Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.6% → 87.3%

Time: 22.0s

Alternatives: 22

Speedup: 24.8×

• 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: 54.6% 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: 87.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + t_1\right)\right)} \leq 2 \cdot 10^{+281}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + t_1\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \left(\frac{k}{\ell} \cdot \frac{t \cdot k}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<=
        (/
         2.0
         (*
          (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
          (+ 1.0 (+ 1.0 t_1))))
        2e+281)
     (*
      l
      (*
       l
       (/
        2.0
        (pow (* (cbrt (* (tan k) (+ 2.0 t_1))) (* t (cbrt (sin k)))) 3.0))))
     (/ 2.0 (* (/ (pow (sin k) 2.0) (cos k)) (* (/ k l) (/ (* t k) l)))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * (1.0 + (1.0 + t_1)))) <= 2e+281) {
		tmp = l * (l * (2.0 / pow((cbrt((tan(k) * (2.0 + t_1))) * (t * cbrt(sin(k)))), 3.0)));
	} else {
		tmp = 2.0 / ((pow(sin(k), 2.0) / cos(k)) * ((k / l) * ((t * k) / l)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * (1.0 + (1.0 + t_1)))) <= 2e+281) {
		tmp = l * (l * (2.0 / Math.pow((Math.cbrt((Math.tan(k) * (2.0 + t_1))) * (t * Math.cbrt(Math.sin(k)))), 3.0)));
	} else {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) / Math.cos(k)) * ((k / l) * ((t * k) / l)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(1.0 + Float64(1.0 + t_1)))) <= 2e+281)
		tmp = Float64(l * Float64(l * Float64(2.0 / (Float64(cbrt(Float64(tan(k) * Float64(2.0 + t_1))) * Float64(t * cbrt(sin(k)))) ^ 3.0))));
	else
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) / cos(k)) * Float64(Float64(k / l) * Float64(Float64(t * k) / l))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[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[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+281], N[(l * N[(l * N[(2.0 / N[Power[N[(N[Power[N[(N[Tan[k], $MachinePrecision] * N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(t * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(N[(t * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 2.0000000000000001e281

   1. Initial program 85.2%

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

      1. associate-/l/ 85.2%

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

      2. associate-*l/ 84.4%

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

      3. associate-*l/ 83.7%

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

      4. associate-/r/ 83.7%

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

      5. *-commutative 83.7%

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

      6. associate-/l/ 83.7%

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

      7. associate-*r* 84.4%

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

      8. *-commutative 84.4%

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

      9. associate-*r* 84.4%

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

      10. *-commutative 84.4%

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

   3. Simplified 84.4%

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

      1. add-cube-cbrt 84.1%

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

      2. pow3 84.1%

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

   5. Applied egg-rr 90.3%

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

      1. associate-*r/ 90.3%

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

   7. Applied egg-rr 90.3%

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

      1. unpow2 90.3%

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

      2. associate-*r/ 90.3%

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

      3. unpow2 90.3%

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

      4. associate-*l* 92.1%

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

   9. Simplified 92.1%

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

   if 2.0000000000000001e281 < (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))

   1. Initial program 17.4%

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

      1. *-commutative 17.4%

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

      2. associate-*l* 17.4%

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

      3. associate-*r* 17.4%

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

      4. +-commutative 17.4%

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

      5. associate-+r+ 17.4%

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

      6. metadata-eval 17.4%

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

   3. Simplified 17.4%

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

   4. Taylor expanded in k around inf 49.9%

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

      1. *-commutative 49.9%

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

      2. associate-*l* 49.9%

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

      3. *-commutative 49.9%

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

      4. unpow2 49.9%

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

      5. associate-*l* 58.3%

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

      6. unpow2 58.3%

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

   6. Simplified 58.3%

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

      1. div-inv 58.2%

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

      2. *-commutative 58.2%

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

      3. *-commutative 58.2%

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

   8. Applied egg-rr 58.2%

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

      1. associate-*r/ 58.3%

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

      2. unpow2 58.3%

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

      3. *-rgt-identity 58.3%

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

      4. *-commutative 58.3%

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

      5. associate-/r* 58.3%

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

      6. *-commutative 58.3%

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

      7. associate-*l/ 58.2%

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

      8. associate-*r/ 58.4%

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

      9. associate-/l* 58.6%

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

      10. unpow2 58.6%

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

      11. times-frac 75.7%

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

   10. Simplified 75.7%

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

       1. frac-times 58.6%

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

       2. associate-/l* 58.4%

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

       3. times-frac 79.5%

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

   12. Applied egg-rr 79.5%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq 2 \cdot 10^{+281}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \left(\frac{k}{\ell} \cdot \frac{t \cdot k}{\ell}\right)}\\ \end{array} \]

Alternative 2: 84.1% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0))
        (t_2
         (*
          (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
          (+ 1.0 (+ 1.0 t_1)))))
   (if (<= t_2 (- INFINITY))
     (* (* l l) (/ 2.0 (pow (* (* t (cbrt (sin k))) (cbrt (* 2.0 k))) 3.0)))
     (if (<= t_2 INFINITY)
       (* (/ 2.0 (sin k)) (/ (* l (* l (pow t -3.0))) (* (tan k) (+ 2.0 t_1))))
       (/ 2.0 (* (/ (pow (sin k) 2.0) (cos k)) (* (/ k l) (/ (* t k) l))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double t_2 = (((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * (1.0 + (1.0 + t_1));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (l * l) * (2.0 / pow(((t * cbrt(sin(k))) * cbrt((2.0 * k))), 3.0));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (2.0 / sin(k)) * ((l * (l * pow(t, -3.0))) / (tan(k) * (2.0 + t_1)));
	} else {
		tmp = 2.0 / ((pow(sin(k), 2.0) / cos(k)) * ((k / l) * ((t * k) / l)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double t_2 = (((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * (1.0 + (1.0 + t_1));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = (l * l) * (2.0 / Math.pow(((t * Math.cbrt(Math.sin(k))) * Math.cbrt((2.0 * k))), 3.0));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = (2.0 / Math.sin(k)) * ((l * (l * Math.pow(t, -3.0))) / (Math.tan(k) * (2.0 + t_1)));
	} else {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) / Math.cos(k)) * ((k / l) * ((t * k) / l)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	t_2 = Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(1.0 + Float64(1.0 + t_1)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(l * l) * Float64(2.0 / (Float64(Float64(t * cbrt(sin(k))) * cbrt(Float64(2.0 * k))) ^ 3.0)));
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(2.0 / sin(k)) * Float64(Float64(l * Float64(l * (t ^ -3.0))) / Float64(tan(k) * Float64(2.0 + t_1))));
	else
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) / cos(k)) * Float64(Float64(k / l) * Float64(Float64(t * k) / l))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = 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[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[Power[N[(N[(t * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[(2.0 * k), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(2.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[(l * N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] * N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(N[(t * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) < -inf.0

   1. Initial program 78.7%

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

      1. associate-/l/ 78.7%

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

      2. associate-*l/ 78.7%

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

      3. associate-*l/ 80.5%

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

      4. associate-/r/ 80.5%

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

      5. *-commutative 80.5%

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

      6. associate-/l/ 80.5%

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

      7. associate-*r* 80.5%

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

      8. *-commutative 80.5%

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

      9. associate-*r* 80.5%

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

      10. *-commutative 80.5%

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

   3. Simplified 80.5%

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

      1. add-cube-cbrt 80.5%

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

      2. pow3 80.5%

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

   5. Applied egg-rr 92.3%

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

   6. Taylor expanded in k around 0 92.3%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) < +inf.0

   1. Initial program 90.1%

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

      1. associate-*l* 90.1%

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

      2. associate-/l/ 90.0%

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

      3. *-commutative 90.0%

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

      4. associate-*r/ 89.1%

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

      5. associate-/l* 90.1%

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

      6. associate-/r/ 83.7%

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

   3. Simplified 85.6%

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

      1. expm1-log1p-u 69.7%

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

      2. expm1-udef 63.8%

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

      3. associate-*l/ 63.8%

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

      4. *-commutative 63.8%

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

      5. div-inv 63.8%

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

      6. pow-flip 63.8%

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

      7. metadata-eval 63.8%

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

      8. *-commutative 63.8%

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

   5. Applied egg-rr 63.8%

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

      1. expm1-def 69.7%

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

      2. expm1-log1p 86.4%

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

      3. associate-*r* 86.4%

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

      4. *-commutative 86.4%

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

      5. *-commutative 86.4%

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

      6. times-frac 92.7%

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

   7. Simplified 92.7%

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

   if +inf.0 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))

   1. Initial program 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*l* 0.0%

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

      3. associate-*r* 0.0%

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

      4. +-commutative 0.0%

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

      5. associate-+r+ 0.0%

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

      6. metadata-eval 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in k around inf 39.8%

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

      1. *-commutative 39.8%

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

      2. associate-*l* 39.8%

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

      3. *-commutative 39.8%

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

      4. unpow2 39.8%

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

      5. associate-*l* 50.0%

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

      6. unpow2 50.0%

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

   6. Simplified 50.0%

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

      1. div-inv 50.0%

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

      2. *-commutative 50.0%

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

      3. *-commutative 50.0%

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

   8. Applied egg-rr 50.0%

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

      1. associate-*r/ 50.0%

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

      2. unpow2 50.0%

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

      3. *-rgt-identity 50.0%

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

      4. *-commutative 50.0%

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

      5. associate-/r* 50.0%

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

      6. *-commutative 50.0%

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

      7. associate-*l/ 50.0%

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

      8. associate-*r/ 50.2%

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

      9. associate-/l* 50.3%

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

      10. unpow2 50.3%

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

      11. times-frac 70.3%

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

   10. Simplified 70.3%

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

       1. frac-times 50.3%

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

       2. associate-/l* 50.2%

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

       3. times-frac 75.0%

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

   12. Applied egg-rr 75.0%

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

4. Final simplification 86.1%

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

Alternative 3: 87.1% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<=
        (/
         2.0
         (*
          (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
          (+ 1.0 (+ 1.0 t_1))))
        INFINITY)
     (*
      l
      (* (/ 2.0 (* (tan k) (+ 2.0 t_1))) (/ l (pow (* t (cbrt (sin k))) 3.0))))
     (/ 2.0 (* (/ (pow (sin k) 2.0) (cos k)) (* (/ k l) (/ (* t k) l)))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * (1.0 + (1.0 + t_1)))) <= ((double) INFINITY)) {
		tmp = l * ((2.0 / (tan(k) * (2.0 + t_1))) * (l / pow((t * cbrt(sin(k))), 3.0)));
	} else {
		tmp = 2.0 / ((pow(sin(k), 2.0) / cos(k)) * ((k / l) * ((t * k) / l)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * (1.0 + (1.0 + t_1)))) <= Double.POSITIVE_INFINITY) {
		tmp = l * ((2.0 / (Math.tan(k) * (2.0 + t_1))) * (l / Math.pow((t * Math.cbrt(Math.sin(k))), 3.0)));
	} else {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) / Math.cos(k)) * ((k / l) * ((t * k) / l)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(1.0 + Float64(1.0 + t_1)))) <= Inf)
		tmp = Float64(l * Float64(Float64(2.0 / Float64(tan(k) * Float64(2.0 + t_1))) * Float64(l / (Float64(t * cbrt(sin(k))) ^ 3.0))));
	else
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) / cos(k)) * Float64(Float64(k / l) * Float64(Float64(t * k) / l))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[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[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(l * N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[N[(t * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(N[(t * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < +inf.0

   1. Initial program 86.5%

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

      1. associate-/l/ 86.5%

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

      2. associate-*l/ 85.9%

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

      3. associate-*l/ 85.2%

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

      4. associate-/r/ 85.2%

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

      5. *-commutative 85.2%

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

      6. associate-/l/ 85.2%

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

      7. associate-*r* 85.8%

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

      8. *-commutative 85.8%

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

      9. associate-*r* 85.8%

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

      10. *-commutative 85.8%

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

   3. Simplified 85.8%

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

      1. add-cube-cbrt 85.6%

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

      2. pow3 85.6%

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

   5. Applied egg-rr 91.0%

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

      1. associate-*r/ 91.0%

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

   7. Applied egg-rr 91.0%

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

      1. unpow2 91.0%

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

      2. associate-*r/ 91.0%

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

      3. unpow2 91.0%

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

      4. associate-*l* 92.5%

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

   9. Simplified 92.5%

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

       1. pow1 92.5%

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

   11. Applied egg-rr 92.5%

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

       1. unpow1 92.5%

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

       2. associate-*r/ 92.5%

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

       3. *-commutative 92.5%

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

       4. cube-prod 89.7%

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

       5. rem-cube-cbrt 89.8%

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

       6. times-frac 91.8%

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

   13. Simplified 91.8%

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

   if +inf.0 < (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))

   1. Initial program 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*l* 0.0%

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

      3. associate-*r* 0.0%

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

      4. +-commutative 0.0%

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

      5. associate-+r+ 0.0%

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

      6. metadata-eval 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in k around inf 39.8%

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

      1. *-commutative 39.8%

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

      2. associate-*l* 39.8%

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

      3. *-commutative 39.8%

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

      4. unpow2 39.8%

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

      5. associate-*l* 50.0%

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

      6. unpow2 50.0%

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

   6. Simplified 50.0%

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

      1. div-inv 50.0%

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

      2. *-commutative 50.0%

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

      3. *-commutative 50.0%

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

   8. Applied egg-rr 50.0%

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

      1. associate-*r/ 50.0%

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

      2. unpow2 50.0%

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

      3. *-rgt-identity 50.0%

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

      4. *-commutative 50.0%

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

      5. associate-/r* 50.0%

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

      6. *-commutative 50.0%

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

      7. associate-*l/ 50.0%

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

      8. associate-*r/ 50.2%

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

      9. associate-/l* 50.3%

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

      10. unpow2 50.3%

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

      11. times-frac 70.3%

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

   10. Simplified 70.3%

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

       1. frac-times 50.3%

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

       2. associate-/l* 50.2%

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

       3. times-frac 75.0%

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

   12. Applied egg-rr 75.0%

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

4. Final simplification 85.6%

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

Alternative 4: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\\ \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot t_1} \leq \infty:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \left(\tan k \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \left(\frac{k}{\ell} \cdot \frac{t \cdot k}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))))
   (if (<=
        (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) t_1))
        INFINITY)
     (/ 2.0 (* (* (/ (pow t 3.0) l) (/ (sin k) l)) (* (tan k) t_1)))
     (/ 2.0 (* (/ (pow (sin k) 2.0) (cos k)) (* (/ k l) (/ (* t k) l)))))))

C

double code(double t, double l, double k) {
	double t_1 = 1.0 + (1.0 + pow((k / t), 2.0));
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * t_1)) <= ((double) INFINITY)) {
		tmp = 2.0 / (((pow(t, 3.0) / l) * (sin(k) / l)) * (tan(k) * t_1));
	} else {
		tmp = 2.0 / ((pow(sin(k), 2.0) / cos(k)) * ((k / l) * ((t * k) / l)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = 1.0 + (1.0 + Math.pow((k / t), 2.0));
	double tmp;
	if ((2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * t_1)) <= Double.POSITIVE_INFINITY) {
		tmp = 2.0 / (((Math.pow(t, 3.0) / l) * (Math.sin(k) / l)) * (Math.tan(k) * t_1));
	} else {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) / Math.cos(k)) * ((k / l) * ((t * k) / l)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = 1.0 + (1.0 + math.pow((k / t), 2.0))
	tmp = 0
	if (2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * t_1)) <= math.inf:
		tmp = 2.0 / (((math.pow(t, 3.0) / l) * (math.sin(k) / l)) * (math.tan(k) * t_1))
	else:
		tmp = 2.0 / ((math.pow(math.sin(k), 2.0) / math.cos(k)) * ((k / l) * ((t * k) / l)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0)))
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * t_1)) <= Inf)
		tmp = Float64(2.0 / Float64(Float64(Float64((t ^ 3.0) / l) * Float64(sin(k) / l)) * Float64(tan(k) * t_1)));
	else
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) / cos(k)) * Float64(Float64(k / l) * Float64(Float64(t * k) / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = 1.0 + (1.0 + ((k / t) ^ 2.0));
	tmp = 0.0;
	if ((2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * t_1)) <= Inf)
		tmp = 2.0 / ((((t ^ 3.0) / l) * (sin(k) / l)) * (tan(k) * t_1));
	else
		tmp = 2.0 / (((sin(k) ^ 2.0) / cos(k)) * ((k / l) * ((t * k) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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] * t$95$1), $MachinePrecision]), $MachinePrecision], Infinity], N[(2.0 / N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(N[(t * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < +inf.0

   1. Initial program 86.5%

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

      1. associate-*l* 86.5%

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

      2. +-commutative 86.5%

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

   3. Simplified 86.5%

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

   4. Taylor expanded in t around 0 85.9%

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

      1. *-commutative 85.9%

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

      2. unpow2 85.9%

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

      3. times-frac 90.6%

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

   6. Simplified 90.6%

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

   if +inf.0 < (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))

   1. Initial program 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*l* 0.0%

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

      3. associate-*r* 0.0%

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

      4. +-commutative 0.0%

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

      5. associate-+r+ 0.0%

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

      6. metadata-eval 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in k around inf 39.8%

