Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.9% → 86.4%

Time: 24.5s

Alternatives: 17

Speedup: 3.8×

• Could not converge on a ground truth

  1. reg0 = (bf "3.376332000550348816302519515519739023801e-246")
  2. reg1 = (bf "2.977671922185852873638774435703955820305e189")
  3. reg2 = (bf "3.566892267235177613763152399391549288045e152")

• 39.8% 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: 34.9% 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: 86.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t_2 := \sqrt[3]{\sin k \cdot \tan k}\\ t_3 := \frac{\sqrt{2}}{k}\\ t_4 := \sqrt[3]{t\_3 \cdot \frac{t}{{\left(\left(t \cdot t\_1\right) \cdot t\_2\right)}^{2}}}\\ t_5 := \frac{t\_3 \cdot \frac{1}{t\_1}}{t\_2}\\ \mathbf{if}\;\ell \cdot \ell \leq 10^{-150}:\\ \;\;\;\;\left(t\_4 \cdot {t\_4}^{2}\right) \cdot \frac{t\_3 \cdot {\left(\sqrt[3]{\ell}\right)}^{2}}{t\_2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+169}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{k \cdot t} \cdot \sqrt[3]{\frac{{\ell}^{4} \cdot {\cos k}^{2}}{{\sin k}^{4}}}\right) \cdot t\_5\\ \mathbf{else}:\\ \;\;\;\;t\_5 \cdot \frac{\sqrt{2} \cdot \left(t \cdot {\left(t \cdot \left(t\_1 \cdot t\_2\right)\right)}^{-2}\right)}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (cbrt l) -2.0))
        (t_2 (cbrt (* (sin k) (tan k))))
        (t_3 (/ (sqrt 2.0) k))
        (t_4 (cbrt (* t_3 (/ t (pow (* (* t t_1) t_2) 2.0)))))
        (t_5 (/ (* t_3 (/ 1.0 t_1)) t_2)))
   (if (<= (* l l) 1e-150)
     (* (* t_4 (pow t_4 2.0)) (/ (* t_3 (pow (cbrt l) 2.0)) t_2))
     (if (<= (* l l) 2e+169)
       (*
        (*
         (/ (sqrt 2.0) (* k t))
         (cbrt (/ (* (pow l 4.0) (pow (cos k) 2.0)) (pow (sin k) 4.0))))
        t_5)
       (* t_5 (/ (* (sqrt 2.0) (* t (pow (* t (* t_1 t_2)) -2.0))) k))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(cbrt(l), -2.0);
	double t_2 = cbrt((sin(k) * tan(k)));
	double t_3 = sqrt(2.0) / k;
	double t_4 = cbrt((t_3 * (t / pow(((t * t_1) * t_2), 2.0))));
	double t_5 = (t_3 * (1.0 / t_1)) / t_2;
	double tmp;
	if ((l * l) <= 1e-150) {
		tmp = (t_4 * pow(t_4, 2.0)) * ((t_3 * pow(cbrt(l), 2.0)) / t_2);
	} else if ((l * l) <= 2e+169) {
		tmp = ((sqrt(2.0) / (k * t)) * cbrt(((pow(l, 4.0) * pow(cos(k), 2.0)) / pow(sin(k), 4.0)))) * t_5;
	} else {
		tmp = t_5 * ((sqrt(2.0) * (t * pow((t * (t_1 * t_2)), -2.0))) / k);
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.cbrt(l), -2.0);
	double t_2 = Math.cbrt((Math.sin(k) * Math.tan(k)));
	double t_3 = Math.sqrt(2.0) / k;
	double t_4 = Math.cbrt((t_3 * (t / Math.pow(((t * t_1) * t_2), 2.0))));
	double t_5 = (t_3 * (1.0 / t_1)) / t_2;
	double tmp;
	if ((l * l) <= 1e-150) {
		tmp = (t_4 * Math.pow(t_4, 2.0)) * ((t_3 * Math.pow(Math.cbrt(l), 2.0)) / t_2);
	} else if ((l * l) <= 2e+169) {
		tmp = ((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)))) * t_5;
	} else {
		tmp = t_5 * ((Math.sqrt(2.0) * (t * Math.pow((t * (t_1 * t_2)), -2.0))) / k);
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cbrt(l) ^ -2.0
	t_2 = cbrt(Float64(sin(k) * tan(k)))
	t_3 = Float64(sqrt(2.0) / k)
	t_4 = cbrt(Float64(t_3 * Float64(t / (Float64(Float64(t * t_1) * t_2) ^ 2.0))))
	t_5 = Float64(Float64(t_3 * Float64(1.0 / t_1)) / t_2)
	tmp = 0.0
	if (Float64(l * l) <= 1e-150)
		tmp = Float64(Float64(t_4 * (t_4 ^ 2.0)) * Float64(Float64(t_3 * (cbrt(l) ^ 2.0)) / t_2));
	elseif (Float64(l * l) <= 2e+169)
		tmp = Float64(Float64(Float64(sqrt(2.0) / Float64(k * t)) * cbrt(Float64(Float64((l ^ 4.0) * (cos(k) ^ 2.0)) / (sin(k) ^ 4.0)))) * t_5);
	else
		tmp = Float64(t_5 * Float64(Float64(sqrt(2.0) * Float64(t * (Float64(t * Float64(t_1 * t_2)) ^ -2.0))) / k));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[(t$95$3 * N[(t / N[Power[N[(N[(t * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$3 * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 1e-150], N[(N[(t$95$4 * N[Power[t$95$4, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$3 * N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 2e+169], N[(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] * t$95$5), $MachinePrecision], N[(t$95$5 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(t * N[Power[N[(t * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 1.00000000000000001e-150

   1. Initial program 27.4%

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

      1. *-commutative 27.4%

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

      2. associate-/r* 27.4%

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

   3. Simplified 36.3%

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

      1. add-sqr-sqrt 36.3%

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

      2. add-cube-cbrt 36.3%

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

      3. times-frac 36.3%

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

   6. Applied egg-rr 82.7%

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

      1. associate-/r/ 82.7%

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

      2. associate-/r* 82.7%

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

      3. associate-/r/ 83.3%

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

   8. Simplified 83.3%

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

      1. add-cube-cbrt 83.3%

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

   10. Applied egg-rr 83.3%

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

       1. associate-/l* 86.6%

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

       2. div-inv 86.6%

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

       3. pow-flip 86.6%

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

       4. metadata-eval 86.6%

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

   12. Applied egg-rr 86.6%

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

       1. associate-/r* 86.6%

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

       2. *-inverses 86.6%

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

   14. Simplified 86.6%

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

       1. associate-*l/ 86.6%

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

       2. pow-flip 86.6%

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

       3. metadata-eval 86.6%

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

   16. Applied egg-rr 86.6%

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

       1. associate-*l/ 86.6%

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

   18. Simplified 86.6%

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

   if 1.00000000000000001e-150 < (*.f64 l l) < 1.99999999999999987e169

   1. Initial program 45.6%

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

      1. *-commutative 45.6%

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

      2. associate-/r* 45.6%

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

   3. Simplified 50.7%

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

      1. add-sqr-sqrt 50.7%

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

      2. add-cube-cbrt 50.7%

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

      3. times-frac 50.7%

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

   6. Applied egg-rr 85.7%

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

      1. associate-/r/ 85.7%

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

      2. associate-/r* 85.7%

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

      3. associate-/r/ 85.7%

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

   8. Simplified 85.7%

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

      1. add-cube-cbrt 85.7%

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

   10. Applied egg-rr 85.8%

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

       1. associate-/l* 85.9%

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

       2. div-inv 85.9%

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

       3. pow-flip 85.9%

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

       4. metadata-eval 85.9%

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

   12. Applied egg-rr 85.9%

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

       1. associate-/r* 85.9%

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

       2. *-inverses 85.9%

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

   14. Simplified 85.9%

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

   15. Taylor expanded in k around inf 97.8%

       \[\leadsto \color{blue}{\left(\frac{\sqrt{2}}{k \cdot t} \cdot \sqrt[3]{\frac{{\ell}^{4} \cdot {\cos k}^{2}}{{\sin k}^{4}}}\right)} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{1}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

   if 1.99999999999999987e169 < (*.f64 l l)

