Toniolo and Linder, Equation (10+)

Percentage Accurate: 53.8% → 88.0%

Time: 20.6s

Alternatives: 12

Speedup: 3.8×

• 26.3% 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: 53.8% 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: 88.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\sqrt[3]{\ell}}\\ t_2 := \frac{\sqrt[3]{\sin k}}{\sqrt[3]{\ell}}\\ t_3 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_4 := \sqrt[3]{\frac{2}{\tan k}}\\ \mathbf{if}\;t \leq -4.8 \cdot 10^{-101}:\\ \;\;\;\;\frac{{\left(\frac{t_4}{\frac{t}{\frac{t_1 \cdot t_1}{t_2}}}\right)}^{3}}{t_3}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-97}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{t_4}{\frac{t}{\frac{\sqrt[3]{\ell}}{t_2}}}\right)}^{3}}{t_3}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (sqrt (cbrt l)))
        (t_2 (/ (cbrt (sin k)) (cbrt l)))
        (t_3 (+ 2.0 (pow (/ k t) 2.0)))
        (t_4 (cbrt (/ 2.0 (tan k)))))
   (if (<= t -4.8e-101)
     (/ (pow (/ t_4 (/ t (/ (* t_1 t_1) t_2))) 3.0) t_3)
     (if (<= t 2.3e-97)
       (* 2.0 (* (/ (cos k) (* k (* t k))) (/ (* l l) (pow (sin k) 2.0))))
       (/ (pow (/ t_4 (/ t (/ (cbrt l) t_2))) 3.0) t_3)))))

C

l = abs(l);
double code(double t, double l, double k) {
	double t_1 = sqrt(cbrt(l));
	double t_2 = cbrt(sin(k)) / cbrt(l);
	double t_3 = 2.0 + pow((k / t), 2.0);
	double t_4 = cbrt((2.0 / tan(k)));
	double tmp;
	if (t <= -4.8e-101) {
		tmp = pow((t_4 / (t / ((t_1 * t_1) / t_2))), 3.0) / t_3;
	} else if (t <= 2.3e-97) {
		tmp = 2.0 * ((cos(k) / (k * (t * k))) * ((l * l) / pow(sin(k), 2.0)));
	} else {
		tmp = pow((t_4 / (t / (cbrt(l) / t_2))), 3.0) / t_3;
	}
	return tmp;
}

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double t_1 = Math.sqrt(Math.cbrt(l));
	double t_2 = Math.cbrt(Math.sin(k)) / Math.cbrt(l);
	double t_3 = 2.0 + Math.pow((k / t), 2.0);
	double t_4 = Math.cbrt((2.0 / Math.tan(k)));
	double tmp;
	if (t <= -4.8e-101) {
		tmp = Math.pow((t_4 / (t / ((t_1 * t_1) / t_2))), 3.0) / t_3;
	} else if (t <= 2.3e-97) {
		tmp = 2.0 * ((Math.cos(k) / (k * (t * k))) * ((l * l) / Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = Math.pow((t_4 / (t / (Math.cbrt(l) / t_2))), 3.0) / t_3;
	}
	return tmp;
}

Julia

l = abs(l)
function code(t, l, k)
	t_1 = sqrt(cbrt(l))
	t_2 = Float64(cbrt(sin(k)) / cbrt(l))
	t_3 = Float64(2.0 + (Float64(k / t) ^ 2.0))
	t_4 = cbrt(Float64(2.0 / tan(k)))
	tmp = 0.0
	if (t <= -4.8e-101)
		tmp = Float64((Float64(t_4 / Float64(t / Float64(Float64(t_1 * t_1) / t_2))) ^ 3.0) / t_3);
	elseif (t <= 2.3e-97)
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * Float64(t * k))) * Float64(Float64(l * l) / (sin(k) ^ 2.0))));
	else
		tmp = Float64((Float64(t_4 / Float64(t / Float64(cbrt(l) / t_2))) ^ 3.0) / t_3);
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[Sqrt[N[Power[l, 1/3], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[t, -4.8e-101], N[(N[Power[N[(t$95$4 / N[(t / N[(N[(t$95$1 * t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t, 2.3e-97], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(t$95$4 / N[(t / N[(N[Power[l, 1/3], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / t$95$3), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.8e-101

   1. Initial program 72.2%

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

      1. associate-/r* 72.2%

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

      2. associate-*l* 69.2%

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

      3. sqr-neg 69.2%

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

      4. associate-*l* 72.2%

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

      5. *-commutative 72.2%

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

      6. sqr-neg 72.2%

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

      7. associate-/r* 72.2%

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

   3. Simplified 72.2%

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

      1. associate-/l/ 72.2%

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

      2. associate-/r/ 72.2%

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

      3. add-cube-cbrt 72.1%

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

      4. pow3 72.1%

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

   5. Applied egg-rr 83.5%

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

      1. cbrt-div 91.9%

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

   7. Applied egg-rr 91.9%

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

      1. cbrt-div 94.8%

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

   9. Applied egg-rr 94.8%

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

       1. add-sqr-sqrt 56.3%

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

   11. Applied egg-rr 56.3%

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

   if -4.8e-101 < t < 2.29999999999999994e-97

   1. Initial program 31.5%

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

      1. associate-/r* 31.5%

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

      2. associate-*l* 31.5%

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

      3. sqr-neg 31.5%

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

      4. associate-*l* 31.5%

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

      5. *-commutative 31.5%

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

      6. sqr-neg 31.5%

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

      7. associate-*l/ 30.4%

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

      8. associate-*r/ 30.4%

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

      9. associate-/r/ 30.4%

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

   3. Simplified 30.4%

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

   4. Taylor expanded in k around inf 69.2%

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

      1. *-commutative 69.2%

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

      2. associate-*r* 69.2%

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

      3. times-frac 69.3%

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

      4. unpow2 69.3%

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

      5. associate-*l* 74.2%

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

      6. unpow2 74.2%

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

   6. Simplified 74.2%

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

   if 2.29999999999999994e-97 < t

   1. Initial program 56.9%

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

      1. associate-/r* 57.8%

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

      2. associate-*l* 57.5%

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

      3. sqr-neg 57.5%

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

      4. associate-*l* 57.8%

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

      5. *-commutative 57.8%

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

      6. sqr-neg 57.8%

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

      7. associate-/r* 57.8%

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

   3. Simplified 57.7%

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

      1. associate-/l/ 57.7%

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

      2. associate-/r/ 57.8%

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

      3. add-cube-cbrt 57.7%

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

      4. pow3 57.7%

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

   5. Applied egg-rr 70.5%

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

      1. cbrt-div 90.3%

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

   7. Applied egg-rr 90.3%

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

      1. cbrt-div 94.3%

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

   9. Applied egg-rr 94.3%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{-101}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{\sqrt{\sqrt[3]{\ell}} \cdot \sqrt{\sqrt[3]{\ell}}}{\frac{\sqrt[3]{\sin k}}{\sqrt[3]{\ell}}}}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-97}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{\sqrt[3]{\ell}}{\frac{\sqrt[3]{\sin k}}{\sqrt[3]{\ell}}}}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]

Alternative 2: 88.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.14 \cdot 10^{-100} \lor \neg \left(t \leq 1.02 \cdot 10^{-97}\right):\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{\sqrt[3]{\ell}}{\frac{\sqrt[3]{\sin k}}{\sqrt[3]{\ell}}}}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (or (<= t -1.14e-100) (not (<= t 1.02e-97)))
   (/
    (pow
     (/ (cbrt (/ 2.0 (tan k))) (/ t (/ (cbrt l) (/ (cbrt (sin k)) (cbrt l)))))
     3.0)
    (+ 2.0 (pow (/ k t) 2.0)))
   (* 2.0 (* (/ (cos k) (* k (* t k))) (/ (* l l) (pow (sin k) 2.0))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.14e-100) || !(t <= 1.02e-97)) {
		tmp = pow((cbrt((2.0 / tan(k))) / (t / (cbrt(l) / (cbrt(sin(k)) / cbrt(l))))), 3.0) / (2.0 + pow((k / t), 2.0));
	} else {
		tmp = 2.0 * ((cos(k) / (k * (t * k))) * ((l * l) / pow(sin(k), 2.0)));
	}
	return tmp;
}

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.14e-100) || !(t <= 1.02e-97)) {
		tmp = Math.pow((Math.cbrt((2.0 / Math.tan(k))) / (t / (Math.cbrt(l) / (Math.cbrt(Math.sin(k)) / Math.cbrt(l))))), 3.0) / (2.0 + Math.pow((k / t), 2.0));
	} else {
		tmp = 2.0 * ((Math.cos(k) / (k * (t * k))) * ((l * l) / Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if ((t <= -1.14e-100) || !(t <= 1.02e-97))
		tmp = Float64((Float64(cbrt(Float64(2.0 / tan(k))) / Float64(t / Float64(cbrt(l) / Float64(cbrt(sin(k)) / cbrt(l))))) ^ 3.0) / Float64(2.0 + (Float64(k / t) ^ 2.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * Float64(t * k))) * Float64(Float64(l * l) / (sin(k) ^ 2.0))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[Or[LessEqual[t, -1.14e-100], N[Not[LessEqual[t, 1.02e-97]], $MachinePrecision]], N[(N[Power[N[(N[Power[N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[(t / N[(N[Power[l, 1/3], $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.13999999999999997e-100 or 1.02000000000000004e-97 < t