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

      1. *-commutative 39.8%

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

      2. associate-*l* 39.8%

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

      3. *-commutative 39.8%

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

      4. unpow2 39.8%

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

      5. associate-*l* 50.0%

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

      6. unpow2 50.0%

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

   6. Simplified 50.0%

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

      1. div-inv 50.0%

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

      2. *-commutative 50.0%

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

      3. *-commutative 50.0%

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

   8. Applied egg-rr 50.0%

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

      1. associate-*r/ 50.0%

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

      2. unpow2 50.0%

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

      3. *-rgt-identity 50.0%

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

      4. *-commutative 50.0%

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

      5. associate-/r* 50.0%

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

      6. *-commutative 50.0%

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

      7. associate-*l/ 50.0%

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

      8. associate-*r/ 50.2%

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

      9. associate-/l* 50.3%

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

      10. unpow2 50.3%

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

      11. times-frac 70.3%

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

   10. Simplified 70.3%

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

       1. frac-times 50.3%

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

       2. associate-/l* 50.2%

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

       3. times-frac 75.0%

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

   12. Applied egg-rr 75.0%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq \infty:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \left(\frac{k}{\ell} \cdot \frac{t \cdot k}{\ell}\right)}\\ \end{array} \]

Alternative 5: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{+21}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{\left(\left(t \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{2 \cdot k}\right)}^{3}}\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \left(\frac{k}{\ell} \cdot \frac{t \cdot k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -5.5e+21)
   (* (* l l) (/ 2.0 (pow (* (* t (cbrt (sin k))) (cbrt (* 2.0 k))) 3.0)))
   (if (<= t 4.1e-83)
     (/ 2.0 (* (/ (pow (sin k) 2.0) (cos k)) (* (/ k l) (/ (* t k) l))))
     (/ 2.0 (* (* (/ (pow t 3.0) l) (/ (sin k) l)) (* 2.0 k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -5.5e+21) {
		tmp = (l * l) * (2.0 / pow(((t * cbrt(sin(k))) * cbrt((2.0 * k))), 3.0));
	} else if (t <= 4.1e-83) {
		tmp = 2.0 / ((pow(sin(k), 2.0) / cos(k)) * ((k / l) * ((t * k) / l)));
	} else {
		tmp = 2.0 / (((pow(t, 3.0) / l) * (sin(k) / l)) * (2.0 * k));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -5.5e+21) {
		tmp = (l * l) * (2.0 / Math.pow(((t * Math.cbrt(Math.sin(k))) * Math.cbrt((2.0 * k))), 3.0));
	} else if (t <= 4.1e-83) {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) / Math.cos(k)) * ((k / l) * ((t * k) / l)));
	} else {
		tmp = 2.0 / (((Math.pow(t, 3.0) / l) * (Math.sin(k) / l)) * (2.0 * k));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -5.5e+21)
		tmp = Float64(Float64(l * l) * Float64(2.0 / (Float64(Float64(t * cbrt(sin(k))) * cbrt(Float64(2.0 * k))) ^ 3.0)));
	elseif (t <= 4.1e-83)
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) / cos(k)) * Float64(Float64(k / l) * Float64(Float64(t * k) / l))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((t ^ 3.0) / l) * Float64(sin(k) / l)) * Float64(2.0 * k)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -5.5e+21], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[Power[N[(N[(t * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[(2.0 * k), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.1e-83], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(N[(t * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.5e21

   1. Initial program 70.2%

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

      1. associate-/l/ 70.2%

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

      2. associate-*l/ 70.2%

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

      3. associate-*l/ 71.9%

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

      4. associate-/r/ 71.8%

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

      5. *-commutative 71.8%

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

      6. associate-/l/ 71.8%

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

      7. associate-*r* 71.8%

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

      8. *-commutative 71.8%

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

      9. associate-*r* 71.8%

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

      10. *-commutative 71.8%

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

   3. Simplified 71.8%

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

      1. add-cube-cbrt 71.7%

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

      2. pow3 71.7%

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

   5. Applied egg-rr 84.0%

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

   6. Taylor expanded in k around 0 84.0%

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

   if -5.5e21 < t < 4.1e-83

   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. *-commutative 30.2%

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

      2. associate-*l* 30.2%

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

      3. associate-*r* 30.2%

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

      4. +-commutative 30.2%

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

      5. associate-+r+ 30.2%

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

      6. metadata-eval 30.2%

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

   3. Simplified 30.2%

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

   4. Taylor expanded in k around inf 62.9%

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

      1. *-commutative 62.9%

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

      2. associate-*l* 62.9%

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

      3. *-commutative 62.9%

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

      4. unpow2 62.9%

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

      5. associate-*l* 72.0%

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

      6. unpow2 72.0%

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

   6. Simplified 72.0%

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

      1. div-inv 71.9%

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

      2. *-commutative 71.9%

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

      3. *-commutative 71.9%

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

   8. Applied egg-rr 71.9%

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

      1. associate-*r/ 72.0%

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

      2. unpow2 72.0%

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

      3. *-rgt-identity 72.0%

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

      4. *-commutative 72.0%

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

      5. associate-/r* 71.9%

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

      6. *-commutative 71.9%

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

      7. associate-*l/ 71.9%

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

      8. associate-*r/ 72.1%

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

      9. associate-/l* 72.2%

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

      10. unpow2 72.2%

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

      11. times-frac 87.1%

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

   10. Simplified 87.1%

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

       1. frac-times 72.2%

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

       2. associate-/l* 72.1%

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

       3. times-frac 91.1%

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

   12. Applied egg-rr 91.1%

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

   if 4.1e-83 < t

   1. Initial program 73.2%

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

      1. associate-*l* 73.2%

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

      2. +-commutative 73.2%

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

   3. Simplified 73.2%

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

   4. Taylor expanded in t around 0 72.1%

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

      1. *-commutative 72.1%

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

      2. unpow2 72.1%

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

      3. times-frac 81.0%

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

   6. Simplified 81.0%

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

   7. Taylor expanded in k around 0 76.1%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{+21}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{\left(\left(t \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{2 \cdot k}\right)}^{3}}\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \left(\frac{k}{\ell} \cdot \frac{t \cdot k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \left(2 \cdot k\right)}\\ \end{array} \]

Alternative 6: 67.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := k \cdot \left(t \cdot k\right)\\ \mathbf{if}\;k \leq 9 \cdot 10^{-65}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 2.06 \cdot 10^{-19}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{t}}{{k}^{4}}\\ \mathbf{elif}\;k \leq 4.4 \cdot 10^{-5}:\\ \;\;\;\;\ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot k\right)}\\ \mathbf{elif}\;k \leq 1.15 \cdot 10^{+196}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot t_1\right)}\\ \mathbf{elif}\;k \leq 5.5 \cdot 10^{+270}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t_1}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* k (* t k))))
   (if (<= k 9e-65)
     (/ 2.0 (* (* (/ (pow t 3.0) l) (/ (sin k) l)) (* 2.0 k)))
     (if (<= k 2.06e-19)
       (* 2.0 (/ (* l (/ l t)) (pow k 4.0)))
       (if (<= k 4.4e-5)
         (* l (/ l (* (pow t 3.0) (* k k))))
         (if (<= k 1.15e+196)
           (* (* l l) (/ 2.0 (* (tan k) (* (sin k) t_1))))
           (if (<= k 5.5e+270)
             (*
              2.0
              (* (* (/ l k) (/ l k)) (/ (/ (cos k) t) (pow (sin k) 2.0))))
             (/ 2.0 (* (/ t_1 (* l l)) (* (sin k) (tan k)))))))))))

C

double code(double t, double l, double k) {
	double t_1 = k * (t * k);
	double tmp;
	if (k <= 9e-65) {
		tmp = 2.0 / (((pow(t, 3.0) / l) * (sin(k) / l)) * (2.0 * k));
	} else if (k <= 2.06e-19) {
		tmp = 2.0 * ((l * (l / t)) / pow(k, 4.0));
	} else if (k <= 4.4e-5) {
		tmp = l * (l / (pow(t, 3.0) * (k * k)));
	} else if (k <= 1.15e+196) {
		tmp = (l * l) * (2.0 / (tan(k) * (sin(k) * t_1)));
	} else if (k <= 5.5e+270) {
		tmp = 2.0 * (((l / k) * (l / k)) * ((cos(k) / t) / pow(sin(k), 2.0)));
	} else {
		tmp = 2.0 / ((t_1 / (l * l)) * (sin(k) * tan(k)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = k * (t * k)
    if (k <= 9d-65) then
        tmp = 2.0d0 / ((((t ** 3.0d0) / l) * (sin(k) / l)) * (2.0d0 * k))
    else if (k <= 2.06d-19) then
        tmp = 2.0d0 * ((l * (l / t)) / (k ** 4.0d0))
    else if (k <= 4.4d-5) then
        tmp = l * (l / ((t ** 3.0d0) * (k * k)))
    else if (k <= 1.15d+196) then
        tmp = (l * l) * (2.0d0 / (tan(k) * (sin(k) * t_1)))
    else if (k <= 5.5d+270) then
        tmp = 2.0d0 * (((l / k) * (l / k)) * ((cos(k) / t) / (sin(k) ** 2.0d0)))
    else
        tmp = 2.0d0 / ((t_1 / (l * l)) * (sin(k) * tan(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = k * (t * k);
	double tmp;
	if (k <= 9e-65) {
		tmp = 2.0 / (((Math.pow(t, 3.0) / l) * (Math.sin(k) / l)) * (2.0 * k));
	} else if (k <= 2.06e-19) {
		tmp = 2.0 * ((l * (l / t)) / Math.pow(k, 4.0));
	} else if (k <= 4.4e-5) {
		tmp = l * (l / (Math.pow(t, 3.0) * (k * k)));
	} else if (k <= 1.15e+196) {
		tmp = (l * l) * (2.0 / (Math.tan(k) * (Math.sin(k) * t_1)));
	} else if (k <= 5.5e+270) {
		tmp = 2.0 * (((l / k) * (l / k)) * ((Math.cos(k) / t) / Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = 2.0 / ((t_1 / (l * l)) * (Math.sin(k) * Math.tan(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = k * (t * k)
	tmp = 0
	if k <= 9e-65:
		tmp = 2.0 / (((math.pow(t, 3.0) / l) * (math.sin(k) / l)) * (2.0 * k))
	elif k <= 2.06e-19:
		tmp = 2.0 * ((l * (l / t)) / math.pow(k, 4.0))
	elif k <= 4.4e-5:
		tmp = l * (l / (math.pow(t, 3.0) * (k * k)))
	elif k <= 1.15e+196:
		tmp = (l * l) * (2.0 / (math.tan(k) * (math.sin(k) * t_1)))
	elif k <= 5.5e+270:
		tmp = 2.0 * (((l / k) * (l / k)) * ((math.cos(k) / t) / math.pow(math.sin(k), 2.0)))
	else:
		tmp = 2.0 / ((t_1 / (l * l)) * (math.sin(k) * math.tan(k)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(k * Float64(t * k))
	tmp = 0.0
	if (k <= 9e-65)
		tmp = Float64(2.0 / Float64(Float64(Float64((t ^ 3.0) / l) * Float64(sin(k) / l)) * Float64(2.0 * k)));
	elseif (k <= 2.06e-19)
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l / t)) / (k ^ 4.0)));
	elseif (k <= 4.4e-5)
		tmp = Float64(l * Float64(l / Float64((t ^ 3.0) * Float64(k * k))));
	elseif (k <= 1.15e+196)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(tan(k) * Float64(sin(k) * t_1))));
	elseif (k <= 5.5e+270)
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(Float64(cos(k) / t) / (sin(k) ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(t_1 / Float64(l * l)) * Float64(sin(k) * tan(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = k * (t * k);
	tmp = 0.0;
	if (k <= 9e-65)
		tmp = 2.0 / ((((t ^ 3.0) / l) * (sin(k) / l)) * (2.0 * k));
	elseif (k <= 2.06e-19)
		tmp = 2.0 * ((l * (l / t)) / (k ^ 4.0));
	elseif (k <= 4.4e-5)
		tmp = l * (l / ((t ^ 3.0) * (k * k)));
	elseif (k <= 1.15e+196)
		tmp = (l * l) * (2.0 / (tan(k) * (sin(k) * t_1)));
	elseif (k <= 5.5e+270)
		tmp = 2.0 * (((l / k) * (l / k)) * ((cos(k) / t) / (sin(k) ^ 2.0)));
	else
		tmp = 2.0 / ((t_1 / (l * l)) * (sin(k) * tan(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 9e-65], N[(2.0 / N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.06e-19], N[(2.0 * N[(N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4.4e-5], N[(l * N[(l / N[(N[Power[t, 3.0], $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.15e+196], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 5.5e+270], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$1 / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if k < 8.9999999999999995e-65

   1. Initial program 55.2%

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

      1. associate-*l* 55.2%

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

      2. +-commutative 55.2%

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

   3. Simplified 55.2%

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

   4. Taylor expanded in t around 0 54.6%

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

      1. *-commutative 54.6%

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

      2. unpow2 54.6%

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

      3. times-frac 65.1%

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

   6. Simplified 65.1%

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

   7. Taylor expanded in k around 0 65.6%

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

   if 8.9999999999999995e-65 < k < 2.06e-19

   1. Initial program 30.7%

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

      1. associate-/l/ 30.7%

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

      2. associate-*l/ 30.6%

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

      3. associate-*l/ 21.1%

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

      4. associate-/r/ 21.1%

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

      5. *-commutative 21.1%

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

      6. associate-/l/ 21.1%

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

      7. associate-*r* 30.6%

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

      8. *-commutative 30.6%

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

      9. associate-*r* 30.6%

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

      10. *-commutative 30.6%

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

   3. Simplified 30.6%

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

   4. Taylor expanded in k around inf 40.2%

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

      1. *-commutative 40.2%

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

      2. associate-*l* 40.2%

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

      3. *-commutative 40.2%

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

      4. unpow2 40.2%

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

      5. associate-*l* 40.2%

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

   6. Simplified 40.2%

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

   7. Taylor expanded in k around 0 40.1%

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

      1. unpow2 40.1%

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

      2. *-commutative 40.1%

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

      3. times-frac 68.1%

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

   9. Simplified 68.1%

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

       1. associate-*r/ 50.4%

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

   11. Applied egg-rr 50.4%

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

   if 2.06e-19 < k < 4.3999999999999999e-5

   1. Initial program 66.7%

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

      1. associate-/l/ 66.7%

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

      2. associate-*l/ 66.7%

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

      3. associate-*l/ 66.7%

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

      4. associate-/r/ 66.7%

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

      5. *-commutative 66.7%

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

      6. associate-/l/ 66.7%

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

      7. associate-*r* 66.7%

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

      8. *-commutative 66.7%

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

      9. associate-*r* 66.7%

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

      10. *-commutative 66.7%

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

   3. Simplified 66.7%

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

      1. add-cube-cbrt 66.7%

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

      2. pow3 66.7%

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

   5. Applied egg-rr 66.7%

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

      1. associate-*r/ 66.7%

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

   7. Applied egg-rr 66.7%

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

      1. unpow2 66.7%

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

      2. associate-*r/ 66.7%

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

      3. unpow2 66.7%

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

      4. associate-*l* 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in k around 0 100.0%

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

       1. *-commutative 100.0%

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

       2. unpow2 100.0%

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

   12. Simplified 100.0%

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

   if 4.3999999999999999e-5 < k < 1.1499999999999999e196

   1. Initial program 51.3%

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

      1. associate-/l/ 51.2%

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

      2. associate-*l/ 51.2%

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

      3. associate-*l/ 51.2%

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

      4. associate-/r/ 51.3%

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

      5. *-commutative 51.3%

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

      6. associate-/l/ 51.4%

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

      7. associate-*r* 51.3%

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

      8. *-commutative 51.3%

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

      9. associate-*r* 51.4%

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

      10. *-commutative 51.4%

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

   3. Simplified 51.4%

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

   4. Taylor expanded in k around inf 77.7%

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

      1. *-commutative 77.7%

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

      2. associate-*l* 77.6%

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

      3. *-commutative 77.6%

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

      4. unpow2 77.6%

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

      5. associate-*l* 82.7%

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

   6. Simplified 82.7%

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

   if 1.1499999999999999e196 < k < 5.50000000000000002e270

   1. Initial program 61.9%

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

      1. associate-/l/ 61.9%

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

      2. associate-*l/ 61.9%

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

      3. associate-*l/ 61.9%

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

      4. associate-/r/ 61.9%

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

      5. *-commutative 61.9%

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

      6. associate-/l/ 61.9%

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

      7. associate-*r* 61.9%

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

      8. *-commutative 61.9%

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

      9. associate-*r* 61.9%

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

      10. *-commutative 61.9%

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

   3. Simplified 61.9%

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

      1. add-cube-cbrt 61.9%

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

      2. pow3 61.9%

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

   5. Applied egg-rr 90.5%

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

   6. Taylor expanded in k around inf 90.5%

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

      1. *-commutative 90.5%

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

      2. times-frac 90.5%

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

      3. unpow2 90.5%

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

      4. unpow2 90.5%

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

      5. times-frac 99.9%

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

      6. *-commutative 99.9%

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

      7. associate-/r* 99.9%

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

   8. Simplified 99.9%

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

   if 5.50000000000000002e270 < k

   1. Initial program 66.7%

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

      1. *-commutative 66.7%

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

      2. associate-*l* 66.7%

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

      3. associate-*r* 66.7%

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

      4. +-commutative 66.7%

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

      5. associate-+r+ 66.7%

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

      6. metadata-eval 66.7%

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

   3. Simplified 66.7%

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

   4. Taylor expanded in k around inf 78.6%

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

      1. unpow2 78.6%

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

      2. associate-*l* 100.0%

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

      3. unpow2 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9 \cdot 10^{-65}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 2.06 \cdot 10^{-19}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{t}}{{k}^{4}}\\ \mathbf{elif}\;k \leq 4.4 \cdot 10^{-5}:\\ \;\;\;\;\ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot k\right)}\\ \mathbf{elif}\;k \leq 1.15 \cdot 10^{+196}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}\\ \mathbf{elif}\;k \leq 5.5 \cdot 10^{+270}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(t \cdot k\right)}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]