   1. Initial program 41.9%

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

      1. *-commutative 41.9%

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

      2. associate-/r* 41.9%

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

   3. Simplified 43.1%

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

      1. add-sqr-sqrt 43.1%

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

      2. add-cube-cbrt 43.1%

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

      3. times-frac 43.1%

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

   6. Applied egg-rr 78.4%

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

      1. associate-/r/ 78.4%

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

      2. associate-/r* 78.4%

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

      3. associate-/r/ 79.5%

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

   8. Simplified 79.5%

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

      1. add-cube-cbrt 79.5%

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

   10. Applied egg-rr 79.5%

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

       1. associate-/l* 86.1%

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

       2. div-inv 86.1%

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

       3. pow-flip 86.1%

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

       4. metadata-eval 86.1%

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

   12. Applied egg-rr 86.1%

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

       1. associate-/r* 86.1%

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

       2. *-inverses 86.1%

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

   14. Simplified 86.1%

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

       1. unpow2 86.1%

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

       2. add-cube-cbrt 86.1%

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

       3. associate-*l/ 86.1%

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

   16. Applied egg-rr 87.4%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-150}:\\ \;\;\;\;\left(\sqrt[3]{\frac{\sqrt{2}}{k} \cdot \frac{t}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}}} \cdot {\left(\sqrt[3]{\frac{\sqrt{2}}{k} \cdot \frac{t}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}}}\right)}^{2}\right) \cdot \frac{\frac{\sqrt{2}}{k} \cdot {\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{\sin k \cdot \tan k}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+169}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{k \cdot t} \cdot \sqrt[3]{\frac{{\ell}^{4} \cdot {\cos k}^{2}}{{\sin k}^{4}}}\right) \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{1}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{k} \cdot \frac{1}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}} \cdot \frac{\sqrt{2} \cdot \left(t \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2}\right)}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t_2 := \sqrt[3]{\sin k \cdot \tan k}\\ t_3 := \frac{\frac{\sqrt{2}}{k} \cdot \frac{1}{t\_1}}{t\_2}\\ \mathbf{if}\;\ell \cdot \ell \leq 10^{-150} \lor \neg \left(\ell \cdot \ell \leq 2 \cdot 10^{+169}\right):\\ \;\;\;\;t\_3 \cdot \frac{\sqrt{2} \cdot \left(t \cdot {\left(t \cdot \left(t\_1 \cdot t\_2\right)\right)}^{-2}\right)}{k}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{k \cdot t} \cdot \sqrt[3]{\frac{{\ell}^{4} \cdot {\cos k}^{2}}{{\sin k}^{4}}}\right) \cdot t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (cbrt l) -2.0))
        (t_2 (cbrt (* (sin k) (tan k))))
        (t_3 (/ (* (/ (sqrt 2.0) k) (/ 1.0 t_1)) t_2)))
   (if (or (<= (* l l) 1e-150) (not (<= (* l l) 2e+169)))
     (* t_3 (/ (* (sqrt 2.0) (* t (pow (* t (* t_1 t_2)) -2.0))) k))
     (*
      (*
       (/ (sqrt 2.0) (* k t))
       (cbrt (/ (* (pow l 4.0) (pow (cos k) 2.0)) (pow (sin k) 4.0))))
      t_3))))

C

double code(double t, double l, double k) {
	double t_1 = pow(cbrt(l), -2.0);
	double t_2 = cbrt((sin(k) * tan(k)));
	double t_3 = ((sqrt(2.0) / k) * (1.0 / t_1)) / t_2;
	double tmp;
	if (((l * l) <= 1e-150) || !((l * l) <= 2e+169)) {
		tmp = t_3 * ((sqrt(2.0) * (t * pow((t * (t_1 * t_2)), -2.0))) / k);
	} else {
		tmp = ((sqrt(2.0) / (k * t)) * cbrt(((pow(l, 4.0) * pow(cos(k), 2.0)) / pow(sin(k), 4.0)))) * t_3;
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.cbrt(l), -2.0);
	double t_2 = Math.cbrt((Math.sin(k) * Math.tan(k)));
	double t_3 = ((Math.sqrt(2.0) / k) * (1.0 / t_1)) / t_2;
	double tmp;
	if (((l * l) <= 1e-150) || !((l * l) <= 2e+169)) {
		tmp = t_3 * ((Math.sqrt(2.0) * (t * Math.pow((t * (t_1 * t_2)), -2.0))) / k);
	} else {
		tmp = ((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)))) * t_3;
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cbrt(l) ^ -2.0
	t_2 = cbrt(Float64(sin(k) * tan(k)))
	t_3 = Float64(Float64(Float64(sqrt(2.0) / k) * Float64(1.0 / t_1)) / t_2)
	tmp = 0.0
	if ((Float64(l * l) <= 1e-150) || !(Float64(l * l) <= 2e+169))
		tmp = Float64(t_3 * Float64(Float64(sqrt(2.0) * Float64(t * (Float64(t * Float64(t_1 * t_2)) ^ -2.0))) / k));
	else
		tmp = Float64(Float64(Float64(sqrt(2.0) / Float64(k * t)) * cbrt(Float64(Float64((l ^ 4.0) * (cos(k) ^ 2.0)) / (sin(k) ^ 4.0)))) * t_3);
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision] * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]}, If[Or[LessEqual[N[(l * l), $MachinePrecision], 1e-150], N[Not[LessEqual[N[(l * l), $MachinePrecision], 2e+169]], $MachinePrecision]], N[(t$95$3 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(t * N[Power[N[(t * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(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] * t$95$3), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.00000000000000001e-150 or 1.99999999999999987e169 < (*.f64 l l)

   1. Initial program 34.3%

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

      1. *-commutative 34.3%

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

      2. associate-/r* 34.3%

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

   3. Simplified 39.5%

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

      1. add-sqr-sqrt 39.5%

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

      2. add-cube-cbrt 39.5%

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

      3. times-frac 39.5%

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

   6. Applied egg-rr 80.6%

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

      1. associate-/r/ 80.6%

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

      2. associate-/r* 80.6%

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

      3. associate-/r/ 81.5%

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

   8. Simplified 81.5%

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

      1. add-cube-cbrt 81.5%

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

   10. Applied egg-rr 81.5%

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

       1. associate-/l* 86.4%

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

       2. div-inv 86.4%

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

       3. pow-flip 86.4%

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

       4. metadata-eval 86.4%

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

   12. Applied egg-rr 86.4%

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

       1. associate-/r* 86.4%

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

       2. *-inverses 86.4%

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

   14. Simplified 86.4%

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

       1. unpow2 86.4%

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

       2. add-cube-cbrt 86.4%

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

       3. associate-*l/ 86.4%

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

   16. Applied egg-rr 86.5%

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

   if 1.00000000000000001e-150 < (*.f64 l l) < 1.99999999999999987e169

   1. Initial program 45.6%

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

      1. *-commutative 45.6%

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

      2. associate-/r* 45.6%

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

   3. Simplified 50.7%

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

      1. add-sqr-sqrt 50.7%

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

      2. add-cube-cbrt 50.7%

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

      3. times-frac 50.7%

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

   6. Applied egg-rr 85.7%

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

      1. associate-/r/ 85.7%

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

      2. associate-/r* 85.7%

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

      3. associate-/r/ 85.7%

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

   8. Simplified 85.7%

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

      1. add-cube-cbrt 85.7%

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

   10. Applied egg-rr 85.8%

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

       1. associate-/l* 85.9%

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

       2. div-inv 85.9%

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

       3. pow-flip 85.9%

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

       4. metadata-eval 85.9%

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

   12. Applied egg-rr 85.9%

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

       1. associate-/r* 85.9%

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

       2. *-inverses 85.9%

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

   14. Simplified 85.9%

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

   15. Taylor expanded in k around inf 97.8%

       \[\leadsto \color{blue}{\left(\frac{\sqrt{2}}{k \cdot t} \cdot \sqrt[3]{\frac{{\ell}^{4} \cdot {\cos k}^{2}}{{\sin k}^{4}}}\right)} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{1}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-150} \lor \neg \left(\ell \cdot \ell \leq 2 \cdot 10^{+169}\right):\\ \;\;\;\;\frac{\frac{\sqrt{2}}{k} \cdot \frac{1}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}} \cdot \frac{\sqrt{2} \cdot \left(t \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2}\right)}{k}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{k \cdot t} \cdot \sqrt[3]{\frac{{\ell}^{4} \cdot {\cos k}^{2}}{{\sin k}^{4}}}\right) \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{1}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt[3]{\sin k \cdot \tan k}\\ t_2 := \frac{\sqrt{2}}{k}\\ \mathbf{if}\;\ell \leq 6.8 \cdot 10^{-162} \lor \neg \left(\ell \leq 1.15 \cdot 10^{+202}\right):\\ \;\;\;\;\left(t\_2 \cdot t\right) \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot t\_1\right)}^{-2} \cdot \left(t\_2 \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t\_1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2} \cdot \left(\cos k \cdot {\left(\sqrt[3]{-1}\right)}^{6}\right)}{t \cdot {\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (cbrt (* (sin k) (tan k)))) (t_2 (/ (sqrt 2.0) k)))
   (if (or (<= l 6.8e-162) (not (<= l 1.15e+202)))
     (*
      (* t_2 t)
      (*
       (pow (* (* t (pow (cbrt l) -2.0)) t_1) -2.0)
       (* t_2 (/ (pow (cbrt l) 2.0) t_1))))
     (*
      (/ 2.0 (pow k 2.0))
      (/
       (* (pow l 2.0) (* (cos k) (pow (cbrt -1.0) 6.0)))
       (* t (pow (sin k) 2.0)))))))