   1. Initial program 64.7%

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

      1. associate-/r* 65.2%

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

      2. associate-*l* 63.5%

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

      3. sqr-neg 63.5%

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

      4. associate-*l* 65.2%

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

      5. *-commutative 65.2%

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

      6. sqr-neg 65.2%

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

      7. associate-/r* 65.2%

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

   3. Simplified 65.1%

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

      1. associate-/l/ 65.1%

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

      2. associate-/r/ 65.2%

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

      3. add-cube-cbrt 65.1%

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

      4. pow3 65.1%

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

   5. Applied egg-rr 77.1%

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

      1. cbrt-div 91.1%

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

   7. Applied egg-rr 91.1%

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

      1. cbrt-div 94.6%

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

   9. Applied egg-rr 94.6%

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

   if -1.13999999999999997e-100 < t < 1.02000000000000004e-97

   1. Initial program 31.5%

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

      1. associate-/r* 31.5%

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

      2. associate-*l* 31.5%

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

      3. sqr-neg 31.5%

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

      4. associate-*l* 31.5%

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

      5. *-commutative 31.5%

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

      6. sqr-neg 31.5%

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

      7. associate-*l/ 30.4%

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

      8. associate-*r/ 30.4%

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

      9. associate-/r/ 30.4%

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

   3. Simplified 30.4%

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

   4. Taylor expanded in k around inf 69.2%

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

      1. *-commutative 69.2%

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

      2. associate-*r* 69.2%

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

      3. times-frac 69.3%

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

      4. unpow2 69.3%

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

      5. associate-*l* 74.2%

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

      6. unpow2 74.2%

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

   6. Simplified 74.2%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.14 \cdot 10^{-100} \lor \neg \left(t \leq 1.02 \cdot 10^{-97}\right):\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{\sqrt[3]{\ell}}{\frac{\sqrt[3]{\sin k}}{\sqrt[3]{\ell}}}}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 3: 85.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{-104} \lor \neg \left(t \leq 2.2 \cdot 10^{-97}\right):\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{\frac{\sin k}{\ell}}}}}\right)}^{3}}{2 + \frac{k}{t} \cdot \frac{k}{t}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (or (<= t -2.8e-104) (not (<= t 2.2e-97)))
   (/
    (pow
     (/ (cbrt (/ 2.0 (tan k))) (/ t (/ (cbrt l) (cbrt (/ (sin k) l)))))
     3.0)
    (+ 2.0 (* (/ k t) (/ k t))))
   (* 2.0 (* (/ (cos k) (* k (* t k))) (/ (* l l) (pow (sin k) 2.0))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if ((t <= -2.8e-104) || !(t <= 2.2e-97)) {
		tmp = pow((cbrt((2.0 / tan(k))) / (t / (cbrt(l) / cbrt((sin(k) / l))))), 3.0) / (2.0 + ((k / t) * (k / t)));
	} else {
		tmp = 2.0 * ((cos(k) / (k * (t * k))) * ((l * l) / pow(sin(k), 2.0)));
	}
	return tmp;
}

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -2.8e-104) || !(t <= 2.2e-97)) {
		tmp = Math.pow((Math.cbrt((2.0 / Math.tan(k))) / (t / (Math.cbrt(l) / Math.cbrt((Math.sin(k) / l))))), 3.0) / (2.0 + ((k / t) * (k / t)));
	} else {
		tmp = 2.0 * ((Math.cos(k) / (k * (t * k))) * ((l * l) / Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if ((t <= -2.8e-104) || !(t <= 2.2e-97))
		tmp = Float64((Float64(cbrt(Float64(2.0 / tan(k))) / Float64(t / Float64(cbrt(l) / cbrt(Float64(sin(k) / l))))) ^ 3.0) / Float64(2.0 + Float64(Float64(k / t) * Float64(k / t))));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * Float64(t * k))) * Float64(Float64(l * l) / (sin(k) ^ 2.0))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[Or[LessEqual[t, -2.8e-104], N[Not[LessEqual[t, 2.2e-97]], $MachinePrecision]], N[(N[Power[N[(N[Power[N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[(t / N[(N[Power[l, 1/3], $MachinePrecision] / N[Power[N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[(2.0 + N[(N[(k / t), $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.8e-104 or 2.1999999999999999e-97 < t

   1. Initial program 64.7%

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

      1. associate-/r* 65.2%

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

      2. associate-*l* 63.5%

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

      3. sqr-neg 63.5%

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

      4. associate-*l* 65.2%

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

      5. *-commutative 65.2%

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

      6. sqr-neg 65.2%

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

      7. associate-/r* 65.2%

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

   3. Simplified 65.1%

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

      1. associate-/l/ 65.1%

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

      2. associate-/r/ 65.2%

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

      3. add-cube-cbrt 65.1%

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

      4. pow3 65.1%

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

   5. Applied egg-rr 77.1%

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

      1. unpow2 74.1%

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

   7. Applied egg-rr 77.1%

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

      1. cbrt-div 91.1%

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

   9. Applied egg-rr 91.1%

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

   if -2.8e-104 < t < 2.1999999999999999e-97

   1. Initial program 31.5%

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

      1. associate-/r* 31.5%

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

      2. associate-*l* 31.5%

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

      3. sqr-neg 31.5%

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

      4. associate-*l* 31.5%

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

      5. *-commutative 31.5%

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

      6. sqr-neg 31.5%

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

      7. associate-*l/ 30.4%

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

      8. associate-*r/ 30.4%

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

      9. associate-/r/ 30.4%

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

   3. Simplified 30.4%

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

   4. Taylor expanded in k around inf 69.2%

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

      1. *-commutative 69.2%

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

      2. associate-*r* 69.2%

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

      3. times-frac 69.3%

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

      4. unpow2 69.3%

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

      5. associate-*l* 74.2%

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

      6. unpow2 74.2%

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

   6. Simplified 74.2%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{-104} \lor \neg \left(t \leq 2.2 \cdot 10^{-97}\right):\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{\frac{\sin k}{\ell}}}}}\right)}^{3}}{2 + \frac{k}{t} \cdot \frac{k}{t}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 4: 83.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -3.4 \cdot 10^{-109} \lor \neg \left(t \leq 6 \cdot 10^{-98}\right):\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot {\left(\frac{t}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{\frac{\sin k}{\ell}}}}\right)}^{3}}}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (or (<= t -3.4e-109) (not (<= t 6e-98)))
   (/
    (/ 2.0 (* (tan k) (pow (/ t (/ (cbrt l) (cbrt (/ (sin k) l)))) 3.0)))
    (+ 1.0 (+ (pow (/ k t) 2.0) 1.0)))
   (* 2.0 (* (/ (cos k) (* k (* t k))) (/ (* l l) (pow (sin k) 2.0))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if ((t <= -3.4e-109) || !(t <= 6e-98)) {
		tmp = (2.0 / (tan(k) * pow((t / (cbrt(l) / cbrt((sin(k) / l)))), 3.0))) / (1.0 + (pow((k / t), 2.0) + 1.0));
	} else {
		tmp = 2.0 * ((cos(k) / (k * (t * k))) * ((l * l) / pow(sin(k), 2.0)));
	}
	return tmp;
}

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -3.4e-109) || !(t <= 6e-98)) {
		tmp = (2.0 / (Math.tan(k) * Math.pow((t / (Math.cbrt(l) / Math.cbrt((Math.sin(k) / l)))), 3.0))) / (1.0 + (Math.pow((k / t), 2.0) + 1.0));
	} else {
		tmp = 2.0 * ((Math.cos(k) / (k * (t * k))) * ((l * l) / Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if ((t <= -3.4e-109) || !(t <= 6e-98))
		tmp = Float64(Float64(2.0 / Float64(tan(k) * (Float64(t / Float64(cbrt(l) / cbrt(Float64(sin(k) / l)))) ^ 3.0))) / Float64(1.0 + Float64((Float64(k / t) ^ 2.0) + 1.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * Float64(t * k))) * Float64(Float64(l * l) / (sin(k) ^ 2.0))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[Or[LessEqual[t, -3.4e-109], N[Not[LessEqual[t, 6e-98]], $MachinePrecision]], N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[Power[N[(t / N[(N[Power[l, 1/3], $MachinePrecision] / N[Power[N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.40000000000000012e-109 or 6e-98 < t