Alternative 7: 74.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \ell \cdot \left(2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{k}}{t \cdot {\sin k}^{2}}\right)\\ t_2 := \frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{if}\;t \leq -7.2 \cdot 10^{+21}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -4.2 \cdot 10^{-267}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-267}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-83}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1
         (* l (* 2.0 (/ (* (/ l k) (/ (cos k) k)) (* t (pow (sin k) 2.0))))))
        (t_2 (/ 2.0 (* (* (/ (pow t 3.0) l) (/ (sin k) l)) (* 2.0 k)))))
   (if (<= t -7.2e+21)
     t_2
     (if (<= t -4.2e-267)
       t_1
       (if (<= t 3.8e-267)
         (* (* l l) (/ 2.0 (* (tan k) (* (sin k) (* k (* t k))))))
         (if (<= t 4.2e-83) t_1 t_2))))))

C

double code(double t, double l, double k) {
	double t_1 = l * (2.0 * (((l / k) * (cos(k) / k)) / (t * pow(sin(k), 2.0))));
	double t_2 = 2.0 / (((pow(t, 3.0) / l) * (sin(k) / l)) * (2.0 * k));
	double tmp;
	if (t <= -7.2e+21) {
		tmp = t_2;
	} else if (t <= -4.2e-267) {
		tmp = t_1;
	} else if (t <= 3.8e-267) {
		tmp = (l * l) * (2.0 / (tan(k) * (sin(k) * (k * (t * k)))));
	} else if (t <= 4.2e-83) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = l * (2.0d0 * (((l / k) * (cos(k) / k)) / (t * (sin(k) ** 2.0d0))))
    t_2 = 2.0d0 / ((((t ** 3.0d0) / l) * (sin(k) / l)) * (2.0d0 * k))
    if (t <= (-7.2d+21)) then
        tmp = t_2
    else if (t <= (-4.2d-267)) then
        tmp = t_1
    else if (t <= 3.8d-267) then
        tmp = (l * l) * (2.0d0 / (tan(k) * (sin(k) * (k * (t * k)))))
    else if (t <= 4.2d-83) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = l * (2.0 * (((l / k) * (Math.cos(k) / k)) / (t * Math.pow(Math.sin(k), 2.0))));
	double t_2 = 2.0 / (((Math.pow(t, 3.0) / l) * (Math.sin(k) / l)) * (2.0 * k));
	double tmp;
	if (t <= -7.2e+21) {
		tmp = t_2;
	} else if (t <= -4.2e-267) {
		tmp = t_1;
	} else if (t <= 3.8e-267) {
		tmp = (l * l) * (2.0 / (Math.tan(k) * (Math.sin(k) * (k * (t * k)))));
	} else if (t <= 4.2e-83) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = l * (2.0 * (((l / k) * (math.cos(k) / k)) / (t * math.pow(math.sin(k), 2.0))))
	t_2 = 2.0 / (((math.pow(t, 3.0) / l) * (math.sin(k) / l)) * (2.0 * k))
	tmp = 0
	if t <= -7.2e+21:
		tmp = t_2
	elif t <= -4.2e-267:
		tmp = t_1
	elif t <= 3.8e-267:
		tmp = (l * l) * (2.0 / (math.tan(k) * (math.sin(k) * (k * (t * k)))))
	elif t <= 4.2e-83:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(l * Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(cos(k) / k)) / Float64(t * (sin(k) ^ 2.0)))))
	t_2 = Float64(2.0 / Float64(Float64(Float64((t ^ 3.0) / l) * Float64(sin(k) / l)) * Float64(2.0 * k)))
	tmp = 0.0
	if (t <= -7.2e+21)
		tmp = t_2;
	elseif (t <= -4.2e-267)
		tmp = t_1;
	elseif (t <= 3.8e-267)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(tan(k) * Float64(sin(k) * Float64(k * Float64(t * k))))));
	elseif (t <= 4.2e-83)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = l * (2.0 * (((l / k) * (cos(k) / k)) / (t * (sin(k) ^ 2.0))));
	t_2 = 2.0 / ((((t ^ 3.0) / l) * (sin(k) / l)) * (2.0 * k));
	tmp = 0.0;
	if (t <= -7.2e+21)
		tmp = t_2;
	elseif (t <= -4.2e-267)
		tmp = t_1;
	elseif (t <= 3.8e-267)
		tmp = (l * l) * (2.0 / (tan(k) * (sin(k) * (k * (t * k)))));
	elseif (t <= 4.2e-83)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(l * N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 / N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -7.2e+21], t$95$2, If[LessEqual[t, -4.2e-267], t$95$1, If[LessEqual[t, 3.8e-267], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.2e-83], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.2e21 or 4.1999999999999998e-83 < t

   1. Initial program 72.0%

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

      1. associate-*l* 72.0%

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

      2. +-commutative 72.0%

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

   3. Simplified 72.0%

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

   4. Taylor expanded in t around 0 71.4%

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

      1. *-commutative 71.4%

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

      2. unpow2 71.4%

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

      3. times-frac 78.2%

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

   6. Simplified 78.2%

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

   7. Taylor expanded in k around 0 75.2%

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

   if -7.2e21 < t < -4.2000000000000003e-267 or 3.80000000000000003e-267 < t < 4.1999999999999998e-83

   1. Initial program 34.2%

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

      1. associate-/l/ 34.1%

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

      2. associate-*l/ 34.1%

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

      3. associate-*l/ 33.0%

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

      4. associate-/r/ 34.2%

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

      5. *-commutative 34.2%

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

      6. associate-/l/ 34.2%

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

      7. associate-*r* 35.3%

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

      8. *-commutative 35.3%

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

      9. associate-*r* 35.3%

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

      10. *-commutative 35.3%

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

   3. Simplified 35.3%

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

      1. add-cube-cbrt 35.1%

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

      2. pow3 35.2%

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

   5. Applied egg-rr 52.0%

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

      1. associate-*r/ 52.0%

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

   7. Applied egg-rr 52.0%

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

      1. unpow2 52.0%

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

      2. associate-*r/ 52.0%

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

      3. unpow2 52.0%

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

      4. associate-*l* 66.8%

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

   9. Simplified 66.8%

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

   10. Taylor expanded in k around inf 84.3%

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

       1. associate-/r* 85.3%

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

       2. *-commutative 85.3%

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

       3. unpow2 85.3%

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

       4. times-frac 89.7%

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

   12. Simplified 89.7%

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

   if -4.2000000000000003e-267 < t < 3.80000000000000003e-267

   1. Initial program 11.1%

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

      1. associate-/l/ 11.1%

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

      2. associate-*l/ 11.1%

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

      3. associate-*l/ 11.1%

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

      4. associate-/r/ 11.1%

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

      5. *-commutative 11.1%

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

      6. associate-/l/ 11.1%

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

      7. associate-*r* 11.1%

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

      8. *-commutative 11.1%

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

      9. associate-*r* 11.1%

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

      10. *-commutative 11.1%

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

   3. Simplified 11.1%

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

   4. Taylor expanded in k around inf 35.5%

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

      1. *-commutative 35.5%

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

      2. associate-*l* 35.4%

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

      3. *-commutative 35.4%

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

      4. unpow2 35.4%

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

      5. associate-*l* 72.5%

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

   6. Simplified 72.5%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{+21}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;t \leq -4.2 \cdot 10^{-267}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{k}}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-267}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-83}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{k}}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \left(2 \cdot k\right)}\\ \end{array} \]

Alternative 8: 70.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-43}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \left(\frac{k}{\ell} \cdot \frac{t \cdot k}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.5e-43)
   (/ 2.0 (* (* (/ (pow t 3.0) l) (/ (sin k) l)) (* 2.0 k)))
   (/ 2.0 (* (/ (pow (sin k) 2.0) (cos k)) (* (/ k l) (/ (* t k) l))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.5e-43) {
		tmp = 2.0 / (((pow(t, 3.0) / l) * (sin(k) / l)) * (2.0 * k));
	} else {
		tmp = 2.0 / ((pow(sin(k), 2.0) / cos(k)) * ((k / l) * ((t * k) / l)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.5d-43) then
        tmp = 2.0d0 / ((((t ** 3.0d0) / l) * (sin(k) / l)) * (2.0d0 * k))
    else
        tmp = 2.0d0 / (((sin(k) ** 2.0d0) / cos(k)) * ((k / l) * ((t * k) / l)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.5e-43) {
		tmp = 2.0 / (((Math.pow(t, 3.0) / l) * (Math.sin(k) / l)) * (2.0 * k));
	} else {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) / Math.cos(k)) * ((k / l) * ((t * k) / l)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.5e-43:
		tmp = 2.0 / (((math.pow(t, 3.0) / l) * (math.sin(k) / l)) * (2.0 * k))
	else:
		tmp = 2.0 / ((math.pow(math.sin(k), 2.0) / math.cos(k)) * ((k / l) * ((t * k) / l)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.5e-43)
		tmp = Float64(2.0 / Float64(Float64(Float64((t ^ 3.0) / l) * Float64(sin(k) / l)) * Float64(2.0 * k)));
	else
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) / cos(k)) * Float64(Float64(k / l) * Float64(Float64(t * k) / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.5e-43)
		tmp = 2.0 / ((((t ^ 3.0) / l) * (sin(k) / l)) * (2.0 * k));
	else
		tmp = 2.0 / (((sin(k) ^ 2.0) / cos(k)) * ((k / l) * ((t * k) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.5e-43], N[(2.0 / N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(N[(t * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.50000000000000002e-43

   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-*l* 54.3%

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

      2. +-commutative 54.3%

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

   3. Simplified 54.3%

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

   4. Taylor expanded in t around 0 53.7%

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

      1. *-commutative 53.7%

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

      2. unpow2 53.7%

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

      3. times-frac 64.0%

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

   6. Simplified 64.0%

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

   7. Taylor expanded in k around 0 64.9%

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

   if 1.50000000000000002e-43 < k

   1. Initial program 55.8%

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

      1. *-commutative 55.8%

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

      2. associate-*l* 55.8%

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

      3. associate-*r* 55.8%

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

      4. +-commutative 55.8%

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

      5. associate-+r+ 55.8%

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

      6. metadata-eval 55.8%

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

   3. Simplified 55.8%

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

   4. Taylor expanded in k around inf 77.1%

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

      1. *-commutative 77.1%

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

      2. associate-*l* 77.1%

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

      3. *-commutative 77.1%

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

      4. unpow2 77.1%

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

      5. associate-*l* 82.1%

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

      6. unpow2 82.1%

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

   6. Simplified 82.1%

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

      1. div-inv 82.1%

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

      2. *-commutative 82.1%

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

      3. *-commutative 82.1%

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

   8. Applied egg-rr 82.1%

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

      1. associate-*r/ 82.1%

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

      2. unpow2 82.1%

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

      3. *-rgt-identity 82.1%

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

      4. *-commutative 82.1%

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

      5. associate-/r* 82.1%

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

      6. *-commutative 82.1%

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

      7. associate-*l/ 82.1%

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

      8. associate-*r/ 82.1%

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

      9. associate-/l* 82.2%

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

      10. unpow2 82.2%

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

      11. times-frac 79.8%

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

   10. Simplified 79.8%

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

       1. frac-times 82.2%

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

       2. associate-/l* 82.1%

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

       3. times-frac 91.1%

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

   12. Applied egg-rr 91.1%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-43}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \left(\frac{k}{\ell} \cdot \frac{t \cdot k}{\ell}\right)}\\ \end{array} \]

Alternative 9: 66.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{-65}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 1.45 \cdot 10^{-18}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{t}}{{k}^{4}}\\ \mathbf{elif}\;k \leq 6.4 \cdot 10^{-7}:\\ \;\;\;\;\ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 9.5e-65)
   (/ 2.0 (* (* (/ (pow t 3.0) l) (/ (sin k) l)) (* 2.0 k)))
   (if (<= k 1.45e-18)
     (* 2.0 (/ (* l (/ l t)) (pow k 4.0)))
     (if (<= k 6.4e-7)
       (* l (/ l (* (pow t 3.0) (* k k))))
       (* (* l l) (/ 2.0 (* (tan k) (* (sin k) (* k (* t k))))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 9.5e-65) {
		tmp = 2.0 / (((pow(t, 3.0) / l) * (sin(k) / l)) * (2.0 * k));
	} else if (k <= 1.45e-18) {
		tmp = 2.0 * ((l * (l / t)) / pow(k, 4.0));
	} else if (k <= 6.4e-7) {
		tmp = l * (l / (pow(t, 3.0) * (k * k)));
	} else {
		tmp = (l * l) * (2.0 / (tan(k) * (sin(k) * (k * (t * k)))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 9.5d-65) then
        tmp = 2.0d0 / ((((t ** 3.0d0) / l) * (sin(k) / l)) * (2.0d0 * k))
    else if (k <= 1.45d-18) then
        tmp = 2.0d0 * ((l * (l / t)) / (k ** 4.0d0))
    else if (k <= 6.4d-7) then
        tmp = l * (l / ((t ** 3.0d0) * (k * k)))
    else
        tmp = (l * l) * (2.0d0 / (tan(k) * (sin(k) * (k * (t * k)))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 9.5e-65) {
		tmp = 2.0 / (((Math.pow(t, 3.0) / l) * (Math.sin(k) / l)) * (2.0 * k));
	} else if (k <= 1.45e-18) {
		tmp = 2.0 * ((l * (l / t)) / Math.pow(k, 4.0));
	} else if (k <= 6.4e-7) {
		tmp = l * (l / (Math.pow(t, 3.0) * (k * k)));
	} else {
		tmp = (l * l) * (2.0 / (Math.tan(k) * (Math.sin(k) * (k * (t * k)))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 9.5e-65:
		tmp = 2.0 / (((math.pow(t, 3.0) / l) * (math.sin(k) / l)) * (2.0 * k))
	elif k <= 1.45e-18:
		tmp = 2.0 * ((l * (l / t)) / math.pow(k, 4.0))
	elif k <= 6.4e-7:
		tmp = l * (l / (math.pow(t, 3.0) * (k * k)))
	else:
		tmp = (l * l) * (2.0 / (math.tan(k) * (math.sin(k) * (k * (t * k)))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 9.5e-65)
		tmp = Float64(2.0 / Float64(Float64(Float64((t ^ 3.0) / l) * Float64(sin(k) / l)) * Float64(2.0 * k)));
	elseif (k <= 1.45e-18)
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l / t)) / (k ^ 4.0)));
	elseif (k <= 6.4e-7)
		tmp = Float64(l * Float64(l / Float64((t ^ 3.0) * Float64(k * k))));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(tan(k) * Float64(sin(k) * Float64(k * Float64(t * k))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 9.5e-65)
		tmp = 2.0 / ((((t ^ 3.0) / l) * (sin(k) / l)) * (2.0 * k));
	elseif (k <= 1.45e-18)
		tmp = 2.0 * ((l * (l / t)) / (k ^ 4.0));
	elseif (k <= 6.4e-7)
		tmp = l * (l / ((t ^ 3.0) * (k * k)));
	else
		tmp = (l * l) * (2.0 / (tan(k) * (sin(k) * (k * (t * k)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 9.5e-65], N[(2.0 / N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.45e-18], N[(2.0 * N[(N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 6.4e-7], N[(l * N[(l / N[(N[Power[t, 3.0], $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if k < 9.5000000000000004e-65