C

double code(double t, double l, double k) {
	double t_1 = cbrt((sin(k) * tan(k)));
	double t_2 = sqrt(2.0) / k;
	double tmp;
	if ((l <= 6.8e-162) || !(l <= 1.15e+202)) {
		tmp = (t_2 * t) * (pow(((t * pow(cbrt(l), -2.0)) * t_1), -2.0) * (t_2 * (pow(cbrt(l), 2.0) / t_1)));
	} else {
		tmp = (2.0 / pow(k, 2.0)) * ((pow(l, 2.0) * (cos(k) * pow(cbrt(-1.0), 6.0))) / (t * pow(sin(k), 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)));
	double t_2 = Math.sqrt(2.0) / k;
	double tmp;
	if ((l <= 6.8e-162) || !(l <= 1.15e+202)) {
		tmp = (t_2 * t) * (Math.pow(((t * Math.pow(Math.cbrt(l), -2.0)) * t_1), -2.0) * (t_2 * (Math.pow(Math.cbrt(l), 2.0) / t_1)));
	} else {
		tmp = (2.0 / Math.pow(k, 2.0)) * ((Math.pow(l, 2.0) * (Math.cos(k) * Math.pow(Math.cbrt(-1.0), 6.0))) / (t * Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cbrt(Float64(sin(k) * tan(k)))
	t_2 = Float64(sqrt(2.0) / k)
	tmp = 0.0
	if ((l <= 6.8e-162) || !(l <= 1.15e+202))
		tmp = Float64(Float64(t_2 * t) * Float64((Float64(Float64(t * (cbrt(l) ^ -2.0)) * t_1) ^ -2.0) * Float64(t_2 * Float64((cbrt(l) ^ 2.0) / t_1))));
	else
		tmp = Float64(Float64(2.0 / (k ^ 2.0)) * Float64(Float64((l ^ 2.0) * Float64(cos(k) * (cbrt(-1.0) ^ 6.0))) / Float64(t * (sin(k) ^ 2.0))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]}, If[Or[LessEqual[l, 6.8e-162], N[Not[LessEqual[l, 1.15e+202]], $MachinePrecision]], N[(N[(t$95$2 * t), $MachinePrecision] * N[(N[Power[N[(N[(t * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], -2.0], $MachinePrecision] * N[(t$95$2 * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * N[Power[N[Power[-1.0, 1/3], $MachinePrecision], 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if l < 6.8e-162 or 1.15e202 < l

   1. Initial program 31.3%

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

      1. *-commutative 31.3%

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

      2. associate-/r* 31.3%

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

   3. Simplified 37.3%

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

      1. add-sqr-sqrt 37.3%

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

      2. add-cube-cbrt 37.3%

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

      3. times-frac 37.3%

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

   6. Applied egg-rr 81.6%

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

      1. associate-/r/ 81.6%

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

      2. associate-/r* 81.6%

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

      3. associate-/r/ 82.0%

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

   8. Simplified 82.0%

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

      1. add-cube-cbrt 82.0%

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

   10. Applied egg-rr 82.0%

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

       1. associate-/l* 85.3%

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

       2. div-inv 85.3%

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

       3. pow-flip 85.3%

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

       4. metadata-eval 85.3%

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

   12. Applied egg-rr 85.3%

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

       1. associate-/r* 85.3%

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

       2. *-inverses 85.3%

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

   14. Simplified 85.3%

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

   16. Applied egg-rr 81.0%

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

       1. associate-*r/ 81.7%

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

       2. associate-*l* 80.9%

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

       3. *-commutative 80.9%

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

       4. associate-*r/ 80.9%

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

   18. Simplified 80.9%

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

   if 6.8e-162 < l < 1.15e202

   1. Initial program 47.9%

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

      1. *-commutative 47.9%

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

      2. associate-/r* 47.9%

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

   3. Simplified 51.5%

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

      1. add-cube-cbrt 51.5%

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

      2. div-inv 51.5%

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

      3. times-frac 51.5%

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

   6. Applied egg-rr 69.3%

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

      1. associate-*l/ 70.3%

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

      2. associate-/l* 70.3%

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

      3. associate-/l/ 67.2%

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

      4. unpow2 67.2%

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

      5. unpow3 67.2%

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

      6. *-commutative 67.2%

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

   8. Simplified 60.5%

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

   9. Taylor expanded in l around -inf 90.2%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   10. Step-by-step derivation

       1. associate-*r* 90.2%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

       2. associate-*r/ 90.2%

          \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)\right)}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

       3. associate-*r* 90.2%

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

       4. times-frac 94.4%

          \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2} \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)}{t \cdot {\sin k}^{2}}} \]

       5. *-commutative 94.4%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\left(\cos k \cdot {\left(\sqrt[3]{-1}\right)}^{6}\right)}}{t \cdot {\sin k}^{2}} \]

   11. Simplified 94.4%

       \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2} \cdot \left(\cos k \cdot {\left(\sqrt[3]{-1}\right)}^{6}\right)}{t \cdot {\sin k}^{2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.8 \cdot 10^{-162} \lor \neg \left(\ell \leq 1.15 \cdot 10^{+202}\right):\\ \;\;\;\;\left(\frac{\sqrt{2}}{k} \cdot t\right) \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{\sin k \cdot \tan k}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2} \cdot \left(\cos k \cdot {\left(\sqrt[3]{-1}\right)}^{6}\right)}{t \cdot {\sin k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt[3]{\sin k \cdot \tan k}\\ t_2 := {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot t\_1\right)}^{-2}\\ t_3 := \frac{\sqrt{2}}{k}\\ t_4 := t\_3 \cdot t\\ t_5 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ \mathbf{if}\;\ell \leq 1.85 \cdot 10^{-161}:\\ \;\;\;\;\frac{\left(t\_3 \cdot t\_5\right) \cdot \left(t\_4 \cdot t\_2\right)}{t\_1}\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{+202}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2} \cdot \left(\cos k \cdot {\left(\sqrt[3]{-1}\right)}^{6}\right)}{t \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_4 \cdot \left(t\_2 \cdot \left(t\_3 \cdot \frac{t\_5}{t\_1}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (cbrt (* (sin k) (tan k))))
        (t_2 (pow (* (* t (pow (cbrt l) -2.0)) t_1) -2.0))
        (t_3 (/ (sqrt 2.0) k))
        (t_4 (* t_3 t))
        (t_5 (pow (cbrt l) 2.0)))
   (if (<= l 1.85e-161)
     (/ (* (* t_3 t_5) (* t_4 t_2)) t_1)
     (if (<= l 1.15e+202)
       (*
        (/ 2.0 (pow k 2.0))
        (/
         (* (pow l 2.0) (* (cos k) (pow (cbrt -1.0) 6.0)))
         (* t (pow (sin k) 2.0))))
       (* t_4 (* t_2 (* t_3 (/ t_5 t_1))))))))

C

double code(double t, double l, double k) {
	double t_1 = cbrt((sin(k) * tan(k)));
	double t_2 = pow(((t * pow(cbrt(l), -2.0)) * t_1), -2.0);
	double t_3 = sqrt(2.0) / k;
	double t_4 = t_3 * t;
	double t_5 = pow(cbrt(l), 2.0);
	double tmp;
	if (l <= 1.85e-161) {
		tmp = ((t_3 * t_5) * (t_4 * t_2)) / t_1;
	} else if (l <= 1.15e+202) {
		tmp = (2.0 / pow(k, 2.0)) * ((pow(l, 2.0) * (cos(k) * pow(cbrt(-1.0), 6.0))) / (t * pow(sin(k), 2.0)));
	} else {
		tmp = t_4 * (t_2 * (t_3 * (t_5 / t_1)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.cbrt((Math.sin(k) * Math.tan(k)));
	double t_2 = Math.pow(((t * Math.pow(Math.cbrt(l), -2.0)) * t_1), -2.0);
	double t_3 = Math.sqrt(2.0) / k;
	double t_4 = t_3 * t;
	double t_5 = Math.pow(Math.cbrt(l), 2.0);
	double tmp;
	if (l <= 1.85e-161) {
		tmp = ((t_3 * t_5) * (t_4 * t_2)) / t_1;
	} else if (l <= 1.15e+202) {
		tmp = (2.0 / Math.pow(k, 2.0)) * ((Math.pow(l, 2.0) * (Math.cos(k) * Math.pow(Math.cbrt(-1.0), 6.0))) / (t * Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = t_4 * (t_2 * (t_3 * (t_5 / t_1)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cbrt(Float64(sin(k) * tan(k)))
	t_2 = Float64(Float64(t * (cbrt(l) ^ -2.0)) * t_1) ^ -2.0
	t_3 = Float64(sqrt(2.0) / k)
	t_4 = Float64(t_3 * t)
	t_5 = cbrt(l) ^ 2.0
	tmp = 0.0
	if (l <= 1.85e-161)
		tmp = Float64(Float64(Float64(t_3 * t_5) * Float64(t_4 * t_2)) / t_1);
	elseif (l <= 1.15e+202)
		tmp = Float64(Float64(2.0 / (k ^ 2.0)) * Float64(Float64((l ^ 2.0) * Float64(cos(k) * (cbrt(-1.0) ^ 6.0))) / Float64(t * (sin(k) ^ 2.0))));
	else
		tmp = Float64(t_4 * Float64(t_2 * Float64(t_3 * Float64(t_5 / t_1))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(N[(t * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], -2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * t), $MachinePrecision]}, Block[{t$95$5 = N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[l, 1.85e-161], N[(N[(N[(t$95$3 * t$95$5), $MachinePrecision] * N[(t$95$4 * t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[l, 1.15e+202], N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * N[Power[N[Power[-1.0, 1/3], $MachinePrecision], 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$4 * N[(t$95$2 * N[(t$95$3 * N[(t$95$5 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.8499999999999999e-161