   1. Initial program 64.7%

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

      1. associate-/r* 65.2%

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

      2. +-commutative 65.2%

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

   3. Simplified 65.2%

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

      1. associate-/r/ 65.1%

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

      2. add-cube-cbrt 65.0%

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

      3. pow3 65.0%

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

      4. cbrt-div 65.0%

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

      5. rem-cbrt-cube 69.8%

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

      6. associate-/l* 74.1%

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

   5. Applied egg-rr 74.1%

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

      1. cbrt-div 91.1%

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

   7. Applied egg-rr 87.5%

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

   if -3.40000000000000012e-109 < t < 6e-98

   1. Initial program 31.5%

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

      1. associate-/r* 31.5%

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

      2. associate-*l* 31.5%

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

      3. sqr-neg 31.5%

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

      4. associate-*l* 31.5%

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

      5. *-commutative 31.5%

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

      6. sqr-neg 31.5%

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

      7. associate-*l/ 30.4%

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

      8. associate-*r/ 30.4%

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

      9. associate-/r/ 30.4%

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

   3. Simplified 30.4%

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

   4. Taylor expanded in k around inf 69.2%

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

      1. *-commutative 69.2%

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

      2. associate-*r* 69.2%

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

      3. times-frac 69.3%

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

      4. unpow2 69.3%

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

      5. associate-*l* 74.2%

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

      6. unpow2 74.2%

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

   6. Simplified 74.2%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.4 \cdot 10^{-109} \lor \neg \left(t \leq 6 \cdot 10^{-98}\right):\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot {\left(\frac{t}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{\frac{\sin k}{\ell}}}}\right)}^{3}}}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 5: 76.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \frac{\ell}{\sin k}\\ t_3 := \tan k \cdot {t}^{3}\\ \mathbf{if}\;t \leq -1.15 \cdot 10^{-100}:\\ \;\;\;\;\frac{2 \cdot \frac{\ell}{t_3}}{t_1} \cdot t_2\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{-97}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{t_2 \cdot \frac{2 \cdot \ell}{t_3}}{t_1}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{+170}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot {\left(t \cdot \sqrt[3]{\sin k}\right)}^{3}}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{k \cdot {t}^{3}}}{k}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (+ 2.0 (pow (/ k t) 2.0)))
        (t_2 (/ l (sin k)))
        (t_3 (* (tan k) (pow t 3.0))))
   (if (<= t -1.15e-100)
     (* (/ (* 2.0 (/ l t_3)) t_1) t_2)
     (if (<= t 2.25e-97)
       (* 2.0 (* (/ (cos k) (* k (* t k))) (/ (* l l) (pow (sin k) 2.0))))
       (if (<= t 5.6e+102)
         (/ (* t_2 (/ (* 2.0 l) t_3)) t_1)
         (if (<= t 1.2e+170)
           (/
            (* (* l l) (/ 2.0 (* (tan k) (pow (* t (cbrt (sin k))) 3.0))))
            t_1)
           (/ (* l (/ l (* k (pow t 3.0)))) k)))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double t_1 = 2.0 + pow((k / t), 2.0);
	double t_2 = l / sin(k);
	double t_3 = tan(k) * pow(t, 3.0);
	double tmp;
	if (t <= -1.15e-100) {
		tmp = ((2.0 * (l / t_3)) / t_1) * t_2;
	} else if (t <= 2.25e-97) {
		tmp = 2.0 * ((cos(k) / (k * (t * k))) * ((l * l) / pow(sin(k), 2.0)));
	} else if (t <= 5.6e+102) {
		tmp = (t_2 * ((2.0 * l) / t_3)) / t_1;
	} else if (t <= 1.2e+170) {
		tmp = ((l * l) * (2.0 / (tan(k) * pow((t * cbrt(sin(k))), 3.0)))) / t_1;
	} else {
		tmp = (l * (l / (k * pow(t, 3.0)))) / k;
	}
	return tmp;
}

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double t_1 = 2.0 + Math.pow((k / t), 2.0);
	double t_2 = l / Math.sin(k);
	double t_3 = Math.tan(k) * Math.pow(t, 3.0);
	double tmp;
	if (t <= -1.15e-100) {
		tmp = ((2.0 * (l / t_3)) / t_1) * t_2;
	} else if (t <= 2.25e-97) {
		tmp = 2.0 * ((Math.cos(k) / (k * (t * k))) * ((l * l) / Math.pow(Math.sin(k), 2.0)));
	} else if (t <= 5.6e+102) {
		tmp = (t_2 * ((2.0 * l) / t_3)) / t_1;
	} else if (t <= 1.2e+170) {
		tmp = ((l * l) * (2.0 / (Math.tan(k) * Math.pow((t * Math.cbrt(Math.sin(k))), 3.0)))) / t_1;
	} else {
		tmp = (l * (l / (k * Math.pow(t, 3.0)))) / k;
	}
	return tmp;
}

Julia

l = abs(l)
function code(t, l, k)
	t_1 = Float64(2.0 + (Float64(k / t) ^ 2.0))
	t_2 = Float64(l / sin(k))
	t_3 = Float64(tan(k) * (t ^ 3.0))
	tmp = 0.0
	if (t <= -1.15e-100)
		tmp = Float64(Float64(Float64(2.0 * Float64(l / t_3)) / t_1) * t_2);
	elseif (t <= 2.25e-97)
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * Float64(t * k))) * Float64(Float64(l * l) / (sin(k) ^ 2.0))));
	elseif (t <= 5.6e+102)
		tmp = Float64(Float64(t_2 * Float64(Float64(2.0 * l) / t_3)) / t_1);
	elseif (t <= 1.2e+170)
		tmp = Float64(Float64(Float64(l * l) * Float64(2.0 / Float64(tan(k) * (Float64(t * cbrt(sin(k))) ^ 3.0)))) / t_1);
	else
		tmp = Float64(Float64(l * Float64(l / Float64(k * (t ^ 3.0)))) / k);
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Tan[k], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.15e-100], N[(N[(N[(2.0 * N[(l / t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t, 2.25e-97], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.6e+102], N[(N[(t$95$2 * N[(N[(2.0 * l), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t, 1.2e+170], N[(N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[Power[N[(t * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(l * N[(l / N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -1.14999999999999997e-100

   1. Initial program 72.2%

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

      1. associate-/r* 72.2%

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

      2. associate-*l* 69.2%

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

      3. sqr-neg 69.2%

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

      4. associate-*l* 72.2%

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

      5. *-commutative 72.2%

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

      6. sqr-neg 72.2%

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

      7. associate-*l/ 73.2%

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

      8. associate-*r/ 73.0%

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

      9. associate-/r/ 72.9%

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

   3. Simplified 72.9%

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

      1. expm1-log1p-u 55.0%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)\right)\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. expm1-udef 51.9%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)\right)} - 1}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-*l/ 52.0%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}\right)} - 1}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. associate-*r* 52.0%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}\right)} - 1}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 52.0%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}\right)} - 1}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   6. Step-by-step derivation

      1. expm1-def 55.1%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}\right)\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. expm1-log1p 73.0%

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

      3. associate-*r* 73.0%

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

      4. times-frac 83.0%

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

      5. *-commutative 83.0%

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

   7. Simplified 83.0%

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

      1. div-inv 83.0%

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

      2. times-frac 82.9%

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

   9. Applied egg-rr 82.9%

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

       1. associate-*r/ 82.9%

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

       2. *-rgt-identity 82.9%

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

       3. associate-*l/ 85.2%

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

       4. times-frac 85.2%

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

       5. *-commutative 85.2%

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

       6. *-rgt-identity 85.2%

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

       7. associate-*r/ 85.2%

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

       8. associate-*l* 85.2%

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

       9. associate-*r/ 85.2%

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

       10. *-rgt-identity 85.2%

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

       11. *-commutative 85.2%

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

   11. Simplified 85.2%

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

   if -1.14999999999999997e-100 < t < 2.25000000000000005e-97

   1. Initial program 31.5%

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

      1. associate-/r* 31.5%

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

      2. associate-*l* 31.5%

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

      3. sqr-neg 31.5%

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

      4. associate-*l* 31.5%

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

      5. *-commutative 31.5%

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

      6. sqr-neg 31.5%

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

      7. associate-*l/ 30.4%

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

      8. associate-*r/ 30.4%

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

      9. associate-/r/ 30.4%

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

   3. Simplified 30.4%

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

   4. Taylor expanded in k around inf 69.2%

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

      1. *-commutative 69.2%

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

      2. associate-*r* 69.2%

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

      3. times-frac 69.3%

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

      4. unpow2 69.3%

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

      5. associate-*l* 74.2%

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

      6. unpow2 74.2%

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

   6. Simplified 74.2%

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

   if 2.25000000000000005e-97 < t < 5.60000000000000037e102

   1. Initial program 60.8%

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

      1. associate-/r* 62.8%

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

      2. associate-*l* 62.5%

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

      3. sqr-neg 62.5%

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

      4. associate-*l* 62.8%

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

      5. *-commutative 62.8%

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

      6. sqr-neg 62.8%

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

      7. associate-*l/ 62.6%

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

      8. associate-*r/ 64.9%

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

      9. associate-/r/ 64.8%

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

   3. Simplified 64.8%

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

      1. expm1-log1p-u 63.8%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)\right)\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. expm1-udef 52.4%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)\right)} - 1}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-*l/ 52.4%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}\right)} - 1}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. associate-*r* 52.4%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}\right)} - 1}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 52.4%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}\right)} - 1}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   6. Step-by-step derivation

      1. expm1-def 63.9%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}\right)\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. expm1-log1p 64.9%