   1. Initial program 55.2%

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

      1. associate-*l* 55.2%

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

      2. +-commutative 55.2%

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

   3. Simplified 55.2%

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

   4. Taylor expanded in t around 0 54.6%

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

      1. *-commutative 54.6%

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

      2. unpow2 54.6%

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

      3. times-frac 65.1%

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

   6. Simplified 65.1%

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

   7. Taylor expanded in k around 0 65.6%

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

   if 9.5000000000000004e-65 < k < 1.45e-18

   1. Initial program 30.7%

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

      1. associate-/l/ 30.7%

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

      2. associate-*l/ 30.6%

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

      3. associate-*l/ 21.1%

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

      4. associate-/r/ 21.1%

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

      5. *-commutative 21.1%

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

      6. associate-/l/ 21.1%

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

      7. associate-*r* 30.6%

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

      8. *-commutative 30.6%

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

      9. associate-*r* 30.6%

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

      10. *-commutative 30.6%

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

   3. Simplified 30.6%

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

   4. Taylor expanded in k around inf 40.2%

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

      1. *-commutative 40.2%

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

      2. associate-*l* 40.2%

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

      3. *-commutative 40.2%

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

      4. unpow2 40.2%

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

      5. associate-*l* 40.2%

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

   6. Simplified 40.2%

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

   7. Taylor expanded in k around 0 40.1%

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

      1. unpow2 40.1%

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

      2. *-commutative 40.1%

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

      3. times-frac 68.1%

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

   9. Simplified 68.1%

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

       1. associate-*r/ 50.4%

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

   11. Applied egg-rr 50.4%

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

   if 1.45e-18 < k < 6.4000000000000001e-7

   1. Initial program 66.7%

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

      1. associate-/l/ 66.7%

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

      2. associate-*l/ 66.7%

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

      3. associate-*l/ 66.7%

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

      4. associate-/r/ 66.7%

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

      5. *-commutative 66.7%

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

      6. associate-/l/ 66.7%

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

      7. associate-*r* 66.7%

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

      8. *-commutative 66.7%

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

      9. associate-*r* 66.7%

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

      10. *-commutative 66.7%

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

   3. Simplified 66.7%

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

      1. add-cube-cbrt 66.7%

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

      2. pow3 66.7%

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

   5. Applied egg-rr 66.7%

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

      1. associate-*r/ 66.7%

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

   7. Applied egg-rr 66.7%

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

      1. unpow2 66.7%

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

      2. associate-*r/ 66.7%

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

      3. unpow2 66.7%

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

      4. associate-*l* 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in k around 0 100.0%

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

       1. *-commutative 100.0%

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

       2. unpow2 100.0%

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

   12. Simplified 100.0%

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

   if 6.4000000000000001e-7 < k

   1. Initial program 56.5%

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

      1. associate-/l/ 56.5%

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

      2. associate-*l/ 56.5%

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

      3. associate-*l/ 56.5%

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

      4. associate-/r/ 56.5%

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

      5. *-commutative 56.5%

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

      6. associate-/l/ 56.6%

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

      7. associate-*r* 56.5%

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

      8. *-commutative 56.5%

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

      9. associate-*r* 56.6%

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

      10. *-commutative 56.6%

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

   3. Simplified 56.6%

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

   4. Taylor expanded in k around inf 81.7%

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

      1. *-commutative 81.7%

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

      2. associate-*l* 81.6%

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

      3. *-commutative 81.6%

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

      4. unpow2 81.6%

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

      5. associate-*l* 87.3%

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

   6. Simplified 87.3%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{-65}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 1.45 \cdot 10^{-18}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{t}}{{k}^{4}}\\ \mathbf{elif}\;k \leq 6.4 \cdot 10^{-7}:\\ \;\;\;\;\ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}\\ \end{array} \]

Alternative 10: 66.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 7 \cdot 10^{-65}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 2.35 \cdot 10^{-20}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{t}}{{k}^{4}}\\ \mathbf{elif}\;k \leq 0.000165:\\ \;\;\;\;\ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(t \cdot k\right)}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 7e-65)
   (/ 2.0 (* (* (/ (pow t 3.0) l) (/ (sin k) l)) (* 2.0 k)))
   (if (<= k 2.35e-20)
     (* 2.0 (/ (* l (/ l t)) (pow k 4.0)))
     (if (<= k 0.000165)
       (* l (/ l (* (pow t 3.0) (* k k))))
       (/ 2.0 (* (/ (* k (* t k)) (* l l)) (* (sin k) (tan k))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 7e-65) {
		tmp = 2.0 / (((pow(t, 3.0) / l) * (sin(k) / l)) * (2.0 * k));
	} else if (k <= 2.35e-20) {
		tmp = 2.0 * ((l * (l / t)) / pow(k, 4.0));
	} else if (k <= 0.000165) {
		tmp = l * (l / (pow(t, 3.0) * (k * k)));
	} else {
		tmp = 2.0 / (((k * (t * k)) / (l * l)) * (sin(k) * tan(k)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 7d-65) then
        tmp = 2.0d0 / ((((t ** 3.0d0) / l) * (sin(k) / l)) * (2.0d0 * k))
    else if (k <= 2.35d-20) then
        tmp = 2.0d0 * ((l * (l / t)) / (k ** 4.0d0))
    else if (k <= 0.000165d0) then
        tmp = l * (l / ((t ** 3.0d0) * (k * k)))
    else
        tmp = 2.0d0 / (((k * (t * k)) / (l * l)) * (sin(k) * tan(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 7e-65) {
		tmp = 2.0 / (((Math.pow(t, 3.0) / l) * (Math.sin(k) / l)) * (2.0 * k));
	} else if (k <= 2.35e-20) {
		tmp = 2.0 * ((l * (l / t)) / Math.pow(k, 4.0));
	} else if (k <= 0.000165) {
		tmp = l * (l / (Math.pow(t, 3.0) * (k * k)));
	} else {
		tmp = 2.0 / (((k * (t * k)) / (l * l)) * (Math.sin(k) * Math.tan(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 7e-65:
		tmp = 2.0 / (((math.pow(t, 3.0) / l) * (math.sin(k) / l)) * (2.0 * k))
	elif k <= 2.35e-20:
		tmp = 2.0 * ((l * (l / t)) / math.pow(k, 4.0))
	elif k <= 0.000165:
		tmp = l * (l / (math.pow(t, 3.0) * (k * k)))
	else:
		tmp = 2.0 / (((k * (t * k)) / (l * l)) * (math.sin(k) * math.tan(k)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 7e-65)
		tmp = Float64(2.0 / Float64(Float64(Float64((t ^ 3.0) / l) * Float64(sin(k) / l)) * Float64(2.0 * k)));
	elseif (k <= 2.35e-20)
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l / t)) / (k ^ 4.0)));
	elseif (k <= 0.000165)
		tmp = Float64(l * Float64(l / Float64((t ^ 3.0) * Float64(k * k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * Float64(t * k)) / Float64(l * l)) * Float64(sin(k) * tan(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 7e-65)
		tmp = 2.0 / ((((t ^ 3.0) / l) * (sin(k) / l)) * (2.0 * k));
	elseif (k <= 2.35e-20)
		tmp = 2.0 * ((l * (l / t)) / (k ^ 4.0));
	elseif (k <= 0.000165)
		tmp = l * (l / ((t ^ 3.0) * (k * k)));
	else
		tmp = 2.0 / (((k * (t * k)) / (l * l)) * (sin(k) * tan(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 7e-65], N[(2.0 / N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.35e-20], N[(2.0 * N[(N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.000165], N[(l * N[(l / N[(N[Power[t, 3.0], $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if k < 7.00000000000000009e-65

   1. Initial program 55.2%

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

      1. associate-*l* 55.2%

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

      2. +-commutative 55.2%

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

   3. Simplified 55.2%

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

   4. Taylor expanded in t around 0 54.6%

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

      1. *-commutative 54.6%

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

      2. unpow2 54.6%

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

      3. times-frac 65.1%

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

   6. Simplified 65.1%

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

   7. Taylor expanded in k around 0 65.6%

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

   if 7.00000000000000009e-65 < k < 2.35000000000000007e-20

   1. Initial program 30.7%

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

      1. associate-/l/ 30.7%

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

      2. associate-*l/ 30.6%

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

      3. associate-*l/ 21.1%

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

      4. associate-/r/ 21.1%

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

      5. *-commutative 21.1%

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

      6. associate-/l/ 21.1%

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

      7. associate-*r* 30.6%

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

      8. *-commutative 30.6%

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

      9. associate-*r* 30.6%

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

      10. *-commutative 30.6%

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

   3. Simplified 30.6%

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

   4. Taylor expanded in k around inf 40.2%

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

      1. *-commutative 40.2%

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

      2. associate-*l* 40.2%

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

      3. *-commutative 40.2%

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

      4. unpow2 40.2%

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

      5. associate-*l* 40.2%

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

   6. Simplified 40.2%

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

   7. Taylor expanded in k around 0 40.1%

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

      1. unpow2 40.1%

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

      2. *-commutative 40.1%

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

      3. times-frac 68.1%

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

   9. Simplified 68.1%

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

       1. associate-*r/ 50.4%

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

   11. Applied egg-rr 50.4%

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

   if 2.35000000000000007e-20 < k < 1.65e-4

   1. Initial program 66.7%

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

      1. associate-/l/ 66.7%

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

      2. associate-*l/ 66.7%

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

      3. associate-*l/ 66.7%

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

      4. associate-/r/ 66.7%

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

      5. *-commutative 66.7%

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

      6. associate-/l/ 66.7%

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

      7. associate-*r* 66.7%

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

      8. *-commutative 66.7%

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

      9. associate-*r* 66.7%

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

      10. *-commutative 66.7%

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

   3. Simplified 66.7%

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

      1. add-cube-cbrt 66.7%

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

      2. pow3 66.7%

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

   5. Applied egg-rr 66.7%

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

      1. associate-*r/ 66.7%

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

   7. Applied egg-rr 66.7%

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

      1. unpow2 66.7%

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

      2. associate-*r/ 66.7%

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

      3. unpow2 66.7%

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

      4. associate-*l* 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in k around 0 100.0%

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

       1. *-commutative 100.0%

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

       2. unpow2 100.0%

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

   12. Simplified 100.0%

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

   if 1.65e-4 < k

   1. Initial program 56.5%

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

      1. *-commutative 56.5%

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

      2. associate-*l* 56.5%

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

      3. associate-*r* 56.5%

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

      4. +-commutative 56.5%

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

      5. associate-+r+ 56.5%

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

      6. metadata-eval 56.5%

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

   3. Simplified 56.5%

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

   4. Taylor expanded in k around inf 81.6%

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

      1. unpow2 81.6%

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

      2. associate-*l* 87.2%

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

      3. unpow2 87.2%

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

   6. Simplified 87.2%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7 \cdot 10^{-65}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 2.35 \cdot 10^{-20}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{t}}{{k}^{4}}\\ \mathbf{elif}\;k \leq 0.000165:\\ \;\;\;\;\ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(t \cdot k\right)}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]

Alternative 11: 68.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.7 \cdot 10^{-38} \lor \neg \left(t \leq 6.6 \cdot 10^{-84}\right):\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{k}{\frac{\ell}{k} \cdot \frac{\ell}{t}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -1.7e-38) (not (<= t 6.6e-84)))
   (/ 2.0 (* (* (/ (pow t 3.0) l) (/ (sin k) l)) (* 2.0 k)))
   (/ 2.0 (* (/ (* k k) (cos k)) (/ k (* (/ l k) (/ l t)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.7e-38) || !(t <= 6.6e-84)) {
		tmp = 2.0 / (((pow(t, 3.0) / l) * (sin(k) / l)) * (2.0 * k));
	} else {
		tmp = 2.0 / (((k * k) / cos(k)) * (k / ((l / k) * (l / t))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-1.7d-38)) .or. (.not. (t <= 6.6d-84))) then
        tmp = 2.0d0 / ((((t ** 3.0d0) / l) * (sin(k) / l)) * (2.0d0 * k))
    else
        tmp = 2.0d0 / (((k * k) / cos(k)) * (k / ((l / k) * (l / t))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.7e-38) || !(t <= 6.6e-84)) {
		tmp = 2.0 / (((Math.pow(t, 3.0) / l) * (Math.sin(k) / l)) * (2.0 * k));
	} else {
		tmp = 2.0 / (((k * k) / Math.cos(k)) * (k / ((l / k) * (l / t))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -1.7e-38) or not (t <= 6.6e-84):
		tmp = 2.0 / (((math.pow(t, 3.0) / l) * (math.sin(k) / l)) * (2.0 * k))
	else:
		tmp = 2.0 / (((k * k) / math.cos(k)) * (k / ((l / k) * (l / t))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -1.7e-38) || !(t <= 6.6e-84))
		tmp = Float64(2.0 / Float64(Float64(Float64((t ^ 3.0) / l) * Float64(sin(k) / l)) * Float64(2.0 * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) / cos(k)) * Float64(k / Float64(Float64(l / k) * Float64(l / t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -1.7e-38) || ~((t <= 6.6e-84)))
		tmp = 2.0 / ((((t ^ 3.0) / l) * (sin(k) / l)) * (2.0 * k));
	else
		tmp = 2.0 / (((k * k) / cos(k)) * (k / ((l / k) * (l / t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -1.7e-38], N[Not[LessEqual[t, 6.6e-84]], $MachinePrecision]], N[(2.0 / N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(k / N[(N[(l / k), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.7000000000000001e-38 or 6.59999999999999968e-84 < t

   1. Initial program 71.7%

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

      1. associate-*l* 71.7%

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

      2. +-commutative 71.7%

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

   3. Simplified 71.7%

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

   4. Taylor expanded in t around 0 71.1%

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

      1. *-commutative 71.1%

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

      2. unpow2 71.1%

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

      3. times-frac 78.0%

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

   6. Simplified 78.0%

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

   7. Taylor expanded in k around 0 74.1%

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

   if -1.7000000000000001e-38 < t < 6.59999999999999968e-84

   1. Initial program 25.5%

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

      1. *-commutative 25.5%

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

      2. associate-*l* 25.5%

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

      3. associate-*r* 25.5%

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

      4. +-commutative 25.5%

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

      5. associate-+r+ 25.5%

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

      6. metadata-eval 25.5%

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

   3. Simplified 25.5%

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

   4. Taylor expanded in k around inf 62.7%

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

      1. *-commutative 62.7%

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

      2. associate-*l* 62.7%

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

      3. *-commutative 62.7%

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

      4. unpow2 62.7%

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

      5. associate-*l* 72.9%

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

      6. unpow2 72.9%

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

   6. Simplified 72.9%

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

      1. div-inv 72.8%

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

      2. *-commutative 72.8%

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

      3. *-commutative 72.8%

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

   8. Applied egg-rr 72.8%

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

      1. associate-*r/ 72.9%

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

      2. unpow2 72.9%

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

      3. *-rgt-identity 72.9%

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

      4. *-commutative 72.9%

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

      5. associate-/r* 72.8%

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

      6. *-commutative 72.8%

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

      7. associate-*l/ 72.8%

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

      8. associate-*r/ 73.0%

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

      9. associate-/l* 73.0%

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

      10. unpow2 73.0%

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

      11. times-frac 88.7%

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

   10. Simplified 88.7%

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

   11. Taylor expanded in k around 0 73.1%

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

       1. unpow2 53.2%

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

   13. Simplified 73.1%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.7 \cdot 10^{-38} \lor \neg \left(t \leq 6.6 \cdot 10^{-84}\right):\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{k}{\frac{\ell}{k} \cdot \frac{\ell}{t}}}\\ \end{array} \]

Alternative 12: 67.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.28 \cdot 10^{-18}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(2 \cdot \left({t}^{3} \cdot k\right)\right)}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{k}{\frac{\ell}{k} \cdot \frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -1.28e-18)
   (* (* l l) (/ 2.0 (* (tan k) (* 2.0 (* (pow t 3.0) k)))))
   (if (<= t 1.6e-83)
     (/ 2.0 (* (/ (* k k) (cos k)) (/ k (* (/ l k) (/ l t)))))
     (/ (* (/ l k) (/ l k)) (pow t 3.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.28e-18) {
		tmp = (l * l) * (2.0 / (tan(k) * (2.0 * (pow(t, 3.0) * k))));
	} else if (t <= 1.6e-83) {
		tmp = 2.0 / (((k * k) / cos(k)) * (k / ((l / k) * (l / t))));
	} else {
		tmp = ((l / k) * (l / k)) / pow(t, 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-1.28d-18)) then
        tmp = (l * l) * (2.0d0 / (tan(k) * (2.0d0 * ((t ** 3.0d0) * k))))
    else if (t <= 1.6d-83) then
        tmp = 2.0d0 / (((k * k) / cos(k)) * (k / ((l / k) * (l / t))))
    else
        tmp = ((l / k) * (l / k)) / (t ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.28e-18) {
		tmp = (l * l) * (2.0 / (Math.tan(k) * (2.0 * (Math.pow(t, 3.0) * k))));
	} else if (t <= 1.6e-83) {
		tmp = 2.0 / (((k * k) / Math.cos(k)) * (k / ((l / k) * (l / t))));
	} else {
		tmp = ((l / k) * (l / k)) / Math.pow(t, 3.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -1.28e-18:
		tmp = (l * l) * (2.0 / (math.tan(k) * (2.0 * (math.pow(t, 3.0) * k))))
	elif t <= 1.6e-83:
		tmp = 2.0 / (((k * k) / math.cos(k)) * (k / ((l / k) * (l / t))))
	else:
		tmp = ((l / k) * (l / k)) / math.pow(t, 3.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -1.28e-18)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(tan(k) * Float64(2.0 * Float64((t ^ 3.0) * k)))));
	elseif (t <= 1.6e-83)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) / cos(k)) * Float64(k / Float64(Float64(l / k) * Float64(l / t)))));
	else
		tmp = Float64(Float64(Float64(l / k) * Float64(l / k)) / (t ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -1.28e-18)
		tmp = (l * l) * (2.0 / (tan(k) * (2.0 * ((t ^ 3.0) * k))));
	elseif (t <= 1.6e-83)
		tmp = 2.0 / (((k * k) / cos(k)) * (k / ((l / k) * (l / t))));
	else
		tmp = ((l / k) * (l / k)) / (t ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -1.28e-18], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(2.0 * N[(N[Power[t, 3.0], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.6e-83], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(k / N[(N[(l / k), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.27999999999999993e-18