   1. Initial program 28.8%

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

      1. *-commutative 28.8%

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

      2. associate-/r* 28.8%

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

   3. Simplified 36.0%

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

      1. add-sqr-sqrt 36.0%

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

      2. add-cube-cbrt 35.9%

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

      3. times-frac 36.0%

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

   6. Applied egg-rr 82.3%

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

      1. associate-/r/ 82.3%

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

      2. associate-/r* 82.3%

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

      3. associate-/r/ 82.8%

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

   8. Simplified 82.8%

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

      1. add-cube-cbrt 82.8%

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

   10. Applied egg-rr 82.8%

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

       1. associate-/l* 86.0%

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

       2. div-inv 86.0%

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

       3. pow-flip 86.0%

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

       4. metadata-eval 86.0%

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

   12. Applied egg-rr 86.0%

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

       1. associate-/r* 86.0%

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

       2. *-inverses 86.0%

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

   14. Simplified 86.0%

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

   16. Applied egg-rr 81.2%

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

   if 1.8499999999999999e-161 < l < 1.15e202

   1. Initial program 47.9%

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

      1. *-commutative 47.9%

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

      2. associate-/r* 47.9%

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

   3. Simplified 51.5%

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

      1. add-cube-cbrt 51.5%

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

      2. div-inv 51.5%

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

      3. times-frac 51.5%

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

   6. Applied egg-rr 69.3%

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

      1. associate-*l/ 70.3%

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

      2. associate-/l* 70.3%

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

      3. associate-/l/ 67.2%

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

      4. unpow2 67.2%

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

      5. unpow3 67.2%

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

      6. *-commutative 67.2%

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

   8. Simplified 60.5%

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

   9. Taylor expanded in l around -inf 90.2%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   10. Step-by-step derivation

       1. associate-*r* 90.2%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

       2. associate-*r/ 90.2%

          \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)\right)}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

       3. associate-*r* 90.2%

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

       4. times-frac 94.4%

          \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2} \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)}{t \cdot {\sin k}^{2}}} \]

       5. *-commutative 94.4%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\left(\cos k \cdot {\left(\sqrt[3]{-1}\right)}^{6}\right)}}{t \cdot {\sin k}^{2}} \]

   11. Simplified 94.4%

       \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2} \cdot \left(\cos k \cdot {\left(\sqrt[3]{-1}\right)}^{6}\right)}{t \cdot {\sin k}^{2}}} \]

   if 1.15e202 < l

   1. Initial program 44.4%

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

      1. *-commutative 44.4%

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

      2. associate-/r* 44.4%

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

   3. Simplified 44.4%

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

      1. add-sqr-sqrt 44.4%

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

      2. add-cube-cbrt 44.4%

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

      3. times-frac 44.4%

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

   6. Applied egg-rr 77.9%

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

      1. associate-/r/ 77.9%

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

      2. associate-/r* 77.9%

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

      3. associate-/r/ 77.9%

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

   8. Simplified 77.9%

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

      1. add-cube-cbrt 78.0%

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

   10. Applied egg-rr 78.0%

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

       1. associate-/l* 81.7%

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

       2. div-inv 81.7%

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

       3. pow-flip 81.7%

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

       4. metadata-eval 81.7%

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

   12. Applied egg-rr 81.7%

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

       1. associate-/r* 81.7%

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

       2. *-inverses 81.7%

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

   14. Simplified 81.7%

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

   16. Applied egg-rr 79.7%

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

       1. associate-*r/ 79.7%

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

       2. associate-*l* 79.7%

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

       3. *-commutative 79.7%

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

       4. associate-*r/ 79.7%

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

   18. Simplified 79.7%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.85 \cdot 10^{-161}:\\ \;\;\;\;\frac{\left(\frac{\sqrt{2}}{k} \cdot {\left(\sqrt[3]{\ell}\right)}^{2}\right) \cdot \left(\left(\frac{\sqrt{2}}{k} \cdot t\right) \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right)}{\sqrt[3]{\sin k \cdot \tan k}}\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{+202}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2} \cdot \left(\cos k \cdot {\left(\sqrt[3]{-1}\right)}^{6}\right)}{t \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{k} \cdot t\right) \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{\sin k \cdot \tan k}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ t_2 := \frac{\sqrt{2}}{k} \cdot t\\ \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{t\_2}{{\left(t\_1 \cdot \sqrt[3]{{k}^{2}}\right)}^{2}} \cdot \frac{\frac{t\_2}{t\_1}}{\sqrt[3]{\sin k \cdot \tan k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2} \cdot \left(\cos k \cdot {\left(\sqrt[3]{-1}\right)}^{6}\right)}{t \cdot {\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ t (pow (cbrt l) 2.0))) (t_2 (* (/ (sqrt 2.0) k) t)))
   (if (<= (* l l) 0.0)
     (*
      (/ t_2 (pow (* t_1 (cbrt (pow k 2.0))) 2.0))
      (/ (/ t_2 t_1) (cbrt (* (sin k) (tan k)))))
     (*
      (/ 2.0 (pow k 2.0))
      (/
       (* (pow l 2.0) (* (cos k) (pow (cbrt -1.0) 6.0)))
       (* t (pow (sin k) 2.0)))))))

C

double code(double t, double l, double k) {
	double t_1 = t / pow(cbrt(l), 2.0);
	double t_2 = (sqrt(2.0) / k) * t;
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = (t_2 / pow((t_1 * cbrt(pow(k, 2.0))), 2.0)) * ((t_2 / t_1) / cbrt((sin(k) * tan(k))));
	} else {
		tmp = (2.0 / pow(k, 2.0)) * ((pow(l, 2.0) * (cos(k) * pow(cbrt(-1.0), 6.0))) / (t * pow(sin(k), 2.0)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = t / Math.pow(Math.cbrt(l), 2.0);
	double t_2 = (Math.sqrt(2.0) / k) * t;
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = (t_2 / Math.pow((t_1 * Math.cbrt(Math.pow(k, 2.0))), 2.0)) * ((t_2 / t_1) / Math.cbrt((Math.sin(k) * Math.tan(k))));
	} else {
		tmp = (2.0 / Math.pow(k, 2.0)) * ((Math.pow(l, 2.0) * (Math.cos(k) * Math.pow(Math.cbrt(-1.0), 6.0))) / (t * Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(t / (cbrt(l) ^ 2.0))
	t_2 = Float64(Float64(sqrt(2.0) / k) * t)
	tmp = 0.0
	if (Float64(l * l) <= 0.0)
		tmp = Float64(Float64(t_2 / (Float64(t_1 * cbrt((k ^ 2.0))) ^ 2.0)) * Float64(Float64(t_2 / t_1) / cbrt(Float64(sin(k) * tan(k)))));
	else
		tmp = Float64(Float64(2.0 / (k ^ 2.0)) * Float64(Float64((l ^ 2.0) * Float64(cos(k) * (cbrt(-1.0) ^ 6.0))) / Float64(t * (sin(k) ^ 2.0))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(N[(t$95$2 / N[Power[N[(t$95$1 * N[Power[N[Power[k, 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$2 / t$95$1), $MachinePrecision] / N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * N[Power[N[Power[-1.0, 1/3], $MachinePrecision], 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 15.7%

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

      1. *-commutative 15.7%

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

      2. associate-/r* 15.7%

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

   3. Simplified 24.8%

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

      1. add-sqr-sqrt 24.8%

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

      2. add-cube-cbrt 24.8%

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

      3. times-frac 24.8%

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

   6. Applied egg-rr 75.8%

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

      1. associate-/r/ 75.8%

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

      2. associate-/r* 75.9%

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

      3. associate-/r/ 76.7%

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

   8. Simplified 76.7%

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

   9. Taylor expanded in k around 0 72.1%

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

   if 0.0 < (*.f64 l l)