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

      3. associate-*r* 64.9%

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

      4. times-frac 87.1%

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

      5. *-commutative 87.1%

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

   7. Simplified 87.1%

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

   if 5.60000000000000037e102 < t < 1.2e170

   1. Initial program 35.2%

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

      1. associate-/r* 35.2%

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

      2. associate-*l* 34.7%

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

      3. sqr-neg 34.7%

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

      4. associate-*l* 35.2%

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

      5. *-commutative 35.2%

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

      6. sqr-neg 35.2%

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

      7. associate-*l/ 35.2%

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

      8. associate-*r/ 35.2%

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

      9. associate-/r/ 35.2%

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

   3. Simplified 35.2%

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

      1. add-cube-cbrt 35.2%

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

      2. pow3 35.2%

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

      3. cbrt-prod 35.2%

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

      4. rem-cbrt-cube 67.3%

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

   5. Applied egg-rr 67.3%

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

   if 1.2e170 < t

   1. Initial program 62.3%

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

      1. associate-/r* 62.3%

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

      2. associate-*l* 62.2%

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

      3. sqr-neg 62.2%

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

      4. associate-*l* 62.3%

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

      5. *-commutative 62.3%

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

      6. sqr-neg 62.3%

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

      7. associate-/r* 62.3%

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

   3. Simplified 62.2%

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

      1. associate-/l/ 62.2%

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

      2. associate-/r/ 62.3%

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

      3. add-cube-cbrt 62.3%

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

      4. pow3 62.3%

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

   5. Applied egg-rr 66.6%

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

   6. Taylor expanded in k around 0 62.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   7. Step-by-step derivation

      1. unpow2 62.2%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 62.2%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. unpow2 62.2%

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

      4. associate-*r* 62.3%

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

      5. *-commutative 62.3%

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

      6. times-frac 80.6%

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot {t}^{3}} \cdot \frac{\ell}{k}} \]

   8. Simplified 80.6%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot {t}^{3}} \cdot \frac{\ell}{k}} \]
   9. Step-by-step derivation

      1. associate-*r/ 80.8%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot {t}^{3}} \cdot \ell}{k}} \]

   10. Applied egg-rr 80.8%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot {t}^{3}} \cdot \ell}{k}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{-100}:\\ \;\;\;\;\frac{2 \cdot \frac{\ell}{\tan k \cdot {t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\sin k}\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{-97}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k} \cdot \frac{2 \cdot \ell}{\tan k \cdot {t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{+170}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot {\left(t \cdot \sqrt[3]{\sin k}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{k \cdot {t}^{3}}}{k}\\ \end{array} \]

Alternative 6: 77.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -1.15 \cdot 10^{-100}:\\ \;\;\;\;\frac{2 \cdot \frac{\ell}{\tan k \cdot {t}^{3}}}{2 + t_1} \cdot \frac{\ell}{\sin k}\\ \mathbf{elif}\;t \leq 7.6 \cdot 10^{-126}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}}{1 + \left(t_1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<= t -1.15e-100)
     (* (/ (* 2.0 (/ l (* (tan k) (pow t 3.0)))) (+ 2.0 t_1)) (/ l (sin k)))
     (if (<= t 7.6e-126)
       (* 2.0 (* (/ (cos k) (* k (* t k))) (/ (* l l) (pow (sin k) 2.0))))
       (/
        (/ 2.0 (* (tan k) (* (sin k) (pow (/ (pow t 1.5) l) 2.0))))
        (+ 1.0 (+ t_1 1.0)))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double tmp;
	if (t <= -1.15e-100) {
		tmp = ((2.0 * (l / (tan(k) * pow(t, 3.0)))) / (2.0 + t_1)) * (l / sin(k));
	} else if (t <= 7.6e-126) {
		tmp = 2.0 * ((cos(k) / (k * (t * k))) * ((l * l) / pow(sin(k), 2.0)));
	} else {
		tmp = (2.0 / (tan(k) * (sin(k) * pow((pow(t, 1.5) / l), 2.0)))) / (1.0 + (t_1 + 1.0));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (k / t) ** 2.0d0
    if (t <= (-1.15d-100)) then
        tmp = ((2.0d0 * (l / (tan(k) * (t ** 3.0d0)))) / (2.0d0 + t_1)) * (l / sin(k))
    else if (t <= 7.6d-126) then
        tmp = 2.0d0 * ((cos(k) / (k * (t * k))) * ((l * l) / (sin(k) ** 2.0d0)))
    else
        tmp = (2.0d0 / (tan(k) * (sin(k) * (((t ** 1.5d0) / l) ** 2.0d0)))) / (1.0d0 + (t_1 + 1.0d0))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double tmp;
	if (t <= -1.15e-100) {
		tmp = ((2.0 * (l / (Math.tan(k) * Math.pow(t, 3.0)))) / (2.0 + t_1)) * (l / Math.sin(k));
	} else if (t <= 7.6e-126) {
		tmp = 2.0 * ((Math.cos(k) / (k * (t * k))) * ((l * l) / Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = (2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow((Math.pow(t, 1.5) / l), 2.0)))) / (1.0 + (t_1 + 1.0));
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	t_1 = math.pow((k / t), 2.0)
	tmp = 0
	if t <= -1.15e-100:
		tmp = ((2.0 * (l / (math.tan(k) * math.pow(t, 3.0)))) / (2.0 + t_1)) * (l / math.sin(k))
	elif t <= 7.6e-126:
		tmp = 2.0 * ((math.cos(k) / (k * (t * k))) * ((l * l) / math.pow(math.sin(k), 2.0)))
	else:
		tmp = (2.0 / (math.tan(k) * (math.sin(k) * math.pow((math.pow(t, 1.5) / l), 2.0)))) / (1.0 + (t_1 + 1.0))
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (t <= -1.15e-100)
		tmp = Float64(Float64(Float64(2.0 * Float64(l / Float64(tan(k) * (t ^ 3.0)))) / Float64(2.0 + t_1)) * Float64(l / sin(k)));
	elseif (t <= 7.6e-126)
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * Float64(t * k))) * Float64(Float64(l * l) / (sin(k) ^ 2.0))));
	else
		tmp = Float64(Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (Float64((t ^ 1.5) / l) ^ 2.0)))) / Float64(1.0 + Float64(t_1 + 1.0)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	t_1 = (k / t) ^ 2.0;
	tmp = 0.0;
	if (t <= -1.15e-100)
		tmp = ((2.0 * (l / (tan(k) * (t ^ 3.0)))) / (2.0 + t_1)) * (l / sin(k));
	elseif (t <= 7.6e-126)
		tmp = 2.0 * ((cos(k) / (k * (t * k))) * ((l * l) / (sin(k) ^ 2.0)));
	else
		tmp = (2.0 / (tan(k) * (sin(k) * (((t ^ 1.5) / l) ^ 2.0)))) / (1.0 + (t_1 + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t, -1.15e-100], N[(N[(N[(2.0 * N[(l / N[(N[Tan[k], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision] * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7.6e-126], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.14999999999999997e-100

   1. Initial program 72.2%

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

      1. associate-/r* 72.2%

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

      2. associate-*l* 69.2%

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

      3. sqr-neg 69.2%

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

      4. associate-*l* 72.2%

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

      5. *-commutative 72.2%

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

      6. sqr-neg 72.2%

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

      7. associate-*l/ 73.2%

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

      8. associate-*r/ 73.0%

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

      9. associate-/r/ 72.9%

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

   3. Simplified 72.9%

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

      1. expm1-log1p-u 55.0%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)\right)\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. expm1-udef 51.9%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)\right)} - 1}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-*l/ 52.0%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}\right)} - 1}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. associate-*r* 52.0%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}\right)} - 1}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 52.0%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}\right)} - 1}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   6. Step-by-step derivation

      1. expm1-def 55.1%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}\right)\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. expm1-log1p 73.0%

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

      3. associate-*r* 73.0%

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

      4. times-frac 83.0%

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

      5. *-commutative 83.0%

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

   7. Simplified 83.0%

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

      1. div-inv 83.0%

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

      2. times-frac 82.9%

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

   9. Applied egg-rr 82.9%

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

       1. associate-*r/ 82.9%

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

       2. *-rgt-identity 82.9%

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

       3. associate-*l/ 85.2%

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

       4. times-frac 85.2%

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

       5. *-commutative 85.2%

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

       6. *-rgt-identity 85.2%

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

       7. associate-*r/ 85.2%

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

       8. associate-*l* 85.2%

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

       9. associate-*r/ 85.2%

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

       10. *-rgt-identity 85.2%

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

       11. *-commutative 85.2%

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

   11. Simplified 85.2%

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

   if -1.14999999999999997e-100 < t < 7.5999999999999997e-126

   1. Initial program 30.2%

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

      1. associate-/r* 30.2%

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

      2. associate-*l* 30.2%

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

      3. sqr-neg 30.2%

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

      4. associate-*l* 30.2%

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

      5. *-commutative 30.2%

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

      6. sqr-neg 30.2%

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

      7. associate-*l/ 30.2%

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

      8. associate-*r/ 30.2%

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

      9. associate-/r/ 30.2%

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

   3. Simplified 30.2%

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

   4. Taylor expanded in k around inf 68.8%

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

      1. *-commutative 68.8%

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

      2. associate-*r* 68.8%

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

      3. times-frac 68.9%

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

      4. unpow2 68.9%

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

      5. associate-*l* 74.3%

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

      6. unpow2 74.3%

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

   6. Simplified 74.3%

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

   if 7.5999999999999997e-126 < t

   1. Initial program 56.0%

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

      1. associate-/r* 56.8%

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

      2. +-commutative 56.8%

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

   3. Simplified 56.8%

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

      1. add-sqr-sqrt 56.8%

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

      2. pow2 56.8%

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

      3. sqrt-div 56.8%

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

      4. sqrt-pow1 63.2%

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

      5. metadata-eval 63.2%

         \[\leadsto \frac{\frac{2}{\left({\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