   1. Initial program 70.9%

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

      1. associate-/l/ 70.9%

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

      2. associate-*l/ 70.9%

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

      3. associate-*l/ 72.4%

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

      4. associate-/r/ 72.3%

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

      5. *-commutative 72.3%

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

      6. associate-/l/ 72.3%

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

      7. associate-*r* 72.3%

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

      8. *-commutative 72.3%

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

      9. associate-*r* 72.3%

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

      10. *-commutative 72.3%

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

   3. Simplified 72.3%

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

   4. Taylor expanded in k around 0 69.4%

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

   if -1.27999999999999993e-18 < t < 1.6000000000000001e-83

   1. Initial program 26.5%

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

      1. *-commutative 26.5%

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

      2. associate-*l* 26.5%

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

      3. associate-*r* 26.5%

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

      4. +-commutative 26.5%

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

      5. associate-+r+ 26.5%

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

      6. metadata-eval 26.5%

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

   3. Simplified 26.5%

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

   4. Taylor expanded in k around inf 61.9%

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

      1. *-commutative 61.9%

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

      2. associate-*l* 62.0%

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

      3. *-commutative 62.0%

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

      4. unpow2 62.0%

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

      5. associate-*l* 71.7%

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

      6. unpow2 71.7%

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

   6. Simplified 71.7%

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

      1. div-inv 71.6%

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

      2. *-commutative 71.6%

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

      3. *-commutative 71.6%

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

   8. Applied egg-rr 71.6%

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

      1. associate-*r/ 71.7%

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

      2. unpow2 71.7%

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

      3. *-rgt-identity 71.7%

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

      4. *-commutative 71.7%

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

      5. associate-/r* 71.7%

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

      6. *-commutative 71.7%

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

      7. associate-*l/ 71.7%

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

      8. associate-*r/ 71.8%

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

      9. associate-/l* 71.9%

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

      10. unpow2 71.9%

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

      11. times-frac 88.0%

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

   10. Simplified 88.0%

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

   11. Taylor expanded in k around 0 72.0%

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

       1. unpow2 52.8%

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

   13. Simplified 72.0%

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

   if 1.6000000000000001e-83 < t

   1. Initial program 73.2%

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

      1. associate-/l/ 73.2%

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

      2. associate-*l/ 72.1%

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

      3. associate-*l/ 71.0%

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

      4. associate-/r/ 71.0%

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

      5. *-commutative 71.0%

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

      6. associate-/l/ 71.0%

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

      7. associate-*r* 71.0%

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

      8. *-commutative 71.0%

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

      9. associate-*r* 71.0%

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

      10. *-commutative 71.0%

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

   3. Simplified 71.0%

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

      1. add-cube-cbrt 70.8%

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

      2. pow3 70.8%

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

   5. Applied egg-rr 73.9%

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

   6. Taylor expanded in k around 0 61.7%

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

      1. associate-/r* 64.7%

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

      2. unpow2 64.7%

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

      3. unpow2 64.7%

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

      4. times-frac 73.6%

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

   8. Simplified 73.6%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.28 \cdot 10^{-18}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(2 \cdot \left({t}^{3} \cdot k\right)\right)}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{k}{\frac{\ell}{k} \cdot \frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\\ \end{array} \]

Alternative 13: 68.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.6 \cdot 10^{-39} \lor \neg \left(t \leq 7.9 \cdot 10^{-84}\right):\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{k}{\frac{\ell}{k} \cdot \frac{\ell}{t}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -7.6e-39) (not (<= t 7.9e-84)))
   (/ (* (/ l k) (/ l k)) (pow t 3.0))
   (/ 2.0 (* (/ (* k k) (cos k)) (/ k (* (/ l k) (/ l t)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -7.6e-39) || !(t <= 7.9e-84)) {
		tmp = ((l / k) * (l / k)) / pow(t, 3.0);
	} else {
		tmp = 2.0 / (((k * k) / cos(k)) * (k / ((l / k) * (l / t))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-7.6d-39)) .or. (.not. (t <= 7.9d-84))) then
        tmp = ((l / k) * (l / k)) / (t ** 3.0d0)
    else
        tmp = 2.0d0 / (((k * k) / cos(k)) * (k / ((l / k) * (l / t))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -7.6e-39) || !(t <= 7.9e-84)) {
		tmp = ((l / k) * (l / k)) / Math.pow(t, 3.0);
	} else {
		tmp = 2.0 / (((k * k) / Math.cos(k)) * (k / ((l / k) * (l / t))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -7.6e-39) or not (t <= 7.9e-84):
		tmp = ((l / k) * (l / k)) / math.pow(t, 3.0)
	else:
		tmp = 2.0 / (((k * k) / math.cos(k)) * (k / ((l / k) * (l / t))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -7.6e-39) || !(t <= 7.9e-84))
		tmp = Float64(Float64(Float64(l / k) * Float64(l / k)) / (t ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) / cos(k)) * Float64(k / Float64(Float64(l / k) * Float64(l / t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -7.6e-39) || ~((t <= 7.9e-84)))
		tmp = ((l / k) * (l / k)) / (t ^ 3.0);
	else
		tmp = 2.0 / (((k * k) / cos(k)) * (k / ((l / k) * (l / t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -7.6e-39], N[Not[LessEqual[t, 7.9e-84]], $MachinePrecision]], N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(k / N[(N[(l / k), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -7.6000000000000004e-39 or 7.89999999999999991e-84 < t

   1. Initial program 71.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 71.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 71.1%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 71.0%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 71.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 71.0%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 71.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 71.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 71.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 71.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 71.0%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 71.0%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 70.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)} \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right) \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}} \]

      2. pow3 70.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)}^{3}}} \]

   5. Applied egg-rr 76.9%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]

   6. Taylor expanded in k around 0 59.1%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   7. Step-by-step derivation

      1. associate-/r* 60.1%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}}} \]

      2. unpow2 60.1%

         \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{{t}^{3}} \]

      3. unpow2 60.1%

         \[\leadsto \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{{t}^{3}} \]

      4. times-frac 71.3%

         \[\leadsto \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{{t}^{3}} \]

   8. Simplified 71.3%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}} \]

   if -7.6000000000000004e-39 < t < 7.89999999999999991e-84

   1. Initial program 25.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 25.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 25.5%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 25.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 25.5%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 25.5%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 25.5%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 25.5%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   4. Taylor expanded in k around inf 62.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}}} \]
   5. Step-by-step derivation

      1. *-commutative 62.7%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left({\sin k}^{2} \cdot t\right) \cdot {k}^{2}}}{\cos k \cdot {\ell}^{2}}} \]

      2. associate-*l* 62.7%

         \[\leadsto \frac{2}{\frac{\color{blue}{{\sin k}^{2} \cdot \left(t \cdot {k}^{2}\right)}}{\cos k \cdot {\ell}^{2}}} \]

      3. *-commutative 62.7%

         \[\leadsto \frac{2}{\frac{{\sin k}^{2} \cdot \color{blue}{\left({k}^{2} \cdot t\right)}}{\cos k \cdot {\ell}^{2}}} \]

      4. unpow2 62.7%

         \[\leadsto \frac{2}{\frac{{\sin k}^{2} \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)}{\cos k \cdot {\ell}^{2}}} \]

      5. associate-*l* 72.9%

         \[\leadsto \frac{2}{\frac{{\sin k}^{2} \cdot \color{blue}{\left(k \cdot \left(k \cdot t\right)\right)}}{\cos k \cdot {\ell}^{2}}} \]

      6. unpow2 72.9%

         \[\leadsto \frac{2}{\frac{{\sin k}^{2} \cdot \left(k \cdot \left(k \cdot t\right)\right)}{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \]

   6. Simplified 72.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{{\sin k}^{2} \cdot \left(k \cdot \left(k \cdot t\right)\right)}{\cos k \cdot \left(\ell \cdot \ell\right)}}} \]
   7. Step-by-step derivation

      1. div-inv 72.8%

         \[\leadsto \frac{2}{\color{blue}{\left({\sin k}^{2} \cdot \left(k \cdot \left(k \cdot t\right)\right)\right) \cdot \frac{1}{\cos k \cdot \left(\ell \cdot \ell\right)}}} \]

      2. *-commutative 72.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot \left(k \cdot t\right)\right) \cdot {\sin k}^{2}\right)} \cdot \frac{1}{\cos k \cdot \left(\ell \cdot \ell\right)}} \]

      3. *-commutative 72.8%

         \[\leadsto \frac{2}{\left(\left(k \cdot \left(k \cdot t\right)\right) \cdot {\sin k}^{2}\right) \cdot \frac{1}{\color{blue}{\left(\ell \cdot \ell\right) \cdot \cos k}}} \]

   8. Applied egg-rr 72.8%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot \left(k \cdot t\right)\right) \cdot {\sin k}^{2}\right) \cdot \frac{1}{\left(\ell \cdot \ell\right) \cdot \cos k}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 72.9%

         \[\leadsto \frac{2}{\color{blue}{\frac{\left(\left(k \cdot \left(k \cdot t\right)\right) \cdot {\sin k}^{2}\right) \cdot 1}{\left(\ell \cdot \ell\right) \cdot \cos k}}} \]

      2. unpow2 72.9%

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot \left(k \cdot t\right)\right) \cdot {\sin k}^{2}\right) \cdot 1}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. *-rgt-identity 72.9%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot \left(k \cdot t\right)\right) \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

      4. *-commutative 72.9%

         \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot t\right)\right) \cdot {\sin k}^{2}}{\color{blue}{\cos k \cdot {\ell}^{2}}}} \]

      5. associate-/r* 72.8%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{\left(k \cdot \left(k \cdot t\right)\right) \cdot {\sin k}^{2}}{\cos k}}{{\ell}^{2}}}} \]

      6. *-commutative 72.8%

         \[\leadsto \frac{2}{\frac{\frac{\color{blue}{{\sin k}^{2} \cdot \left(k \cdot \left(k \cdot t\right)\right)}}{\cos k}}{{\ell}^{2}}} \]

      7. associate-*l/ 72.8%

         \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{\sin k}^{2}}{\cos k} \cdot \left(k \cdot \left(k \cdot t\right)\right)}}{{\ell}^{2}}} \]

      8. associate-*r/ 73.0%

         \[\leadsto \frac{2}{\color{blue}{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{k \cdot \left(k \cdot t\right)}{{\ell}^{2}}}} \]

      9. associate-/l* 73.0%

         \[\leadsto \frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{k}{\frac{{\ell}^{2}}{k \cdot t}}}} \]

      10. unpow2 73.0%

          \[\leadsto \frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{k}{\frac{\color{blue}{\ell \cdot \ell}}{k \cdot t}}} \]

      11. times-frac 88.7%

          \[\leadsto \frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{k}{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{t}}}} \]

   10. Simplified 88.7%

       \[\leadsto \frac{2}{\color{blue}{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{k}{\frac{\ell}{k} \cdot \frac{\ell}{t}}}} \]

   11. Taylor expanded in k around 0 73.1%

       \[\leadsto \frac{2}{\frac{\color{blue}{{k}^{2}}}{\cos k} \cdot \frac{k}{\frac{\ell}{k} \cdot \frac{\ell}{t}}} \]
   12. Step-by-step derivation

       1. unpow2 53.2%

          \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot \left(k \cdot \left(k \cdot t\right)\right)}{\cos k \cdot \left(\ell \cdot \ell\right)}} \]

   13. Simplified 73.1%

       \[\leadsto \frac{2}{\frac{\color{blue}{k \cdot k}}{\cos k} \cdot \frac{k}{\frac{\ell}{k} \cdot \frac{\ell}{t}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.6 \cdot 10^{-39} \lor \neg \left(t \leq 7.9 \cdot 10^{-84}\right):\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{k}{\frac{\ell}{k} \cdot \frac{\ell}{t}}}\\ \end{array} \]

Alternative 14: 62.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot k\right)}\\ \mathbf{if}\;t \leq -4.2 \cdot 10^{+182}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -5.8 \cdot 10^{+102}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \frac{\ell}{k}}{k} \cdot \frac{1}{k \cdot \left(t \cdot k\right)}\right)\\ \mathbf{elif}\;t \leq -1.4 \cdot 10^{-26}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-83}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k \cdot k}}{{t}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* l (/ l (* (pow t 3.0) (* k k))))))
   (if (<= t -4.2e+182)
     t_1
     (if (<= t -5.8e+102)
       (* 2.0 (* (/ (* l (/ l k)) k) (/ 1.0 (* k (* t k)))))
       (if (<= t -1.4e-26)
         t_1
         (if (<= t 4.2e-83)
           (* 2.0 (* (/ l t) (/ l (pow k 4.0))))
           (* l (/ (/ l (* k k)) (pow t 3.0)))))))))

C

double code(double t, double l, double k) {
	double t_1 = l * (l / (pow(t, 3.0) * (k * k)));
	double tmp;
	if (t <= -4.2e+182) {
		tmp = t_1;
	} else if (t <= -5.8e+102) {
		tmp = 2.0 * (((l * (l / k)) / k) * (1.0 / (k * (t * k))));
	} else if (t <= -1.4e-26) {
		tmp = t_1;
	} else if (t <= 4.2e-83) {
		tmp = 2.0 * ((l / t) * (l / pow(k, 4.0)));
	} else {
		tmp = l * ((l / (k * k)) / pow(t, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = l * (l / ((t ** 3.0d0) * (k * k)))
    if (t <= (-4.2d+182)) then
        tmp = t_1
    else if (t <= (-5.8d+102)) then
        tmp = 2.0d0 * (((l * (l / k)) / k) * (1.0d0 / (k * (t * k))))
    else if (t <= (-1.4d-26)) then
        tmp = t_1
    else if (t <= 4.2d-83) then
        tmp = 2.0d0 * ((l / t) * (l / (k ** 4.0d0)))
    else
        tmp = l * ((l / (k * k)) / (t ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = l * (l / (Math.pow(t, 3.0) * (k * k)));
	double tmp;
	if (t <= -4.2e+182) {
		tmp = t_1;
	} else if (t <= -5.8e+102) {
		tmp = 2.0 * (((l * (l / k)) / k) * (1.0 / (k * (t * k))));
	} else if (t <= -1.4e-26) {
		tmp = t_1;
	} else if (t <= 4.2e-83) {
		tmp = 2.0 * ((l / t) * (l / Math.pow(k, 4.0)));
	} else {
		tmp = l * ((l / (k * k)) / Math.pow(t, 3.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = l * (l / (math.pow(t, 3.0) * (k * k)))
	tmp = 0
	if t <= -4.2e+182:
		tmp = t_1
	elif t <= -5.8e+102:
		tmp = 2.0 * (((l * (l / k)) / k) * (1.0 / (k * (t * k))))
	elif t <= -1.4e-26:
		tmp = t_1
	elif t <= 4.2e-83:
		tmp = 2.0 * ((l / t) * (l / math.pow(k, 4.0)))
	else:
		tmp = l * ((l / (k * k)) / math.pow(t, 3.0))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(l * Float64(l / Float64((t ^ 3.0) * Float64(k * k))))
	tmp = 0.0
	if (t <= -4.2e+182)
		tmp = t_1;
	elseif (t <= -5.8e+102)
		tmp = Float64(2.0 * Float64(Float64(Float64(l * Float64(l / k)) / k) * Float64(1.0 / Float64(k * Float64(t * k)))));
	elseif (t <= -1.4e-26)
		tmp = t_1;
	elseif (t <= 4.2e-83)
		tmp = Float64(2.0 * Float64(Float64(l / t) * Float64(l / (k ^ 4.0))));
	else
		tmp = Float64(l * Float64(Float64(l / Float64(k * k)) / (t ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = l * (l / ((t ^ 3.0) * (k * k)));
	tmp = 0.0;
	if (t <= -4.2e+182)
		tmp = t_1;
	elseif (t <= -5.8e+102)
		tmp = 2.0 * (((l * (l / k)) / k) * (1.0 / (k * (t * k))));
	elseif (t <= -1.4e-26)
		tmp = t_1;
	elseif (t <= 4.2e-83)
		tmp = 2.0 * ((l / t) * (l / (k ^ 4.0)));
	else
		tmp = l * ((l / (k * k)) / (t ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(l * N[(l / N[(N[Power[t, 3.0], $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4.2e+182], t$95$1, If[LessEqual[t, -5.8e+102], N[(2.0 * N[(N[(N[(l * N[(l / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(1.0 / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.4e-26], t$95$1, If[LessEqual[t, 4.2e-83], N[(2.0 * N[(N[(l / t), $MachinePrecision] * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -4.1999999999999998e182 or -5.8000000000000005e102 < t < -1.4000000000000001e-26