   1. Initial program 44.7%

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

      1. *-commutative 44.7%

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

      2. associate-/r* 44.7%

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

   3. Simplified 48.5%

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

      1. add-cube-cbrt 48.5%

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

      2. div-inv 48.5%

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

      3. times-frac 48.5%

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

   6. Applied egg-rr 71.7%

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

      1. associate-*l/ 72.1%

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

      2. associate-/l* 72.1%

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

      3. associate-/l/ 69.4%

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

      4. unpow2 69.4%

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

      5. unpow3 69.4%

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

      6. *-commutative 69.4%

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

   8. Simplified 61.0%

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

   9. Taylor expanded in l around -inf 82.7%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   10. Step-by-step derivation

       1. associate-*r* 82.7%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

       2. associate-*r/ 82.7%

          \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)\right)}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

       3. associate-*r* 82.7%

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

       4. times-frac 86.3%

          \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2} \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)}{t \cdot {\sin k}^{2}}} \]

       5. *-commutative 86.3%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\left(\cos k \cdot {\left(\sqrt[3]{-1}\right)}^{6}\right)}}{t \cdot {\sin k}^{2}} \]

   11. Simplified 86.3%

       \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2} \cdot \left(\cos k \cdot {\left(\sqrt[3]{-1}\right)}^{6}\right)}{t \cdot {\sin k}^{2}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 73.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ \mathbf{if}\;k \leq 3.6 \cdot 10^{-136}:\\ \;\;\;\;\frac{2}{{t\_1}^{2}} \cdot \frac{{\left(\frac{k}{t}\right)}^{-2}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2} \cdot \left(\cos k \cdot {\left(\sqrt[3]{-1}\right)}^{6}\right)}{t \cdot {\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (cbrt (* (sin k) (tan k))) (/ t (pow (cbrt l) 2.0)))))
   (if (<= k 3.6e-136)
     (* (/ 2.0 (pow t_1 2.0)) (/ (pow (/ k t) -2.0) t_1))
     (*
      (/ 2.0 (pow k 2.0))
      (/
       (* (pow l 2.0) (* (cos k) (pow (cbrt -1.0) 6.0)))
       (* t (pow (sin k) 2.0)))))))

C

double code(double t, double l, double k) {
	double t_1 = cbrt((sin(k) * tan(k))) * (t / pow(cbrt(l), 2.0));
	double tmp;
	if (k <= 3.6e-136) {
		tmp = (2.0 / pow(t_1, 2.0)) * (pow((k / t), -2.0) / t_1);
	} else {
		tmp = (2.0 / pow(k, 2.0)) * ((pow(l, 2.0) * (cos(k) * pow(cbrt(-1.0), 6.0))) / (t * pow(sin(k), 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))) * (t / Math.pow(Math.cbrt(l), 2.0));
	double tmp;
	if (k <= 3.6e-136) {
		tmp = (2.0 / Math.pow(t_1, 2.0)) * (Math.pow((k / t), -2.0) / t_1);
	} else {
		tmp = (2.0 / Math.pow(k, 2.0)) * ((Math.pow(l, 2.0) * (Math.cos(k) * Math.pow(Math.cbrt(-1.0), 6.0))) / (t * Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(cbrt(Float64(sin(k) * tan(k))) * Float64(t / (cbrt(l) ^ 2.0)))
	tmp = 0.0
	if (k <= 3.6e-136)
		tmp = Float64(Float64(2.0 / (t_1 ^ 2.0)) * Float64((Float64(k / t) ^ -2.0) / t_1));
	else
		tmp = Float64(Float64(2.0 / (k ^ 2.0)) * Float64(Float64((l ^ 2.0) * Float64(cos(k) * (cbrt(-1.0) ^ 6.0))) / Float64(t * (sin(k) ^ 2.0))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 3.6e-136], N[(N[(2.0 / N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(k / t), $MachinePrecision], -2.0], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * N[Power[N[Power[-1.0, 1/3], $MachinePrecision], 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 3.5999999999999998e-136

   1. Initial program 37.7%

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

      1. *-commutative 37.7%

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

      2. associate-/r* 37.7%

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

   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-cube-cbrt 42.1%

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

      2. div-inv 42.1%

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

      3. times-frac 42.1%

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

   6. Applied egg-rr 71.7%

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

   if 3.5999999999999998e-136 < k

   1. Initial program 35.8%

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

      1. *-commutative 35.8%

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

      2. associate-/r* 35.8%

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

   3. Simplified 42.3%

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

      1. add-cube-cbrt 42.3%

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

      2. div-inv 42.3%

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

      3. times-frac 42.3%

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

   6. Applied egg-rr 63.6%

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

      1. associate-*l/ 63.6%

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

      2. associate-/l* 63.6%

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

      3. associate-/l/ 61.3%

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

      4. unpow2 61.3%

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

      5. unpow3 61.3%

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

      6. *-commutative 61.3%

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

   8. Simplified 59.3%

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

   9. Taylor expanded in l around -inf 72.3%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   10. Step-by-step derivation

       1. associate-*r* 72.3%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

       2. associate-*r/ 72.3%

          \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)\right)}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

       3. associate-*r* 72.3%

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

       4. times-frac 74.4%

          \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2} \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)}{t \cdot {\sin k}^{2}}} \]

       5. *-commutative 74.4%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\left(\cos k \cdot {\left(\sqrt[3]{-1}\right)}^{6}\right)}}{t \cdot {\sin k}^{2}} \]

   11. Simplified 74.4%

       \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2} \cdot \left(\cos k \cdot {\left(\sqrt[3]{-1}\right)}^{6}\right)}{t \cdot {\sin k}^{2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.6 \cdot 10^{-136}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{{\left(\frac{k}{t}\right)}^{-2}}{\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2} \cdot \left(\cos k \cdot {\left(\sqrt[3]{-1}\right)}^{6}\right)}{t \cdot {\sin k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t_2 := \sqrt[3]{\sin k \cdot \tan k}\\ \mathbf{if}\;k \leq 1.6 \cdot 10^{-136}:\\ \;\;\;\;\frac{\frac{\frac{2 \cdot \left(\frac{t}{k} \cdot \frac{t}{k}\right)}{t \cdot t\_1}}{t\_2}}{{\left(t \cdot \left(t\_1 \cdot t\_2\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2} \cdot \left(\cos k \cdot {\left(\sqrt[3]{-1}\right)}^{6}\right)}{t \cdot {\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (cbrt l) -2.0)) (t_2 (cbrt (* (sin k) (tan k)))))
   (if (<= k 1.6e-136)
     (/
      (/ (/ (* 2.0 (* (/ t k) (/ t k))) (* t t_1)) t_2)
      (pow (* t (* t_1 t_2)) 2.0))
     (*
      (/ 2.0 (pow k 2.0))
      (/
       (* (pow l 2.0) (* (cos k) (pow (cbrt -1.0) 6.0)))
       (* t (pow (sin k) 2.0)))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(cbrt(l), -2.0);
	double t_2 = cbrt((sin(k) * tan(k)));
	double tmp;
	if (k <= 1.6e-136) {
		tmp = (((2.0 * ((t / k) * (t / k))) / (t * t_1)) / t_2) / pow((t * (t_1 * t_2)), 2.0);
	} else {
		tmp = (2.0 / pow(k, 2.0)) * ((pow(l, 2.0) * (cos(k) * pow(cbrt(-1.0), 6.0))) / (t * pow(sin(k), 2.0)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.cbrt(l), -2.0);
	double t_2 = Math.cbrt((Math.sin(k) * Math.tan(k)));
	double tmp;
	if (k <= 1.6e-136) {
		tmp = (((2.0 * ((t / k) * (t / k))) / (t * t_1)) / t_2) / Math.pow((t * (t_1 * t_2)), 2.0);
	} else {
		tmp = (2.0 / Math.pow(k, 2.0)) * ((Math.pow(l, 2.0) * (Math.cos(k) * Math.pow(Math.cbrt(-1.0), 6.0))) / (t * Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cbrt(l) ^ -2.0
	t_2 = cbrt(Float64(sin(k) * tan(k)))
	tmp = 0.0
	if (k <= 1.6e-136)
		tmp = Float64(Float64(Float64(Float64(2.0 * Float64(Float64(t / k) * Float64(t / k))) / Float64(t * t_1)) / t_2) / (Float64(t * Float64(t_1 * t_2)) ^ 2.0));
	else
		tmp = Float64(Float64(2.0 / (k ^ 2.0)) * Float64(Float64((l ^ 2.0) * Float64(cos(k) * (cbrt(-1.0) ^ 6.0))) / Float64(t * (sin(k) ^ 2.0))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[k, 1.6e-136], N[(N[(N[(N[(2.0 * N[(N[(t / k), $MachinePrecision] * N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / N[Power[N[(t * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * N[Power[N[Power[-1.0, 1/3], $MachinePrecision], 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.59999999999999996e-136