      6. sqrt-prod 41.0%

         \[\leadsto \frac{\frac{2}{\left({\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \sin k\right) \cdot \tan k}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

      7. add-sqr-sqrt 84.1%

         \[\leadsto \frac{\frac{2}{\left({\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   5. Applied egg-rr 84.1%

      \[\leadsto \frac{\frac{2}{\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \tan k}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{-100}:\\ \;\;\;\;\frac{2 \cdot \frac{\ell}{\tan k \cdot {t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\sin k}\\ \mathbf{elif}\;t \leq 7.6 \cdot 10^{-126}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}\\ \end{array} \]

Alternative 7: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \frac{\ell}{\sin k}\\ t_3 := \tan k \cdot {t}^{3}\\ \mathbf{if}\;t \leq -3.2 \cdot 10^{-104}:\\ \;\;\;\;\frac{2 \cdot \frac{\ell}{t_3}}{t_1} \cdot t_2\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-98}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{t_2 \cdot \frac{2 \cdot \ell}{t_3}}{t_1}\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{+170}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot {\left(\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}\right)}^{3}}}{1 + \left(\frac{k}{t} \cdot \frac{k}{t} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{k \cdot {t}^{3}}}{k}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (+ 2.0 (pow (/ k t) 2.0)))
        (t_2 (/ l (sin k)))
        (t_3 (* (tan k) (pow t 3.0))))
   (if (<= t -3.2e-104)
     (* (/ (* 2.0 (/ l t_3)) t_1) t_2)
     (if (<= t 6e-98)
       (* 2.0 (* (/ (cos k) (* k (* t k))) (/ (* l l) (pow (sin k) 2.0))))
       (if (<= t 5.5e+102)
         (/ (* t_2 (/ (* 2.0 l) t_3)) t_1)
         (if (<= t 8.8e+170)
           (/
            (/ 2.0 (* (tan k) (pow (/ t (cbrt (/ l (/ (sin k) l)))) 3.0)))
            (+ 1.0 (+ (* (/ k t) (/ k t)) 1.0)))
           (/ (* l (/ l (* k (pow t 3.0)))) k)))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double t_1 = 2.0 + pow((k / t), 2.0);
	double t_2 = l / sin(k);
	double t_3 = tan(k) * pow(t, 3.0);
	double tmp;
	if (t <= -3.2e-104) {
		tmp = ((2.0 * (l / t_3)) / t_1) * t_2;
	} else if (t <= 6e-98) {
		tmp = 2.0 * ((cos(k) / (k * (t * k))) * ((l * l) / pow(sin(k), 2.0)));
	} else if (t <= 5.5e+102) {
		tmp = (t_2 * ((2.0 * l) / t_3)) / t_1;
	} else if (t <= 8.8e+170) {
		tmp = (2.0 / (tan(k) * pow((t / cbrt((l / (sin(k) / l)))), 3.0))) / (1.0 + (((k / t) * (k / t)) + 1.0));
	} else {
		tmp = (l * (l / (k * pow(t, 3.0)))) / k;
	}
	return tmp;
}

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double t_1 = 2.0 + Math.pow((k / t), 2.0);
	double t_2 = l / Math.sin(k);
	double t_3 = Math.tan(k) * Math.pow(t, 3.0);
	double tmp;
	if (t <= -3.2e-104) {
		tmp = ((2.0 * (l / t_3)) / t_1) * t_2;
	} else if (t <= 6e-98) {
		tmp = 2.0 * ((Math.cos(k) / (k * (t * k))) * ((l * l) / Math.pow(Math.sin(k), 2.0)));
	} else if (t <= 5.5e+102) {
		tmp = (t_2 * ((2.0 * l) / t_3)) / t_1;
	} else if (t <= 8.8e+170) {
		tmp = (2.0 / (Math.tan(k) * Math.pow((t / Math.cbrt((l / (Math.sin(k) / l)))), 3.0))) / (1.0 + (((k / t) * (k / t)) + 1.0));
	} else {
		tmp = (l * (l / (k * Math.pow(t, 3.0)))) / k;
	}
	return tmp;
}

Julia

l = abs(l)
function code(t, l, k)
	t_1 = Float64(2.0 + (Float64(k / t) ^ 2.0))
	t_2 = Float64(l / sin(k))
	t_3 = Float64(tan(k) * (t ^ 3.0))
	tmp = 0.0
	if (t <= -3.2e-104)
		tmp = Float64(Float64(Float64(2.0 * Float64(l / t_3)) / t_1) * t_2);
	elseif (t <= 6e-98)
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * Float64(t * k))) * Float64(Float64(l * l) / (sin(k) ^ 2.0))));
	elseif (t <= 5.5e+102)
		tmp = Float64(Float64(t_2 * Float64(Float64(2.0 * l) / t_3)) / t_1);
	elseif (t <= 8.8e+170)
		tmp = Float64(Float64(2.0 / Float64(tan(k) * (Float64(t / cbrt(Float64(l / Float64(sin(k) / l)))) ^ 3.0))) / Float64(1.0 + Float64(Float64(Float64(k / t) * Float64(k / t)) + 1.0)));
	else
		tmp = Float64(Float64(l * Float64(l / Float64(k * (t ^ 3.0)))) / k);
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Tan[k], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -3.2e-104], N[(N[(N[(2.0 * N[(l / t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t, 6e-98], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.5e+102], N[(N[(t$95$2 * N[(N[(2.0 * l), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t, 8.8e+170], N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[Power[N[(t / N[Power[N[(l / N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(N[(k / t), $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[(l / N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -3.19999999999999989e-104

   1. Initial program 72.2%

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

      1. associate-/r* 72.2%

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

      2. associate-*l* 69.2%

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

      3. sqr-neg 69.2%

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

      4. associate-*l* 72.2%

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

      5. *-commutative 72.2%

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

      6. sqr-neg 72.2%

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

      7. associate-*l/ 73.2%

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

      8. associate-*r/ 73.0%

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

      9. associate-/r/ 72.9%

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

   3. Simplified 72.9%

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

      1. expm1-log1p-u 55.0%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)\right)\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. expm1-udef 51.9%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)\right)} - 1}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-*l/ 52.0%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}\right)} - 1}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. associate-*r* 52.0%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}\right)} - 1}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 52.0%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}\right)} - 1}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   6. Step-by-step derivation

      1. expm1-def 55.1%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}\right)\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. expm1-log1p 73.0%

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

      3. associate-*r* 73.0%

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

      4. times-frac 83.0%

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

      5. *-commutative 83.0%

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

   7. Simplified 83.0%

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

      1. div-inv 83.0%

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

      2. times-frac 82.9%

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

   9. Applied egg-rr 82.9%

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

       1. associate-*r/ 82.9%

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

       2. *-rgt-identity 82.9%

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

       3. associate-*l/ 85.2%

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

       4. times-frac 85.2%

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

       5. *-commutative 85.2%

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

       6. *-rgt-identity 85.2%

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

       7. associate-*r/ 85.2%

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

       8. associate-*l* 85.2%

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

       9. associate-*r/ 85.2%

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

       10. *-rgt-identity 85.2%

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

       11. *-commutative 85.2%

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

   11. Simplified 85.2%

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

   if -3.19999999999999989e-104 < t < 6e-98

   1. Initial program 31.5%

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

      1. associate-/r* 31.5%

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

      2. associate-*l* 31.5%

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

      3. sqr-neg 31.5%

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

      4. associate-*l* 31.5%

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

      5. *-commutative 31.5%

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

      6. sqr-neg 31.5%

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

      7. associate-*l/ 30.4%

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

      8. associate-*r/ 30.4%

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

      9. associate-/r/ 30.4%

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

   3. Simplified 30.4%

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

   4. Taylor expanded in k around inf 69.2%

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

      1. *-commutative 69.2%

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

      2. associate-*r* 69.2%

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

      3. times-frac 69.3%

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

      4. unpow2 69.3%

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

      5. associate-*l* 74.2%

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

      6. unpow2 74.2%

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

   6. Simplified 74.2%

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

   if 6e-98 < t < 5.49999999999999981e102

   1. Initial program 60.8%

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

      1. associate-/r* 62.8%

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

      2. associate-*l* 62.5%

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

      3. sqr-neg 62.5%

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

      4. associate-*l* 62.8%

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

      5. *-commutative 62.8%

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

      6. sqr-neg 62.8%

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

      7. associate-*l/ 62.6%

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

      8. associate-*r/ 64.9%

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

      9. associate-/r/ 64.8%

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

   3. Simplified 64.8%

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

      1. expm1-log1p-u 63.8%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)\right)\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. expm1-udef 52.4%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)\right)} - 1}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-*l/ 52.4%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}\right)} - 1}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. associate-*r* 52.4%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}\right)} - 1}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 52.4%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}\right)} - 1}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   6. Step-by-step derivation

      1. expm1-def 63.9%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}\right)\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. expm1-log1p 64.9%