   1. Initial program 79.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 79.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 79.7%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 81.7%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 81.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 81.7%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 81.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 81.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 81.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 81.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 81.7%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 81.7%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 81.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)} \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right) \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}} \]

      2. pow3 81.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)}^{3}}} \]

   5. Applied egg-rr 85.3%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]
   6. Step-by-step derivation

      1. associate-*r/ 85.3%

         \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]

   7. Applied egg-rr 85.3%

      \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]
   8. Step-by-step derivation

      1. unpow2 85.3%

         \[\leadsto \frac{\color{blue}{{\ell}^{2}} \cdot 2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}} \]

      2. associate-*r/ 85.3%

         \[\leadsto \color{blue}{{\ell}^{2} \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]

      3. unpow2 85.3%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}} \]

      4. associate-*l* 91.6%

         \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}\right)} \]

   9. Simplified 91.6%

      \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}\right)} \]

   10. Taylor expanded in k around 0 77.2%

       \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]
   11. Step-by-step derivation

       1. *-commutative 77.2%

          \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

       2. unpow2 77.2%

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{\left(k \cdot k\right)}} \]

   12. Simplified 77.2%

       \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{t}^{3} \cdot \left(k \cdot k\right)}} \]

   if -4.1999999999999998e182 < t < -5.8000000000000005e102

   1. Initial program 45.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 45.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 45.9%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 45.9%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 45.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 45.9%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 45.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 45.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 45.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 45.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 45.9%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 45.9%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 45.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)} \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right) \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}} \]

      2. pow3 45.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)}^{3}}} \]

   5. Applied egg-rr 76.5%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]

   6. Taylor expanded in k around inf 31.0%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 31.0%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 31.0%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 31.0%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 31.0%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 31.4%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      6. *-commutative 31.4%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

      7. associate-/r* 31.4%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{\sin k}^{2}}}\right) \]

   8. Simplified 31.4%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)} \]

   9. Taylor expanded in k around 0 31.0%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\frac{1}{{k}^{2} \cdot t}}\right) \]
   10. Step-by-step derivation

       1. unpow2 31.0%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{\color{blue}{\left(k \cdot k\right)} \cdot t}\right) \]

       2. associate-*r* 31.1%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{\color{blue}{k \cdot \left(k \cdot t\right)}}\right) \]

   11. Simplified 31.1%

       \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\frac{1}{k \cdot \left(k \cdot t\right)}}\right) \]
   12. Step-by-step derivation

       1. associate-*r/ 42.6%

          \[\leadsto 2 \cdot \left(\color{blue}{\frac{\frac{\ell}{k} \cdot \ell}{k}} \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\right) \]

   13. Applied egg-rr 42.6%

       \[\leadsto 2 \cdot \left(\color{blue}{\frac{\frac{\ell}{k} \cdot \ell}{k}} \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\right) \]

   if -1.4000000000000001e-26 < t < 4.1999999999999998e-83

   1. Initial program 26.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 26.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 26.5%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 25.5%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 26.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 26.5%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 26.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 27.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 27.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 27.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 27.5%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 27.5%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 61.9%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 61.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(\sin k \cdot t\right) \cdot {k}^{2}\right)}} \]

      2. associate-*l* 61.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(t \cdot {k}^{2}\right)\right)}} \]

      3. *-commutative 61.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)} \]

      4. unpow2 61.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)} \]

      5. associate-*l* 71.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left(k \cdot \left(k \cdot t\right)\right)}\right)} \]

   6. Simplified 71.7%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}} \]

   7. Taylor expanded in k around 0 47.7%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   8. Step-by-step derivation

      1. unpow2 47.7%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      2. *-commutative 47.7%

         \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

      3. times-frac 66.2%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

   9. Simplified 66.2%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

   if 4.1999999999999998e-83 < t

   1. Initial program 73.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 73.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 72.1%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 71.0%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 71.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 71.0%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 71.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 71.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 71.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 71.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 71.0%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 71.0%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 70.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)} \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right) \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}} \]

      2. pow3 70.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)}^{3}}} \]

   5. Applied egg-rr 73.9%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]
   6. Step-by-step derivation

      1. associate-*r/ 73.9%

         \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]

   7. Applied egg-rr 73.9%

      \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]
   8. Step-by-step derivation

      1. unpow2 73.9%

         \[\leadsto \frac{\color{blue}{{\ell}^{2}} \cdot 2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}} \]

      2. associate-*r/ 73.9%

         \[\leadsto \color{blue}{{\ell}^{2} \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]

      3. unpow2 73.9%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}} \]

      4. associate-*l* 79.1%

         \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}\right)} \]

   9. Simplified 79.1%

      \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}\right)} \]

   10. Taylor expanded in k around 0 65.0%

       \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]
   11. Step-by-step derivation

       1. associate-/r* 67.0%

          \[\leadsto \ell \cdot \color{blue}{\frac{\frac{\ell}{{k}^{2}}}{{t}^{3}}} \]

       2. unpow2 67.0%

          \[\leadsto \ell \cdot \frac{\frac{\ell}{\color{blue}{k \cdot k}}}{{t}^{3}} \]

   12. Simplified 67.0%

       \[\leadsto \ell \cdot \color{blue}{\frac{\frac{\ell}{k \cdot k}}{{t}^{3}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{+182}:\\ \;\;\;\;\ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot k\right)}\\ \mathbf{elif}\;t \leq -5.8 \cdot 10^{+102}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \frac{\ell}{k}}{k} \cdot \frac{1}{k \cdot \left(t \cdot k\right)}\right)\\ \mathbf{elif}\;t \leq -1.4 \cdot 10^{-26}:\\ \;\;\;\;\ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot k\right)}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-83}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k \cdot k}}{{t}^{3}}\\ \end{array} \]

Alternative 15: 62.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot k\right)}\\ \mathbf{if}\;t \leq -3.8 \cdot 10^{+182}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -5.5 \cdot 10^{+102}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \frac{\ell}{k}}{k} \cdot \frac{1}{k \cdot \left(t \cdot k\right)}\right)\\ \mathbf{elif}\;t \leq -2.9 \cdot 10^{-27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 6.6 \cdot 10^{-84}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot {t}^{-3}\right) \cdot \frac{\ell}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* l (/ l (* (pow t 3.0) (* k k))))))
   (if (<= t -3.8e+182)
     t_1
     (if (<= t -5.5e+102)
       (* 2.0 (* (/ (* l (/ l k)) k) (/ 1.0 (* k (* t k)))))
       (if (<= t -2.9e-27)
         t_1
         (if (<= t 6.6e-84)
           (* 2.0 (* (/ l t) (/ l (pow k 4.0))))
           (* (* l (pow t -3.0)) (/ l (* k k)))))))))

C

double code(double t, double l, double k) {
	double t_1 = l * (l / (pow(t, 3.0) * (k * k)));
	double tmp;
	if (t <= -3.8e+182) {
		tmp = t_1;
	} else if (t <= -5.5e+102) {
		tmp = 2.0 * (((l * (l / k)) / k) * (1.0 / (k * (t * k))));
	} else if (t <= -2.9e-27) {
		tmp = t_1;
	} else if (t <= 6.6e-84) {
		tmp = 2.0 * ((l / t) * (l / pow(k, 4.0)));
	} else {
		tmp = (l * pow(t, -3.0)) * (l / (k * k));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = l * (l / ((t ** 3.0d0) * (k * k)))
    if (t <= (-3.8d+182)) then
        tmp = t_1
    else if (t <= (-5.5d+102)) then
        tmp = 2.0d0 * (((l * (l / k)) / k) * (1.0d0 / (k * (t * k))))
    else if (t <= (-2.9d-27)) then
        tmp = t_1
    else if (t <= 6.6d-84) then
        tmp = 2.0d0 * ((l / t) * (l / (k ** 4.0d0)))
    else
        tmp = (l * (t ** (-3.0d0))) * (l / (k * k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = l * (l / (Math.pow(t, 3.0) * (k * k)));
	double tmp;
	if (t <= -3.8e+182) {
		tmp = t_1;
	} else if (t <= -5.5e+102) {
		tmp = 2.0 * (((l * (l / k)) / k) * (1.0 / (k * (t * k))));
	} else if (t <= -2.9e-27) {
		tmp = t_1;
	} else if (t <= 6.6e-84) {
		tmp = 2.0 * ((l / t) * (l / Math.pow(k, 4.0)));
	} else {
		tmp = (l * Math.pow(t, -3.0)) * (l / (k * k));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = l * (l / (math.pow(t, 3.0) * (k * k)))
	tmp = 0
	if t <= -3.8e+182:
		tmp = t_1
	elif t <= -5.5e+102:
		tmp = 2.0 * (((l * (l / k)) / k) * (1.0 / (k * (t * k))))
	elif t <= -2.9e-27:
		tmp = t_1
	elif t <= 6.6e-84:
		tmp = 2.0 * ((l / t) * (l / math.pow(k, 4.0)))
	else:
		tmp = (l * math.pow(t, -3.0)) * (l / (k * k))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(l * Float64(l / Float64((t ^ 3.0) * Float64(k * k))))
	tmp = 0.0
	if (t <= -3.8e+182)
		tmp = t_1;
	elseif (t <= -5.5e+102)
		tmp = Float64(2.0 * Float64(Float64(Float64(l * Float64(l / k)) / k) * Float64(1.0 / Float64(k * Float64(t * k)))));
	elseif (t <= -2.9e-27)
		tmp = t_1;
	elseif (t <= 6.6e-84)
		tmp = Float64(2.0 * Float64(Float64(l / t) * Float64(l / (k ^ 4.0))));
	else
		tmp = Float64(Float64(l * (t ^ -3.0)) * Float64(l / Float64(k * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = l * (l / ((t ^ 3.0) * (k * k)));
	tmp = 0.0;
	if (t <= -3.8e+182)
		tmp = t_1;
	elseif (t <= -5.5e+102)
		tmp = 2.0 * (((l * (l / k)) / k) * (1.0 / (k * (t * k))));
	elseif (t <= -2.9e-27)
		tmp = t_1;
	elseif (t <= 6.6e-84)
		tmp = 2.0 * ((l / t) * (l / (k ^ 4.0)));
	else
		tmp = (l * (t ^ -3.0)) * (l / (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(l * N[(l / N[(N[Power[t, 3.0], $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -3.8e+182], t$95$1, If[LessEqual[t, -5.5e+102], N[(2.0 * N[(N[(N[(l * N[(l / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(1.0 / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -2.9e-27], t$95$1, If[LessEqual[t, 6.6e-84], N[(2.0 * N[(N[(l / t), $MachinePrecision] * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -3.80000000000000013e182 or -5.49999999999999981e102 < t < -2.90000000000000004e-27

   1. Initial program 79.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 79.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 79.7%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 81.7%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 81.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 81.7%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 81.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 81.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 81.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 81.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 81.7%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 81.7%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 81.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)} \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right) \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}} \]

      2. pow3 81.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)}^{3}}} \]

   5. Applied egg-rr 85.3%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]
   6. Step-by-step derivation

      1. associate-*r/ 85.3%

         \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]

   7. Applied egg-rr 85.3%

      \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]
   8. Step-by-step derivation

      1. unpow2 85.3%

         \[\leadsto \frac{\color{blue}{{\ell}^{2}} \cdot 2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}} \]

      2. associate-*r/ 85.3%

         \[\leadsto \color{blue}{{\ell}^{2} \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]

      3. unpow2 85.3%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}} \]

      4. associate-*l* 91.6%

         \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}\right)} \]

   9. Simplified 91.6%

      \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}\right)} \]

   10. Taylor expanded in k around 0 77.2%

       \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]
   11. Step-by-step derivation

       1. *-commutative 77.2%

          \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

       2. unpow2 77.2%

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{\left(k \cdot k\right)}} \]

   12. Simplified 77.2%

       \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{t}^{3} \cdot \left(k \cdot k\right)}} \]

   if -3.80000000000000013e182 < t < -5.49999999999999981e102

   1. Initial program 45.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 45.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 45.9%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 45.9%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 45.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 45.9%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 45.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 45.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 45.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 45.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 45.9%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 45.9%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 45.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)} \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right) \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}} \]

      2. pow3 45.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)}^{3}}} \]

   5. Applied egg-rr 76.5%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]

   6. Taylor expanded in k around inf 31.0%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 31.0%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 31.0%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 31.0%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 31.0%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 31.4%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      6. *-commutative 31.4%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

      7. associate-/r* 31.4%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{\sin k}^{2}}}\right) \]

   8. Simplified 31.4%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)} \]

   9. Taylor expanded in k around 0 31.0%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\frac{1}{{k}^{2} \cdot t}}\right) \]
   10. Step-by-step derivation

       1. unpow2 31.0%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{\color{blue}{\left(k \cdot k\right)} \cdot t}\right) \]

       2. associate-*r* 31.1%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{\color{blue}{k \cdot \left(k \cdot t\right)}}\right) \]

   11. Simplified 31.1%

       \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\frac{1}{k \cdot \left(k \cdot t\right)}}\right) \]
   12. Step-by-step derivation

       1. associate-*r/ 42.6%

          \[\leadsto 2 \cdot \left(\color{blue}{\frac{\frac{\ell}{k} \cdot \ell}{k}} \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\right) \]

   13. Applied egg-rr 42.6%

       \[\leadsto 2 \cdot \left(\color{blue}{\frac{\frac{\ell}{k} \cdot \ell}{k}} \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\right) \]

   if -2.90000000000000004e-27 < t < 6.59999999999999968e-84

   1. Initial program 26.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 26.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 26.5%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 25.5%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 26.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 26.5%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 26.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 27.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 27.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 27.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 27.5%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 27.5%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 61.9%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 61.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(\sin k \cdot t\right) \cdot {k}^{2}\right)}} \]

      2. associate-*l* 61.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(t \cdot {k}^{2}\right)\right)}} \]

      3. *-commutative 61.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)} \]

      4. unpow2 61.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)} \]

      5. associate-*l* 71.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left(k \cdot \left(k \cdot t\right)\right)}\right)} \]

   6. Simplified 71.7%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}} \]

   7. Taylor expanded in k around 0 47.7%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   8. Step-by-step derivation

      1. unpow2 47.7%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      2. *-commutative 47.7%

         \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

      3. times-frac 66.2%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

   9. Simplified 66.2%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

   if 6.59999999999999968e-84 < t

   1. Initial program 73.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 73.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 72.1%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 71.0%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 71.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 71.0%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 71.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 71.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 71.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 71.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 71.0%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 71.0%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around 0 61.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 61.7%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 61.7%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac 67.9%

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. unpow2 67.9%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   6. Simplified 67.9%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 58.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell}{{t}^{3}}\right)\right)} \cdot \frac{\ell}{k \cdot k} \]

      2. expm1-udef 54.1%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\ell}{{t}^{3}}\right)} - 1\right)} \cdot \frac{\ell}{k \cdot k} \]

      3. div-inv 54.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\ell \cdot \frac{1}{{t}^{3}}}\right)} - 1\right) \cdot \frac{\ell}{k \cdot k} \]

      4. pow-flip 54.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\ell \cdot \color{blue}{{t}^{\left(-3\right)}}\right)} - 1\right) \cdot \frac{\ell}{k \cdot k} \]

      5. metadata-eval 54.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\ell \cdot {t}^{\color{blue}{-3}}\right)} - 1\right) \cdot \frac{\ell}{k \cdot k} \]

   8. Applied egg-rr 54.1%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\ell \cdot {t}^{-3}\right)} - 1\right)} \cdot \frac{\ell}{k \cdot k} \]
   9. Step-by-step derivation

      1. expm1-def 58.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot {t}^{-3}\right)\right)} \cdot \frac{\ell}{k \cdot k} \]

      2. expm1-log1p 67.9%

         \[\leadsto \color{blue}{\left(\ell \cdot {t}^{-3}\right)} \cdot \frac{\ell}{k \cdot k} \]

   10. Simplified 67.9%

       \[\leadsto \color{blue}{\left(\ell \cdot {t}^{-3}\right)} \cdot \frac{\ell}{k \cdot k} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{+182}:\\ \;\;\;\;\ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot k\right)}\\ \mathbf{elif}\;t \leq -5.5 \cdot 10^{+102}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \frac{\ell}{k}}{k} \cdot \frac{1}{k \cdot \left(t \cdot k\right)}\right)\\ \mathbf{elif}\;t \leq -2.9 \cdot 10^{-27}:\\ \;\;\;\;\ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot k\right)}\\ \mathbf{elif}\;t \leq 6.6 \cdot 10^{-84}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot {t}^{-3}\right) \cdot \frac{\ell}{k \cdot k}\\ \end{array} \]