   1. Initial program 37.7%

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

      1. *-commutative 37.7%

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

      2. associate-/r* 37.7%

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

   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 82.5%

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

      1. associate-/r/ 82.5%

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

      2. associate-/r* 82.5%

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

      3. associate-/r/ 83.1%

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

   8. Simplified 83.1%

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

      1. associate-*l/ 81.3%

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

      2. associate-*l/ 81.3%

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

      3. associate-*l/ 81.3%

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

      4. div-inv 81.3%

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

      5. pow-flip 81.3%

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

      6. metadata-eval 81.3%

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

   10. Applied egg-rr 81.3%

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

       1. associate-*r/ 81.1%

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

       2. associate-*r/ 72.9%

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

       3. associate-/l* 72.9%

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

       4. associate-/l* 72.9%

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

       5. swap-sqr 72.9%

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

       6. rem-square-sqrt 72.9%

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

       7. associate-*l* 72.9%

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

   12. Simplified 72.9%

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

   if 1.59999999999999996e-136 < k

   1. Initial program 35.8%

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

      1. *-commutative 35.8%

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

      2. associate-/r* 35.8%

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

   3. Simplified 42.3%

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

      1. add-cube-cbrt 42.3%

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

      2. div-inv 42.3%

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

      3. times-frac 42.3%

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

   6. Applied egg-rr 63.6%

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

      1. associate-*l/ 63.6%

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

      2. associate-/l* 63.6%

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

      3. associate-/l/ 61.3%

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

      4. unpow2 61.3%

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

      5. unpow3 61.3%

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

      6. *-commutative 61.3%

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

   8. Simplified 59.3%

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

   9. Taylor expanded in l around -inf 72.3%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   10. Step-by-step derivation

       1. associate-*r* 72.3%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

       2. associate-*r/ 72.3%

          \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)\right)}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

       3. associate-*r* 72.3%

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

       4. times-frac 74.4%

          \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2} \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)}{t \cdot {\sin k}^{2}}} \]

       5. *-commutative 74.4%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\left(\cos k \cdot {\left(\sqrt[3]{-1}\right)}^{6}\right)}}{t \cdot {\sin k}^{2}} \]

   11. Simplified 74.4%

       \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2} \cdot \left(\cos k \cdot {\left(\sqrt[3]{-1}\right)}^{6}\right)}{t \cdot {\sin k}^{2}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 72.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{-137}:\\ \;\;\;\;\frac{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}{{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2} \cdot \left(\cos k \cdot {\left(\sqrt[3]{-1}\right)}^{6}\right)}{t \cdot {\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 5.2e-137)
   (/
    (* 2.0 (pow (/ k t) -2.0))
    (pow (* t (* (pow (cbrt l) -2.0) (cbrt (* (sin k) (tan k))))) 3.0))
   (*
    (/ 2.0 (pow k 2.0))
    (/
     (* (pow l 2.0) (* (cos k) (pow (cbrt -1.0) 6.0)))
     (* t (pow (sin k) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.2e-137) {
		tmp = (2.0 * pow((k / t), -2.0)) / pow((t * (pow(cbrt(l), -2.0) * cbrt((sin(k) * tan(k))))), 3.0);
	} else {
		tmp = (2.0 / pow(k, 2.0)) * ((pow(l, 2.0) * (cos(k) * pow(cbrt(-1.0), 6.0))) / (t * pow(sin(k), 2.0)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.2e-137) {
		tmp = (2.0 * Math.pow((k / t), -2.0)) / Math.pow((t * (Math.pow(Math.cbrt(l), -2.0) * Math.cbrt((Math.sin(k) * Math.tan(k))))), 3.0);
	} else {
		tmp = (2.0 / Math.pow(k, 2.0)) * ((Math.pow(l, 2.0) * (Math.cos(k) * Math.pow(Math.cbrt(-1.0), 6.0))) / (t * Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 5.2e-137)
		tmp = Float64(Float64(2.0 * (Float64(k / t) ^ -2.0)) / (Float64(t * Float64((cbrt(l) ^ -2.0) * cbrt(Float64(sin(k) * tan(k))))) ^ 3.0));
	else
		tmp = Float64(Float64(2.0 / (k ^ 2.0)) * Float64(Float64((l ^ 2.0) * Float64(cos(k) * (cbrt(-1.0) ^ 6.0))) / Float64(t * (sin(k) ^ 2.0))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 5.2e-137], N[(N[(2.0 * N[Power[N[(k / t), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[(t * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * N[Power[N[Power[-1.0, 1/3], $MachinePrecision], 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.1999999999999999e-137

   1. Initial program 37.7%

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

      1. *-commutative 37.7%

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

      2. associate-/r* 37.7%

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

   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-cube-cbrt 42.1%

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

      2. div-inv 42.1%

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

      3. times-frac 42.1%

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

   6. Applied egg-rr 71.7%

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

      1. associate-*l/ 72.9%

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

      2. associate-/l* 72.9%

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

      3. associate-/l/ 68.2%

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

      4. unpow2 68.2%

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

      5. unpow3 68.2%

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

      6. *-commutative 68.2%

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

   8. Simplified 58.4%

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

      1. add-cube-cbrt 58.4%

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

      2. pow3 58.4%

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

      3. cbrt-prod 58.4%

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

      4. unpow3 58.4%

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

      5. add-cbrt-cube 68.2%

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

      6. *-commutative 68.2%

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

      7. div-inv 68.2%

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

      8. pow-flip 68.2%

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

      9. metadata-eval 68.2%

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

      10. associate-*l* 68.3%

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

   10. Applied egg-rr 68.3%

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

   if 5.1999999999999999e-137 < k

   1. Initial program 35.8%

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

      1. *-commutative 35.8%

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

      2. associate-/r* 35.8%

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

   3. Simplified 42.3%

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

      1. add-cube-cbrt 42.3%

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

      2. div-inv 42.3%

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

      3. times-frac 42.3%

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

   6. Applied egg-rr 63.6%

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

      1. associate-*l/ 63.6%

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

      2. associate-/l* 63.6%

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

      3. associate-/l/ 61.3%

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

      4. unpow2 61.3%

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

      5. unpow3 61.3%

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

      6. *-commutative 61.3%

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

   8. Simplified 59.3%

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

   9. Taylor expanded in l around -inf 72.3%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   10. Step-by-step derivation

       1. associate-*r* 72.3%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

       2. associate-*r/ 72.3%

          \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)\right)}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

       3. associate-*r* 72.3%

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

       4. times-frac 74.4%

          \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2} \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)}{t \cdot {\sin k}^{2}}} \]

       5. *-commutative 74.4%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\left(\cos k \cdot {\left(\sqrt[3]{-1}\right)}^{6}\right)}}{t \cdot {\sin k}^{2}} \]

   11. Simplified 74.4%

       \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2} \cdot \left(\cos k \cdot {\left(\sqrt[3]{-1}\right)}^{6}\right)}{t \cdot {\sin k}^{2}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 72.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{-136}:\\ \;\;\;\;\frac{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}{{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 3e-136)
   (/
    (* 2.0 (pow (/ k t) -2.0))
    (pow (* t (* (pow (cbrt l) -2.0) (cbrt (* (sin k) (tan k))))) 3.0))
   (*
    2.0
    (* (/ (pow l 2.0) (pow k 2.0)) (/ (cos k) (* t (pow (sin k) 2.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 3e-136) {
		tmp = (2.0 * pow((k / t), -2.0)) / pow((t * (pow(cbrt(l), -2.0) * cbrt((sin(k) * tan(k))))), 3.0);
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k, 2.0)) * (cos(k) / (t * pow(sin(k), 2.0))));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 3e-136) {
		tmp = (2.0 * Math.pow((k / t), -2.0)) / Math.pow((t * (Math.pow(Math.cbrt(l), -2.0) * Math.cbrt((Math.sin(k) * Math.tan(k))))), 3.0);
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k, 2.0)) * (Math.cos(k) / (t * Math.pow(Math.sin(k), 2.0))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 3e-136)
		tmp = Float64(Float64(2.0 * (Float64(k / t) ^ -2.0)) / (Float64(t * Float64((cbrt(l) ^ -2.0) * cbrt(Float64(sin(k) * tan(k))))) ^ 3.0));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(cos(k) / Float64(t * (sin(k) ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 3e-136], N[(N[(2.0 * N[Power[N[(k / t), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[(t * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $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 k < 2.9999999999999998e-136

