t + Float64(Float64(t * (k ^ 2.0)) * 0.16666666666666666)))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l * l) * (2.0 / ((k ^ 4.0) * (t + ((t * (k ^ 2.0)) * 0.16666666666666666))));
end

Wolfram

code[t_, l_, k_] := N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Power[k, 4.0], $MachinePrecision] * N[(t + N[(N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.6%

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

3. Simplified 42.7%

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

5. Taylor expanded in k around 0 64.7%

   \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \left(t + 0.16666666666666666 \cdot \left({k}^{2} \cdot t\right)\right)}} \cdot \left(\ell \cdot \ell\right) \]

6. Final simplification 64.7%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{4} \cdot \left(t + \left(t \cdot {k}^{2}\right) \cdot 0.16666666666666666\right)} \]
7. Add Preprocessing

Alternative 12: 63.0% 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 35.6%

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

3. Simplified 42.7%

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

5. Taylor expanded in k around 0 64.6%

   \[\leadsto \color{blue}{\frac{2}{{k}^{4} \cdot t}} \cdot \left(\ell \cdot \ell\right) \]

6. Final simplification 64.6%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot {k}^{4}} \]
7. Add Preprocessing

Alternative 13: 29.0% accurate, 35.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.25 \cdot 10^{+193}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.11666666666666667}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 1.25e+193) 0.0 (* (* l l) (/ -0.11666666666666667 t))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 1.25e+193) {
		tmp = 0.0;
	} else {
		tmp = (l * l) * (-0.11666666666666667 / 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 (l <= 1.25d+193) then
        tmp = 0.0d0
    else
        tmp = (l * l) * ((-0.11666666666666667d0) / t)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 1.25e+193) {
		tmp = 0.0;
	} else {
		tmp = (l * l) * (-0.11666666666666667 / t);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 1.25e+193:
		tmp = 0.0
	else:
		tmp = (l * l) * (-0.11666666666666667 / t)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 1.25e+193)
		tmp = 0.0;
	else
		tmp = Float64(Float64(l * l) * Float64(-0.11666666666666667 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 1.25e+193)
		tmp = 0.0;
	else
		tmp = (l * l) * (-0.11666666666666667 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 1.25e+193], 0.0, N[(N[(l * l), $MachinePrecision] * N[(-0.11666666666666667 / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.24999999999999993e193

   1. Initial program 37.0%

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

   3. Simplified 44.8%

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

   5. Taylor expanded in k around 0 67.5%

      \[\leadsto \color{blue}{\frac{2}{{k}^{4} \cdot t}} \cdot \left(\ell \cdot \ell\right) \]
   6. Step-by-step derivation

      1. add-log-exp 65.0%

         \[\leadsto \color{blue}{\log \left(e^{\frac{2}{{k}^{4} \cdot t} \cdot \left(\ell \cdot \ell\right)}\right)} \]

      2. *-commutative 65.0%

         \[\leadsto \log \left(e^{\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{4} \cdot t}}}\right) \]

      3. exp-prod 55.6%

         \[\leadsto \log \color{blue}{\left({\left(e^{\ell \cdot \ell}\right)}^{\left(\frac{2}{{k}^{4} \cdot t}\right)}\right)} \]

      4. pow2 55.6%

         \[\leadsto \log \left({\left(e^{\color{blue}{{\ell}^{2}}}\right)}^{\left(\frac{2}{{k}^{4} \cdot t}\right)}\right) \]

      5. associate-/r* 55.6%

         \[\leadsto \log \left({\left(e^{{\ell}^{2}}\right)}^{\color{blue}{\left(\frac{\frac{2}{{k}^{4}}}{t}\right)}}\right) \]

   7. Applied egg-rr 55.6%

      \[\leadsto \color{blue}{\log \left({\left(e^{{\ell}^{2}}\right)}^{\left(\frac{\frac{2}{{k}^{4}}}{t}\right)}\right)} \]

   8. Taylor expanded in l around 0 35.2%

      \[\leadsto \log \color{blue}{1} \]

   if 1.24999999999999993e193 < l

   1. Initial program 21.7%

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

   3. Simplified 21.7%

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

   5. Taylor expanded in k around 0 39.2%

      \[\leadsto \color{blue}{\frac{{k}^{2} \cdot \left(-0.11666666666666667 \cdot \frac{{k}^{2}}{t} - 0.3333333333333333 \cdot \frac{1}{t}\right) + 2 \cdot \frac{1}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

   6. Taylor expanded in k around inf 17.9%

      \[\leadsto \color{blue}{\frac{-0.11666666666666667}{t}} \cdot \left(\ell \cdot \ell\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.25 \cdot 10^{+193}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.11666666666666667}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 20.1% accurate, 60.1× speedup

Math

\[\begin{array}{l} \\ \left(\ell \cdot \ell\right) \cdot \frac{-0.11666666666666667}{t} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* (* l l) (/ -0.11666666666666667 t)))

C

double code(double t, double l, double k) {
	return (l * l) * (-0.11666666666666667 / 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 * l) * ((-0.11666666666666667d0) / t)
end function

Java

public static double code(double t, double l, double k) {
	return (l * l) * (-0.11666666666666667 / t);
}

Python

def code(t, l, k):
	return (l * l) * (-0.11666666666666667 / t)

Julia

function code(t, l, k)
	return Float64(Float64(l * l) * Float64(-0.11666666666666667 / t))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l * l) * (-0.11666666666666667 / t);
end

Wolfram

code[t_, l_, k_] := N[(N[(l * l), $MachinePrecision] * N[(-0.11666666666666667 / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.6%

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

3. Simplified 42.7%

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

5. Taylor expanded in k around 0 42.8%

   \[\leadsto \color{blue}{\frac{{k}^{2} \cdot \left(-0.11666666666666667 \cdot \frac{{k}^{2}}{t} - 0.3333333333333333 \cdot \frac{1}{t}\right) + 2 \cdot \frac{1}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

6. Taylor expanded in k around inf 26.7%

   \[\leadsto \color{blue}{\frac{-0.11666666666666667}{t}} \cdot \left(\ell \cdot \ell\right) \]

7. Final simplification 26.7%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{-0.11666666666666667}{t} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024146 
(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))))