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

      3. associate-*r* 64.9%

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

      4. times-frac 87.1%

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

      5. *-commutative 87.1%

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

   7. Simplified 87.1%

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

   if 5.49999999999999981e102 < t < 8.79999999999999955e170

   1. Initial program 33.8%

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

      1. associate-/r* 33.8%

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

      2. +-commutative 33.8%

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

   3. Simplified 33.8%

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

      1. associate-/r/ 33.5%

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

      2. add-cube-cbrt 33.5%

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

      3. pow3 33.5%

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

      4. cbrt-div 33.5%

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

      5. rem-cbrt-cube 63.5%

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

      6. associate-/l* 63.5%

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

   5. Applied egg-rr 63.5%

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

      1. unpow2 63.5%

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

   7. Applied egg-rr 63.5%

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

   if 8.79999999999999955e170 < t

   1. Initial program 64.0%

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

      1. associate-/r* 64.0%

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

      2. associate-*l* 63.8%

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

      3. sqr-neg 63.8%

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

      4. associate-*l* 64.0%

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

      5. *-commutative 64.0%

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

      6. sqr-neg 64.0%

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

      7. associate-/r* 64.0%

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

   3. Simplified 63.8%

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

      1. associate-/l/ 63.8%

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

      2. associate-/r/ 64.0%

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

      3. add-cube-cbrt 64.0%

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

      4. pow3 64.0%

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

   5. Applied egg-rr 65.6%

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

   6. Taylor expanded in k around 0 63.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   7. Step-by-step derivation

      1. unpow2 63.8%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 63.8%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. unpow2 63.8%

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

      4. associate-*r* 64.0%

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

      5. *-commutative 64.0%

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

      6. times-frac 82.8%

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot {t}^{3}} \cdot \frac{\ell}{k}} \]

   8. Simplified 82.8%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot {t}^{3}} \cdot \frac{\ell}{k}} \]
   9. Step-by-step derivation

      1. associate-*r/ 83.0%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot {t}^{3}} \cdot \ell}{k}} \]

   10. Applied egg-rr 83.0%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot {t}^{3}} \cdot \ell}{k}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{-104}:\\ \;\;\;\;\frac{2 \cdot \frac{\ell}{\tan k \cdot {t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\sin k}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-98}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k} \cdot \frac{2 \cdot \ell}{\tan k \cdot {t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{+170}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot {\left(\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}\right)}^{3}}}{1 + \left(\frac{k}{t} \cdot \frac{k}{t} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{k \cdot {t}^{3}}}{k}\\ \end{array} \]

Alternative 8: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{-103} \lor \neg \left(t \leq 2.3 \cdot 10^{-97}\right):\\ \;\;\;\;\frac{2 \cdot \frac{\ell}{\tan k \cdot {t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (or (<= t -2.9e-103) (not (<= t 2.3e-97)))
   (*
    (/ (* 2.0 (/ l (* (tan k) (pow t 3.0)))) (+ 2.0 (pow (/ k t) 2.0)))
    (/ l (sin k)))
   (* 2.0 (* (/ (cos k) (* k (* t k))) (/ (* l l) (pow (sin k) 2.0))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if ((t <= -2.9e-103) || !(t <= 2.3e-97)) {
		tmp = ((2.0 * (l / (tan(k) * pow(t, 3.0)))) / (2.0 + pow((k / t), 2.0))) * (l / sin(k));
	} else {
		tmp = 2.0 * ((cos(k) / (k * (t * k))) * ((l * l) / pow(sin(k), 2.0)));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-2.9d-103)) .or. (.not. (t <= 2.3d-97))) then
        tmp = ((2.0d0 * (l / (tan(k) * (t ** 3.0d0)))) / (2.0d0 + ((k / t) ** 2.0d0))) * (l / sin(k))
    else
        tmp = 2.0d0 * ((cos(k) / (k * (t * k))) * ((l * l) / (sin(k) ** 2.0d0)))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -2.9e-103) || !(t <= 2.3e-97)) {
		tmp = ((2.0 * (l / (Math.tan(k) * Math.pow(t, 3.0)))) / (2.0 + Math.pow((k / t), 2.0))) * (l / Math.sin(k));
	} else {
		tmp = 2.0 * ((Math.cos(k) / (k * (t * k))) * ((l * l) / Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	tmp = 0
	if (t <= -2.9e-103) or not (t <= 2.3e-97):
		tmp = ((2.0 * (l / (math.tan(k) * math.pow(t, 3.0)))) / (2.0 + math.pow((k / t), 2.0))) * (l / math.sin(k))
	else:
		tmp = 2.0 * ((math.cos(k) / (k * (t * k))) * ((l * l) / math.pow(math.sin(k), 2.0)))
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if ((t <= -2.9e-103) || !(t <= 2.3e-97))
		tmp = Float64(Float64(Float64(2.0 * Float64(l / Float64(tan(k) * (t ^ 3.0)))) / Float64(2.0 + (Float64(k / t) ^ 2.0))) * Float64(l / sin(k)));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * Float64(t * k))) * Float64(Float64(l * l) / (sin(k) ^ 2.0))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -2.9e-103) || ~((t <= 2.3e-97)))
		tmp = ((2.0 * (l / (tan(k) * (t ^ 3.0)))) / (2.0 + ((k / t) ^ 2.0))) * (l / sin(k));
	else
		tmp = 2.0 * ((cos(k) / (k * (t * k))) * ((l * l) / (sin(k) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[Or[LessEqual[t, -2.9e-103], N[Not[LessEqual[t, 2.3e-97]], $MachinePrecision]], N[(N[(N[(2.0 * N[(l / N[(N[Tan[k], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.8999999999999999e-103 or 2.29999999999999994e-97 < t

   1. Initial program 64.7%

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

      1. associate-/r* 65.2%

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

      2. associate-*l* 63.5%

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

      3. sqr-neg 63.5%

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

      4. associate-*l* 65.2%

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

      5. *-commutative 65.2%

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

      6. sqr-neg 65.2%

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

      7. associate-*l/ 65.6%

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

      8. associate-*r/ 66.0%

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

      9. associate-/r/ 66.0%

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

   3. Simplified 66.0%

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

      1. expm1-log1p-u 56.6%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)\right)\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. expm1-udef 52.5%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)\right)} - 1}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-*l/ 52.5%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}\right)} - 1}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. associate-*r* 52.5%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}\right)} - 1}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 52.5%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}\right)} - 1}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   6. Step-by-step derivation

      1. expm1-def 56.7%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}\right)\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. expm1-log1p 66.0%

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

      3. associate-*r* 66.0%

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

      4. times-frac 79.4%

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

      5. *-commutative 79.4%

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

   7. Simplified 79.4%

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

      1. div-inv 79.4%

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

      2. times-frac 79.9%

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

   9. Applied egg-rr 79.9%

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

       1. associate-*r/ 79.9%

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

       2. *-rgt-identity 79.9%

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

       3. associate-*l/ 80.8%

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

       4. times-frac 80.3%

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

       5. *-commutative 80.3%

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

       6. *-rgt-identity 80.3%

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

       7. associate-*r/ 80.3%

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

       8. associate-*l* 80.3%

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

       9. associate-*r/ 80.3%

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

       10. *-rgt-identity 80.3%

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

       11. *-commutative 80.3%

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

   11. Simplified 80.3%

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

   if -2.8999999999999999e-103 < t < 2.29999999999999994e-97

   1. Initial program 31.5%

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

      1. associate-/r* 31.5%

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

      2. associate-*l* 31.5%

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

      3. sqr-neg 31.5%

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

      4. associate-*l* 31.5%

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

      5. *-commutative 31.5%

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

      6. sqr-neg 31.5%

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

      7. associate-*l/ 30.4%

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

      8. associate-*r/ 30.4%

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

      9. associate-/r/ 30.4%

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

   3. Simplified 30.4%

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

   4. Taylor expanded in k around inf 69.2%

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

      1. *-commutative 69.2%

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

      2. associate-*r* 69.2%

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

      3. times-frac 69.3%

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

      4. unpow2 69.3%

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

      5. associate-*l* 74.2%

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

      6. unpow2 74.2%

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

   6. Simplified 74.2%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{-103} \lor \neg \left(t \leq 2.3 \cdot 10^{-97}\right):\\ \;\;\;\;\frac{2 \cdot \frac{\ell}{\tan k \cdot {t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 9: 75.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \frac{\ell}{\sin k}\\ t_3 := \tan k \cdot {t}^{3}\\ \mathbf{if}\;t \leq -1.04 \cdot 10^{-100}:\\ \;\;\;\;\frac{2 \cdot \frac{\ell}{t_3}}{t_1} \cdot t_2\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-97}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2 \cdot \frac{2 \cdot \ell}{t_3}}{t_1}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (+ 2.0 (pow (/ k t) 2.0)))
        (t_2 (/ l (sin k)))
        (t_3 (* (tan k) (pow t 3.0))))
   (if (<= t -1.04e-100)
     (* (/ (* 2.0 (/ l t_3)) t_1) t_2)
     (if (<= t 2.1e-97)
       (* 2.0 (* (/ (cos k) (* k (* t k))) (/ (* l l) (pow (sin k) 2.0))))
       (/ (* t_2 (/ (* 2.0 l) t_3)) t_1)))))