   1. Initial program 37.7%

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

      1. *-commutative 37.7%

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

      2. associate-/r* 37.7%

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

   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-cube-cbrt 42.1%

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

      2. div-inv 42.1%

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

      3. times-frac 42.1%

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

   6. Applied egg-rr 71.7%

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

      1. associate-*l/ 72.9%

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

      2. associate-/l* 72.9%

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

      3. associate-/l/ 68.2%

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

      4. unpow2 68.2%

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

      5. unpow3 68.2%

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

      6. *-commutative 68.2%

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

   8. Simplified 58.4%

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

      1. add-cube-cbrt 58.4%

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

      2. pow3 58.4%

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

      3. cbrt-prod 58.4%

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

      4. unpow3 58.4%

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

      5. add-cbrt-cube 68.2%

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

      6. *-commutative 68.2%

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

      7. div-inv 68.2%

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

      8. pow-flip 68.2%

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

      9. metadata-eval 68.2%

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

      10. associate-*l* 68.3%

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

   10. Applied egg-rr 68.3%

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

   if 2.9999999999999998e-136 < k

   1. Initial program 35.8%

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

   3. Simplified 43.3%

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

   5. Taylor expanded in t around 0 72.3%

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

      1. times-frac 73.5%

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

   7. Simplified 73.5%

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

Alternative 10: 48.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{-159}:\\ \;\;\;\;\frac{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}{\log \left({\left(e^{\sin k}\right)}^{\left(\frac{k \cdot {t}^{3}}{{\ell}^{2}}\right)}\right)}\\ \mathbf{elif}\;k \leq 9.2 \cdot 10^{-60}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{\frac{{t}^{1.5}}{\ell}}}{\sqrt{\sin k \cdot \tan k}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 6e-159)
   (/
    (* 2.0 (pow (/ k t) -2.0))
    (log (pow (exp (sin k)) (/ (* k (pow t 3.0)) (pow l 2.0)))))
   (if (<= k 9.2e-60)
     (pow
      (/
       (* (/ (sqrt 2.0) k) (/ t (/ (pow t 1.5) l)))
       (sqrt (* (sin k) (tan k))))
      2.0)
     (*
      2.0
      (* (/ (pow l 2.0) (pow k 2.0)) (/ (cos k) (* t (pow (sin k) 2.0))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 6e-159) {
		tmp = (2.0 * pow((k / t), -2.0)) / log(pow(exp(sin(k)), ((k * pow(t, 3.0)) / pow(l, 2.0))));
	} else if (k <= 9.2e-60) {
		tmp = pow((((sqrt(2.0) / k) * (t / (pow(t, 1.5) / l))) / sqrt((sin(k) * tan(k)))), 2.0);
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k, 2.0)) * (cos(k) / (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 (k <= 6d-159) then
        tmp = (2.0d0 * ((k / t) ** (-2.0d0))) / log((exp(sin(k)) ** ((k * (t ** 3.0d0)) / (l ** 2.0d0))))
    else if (k <= 9.2d-60) then
        tmp = (((sqrt(2.0d0) / k) * (t / ((t ** 1.5d0) / l))) / sqrt((sin(k) * tan(k)))) ** 2.0d0
    else
        tmp = 2.0d0 * (((l ** 2.0d0) / (k ** 2.0d0)) * (cos(k) / (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 (k <= 6e-159) {
		tmp = (2.0 * Math.pow((k / t), -2.0)) / Math.log(Math.pow(Math.exp(Math.sin(k)), ((k * Math.pow(t, 3.0)) / Math.pow(l, 2.0))));
	} else if (k <= 9.2e-60) {
		tmp = Math.pow((((Math.sqrt(2.0) / k) * (t / (Math.pow(t, 1.5) / l))) / Math.sqrt((Math.sin(k) * Math.tan(k)))), 2.0);
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k, 2.0)) * (Math.cos(k) / (t * Math.pow(Math.sin(k), 2.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 6e-159:
		tmp = (2.0 * math.pow((k / t), -2.0)) / math.log(math.pow(math.exp(math.sin(k)), ((k * math.pow(t, 3.0)) / math.pow(l, 2.0))))
	elif k <= 9.2e-60:
		tmp = math.pow((((math.sqrt(2.0) / k) * (t / (math.pow(t, 1.5) / l))) / math.sqrt((math.sin(k) * math.tan(k)))), 2.0)
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k, 2.0)) * (math.cos(k) / (t * math.pow(math.sin(k), 2.0))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 6e-159)
		tmp = Float64(Float64(2.0 * (Float64(k / t) ^ -2.0)) / log((exp(sin(k)) ^ Float64(Float64(k * (t ^ 3.0)) / (l ^ 2.0)))));
	elseif (k <= 9.2e-60)
		tmp = Float64(Float64(Float64(sqrt(2.0) / k) * Float64(t / Float64((t ^ 1.5) / l))) / sqrt(Float64(sin(k) * tan(k)))) ^ 2.0;
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(cos(k) / Float64(t * (sin(k) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 6e-159)
		tmp = (2.0 * ((k / t) ^ -2.0)) / log((exp(sin(k)) ^ ((k * (t ^ 3.0)) / (l ^ 2.0))));
	elseif (k <= 9.2e-60)
		tmp = (((sqrt(2.0) / k) * (t / ((t ^ 1.5) / l))) / sqrt((sin(k) * tan(k)))) ^ 2.0;
	else
		tmp = 2.0 * (((l ^ 2.0) / (k ^ 2.0)) * (cos(k) / (t * (sin(k) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 6e-159], N[(N[(2.0 * N[Power[N[(k / t), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / N[Log[N[Power[N[Exp[N[Sin[k], $MachinePrecision]], $MachinePrecision], N[(N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 9.2e-60], N[Power[N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision] * N[(t / N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 6.00000000000000018e-159