Alternative 16: 62.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot k\right)}\\ \mathbf{if}\;t \leq -4.2 \cdot 10^{+182}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -6.3 \cdot 10^{+102}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \frac{\ell}{k}}{k} \cdot \frac{1}{k \cdot \left(t \cdot k\right)}\right)\\ \mathbf{elif}\;t \leq -1.3 \cdot 10^{-16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-84}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{\ell} \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot {t}^{-3}\right) \cdot \frac{\ell}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* l (/ l (* (pow t 3.0) (* k k))))))
   (if (<= t -4.2e+182)
     t_1
     (if (<= t -6.3e+102)
       (* 2.0 (* (/ (* l (/ l k)) k) (/ 1.0 (* k (* t k)))))
       (if (<= t -1.3e-16)
         t_1
         (if (<= t 6e-84)
           (/ 2.0 (* (/ (pow k 4.0) l) (/ t l)))
           (* (* l (pow t -3.0)) (/ l (* k k)))))))))

C

double code(double t, double l, double k) {
	double t_1 = l * (l / (pow(t, 3.0) * (k * k)));
	double tmp;
	if (t <= -4.2e+182) {
		tmp = t_1;
	} else if (t <= -6.3e+102) {
		tmp = 2.0 * (((l * (l / k)) / k) * (1.0 / (k * (t * k))));
	} else if (t <= -1.3e-16) {
		tmp = t_1;
	} else if (t <= 6e-84) {
		tmp = 2.0 / ((pow(k, 4.0) / l) * (t / l));
	} else {
		tmp = (l * pow(t, -3.0)) * (l / (k * k));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = l * (l / ((t ** 3.0d0) * (k * k)))
    if (t <= (-4.2d+182)) then
        tmp = t_1
    else if (t <= (-6.3d+102)) then
        tmp = 2.0d0 * (((l * (l / k)) / k) * (1.0d0 / (k * (t * k))))
    else if (t <= (-1.3d-16)) then
        tmp = t_1
    else if (t <= 6d-84) then
        tmp = 2.0d0 / (((k ** 4.0d0) / l) * (t / l))
    else
        tmp = (l * (t ** (-3.0d0))) * (l / (k * k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = l * (l / (Math.pow(t, 3.0) * (k * k)));
	double tmp;
	if (t <= -4.2e+182) {
		tmp = t_1;
	} else if (t <= -6.3e+102) {
		tmp = 2.0 * (((l * (l / k)) / k) * (1.0 / (k * (t * k))));
	} else if (t <= -1.3e-16) {
		tmp = t_1;
	} else if (t <= 6e-84) {
		tmp = 2.0 / ((Math.pow(k, 4.0) / l) * (t / l));
	} else {
		tmp = (l * Math.pow(t, -3.0)) * (l / (k * k));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = l * (l / (math.pow(t, 3.0) * (k * k)))
	tmp = 0
	if t <= -4.2e+182:
		tmp = t_1
	elif t <= -6.3e+102:
		tmp = 2.0 * (((l * (l / k)) / k) * (1.0 / (k * (t * k))))
	elif t <= -1.3e-16:
		tmp = t_1
	elif t <= 6e-84:
		tmp = 2.0 / ((math.pow(k, 4.0) / l) * (t / l))
	else:
		tmp = (l * math.pow(t, -3.0)) * (l / (k * k))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(l * Float64(l / Float64((t ^ 3.0) * Float64(k * k))))
	tmp = 0.0
	if (t <= -4.2e+182)
		tmp = t_1;
	elseif (t <= -6.3e+102)
		tmp = Float64(2.0 * Float64(Float64(Float64(l * Float64(l / k)) / k) * Float64(1.0 / Float64(k * Float64(t * k)))));
	elseif (t <= -1.3e-16)
		tmp = t_1;
	elseif (t <= 6e-84)
		tmp = Float64(2.0 / Float64(Float64((k ^ 4.0) / l) * Float64(t / l)));
	else
		tmp = Float64(Float64(l * (t ^ -3.0)) * Float64(l / Float64(k * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = l * (l / ((t ^ 3.0) * (k * k)));
	tmp = 0.0;
	if (t <= -4.2e+182)
		tmp = t_1;
	elseif (t <= -6.3e+102)
		tmp = 2.0 * (((l * (l / k)) / k) * (1.0 / (k * (t * k))));
	elseif (t <= -1.3e-16)
		tmp = t_1;
	elseif (t <= 6e-84)
		tmp = 2.0 / (((k ^ 4.0) / l) * (t / l));
	else
		tmp = (l * (t ^ -3.0)) * (l / (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(l * N[(l / N[(N[Power[t, 3.0], $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4.2e+182], t$95$1, If[LessEqual[t, -6.3e+102], N[(2.0 * N[(N[(N[(l * N[(l / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(1.0 / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.3e-16], t$95$1, If[LessEqual[t, 6e-84], N[(2.0 / N[(N[(N[Power[k, 4.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -4.1999999999999998e182 or -6.30000000000000029e102 < t < -1.2999999999999999e-16

   1. Initial program 81.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 81.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 81.4%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 83.5%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 83.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 83.4%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 83.4%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 83.4%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 83.4%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 83.4%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 83.4%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 83.4%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 83.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)} \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right) \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}} \]

      2. pow3 83.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)}^{3}}} \]

   5. Applied egg-rr 87.1%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]
   6. Step-by-step derivation

      1. associate-*r/ 87.1%

         \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]

   7. Applied egg-rr 87.1%

      \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]
   8. Step-by-step derivation

      1. unpow2 87.1%

         \[\leadsto \frac{\color{blue}{{\ell}^{2}} \cdot 2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}} \]

      2. associate-*r/ 87.1%

         \[\leadsto \color{blue}{{\ell}^{2} \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]

      3. unpow2 87.1%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}} \]

      4. associate-*l* 93.5%

         \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}\right)} \]

   9. Simplified 93.5%

      \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}\right)} \]

   10. Taylor expanded in k around 0 78.8%

       \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]
   11. Step-by-step derivation

       1. *-commutative 78.8%

          \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

       2. unpow2 78.8%

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{\left(k \cdot k\right)}} \]

   12. Simplified 78.8%

       \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{t}^{3} \cdot \left(k \cdot k\right)}} \]

   if -4.1999999999999998e182 < t < -6.30000000000000029e102

   1. Initial program 45.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 45.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 45.9%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 45.9%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 45.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 45.9%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 45.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 45.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 45.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 45.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 45.9%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 45.9%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 45.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)} \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right) \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}} \]

      2. pow3 45.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)}^{3}}} \]

   5. Applied egg-rr 76.5%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]

   6. Taylor expanded in k around inf 31.0%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 31.0%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 31.0%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 31.0%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 31.0%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 31.4%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      6. *-commutative 31.4%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

      7. associate-/r* 31.4%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{\sin k}^{2}}}\right) \]

   8. Simplified 31.4%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)} \]

   9. Taylor expanded in k around 0 31.0%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\frac{1}{{k}^{2} \cdot t}}\right) \]
   10. Step-by-step derivation

       1. unpow2 31.0%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{\color{blue}{\left(k \cdot k\right)} \cdot t}\right) \]

       2. associate-*r* 31.1%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{\color{blue}{k \cdot \left(k \cdot t\right)}}\right) \]

   11. Simplified 31.1%

       \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\frac{1}{k \cdot \left(k \cdot t\right)}}\right) \]
   12. Step-by-step derivation

       1. associate-*r/ 42.6%

          \[\leadsto 2 \cdot \left(\color{blue}{\frac{\frac{\ell}{k} \cdot \ell}{k}} \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\right) \]

   13. Applied egg-rr 42.6%

       \[\leadsto 2 \cdot \left(\color{blue}{\frac{\frac{\ell}{k} \cdot \ell}{k}} \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\right) \]

   if -1.2999999999999999e-16 < t < 6.0000000000000002e-84

   1. Initial program 26.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 26.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 26.2%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 26.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 26.2%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 26.2%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 26.2%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 26.2%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   4. Taylor expanded in k around inf 61.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}}} \]
   5. Step-by-step derivation

      1. *-commutative 61.3%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left({\sin k}^{2} \cdot t\right) \cdot {k}^{2}}}{\cos k \cdot {\ell}^{2}}} \]

      2. associate-*l* 61.3%

         \[\leadsto \frac{2}{\frac{\color{blue}{{\sin k}^{2} \cdot \left(t \cdot {k}^{2}\right)}}{\cos k \cdot {\ell}^{2}}} \]

      3. *-commutative 61.3%

         \[\leadsto \frac{2}{\frac{{\sin k}^{2} \cdot \color{blue}{\left({k}^{2} \cdot t\right)}}{\cos k \cdot {\ell}^{2}}} \]

      4. unpow2 61.3%

         \[\leadsto \frac{2}{\frac{{\sin k}^{2} \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)}{\cos k \cdot {\ell}^{2}}} \]

      5. associate-*l* 71.0%

         \[\leadsto \frac{2}{\frac{{\sin k}^{2} \cdot \color{blue}{\left(k \cdot \left(k \cdot t\right)\right)}}{\cos k \cdot {\ell}^{2}}} \]

      6. unpow2 71.0%

         \[\leadsto \frac{2}{\frac{{\sin k}^{2} \cdot \left(k \cdot \left(k \cdot t\right)\right)}{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \]

   6. Simplified 71.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{{\sin k}^{2} \cdot \left(k \cdot \left(k \cdot t\right)\right)}{\cos k \cdot \left(\ell \cdot \ell\right)}}} \]

   7. Taylor expanded in k around 0 52.3%

      \[\leadsto \frac{2}{\frac{\color{blue}{{k}^{2}} \cdot \left(k \cdot \left(k \cdot t\right)\right)}{\cos k \cdot \left(\ell \cdot \ell\right)}} \]
   8. Step-by-step derivation

      1. unpow2 52.3%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot \left(k \cdot \left(k \cdot t\right)\right)}{\cos k \cdot \left(\ell \cdot \ell\right)}} \]

   9. Simplified 52.3%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot \left(k \cdot \left(k \cdot t\right)\right)}{\cos k \cdot \left(\ell \cdot \ell\right)}} \]

   10. Taylor expanded in k around 0 47.2%

       \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   11. Step-by-step derivation

       1. unpow2 47.2%

          \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

       2. times-frac 65.6%

          \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{\ell} \cdot \frac{t}{\ell}}} \]

   12. Simplified 65.6%

       \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{\ell} \cdot \frac{t}{\ell}}} \]

   if 6.0000000000000002e-84 < t

   1. Initial program 73.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 73.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 72.1%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 71.0%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 71.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 71.0%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 71.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 71.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 71.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 71.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 71.0%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 71.0%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around 0 61.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 61.7%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 61.7%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac 67.9%

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. unpow2 67.9%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   6. Simplified 67.9%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 58.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell}{{t}^{3}}\right)\right)} \cdot \frac{\ell}{k \cdot k} \]

      2. expm1-udef 54.1%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\ell}{{t}^{3}}\right)} - 1\right)} \cdot \frac{\ell}{k \cdot k} \]

      3. div-inv 54.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\ell \cdot \frac{1}{{t}^{3}}}\right)} - 1\right) \cdot \frac{\ell}{k \cdot k} \]

      4. pow-flip 54.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\ell \cdot \color{blue}{{t}^{\left(-3\right)}}\right)} - 1\right) \cdot \frac{\ell}{k \cdot k} \]

      5. metadata-eval 54.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\ell \cdot {t}^{\color{blue}{-3}}\right)} - 1\right) \cdot \frac{\ell}{k \cdot k} \]

   8. Applied egg-rr 54.1%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\ell \cdot {t}^{-3}\right)} - 1\right)} \cdot \frac{\ell}{k \cdot k} \]
   9. Step-by-step derivation

      1. expm1-def 58.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot {t}^{-3}\right)\right)} \cdot \frac{\ell}{k \cdot k} \]

      2. expm1-log1p 67.9%

         \[\leadsto \color{blue}{\left(\ell \cdot {t}^{-3}\right)} \cdot \frac{\ell}{k \cdot k} \]

   10. Simplified 67.9%

       \[\leadsto \color{blue}{\left(\ell \cdot {t}^{-3}\right)} \cdot \frac{\ell}{k \cdot k} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{+182}:\\ \;\;\;\;\ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot k\right)}\\ \mathbf{elif}\;t \leq -6.3 \cdot 10^{+102}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \frac{\ell}{k}}{k} \cdot \frac{1}{k \cdot \left(t \cdot k\right)}\right)\\ \mathbf{elif}\;t \leq -1.3 \cdot 10^{-16}:\\ \;\;\;\;\ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot k\right)}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-84}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{\ell} \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot {t}^{-3}\right) \cdot \frac{\ell}{k \cdot k}\\ \end{array} \]

Alternative 17: 66.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.4 \cdot 10^{-42} \lor \neg \left(t \leq 7.2 \cdot 10^{-84}\right):\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{\ell} \cdot \frac{t}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -3.4e-42) (not (<= t 7.2e-84)))
   (/ (* (/ l k) (/ l k)) (pow t 3.0))
   (/ 2.0 (* (/ (pow k 4.0) l) (/ t l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -3.4e-42) || !(t <= 7.2e-84)) {
		tmp = ((l / k) * (l / k)) / pow(t, 3.0);
	} else {
		tmp = 2.0 / ((pow(k, 4.0) / l) * (t / l));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-3.4d-42)) .or. (.not. (t <= 7.2d-84))) then
        tmp = ((l / k) * (l / k)) / (t ** 3.0d0)
    else
        tmp = 2.0d0 / (((k ** 4.0d0) / l) * (t / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -3.4e-42) || !(t <= 7.2e-84)) {
		tmp = ((l / k) * (l / k)) / Math.pow(t, 3.0);
	} else {
		tmp = 2.0 / ((Math.pow(k, 4.0) / l) * (t / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -3.4e-42) or not (t <= 7.2e-84):
		tmp = ((l / k) * (l / k)) / math.pow(t, 3.0)
	else:
		tmp = 2.0 / ((math.pow(k, 4.0) / l) * (t / l))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -3.4e-42) || !(t <= 7.2e-84))
		tmp = Float64(Float64(Float64(l / k) * Float64(l / k)) / (t ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64((k ^ 4.0) / l) * Float64(t / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -3.4e-42) || ~((t <= 7.2e-84)))
		tmp = ((l / k) * (l / k)) / (t ^ 3.0);
	else
		tmp = 2.0 / (((k ^ 4.0) / l) * (t / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -3.4e-42], N[Not[LessEqual[t, 7.2e-84]], $MachinePrecision]], N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[k, 4.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.40000000000000022e-42 or 7.20000000000000007e-84 < t

   1. Initial program 71.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 71.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 71.1%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 71.0%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 71.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 71.0%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 71.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 71.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 71.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 71.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 71.0%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 71.0%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 70.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)} \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right) \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}} \]

      2. pow3 70.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)}^{3}}} \]

   5. Applied egg-rr 76.9%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]

   6. Taylor expanded in k around 0 59.1%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   7. Step-by-step derivation

      1. associate-/r* 60.1%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}}} \]

      2. unpow2 60.1%

         \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{{t}^{3}} \]

      3. unpow2 60.1%

         \[\leadsto \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{{t}^{3}} \]

      4. times-frac 71.3%

         \[\leadsto \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{{t}^{3}} \]

   8. Simplified 71.3%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}} \]

   if -3.40000000000000022e-42 < t < 7.20000000000000007e-84

   1. Initial program 25.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 25.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 25.5%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 25.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 25.5%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 25.5%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 25.5%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 25.5%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   4. Taylor expanded in k around inf 62.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}}} \]
   5. Step-by-step derivation

      1. *-commutative 62.7%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left({\sin k}^{2} \cdot t\right) \cdot {k}^{2}}}{\cos k \cdot {\ell}^{2}}} \]

      2. associate-*l* 62.7%

         \[\leadsto \frac{2}{\frac{\color{blue}{{\sin k}^{2} \cdot \left(t \cdot {k}^{2}\right)}}{\cos k \cdot {\ell}^{2}}} \]

      3. *-commutative 62.7%

         \[\leadsto \frac{2}{\frac{{\sin k}^{2} \cdot \color{blue}{\left({k}^{2} \cdot t\right)}}{\cos k \cdot {\ell}^{2}}} \]

      4. unpow2 62.7%

         \[\leadsto \frac{2}{\frac{{\sin k}^{2} \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)}{\cos k \cdot {\ell}^{2}}} \]

      5. associate-*l* 72.9%

         \[\leadsto \frac{2}{\frac{{\sin k}^{2} \cdot \color{blue}{\left(k \cdot \left(k \cdot t\right)\right)}}{\cos k \cdot {\ell}^{2}}} \]

      6. unpow2 72.9%

         \[\leadsto \frac{2}{\frac{{\sin k}^{2} \cdot \left(k \cdot \left(k \cdot t\right)\right)}{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \]

   6. Simplified 72.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{{\sin k}^{2} \cdot \left(k \cdot \left(k \cdot t\right)\right)}{\cos k \cdot \left(\ell \cdot \ell\right)}}} \]

   7. Taylor expanded in k around 0 53.2%

      \[\leadsto \frac{2}{\frac{\color{blue}{{k}^{2}} \cdot \left(k \cdot \left(k \cdot t\right)\right)}{\cos k \cdot \left(\ell \cdot \ell\right)}} \]
   8. Step-by-step derivation

      1. unpow2 53.2%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot \left(k \cdot \left(k \cdot t\right)\right)}{\cos k \cdot \left(\ell \cdot \ell\right)}} \]