C

l = abs(l);
double code(double t, double l, double k) {
	double t_1 = 2.0 + pow((k / t), 2.0);
	double t_2 = l / sin(k);
	double t_3 = tan(k) * pow(t, 3.0);
	double tmp;
	if (t <= -1.04e-100) {
		tmp = ((2.0 * (l / t_3)) / t_1) * t_2;
	} else if (t <= 2.1e-97) {
		tmp = 2.0 * ((cos(k) / (k * (t * k))) * ((l * l) / pow(sin(k), 2.0)));
	} else {
		tmp = (t_2 * ((2.0 * l) / t_3)) / t_1;
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = 2.0d0 + ((k / t) ** 2.0d0)
    t_2 = l / sin(k)
    t_3 = tan(k) * (t ** 3.0d0)
    if (t <= (-1.04d-100)) then
        tmp = ((2.0d0 * (l / t_3)) / t_1) * t_2
    else if (t <= 2.1d-97) then
        tmp = 2.0d0 * ((cos(k) / (k * (t * k))) * ((l * l) / (sin(k) ** 2.0d0)))
    else
        tmp = (t_2 * ((2.0d0 * l) / t_3)) / t_1
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double t_1 = 2.0 + Math.pow((k / t), 2.0);
	double t_2 = l / Math.sin(k);
	double t_3 = Math.tan(k) * Math.pow(t, 3.0);
	double tmp;
	if (t <= -1.04e-100) {
		tmp = ((2.0 * (l / t_3)) / t_1) * t_2;
	} else if (t <= 2.1e-97) {
		tmp = 2.0 * ((Math.cos(k) / (k * (t * k))) * ((l * l) / Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = (t_2 * ((2.0 * l) / t_3)) / t_1;
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	t_1 = 2.0 + math.pow((k / t), 2.0)
	t_2 = l / math.sin(k)
	t_3 = math.tan(k) * math.pow(t, 3.0)
	tmp = 0
	if t <= -1.04e-100:
		tmp = ((2.0 * (l / t_3)) / t_1) * t_2
	elif t <= 2.1e-97:
		tmp = 2.0 * ((math.cos(k) / (k * (t * k))) * ((l * l) / math.pow(math.sin(k), 2.0)))
	else:
		tmp = (t_2 * ((2.0 * l) / t_3)) / t_1
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	t_1 = Float64(2.0 + (Float64(k / t) ^ 2.0))
	t_2 = Float64(l / sin(k))
	t_3 = Float64(tan(k) * (t ^ 3.0))
	tmp = 0.0
	if (t <= -1.04e-100)
		tmp = Float64(Float64(Float64(2.0 * Float64(l / t_3)) / t_1) * t_2);
	elseif (t <= 2.1e-97)
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * Float64(t * k))) * Float64(Float64(l * l) / (sin(k) ^ 2.0))));
	else
		tmp = Float64(Float64(t_2 * Float64(Float64(2.0 * l) / t_3)) / t_1);
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	t_1 = 2.0 + ((k / t) ^ 2.0);
	t_2 = l / sin(k);
	t_3 = tan(k) * (t ^ 3.0);
	tmp = 0.0;
	if (t <= -1.04e-100)
		tmp = ((2.0 * (l / t_3)) / t_1) * t_2;
	elseif (t <= 2.1e-97)
		tmp = 2.0 * ((cos(k) / (k * (t * k))) * ((l * l) / (sin(k) ^ 2.0)));
	else
		tmp = (t_2 * ((2.0 * l) / t_3)) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Tan[k], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.04e-100], N[(N[(N[(2.0 * N[(l / t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t, 2.1e-97], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * N[(N[(2.0 * l), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.04e-100

   1. Initial program 72.2%

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

      1. associate-/r* 72.2%

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

      2. associate-*l* 69.2%

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

      3. sqr-neg 69.2%

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

      4. associate-*l* 72.2%

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

      5. *-commutative 72.2%

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

      6. sqr-neg 72.2%

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

      7. associate-*l/ 73.2%

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

      8. associate-*r/ 73.0%

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

      9. associate-/r/ 72.9%

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

   3. Simplified 72.9%

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

      1. expm1-log1p-u 55.0%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)\right)\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. expm1-udef 51.9%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)\right)} - 1}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-*l/ 52.0%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}\right)} - 1}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. associate-*r* 52.0%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}\right)} - 1}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 52.0%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}\right)} - 1}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   6. Step-by-step derivation

      1. expm1-def 55.1%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}\right)\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. expm1-log1p 73.0%

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

      3. associate-*r* 73.0%

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

      4. times-frac 83.0%

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

      5. *-commutative 83.0%

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

   7. Simplified 83.0%

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

      1. div-inv 83.0%

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

      2. times-frac 82.9%

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

   9. Applied egg-rr 82.9%

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

       1. associate-*r/ 82.9%

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

       2. *-rgt-identity 82.9%

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

       3. associate-*l/ 85.2%

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

       4. times-frac 85.2%

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

       5. *-commutative 85.2%

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

       6. *-rgt-identity 85.2%

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

       7. associate-*r/ 85.2%

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

       8. associate-*l* 85.2%

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

       9. associate-*r/ 85.2%

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

       10. *-rgt-identity 85.2%

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

       11. *-commutative 85.2%

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

   11. Simplified 85.2%

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

   if -1.04e-100 < t < 2.1000000000000001e-97

   1. Initial program 31.5%

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

      1. associate-/r* 31.5%

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

      2. associate-*l* 31.5%

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

      3. sqr-neg 31.5%

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

      4. associate-*l* 31.5%

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

      5. *-commutative 31.5%

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

      6. sqr-neg 31.5%

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

      7. associate-*l/ 30.4%

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

      8. associate-*r/ 30.4%

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

      9. associate-/r/ 30.4%

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

   3. Simplified 30.4%

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

   4. Taylor expanded in k around inf 69.2%

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

      1. *-commutative 69.2%

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

      2. associate-*r* 69.2%

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

      3. times-frac 69.3%

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

      4. unpow2 69.3%

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

      5. associate-*l* 74.2%

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

      6. unpow2 74.2%

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

   6. Simplified 74.2%

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

   if 2.1000000000000001e-97 < t

   1. Initial program 56.9%

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

      1. associate-/r* 57.8%

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

      2. associate-*l* 57.5%

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

      3. sqr-neg 57.5%

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

      4. associate-*l* 57.8%

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

      5. *-commutative 57.8%

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

      6. sqr-neg 57.8%

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

      7. associate-*l/ 57.7%

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

      8. associate-*r/ 58.7%

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

      9. associate-/r/ 58.7%

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

   3. Simplified 58.7%

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

      1. expm1-log1p-u 58.3%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)\right)\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. expm1-udef 53.1%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)\right)} - 1}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-*l/ 53.1%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}\right)} - 1}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. associate-*r* 53.1%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}\right)} - 1}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 53.1%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}\right)} - 1}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   6. Step-by-step derivation

      1. expm1-def 58.3%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}\right)\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. expm1-log1p 58.7%

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

      3. associate-*r* 58.7%

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

      4. times-frac 75.7%

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

      5. *-commutative 75.7%

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

   7. Simplified 75.7%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.04 \cdot 10^{-100}:\\ \;\;\;\;\frac{2 \cdot \frac{\ell}{\tan k \cdot {t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\sin k}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-97}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k} \cdot \frac{2 \cdot \ell}{\tan k \cdot {t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]

Alternative 10: 66.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 8000000000000:\\ \;\;\;\;\frac{\ell}{k \cdot {t}^{3}} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 8000000000000.0)
   (* (/ l (* k (pow t 3.0))) (/ l k))
   (* 2.0 (* (/ (cos k) (* k (* t k))) (/ (* l l) (pow (sin k) 2.0))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 8000000000000.0) {
		tmp = (l / (k * pow(t, 3.0))) * (l / k);
	} else {
		tmp = 2.0 * ((cos(k) / (k * (t * k))) * ((l * l) / pow(sin(k), 2.0)));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
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 <= 8000000000000.0d0) then
        tmp = (l / (k * (t ** 3.0d0))) * (l / k)
    else
        tmp = 2.0d0 * ((cos(k) / (k * (t * k))) * ((l * l) / (sin(k) ** 2.0d0)))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 8000000000000.0) {
		tmp = (l / (k * Math.pow(t, 3.0))) * (l / k);
	} else {
		tmp = 2.0 * ((Math.cos(k) / (k * (t * k))) * ((l * l) / Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	tmp = 0
	if k <= 8000000000000.0:
		tmp = (l / (k * math.pow(t, 3.0))) * (l / k)
	else:
		tmp = 2.0 * ((math.cos(k) / (k * (t * k))) * ((l * l) / math.pow(math.sin(k), 2.0)))
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if (k <= 8000000000000.0)
		tmp = Float64(Float64(l / Float64(k * (t ^ 3.0))) * Float64(l / k));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * Float64(t * k))) * Float64(Float64(l * l) / (sin(k) ^ 2.0))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 8000000000000.0)
		tmp = (l / (k * (t ^ 3.0))) * (l / k);
	else
		tmp = 2.0 * ((cos(k) / (k * (t * k))) * ((l * l) / (sin(k) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 8000000000000.0], N[(N[(l / N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8e12