   1. Initial program 38.4%

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

      1. *-commutative 38.4%

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

      2. associate-/r* 38.4%

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

   3. Simplified 43.0%

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

      1. add-cube-cbrt 43.0%

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

      2. div-inv 43.0%

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

      3. times-frac 43.0%

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

   6. Applied egg-rr 71.2%

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

      1. associate-*l/ 72.4%

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

      2. associate-/l* 72.4%

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

      3. associate-/l/ 67.8%

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

      4. unpow2 67.8%

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

      5. unpow3 67.8%

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

      6. *-commutative 67.8%

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

   8. Simplified 60.0%

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

      1. add-log-exp 30.5%

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

      2. associate-*l* 30.5%

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

      3. exp-prod 33.5%

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

      4. cube-div 31.5%

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

      5. pow3 31.5%

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

      6. pow2 31.5%

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

      7. unpow-prod-down 31.5%

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

      8. unpow2 31.5%

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

      9. add-cube-cbrt 31.5%

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

   10. Applied egg-rr 31.5%

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

   11. Taylor expanded in k around 0 32.8%

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

   if 6.00000000000000018e-159 < k < 9.2000000000000005e-60

   1. Initial program 39.3%

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

      1. *-commutative 39.3%

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

      2. associate-/r* 39.3%

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

   3. Simplified 39.4%

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

      1. add-sqr-sqrt 18.0%

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

   6. Applied egg-rr 42.5%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 42.5%

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

      2. associate-/r* 42.6%

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

      3. associate-/r/ 42.6%

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

      4. associate-/l* 49.8%

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

   8. Simplified 49.8%

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

   if 9.2000000000000005e-60 < k

   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 42.9%

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

   5. Taylor expanded in t around 0 68.9%

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

      1. times-frac 70.3%

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

   7. Simplified 70.3%

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

Alternative 11: 62.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.4 \cdot 10^{-250}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(\frac{2}{t} \cdot {k}^{-4}\right)\\ \mathbf{elif}\;k \leq 9.2 \cdot 10^{-60}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{\frac{{t}^{1.5}}{\ell}}}{\sqrt{\sin k \cdot \tan k}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 4.4e-250)
   (* (* l l) (* (/ 2.0 t) (pow k -4.0)))
   (if (<= k 9.2e-60)
     (pow
      (/
       (* (/ (sqrt 2.0) k) (/ t (/ (pow t 1.5) l)))
       (sqrt (* (sin k) (tan k))))
      2.0)
     (*
      2.0
      (* (/ (pow l 2.0) (pow k 2.0)) (/ (cos k) (* t (pow (sin k) 2.0))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.4e-250) {
		tmp = (l * l) * ((2.0 / t) * pow(k, -4.0));
	} else if (k <= 9.2e-60) {
		tmp = pow((((sqrt(2.0) / k) * (t / (pow(t, 1.5) / l))) / sqrt((sin(k) * tan(k)))), 2.0);
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k, 2.0)) * (cos(k) / (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 (k <= 4.4d-250) then
        tmp = (l * l) * ((2.0d0 / t) * (k ** (-4.0d0)))
    else if (k <= 9.2d-60) then
        tmp = (((sqrt(2.0d0) / k) * (t / ((t ** 1.5d0) / l))) / sqrt((sin(k) * tan(k)))) ** 2.0d0
    else
        tmp = 2.0d0 * (((l ** 2.0d0) / (k ** 2.0d0)) * (cos(k) / (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 (k <= 4.4e-250) {
		tmp = (l * l) * ((2.0 / t) * Math.pow(k, -4.0));
	} else if (k <= 9.2e-60) {
		tmp = Math.pow((((Math.sqrt(2.0) / k) * (t / (Math.pow(t, 1.5) / l))) / Math.sqrt((Math.sin(k) * Math.tan(k)))), 2.0);
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k, 2.0)) * (Math.cos(k) / (t * Math.pow(Math.sin(k), 2.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 4.4e-250:
		tmp = (l * l) * ((2.0 / t) * math.pow(k, -4.0))
	elif k <= 9.2e-60:
		tmp = math.pow((((math.sqrt(2.0) / k) * (t / (math.pow(t, 1.5) / l))) / math.sqrt((math.sin(k) * math.tan(k)))), 2.0)
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k, 2.0)) * (math.cos(k) / (t * math.pow(math.sin(k), 2.0))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 4.4e-250)
		tmp = Float64(Float64(l * l) * Float64(Float64(2.0 / t) * (k ^ -4.0)));
	elseif (k <= 9.2e-60)
		tmp = Float64(Float64(Float64(sqrt(2.0) / k) * Float64(t / Float64((t ^ 1.5) / l))) / sqrt(Float64(sin(k) * tan(k)))) ^ 2.0;
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(cos(k) / Float64(t * (sin(k) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 4.4e-250)
		tmp = (l * l) * ((2.0 / t) * (k ^ -4.0));
	elseif (k <= 9.2e-60)
		tmp = (((sqrt(2.0) / k) * (t / ((t ^ 1.5) / l))) / sqrt((sin(k) * tan(k)))) ^ 2.0;
	else
		tmp = 2.0 * (((l ^ 2.0) / (k ^ 2.0)) * (cos(k) / (t * (sin(k) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 4.4e-250], N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / t), $MachinePrecision] * N[Power[k, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 9.2e-60], N[Power[N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision] * N[(t / N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 4.4e-250

   1. Initial program 38.0%

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

   3. Simplified 42.9%

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

   5. Taylor expanded in k around 0 67.4%

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

      1. *-commutative 67.4%

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

      2. associate-/r* 67.4%

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

   7. Simplified 67.4%

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

      1. div-inv 67.4%

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

      2. pow-flip 67.4%

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

      3. metadata-eval 67.4%

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

   9. Applied egg-rr 67.4%

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

   if 4.4e-250 < k < 9.2000000000000005e-60

   1. Initial program 40.0%

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

      1. *-commutative 40.0%

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

      2. associate-/r* 40.0%

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

   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. add-sqr-sqrt 17.9%

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

   6. Applied egg-rr 37.8%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 37.8%

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

      2. associate-/r* 37.9%

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

      3. associate-/r/ 37.9%

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

      4. associate-/l* 42.4%

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

   8. Simplified 42.4%

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

   if 9.2000000000000005e-60 < k

   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 42.9%

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

   5. Taylor expanded in t around 0 68.9%

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

      1. times-frac 70.3%

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

   7. Simplified 70.3%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.4 \cdot 10^{-250}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(\frac{2}{t} \cdot {k}^{-4}\right)\\ \mathbf{elif}\;k \leq 9.2 \cdot 10^{-60}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{\frac{{t}^{1.5}}{\ell}}}{\sqrt{\sin k \cdot \tan k}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 74.1% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (* 2.0 (* (/ (pow l 2.0) (pow k 2.0)) (/ (cos k) (* t (pow (sin k) 2.0))))))

C

double code(double t, double l, double k) {
	return 2.0 * ((pow(l, 2.0) / pow(k, 2.0)) * (cos(k) / (t * pow(sin(k), 2.0))));
}

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 * (((l ** 2.0d0) / (k ** 2.0d0)) * (cos(k) / (t * (sin(k) ** 2.0d0))))
end function

Java

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

Python

def code(t, l, k):
	return 2.0 * ((math.pow(l, 2.0) / math.pow(k, 2.0)) * (math.cos(k) / (t * math.pow(math.sin(k), 2.0))))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(cos(k) / Float64(t * (sin(k) ^ 2.0)))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * (((l ^ 2.0) / (k ^ 2.0)) * (cos(k) / (t * (sin(k) ^ 2.0))));
end

Wolfram

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

Derivation

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 43.2%

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

5. Taylor expanded in t around 0 73.4%

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

   1. times-frac 75.2%

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

7. Simplified 75.2%

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

Alternative 13: 73.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{t \cdot {k}^{2}} \cdot \frac{{\ell}^{2}}{\sin k \cdot \tan k} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* (/ 2.0 (* t (pow k 2.0))) (/ (pow l 2.0) (* (sin k) (tan k)))))

C

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

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 * (k ** 2.0d0))) * ((l ** 2.0d0) / (sin(k) * tan(k)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.0%

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

   1. *-commutative 37.0%

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

   2. associate-/r* 37.0%

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

3. Simplified 42.2%

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

   1. add-cube-cbrt 42.2%

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

   2. div-inv 42.2%

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

   3. times-frac 42.2%

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

6. Applied egg-rr 68.7%

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

   1. associate-*l/ 69.4%

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

   2. associate-/l* 69.4%

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

   3. associate-/l/ 65.7%

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

   4. unpow2 65.7%

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

   5. unpow3 65.7%

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

   6. *-commutative 65.7%

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

8. Simplified 58.7%

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

   1. add-log-exp 32.5%

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

   2. associate-*l* 32.5%

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

   3. exp-prod 35.1%

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

   4. cube-div 33.4%

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

   5. pow3 33.4%

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

   6. pow2 33.4%

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

   7. unpow-prod-down 33.4%

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

   8. unpow2 33.4%

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

   9. add-cube-cbrt 33.4%

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

10. Applied egg-rr 33.4%

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

11. Taylor expanded in t around inf 73.4%

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

    1. associate-*r/ 73.4%

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

    2. associate-*r* 73.4%

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

    3. times-frac 74.3%

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

13. Simplified 74.3%

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

14. Final simplification 74.3%

    \[\leadsto \frac{2}{t \cdot {k}^{2}} \cdot \frac{{\ell}^{2}}{\sin k \cdot \tan k} \]
15. Add Preprocessing

Alternative 14: 72.8% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (* (* l l) (/ 2.0 (* (pow k 2.0) (* (tan k) (* t (sin k)))))))

C

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

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 ** 2.0d0) * (tan(k) * (t * sin(k)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 43.2%

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

   1. add-log-exp 27.7%

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

   2. exp-prod 31.8%

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

   3. associate-*r* 31.8%

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

   4. *-commutative 31.8%

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

   5. associate-*l* 31.8%

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

6. Applied egg-rr 31.8%

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

7. Taylor expanded in k around inf 73.1%

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

   1. associate-*r* 73.1%

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

9. Simplified 73.1%

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

10. Final simplification 73.1%

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

Alternative 15: 64.3% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (* (* l l) (/ 2.0 (* (pow (sin k) 2.0) (* t (pow k 2.0))))))

C

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

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 / ((sin(k) ** 2.0d0) * (t * (k ** 2.0d0))))
end function

Java

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

Python

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

Julia

function code(t, l, k)
	return Float64(Float64(l * l) * Float64(2.0 / Float64((sin(k) ^ 2.0) * Float64(t * (k ^ 2.0)))))
end

MATLAB

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

Wolfram

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

Derivation

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 43.2%

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

5. Taylor expanded in t around 0 73.1%

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

   1. associate-*r/ 73.1%

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

   2. associate-*r* 73.1%

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

7. Simplified 73.1%

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

8. Taylor expanded in k around 0 66.9%

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

9. Final simplification 66.9%

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

Alternative 16: 60.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 43.2%

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

5. Taylor expanded in k around 0 65.3%

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

   1. *-commutative 65.3%

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

   2. associate-/r* 65.7%

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

7. Simplified 65.7%

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

Alternative 17: 62.0% accurate, 3.8× speedup

Math

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

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(Float64(2.0 / 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[(N[(2.0 / t), $MachinePrecision] * N[Power[k, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 43.2%

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

5. Taylor expanded in k around 0 65.0%

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

   1. *-commutative 65.0%

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

   2. associate-/r* 65.0%

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

7. Simplified 65.0%

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

   1. div-inv 65.0%

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

   2. pow-flip 65.0%

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

   3. metadata-eval 65.0%

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

9. Applied egg-rr 65.0%

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

10. Final simplification 65.0%

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

Reproduce

herbie shell --seed 2024149 
(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))))