   1. Initial program 54.2%

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

      1. associate-/r* 54.1%

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

      2. associate-*l* 52.6%

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

      3. sqr-neg 52.6%

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

      4. associate-*l* 54.1%

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

      5. *-commutative 54.1%

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

      6. sqr-neg 54.1%

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

      7. associate-/r* 54.2%

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

   3. Simplified 54.1%

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

      1. associate-/l/ 54.1%

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

      2. associate-/r/ 54.1%

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

      3. add-cube-cbrt 54.1%

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

      4. pow3 54.1%

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

   5. Applied egg-rr 70.3%

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

   6. Taylor expanded in k around 0 55.9%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   7. Step-by-step derivation

      1. unpow2 55.9%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 55.9%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. unpow2 55.9%

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

      4. associate-*r* 57.4%

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

      5. *-commutative 57.4%

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

      6. times-frac 67.3%

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot {t}^{3}} \cdot \frac{\ell}{k}} \]

   8. Simplified 67.3%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot {t}^{3}} \cdot \frac{\ell}{k}} \]

   if 8e12 < k

   1. Initial program 54.9%

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

      1. associate-/r* 56.0%

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

      2. associate-*l* 56.0%

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

      3. sqr-neg 56.0%

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

      4. associate-*l* 56.0%

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

      5. *-commutative 56.0%

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

      6. sqr-neg 56.0%

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

      7. associate-*l/ 56.0%

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

      8. associate-*r/ 56.0%

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

      9. associate-/r/ 56.0%

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

   3. Simplified 56.0%

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

   4. Taylor expanded in k around inf 69.5%

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

      1. *-commutative 69.5%

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

      2. associate-*r* 69.5%

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

      3. times-frac 69.5%

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

      4. unpow2 69.5%

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

      5. associate-*l* 70.9%

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

      6. unpow2 70.9%

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

   6. Simplified 70.9%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8000000000000:\\ \;\;\;\;\frac{\ell}{k \cdot {t}^{3}} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 11: 60.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 4 \cdot 10^{-197}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{\ell}{k \cdot {t}^{3}} \cdot \frac{\ell}{k}\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\frac{{t}^{3}}{\ell}} \cdot \frac{\cos k}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 4e-197)
   (/ (* 2.0 (* (/ l (* k (pow t 3.0))) (/ l k))) (+ 2.0 (pow (/ k t) 2.0)))
   (* (/ l (/ (pow t 3.0) l)) (/ (cos k) (* k k)))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 4e-197) {
		tmp = (2.0 * ((l / (k * pow(t, 3.0))) * (l / k))) / (2.0 + pow((k / t), 2.0));
	} else {
		tmp = (l / (pow(t, 3.0) / l)) * (cos(k) / (k * k));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 4d-197) then
        tmp = (2.0d0 * ((l / (k * (t ** 3.0d0))) * (l / k))) / (2.0d0 + ((k / t) ** 2.0d0))
    else
        tmp = (l / ((t ** 3.0d0) / l)) * (cos(k) / (k * k))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 4e-197) {
		tmp = (2.0 * ((l / (k * Math.pow(t, 3.0))) * (l / k))) / (2.0 + Math.pow((k / t), 2.0));
	} else {
		tmp = (l / (Math.pow(t, 3.0) / l)) * (Math.cos(k) / (k * k));
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	tmp = 0
	if (l * l) <= 4e-197:
		tmp = (2.0 * ((l / (k * math.pow(t, 3.0))) * (l / k))) / (2.0 + math.pow((k / t), 2.0))
	else:
		tmp = (l / (math.pow(t, 3.0) / l)) * (math.cos(k) / (k * k))
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 4e-197)
		tmp = Float64(Float64(2.0 * Float64(Float64(l / Float64(k * (t ^ 3.0))) * Float64(l / k))) / Float64(2.0 + (Float64(k / t) ^ 2.0)));
	else
		tmp = Float64(Float64(l / Float64((t ^ 3.0) / l)) * Float64(cos(k) / Float64(k * k)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 4e-197)
		tmp = (2.0 * ((l / (k * (t ^ 3.0))) * (l / k))) / (2.0 + ((k / t) ^ 2.0));
	else
		tmp = (l / ((t ^ 3.0) / l)) * (cos(k) / (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 4e-197], N[(N[(2.0 * N[(N[(l / N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 3.9999999999999999e-197

   1. Initial program 57.2%

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

      1. associate-/r* 57.2%

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

      2. associate-*l* 55.5%

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

      3. sqr-neg 55.5%

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

      4. associate-*l* 57.2%

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

      5. *-commutative 57.2%

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

      6. sqr-neg 57.2%

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

      7. associate-/r* 57.2%

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

   3. Simplified 58.3%

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

   4. Taylor expanded in k around 0 58.3%

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

      1. unpow2 58.3%

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

   6. Simplified 58.3%

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

   7. Taylor expanded in k around 0 56.6%

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

      1. unpow2 56.6%

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

      2. *-commutative 56.6%

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

      3. unpow2 56.6%

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

      4. associate-*r* 57.9%

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

      5. *-commutative 57.9%

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

      6. times-frac 77.4%

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

   9. Simplified 77.4%

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

   if 3.9999999999999999e-197 < (*.f64 l l)

   1. Initial program 52.9%

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

      1. associate-/r* 53.3%

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

      2. associate-*l* 52.4%

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

      3. sqr-neg 52.4%

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

      4. associate-*l* 53.3%

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

      5. *-commutative 53.3%

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

      6. sqr-neg 53.3%

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

      7. associate-/r* 53.3%

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

   3. Simplified 52.7%

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

      1. associate-/l/ 52.7%

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

      2. associate-/r/ 53.3%

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

      3. add-cube-cbrt 53.3%

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

      4. pow3 53.3%

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

   5. Applied egg-rr 59.1%

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

   6. Taylor expanded in t around inf 55.0%

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

      1. times-frac 55.0%

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{t}^{3}} \cdot \frac{\cos k}{{\sin k}^{2}}} \]

      2. unpow2 55.0%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{t}^{3}} \cdot \frac{\cos k}{{\sin k}^{2}} \]

      3. associate-/l* 61.0%

         \[\leadsto \color{blue}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \frac{\cos k}{{\sin k}^{2}} \]

   8. Simplified 61.0%

      \[\leadsto \color{blue}{\frac{\ell}{\frac{{t}^{3}}{\ell}} \cdot \frac{\cos k}{{\sin k}^{2}}} \]

   9. Taylor expanded in k around 0 62.1%

      \[\leadsto \frac{\ell}{\frac{{t}^{3}}{\ell}} \cdot \frac{\cos k}{\color{blue}{{k}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 62.1%

          \[\leadsto \frac{\ell}{\frac{{t}^{3}}{\ell}} \cdot \frac{\cos k}{\color{blue}{k \cdot k}} \]

   11. Simplified 62.1%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 4 \cdot 10^{-197}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{\ell}{k \cdot {t}^{3}} \cdot \frac{\ell}{k}\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\frac{{t}^{3}}{\ell}} \cdot \frac{\cos k}{k \cdot k}\\ \end{array} \]

Alternative 12: 59.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \frac{\ell}{k \cdot {t}^{3}} \cdot \frac{\ell}{k} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k) :precision binary64 (* (/ l (* k (pow t 3.0))) (/ l k)))

C

l = abs(l);
double code(double t, double l, double k) {
	return (l / (k * pow(t, 3.0))) * (l / k);
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (l / (k * (t ** 3.0d0))) * (l / k)
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	return (l / (k * Math.pow(t, 3.0))) * (l / k);
}

Python

l = abs(l)
def code(t, l, k):
	return (l / (k * math.pow(t, 3.0))) * (l / k)

Julia

l = abs(l)
function code(t, l, k)
	return Float64(Float64(l / Float64(k * (t ^ 3.0))) * Float64(l / k))
end

MATLAB

l = abs(l)
function tmp = code(t, l, k)
	tmp = (l / (k * (t ^ 3.0))) * (l / k);
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := N[(N[(l / N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.3%

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

   1. associate-/r* 54.6%

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

   2. associate-*l* 53.5%

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

   3. sqr-neg 53.5%

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

   4. associate-*l* 54.6%

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

   5. *-commutative 54.6%

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

   6. sqr-neg 54.6%

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

   7. associate-/r* 54.6%

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

3. Simplified 54.6%

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

   1. associate-/l/ 54.6%

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

   2. associate-/r/ 54.6%

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

   3. add-cube-cbrt 54.6%

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

   4. pow3 54.6%

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

5. Applied egg-rr 68.1%

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

6. Taylor expanded in k around 0 55.0%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
7. Step-by-step derivation

   1. unpow2 55.0%

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   2. *-commutative 55.0%

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   3. unpow2 55.0%

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

   4. associate-*r* 56.1%

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

   5. *-commutative 56.1%

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

   6. times-frac 65.3%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot {t}^{3}} \cdot \frac{\ell}{k}} \]

8. Simplified 65.3%

   \[\leadsto \color{blue}{\frac{\ell}{k \cdot {t}^{3}} \cdot \frac{\ell}{k}} \]

9. Final simplification 65.3%

   \[\leadsto \frac{\ell}{k \cdot {t}^{3}} \cdot \frac{\ell}{k} \]

Reproduce

herbie shell --seed 2023272 
(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))))