   9. Simplified 53.2%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot \left(k \cdot \left(k \cdot t\right)\right)}{\cos k \cdot \left(\ell \cdot \ell\right)}} \]

   10. Taylor expanded in k around 0 47.8%

       \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   11. Step-by-step derivation

       1. unpow2 47.8%

          \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

       2. times-frac 67.1%

          \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{\ell} \cdot \frac{t}{\ell}}} \]

   12. Simplified 67.1%

       \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{\ell} \cdot \frac{t}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.4 \cdot 10^{-42} \lor \neg \left(t \leq 7.2 \cdot 10^{-84}\right):\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{\ell} \cdot \frac{t}{\ell}}\\ \end{array} \]

Alternative 18: 58.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6.6 \cdot 10^{-188}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \frac{\ell}{k}}{k} \cdot \frac{1}{k \cdot \left(t \cdot k\right)}\right)\\ \mathbf{elif}\;k \leq 7.6 \cdot 10^{-65}:\\ \;\;\;\;\ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 6.6e-188)
   (* 2.0 (* (/ (* l (/ l k)) k) (/ 1.0 (* k (* t k)))))
   (if (<= k 7.6e-65)
     (* l (/ l (* (pow t 3.0) (* k k))))
     (* 2.0 (* (/ l t) (/ l (pow k 4.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.6e-188) {
		tmp = 2.0 * (((l * (l / k)) / k) * (1.0 / (k * (t * k))));
	} else if (k <= 7.6e-65) {
		tmp = l * (l / (pow(t, 3.0) * (k * k)));
	} else {
		tmp = 2.0 * ((l / t) * (l / pow(k, 4.0)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 6.6d-188) then
        tmp = 2.0d0 * (((l * (l / k)) / k) * (1.0d0 / (k * (t * k))))
    else if (k <= 7.6d-65) then
        tmp = l * (l / ((t ** 3.0d0) * (k * k)))
    else
        tmp = 2.0d0 * ((l / t) * (l / (k ** 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.6e-188) {
		tmp = 2.0 * (((l * (l / k)) / k) * (1.0 / (k * (t * k))));
	} else if (k <= 7.6e-65) {
		tmp = l * (l / (Math.pow(t, 3.0) * (k * k)));
	} else {
		tmp = 2.0 * ((l / t) * (l / Math.pow(k, 4.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 6.6e-188:
		tmp = 2.0 * (((l * (l / k)) / k) * (1.0 / (k * (t * k))))
	elif k <= 7.6e-65:
		tmp = l * (l / (math.pow(t, 3.0) * (k * k)))
	else:
		tmp = 2.0 * ((l / t) * (l / math.pow(k, 4.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 6.6e-188)
		tmp = Float64(2.0 * Float64(Float64(Float64(l * Float64(l / k)) / k) * Float64(1.0 / Float64(k * Float64(t * k)))));
	elseif (k <= 7.6e-65)
		tmp = Float64(l * Float64(l / Float64((t ^ 3.0) * Float64(k * k))));
	else
		tmp = Float64(2.0 * Float64(Float64(l / t) * Float64(l / (k ^ 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 6.6e-188)
		tmp = 2.0 * (((l * (l / k)) / k) * (1.0 / (k * (t * k))));
	elseif (k <= 7.6e-65)
		tmp = l * (l / ((t ^ 3.0) * (k * k)));
	else
		tmp = 2.0 * ((l / t) * (l / (k ^ 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 6.6e-188], N[(2.0 * N[(N[(N[(l * N[(l / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(1.0 / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 7.6e-65], N[(l * N[(l / N[(N[Power[t, 3.0], $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / t), $MachinePrecision] * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 6.6000000000000005e-188

   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-/l/ 54.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 54.2%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 53.5%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 54.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 54.1%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 54.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 54.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 54.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 54.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 54.1%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 54.1%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 54.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)} \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right) \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}} \]

      2. pow3 54.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)}^{3}}} \]

   5. Applied egg-rr 63.7%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]

   6. Taylor expanded in k around inf 51.2%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 51.2%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 49.5%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 49.5%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 49.5%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 59.1%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      6. *-commutative 59.1%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

      7. associate-/r* 59.1%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{\sin k}^{2}}}\right) \]

   8. Simplified 59.1%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)} \]

   9. Taylor expanded in k around 0 50.7%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\frac{1}{{k}^{2} \cdot t}}\right) \]
   10. Step-by-step derivation

       1. unpow2 50.7%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{\color{blue}{\left(k \cdot k\right)} \cdot t}\right) \]

       2. associate-*r* 50.7%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{\color{blue}{k \cdot \left(k \cdot t\right)}}\right) \]

   11. Simplified 50.7%

       \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\frac{1}{k \cdot \left(k \cdot t\right)}}\right) \]
   12. Step-by-step derivation

       1. associate-*r/ 51.2%

          \[\leadsto 2 \cdot \left(\color{blue}{\frac{\frac{\ell}{k} \cdot \ell}{k}} \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\right) \]

   13. Applied egg-rr 51.2%

       \[\leadsto 2 \cdot \left(\color{blue}{\frac{\frac{\ell}{k} \cdot \ell}{k}} \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\right) \]

   if 6.6000000000000005e-188 < k < 7.6000000000000003e-65

   1. Initial program 61.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 61.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 57.4%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 61.3%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 61.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 61.3%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 61.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 61.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 61.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 61.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 61.3%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 61.3%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 61.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)} \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right) \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}} \]

      2. pow3 61.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)}^{3}}} \]

   5. Applied egg-rr 61.0%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]
   6. Step-by-step derivation

      1. associate-*r/ 61.0%

         \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]

   7. Applied egg-rr 61.0%

      \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]
   8. Step-by-step derivation

      1. unpow2 61.0%

         \[\leadsto \frac{\color{blue}{{\ell}^{2}} \cdot 2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}} \]

      2. associate-*r/ 61.0%

         \[\leadsto \color{blue}{{\ell}^{2} \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]

      3. unpow2 61.0%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}} \]

      4. associate-*l* 75.1%

         \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}\right)} \]

   9. Simplified 75.1%

      \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}\right)} \]

   10. Taylor expanded in k around 0 75.5%

       \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]
   11. Step-by-step derivation

       1. *-commutative 75.5%

          \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

       2. unpow2 75.5%

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{\left(k \cdot k\right)}} \]

   12. Simplified 75.5%

       \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{t}^{3} \cdot \left(k \cdot k\right)}} \]

   if 7.6000000000000003e-65 < k

   1. Initial program 53.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 53.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 53.7%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 52.6%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 52.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 52.6%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 52.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 53.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 53.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 53.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 53.8%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 53.8%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 74.9%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 74.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(\sin k \cdot t\right) \cdot {k}^{2}\right)}} \]

      2. associate-*l* 74.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(t \cdot {k}^{2}\right)\right)}} \]

      3. *-commutative 74.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)} \]

      4. unpow2 74.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)} \]

      5. associate-*l* 79.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left(k \cdot \left(k \cdot t\right)\right)}\right)} \]

   6. Simplified 79.6%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}} \]

   7. Taylor expanded in k around 0 64.9%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   8. Step-by-step derivation

      1. unpow2 64.9%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      2. *-commutative 64.9%

         \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

      3. times-frac 69.6%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

   9. Simplified 69.6%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.6 \cdot 10^{-188}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \frac{\ell}{k}}{k} \cdot \frac{1}{k \cdot \left(t \cdot k\right)}\right)\\ \mathbf{elif}\;k \leq 7.6 \cdot 10^{-65}:\\ \;\;\;\;\ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \end{array} \]

Alternative 19: 55.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 9.2 \cdot 10^{+181}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{k \cdot \left(t \cdot k\right)} - \frac{0.16666666666666666}{t}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 9.2e+181)
   (* 2.0 (* (/ l t) (/ l (pow k 4.0))))
   (*
    2.0
    (*
     (* (/ l k) (/ l k))
     (- (/ 1.0 (* k (* t k))) (/ 0.16666666666666666 t))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 9.2e+181) {
		tmp = 2.0 * ((l / t) * (l / pow(k, 4.0)));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * ((1.0 / (k * (t * k))) - (0.16666666666666666 / t)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 9.2d+181) then
        tmp = 2.0d0 * ((l / t) * (l / (k ** 4.0d0)))
    else
        tmp = 2.0d0 * (((l / k) * (l / k)) * ((1.0d0 / (k * (t * k))) - (0.16666666666666666d0 / t)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 9.2e+181) {
		tmp = 2.0 * ((l / t) * (l / Math.pow(k, 4.0)));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * ((1.0 / (k * (t * k))) - (0.16666666666666666 / t)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 9.2e+181:
		tmp = 2.0 * ((l / t) * (l / math.pow(k, 4.0)))
	else:
		tmp = 2.0 * (((l / k) * (l / k)) * ((1.0 / (k * (t * k))) - (0.16666666666666666 / t)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 9.2e+181)
		tmp = Float64(2.0 * Float64(Float64(l / t) * Float64(l / (k ^ 4.0))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(Float64(1.0 / Float64(k * Float64(t * k))) - Float64(0.16666666666666666 / t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 9.2e+181)
		tmp = 2.0 * ((l / t) * (l / (k ^ 4.0)));
	else
		tmp = 2.0 * (((l / k) * (l / k)) * ((1.0 / (k * (t * k))) - (0.16666666666666666 / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 9.2e+181], N[(2.0 * N[(N[(l / t), $MachinePrecision] * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.16666666666666666 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 9.1999999999999995e181

   1. Initial program 56.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 56.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 55.7%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 55.3%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 55.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 55.7%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 55.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 56.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 56.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 56.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 56.1%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 56.1%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 56.7%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 56.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(\sin k \cdot t\right) \cdot {k}^{2}\right)}} \]

      2. associate-*l* 56.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(t \cdot {k}^{2}\right)\right)}} \]

      3. *-commutative 56.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)} \]

      4. unpow2 56.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)} \]

      5. associate-*l* 60.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left(k \cdot \left(k \cdot t\right)\right)}\right)} \]

   6. Simplified 60.7%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}} \]

   7. Taylor expanded in k around 0 50.8%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   8. Step-by-step derivation

      1. unpow2 50.8%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      2. *-commutative 50.8%

         \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

      3. times-frac 58.7%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

   9. Simplified 58.7%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

   if 9.1999999999999995e181 < l

   1. Initial program 35.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 35.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 35.5%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 35.5%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 35.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 35.3%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 35.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 35.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 35.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 35.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 35.3%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 35.3%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 35.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)} \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right) \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}} \]

      2. pow3 35.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)}^{3}}} \]

   5. Applied egg-rr 35.3%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]

   6. Taylor expanded in k around inf 48.1%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 48.1%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 48.1%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 48.1%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 48.1%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 60.4%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      6. *-commutative 60.4%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

      7. associate-/r* 60.4%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{\sin k}^{2}}}\right) \]

   8. Simplified 60.4%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)} \]

   9. Taylor expanded in k around 0 42.8%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}\right)}\right) \]
   10. Step-by-step derivation

       1. unpow2 42.8%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{\color{blue}{\left(k \cdot k\right)} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

       2. associate-*r* 42.8%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{\color{blue}{k \cdot \left(k \cdot t\right)}} - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

       3. associate-*r/ 42.8%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{k \cdot \left(k \cdot t\right)} - \color{blue}{\frac{0.16666666666666666 \cdot 1}{t}}\right)\right) \]

       4. metadata-eval 42.8%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{k \cdot \left(k \cdot t\right)} - \frac{\color{blue}{0.16666666666666666}}{t}\right)\right) \]

   11. Simplified 42.8%

       \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\left(\frac{1}{k \cdot \left(k \cdot t\right)} - \frac{0.16666666666666666}{t}\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 9.2 \cdot 10^{+181}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{k \cdot \left(t \cdot k\right)} - \frac{0.16666666666666666}{t}\right)\right)\\ \end{array} \]

Alternative 20: 56.1% accurate, 18.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{k \cdot \left(t \cdot k\right)}\\ \mathbf{if}\;\ell \leq 8 \cdot 10^{+177}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \frac{\ell}{k}}{k} \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(t_1 - \frac{0.16666666666666666}{t}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ 1.0 (* k (* t k)))))
   (if (<= l 8e+177)
     (* 2.0 (* (/ (* l (/ l k)) k) t_1))
     (* 2.0 (* (* (/ l k) (/ l k)) (- t_1 (/ 0.16666666666666666 t)))))))

C

double code(double t, double l, double k) {
	double t_1 = 1.0 / (k * (t * k));
	double tmp;
	if (l <= 8e+177) {
		tmp = 2.0 * (((l * (l / k)) / k) * t_1);
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (t_1 - (0.16666666666666666 / t)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 / (k * (t * k))
    if (l <= 8d+177) then
        tmp = 2.0d0 * (((l * (l / k)) / k) * t_1)
    else
        tmp = 2.0d0 * (((l / k) * (l / k)) * (t_1 - (0.16666666666666666d0 / t)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = 1.0 / (k * (t * k));
	double tmp;
	if (l <= 8e+177) {
		tmp = 2.0 * (((l * (l / k)) / k) * t_1);
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (t_1 - (0.16666666666666666 / t)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = 1.0 / (k * (t * k))
	tmp = 0
	if l <= 8e+177:
		tmp = 2.0 * (((l * (l / k)) / k) * t_1)
	else:
		tmp = 2.0 * (((l / k) * (l / k)) * (t_1 - (0.16666666666666666 / t)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(1.0 / Float64(k * Float64(t * k)))
	tmp = 0.0
	if (l <= 8e+177)
		tmp = Float64(2.0 * Float64(Float64(Float64(l * Float64(l / k)) / k) * t_1));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(t_1 - Float64(0.16666666666666666 / t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = 1.0 / (k * (t * k));
	tmp = 0.0;
	if (l <= 8e+177)
		tmp = 2.0 * (((l * (l / k)) / k) * t_1);
	else
		tmp = 2.0 * (((l / k) * (l / k)) * (t_1 - (0.16666666666666666 / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(1.0 / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 8e+177], N[(2.0 * N[(N[(N[(l * N[(l / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 - N[(0.16666666666666666 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 8.0000000000000001e177

   1. Initial program 56.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 56.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 55.7%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 55.3%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 55.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 55.7%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 55.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 56.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 56.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 56.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 56.1%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 56.1%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 55.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)} \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right) \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}} \]

      2. pow3 55.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)}^{3}}} \]

   5. Applied egg-rr 66.4%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]

   6. Taylor expanded in k around inf 56.7%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 56.7%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 54.4%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 54.4%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 54.4%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 62.1%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      6. *-commutative 62.1%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

      7. associate-/r* 62.1%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{\sin k}^{2}}}\right) \]

   8. Simplified 62.1%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)} \]

   9. Taylor expanded in k around 0 55.9%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\frac{1}{{k}^{2} \cdot t}}\right) \]
   10. Step-by-step derivation

       1. unpow2 55.9%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{\color{blue}{\left(k \cdot k\right)} \cdot t}\right) \]

       2. associate-*r* 55.9%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{\color{blue}{k \cdot \left(k \cdot t\right)}}\right) \]

   11. Simplified 55.9%

       \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\frac{1}{k \cdot \left(k \cdot t\right)}}\right) \]
   12. Step-by-step derivation

       1. associate-*r/ 56.6%

          \[\leadsto 2 \cdot \left(\color{blue}{\frac{\frac{\ell}{k} \cdot \ell}{k}} \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\right) \]

   13. Applied egg-rr 56.6%

       \[\leadsto 2 \cdot \left(\color{blue}{\frac{\frac{\ell}{k} \cdot \ell}{k}} \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\right) \]

   if 8.0000000000000001e177 < l

   1. Initial program 35.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 35.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 35.5%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 35.5%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 35.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 35.3%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 35.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 35.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 35.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 35.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 35.3%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 35.3%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 35.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)} \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right) \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}} \]

      2. pow3 35.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)}^{3}}} \]

   5. Applied egg-rr 35.3%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]

   6. Taylor expanded in k around inf 48.1%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 48.1%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 48.1%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 48.1%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 48.1%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 60.4%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      6. *-commutative 60.4%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

      7. associate-/r* 60.4%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{\sin k}^{2}}}\right) \]

   8. Simplified 60.4%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)} \]

   9. Taylor expanded in k around 0 42.8%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}\right)}\right) \]
   10. Step-by-step derivation

       1. unpow2 42.8%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{\color{blue}{\left(k \cdot k\right)} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

       2. associate-*r* 42.8%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{\color{blue}{k \cdot \left(k \cdot t\right)}} - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

       3. associate-*r/ 42.8%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{k \cdot \left(k \cdot t\right)} - \color{blue}{\frac{0.16666666666666666 \cdot 1}{t}}\right)\right) \]

       4. metadata-eval 42.8%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{k \cdot \left(k \cdot t\right)} - \frac{\color{blue}{0.16666666666666666}}{t}\right)\right) \]

   11. Simplified 42.8%

       \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\left(\frac{1}{k \cdot \left(k \cdot t\right)} - \frac{0.16666666666666666}{t}\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 8 \cdot 10^{+177}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \frac{\ell}{k}}{k} \cdot \frac{1}{k \cdot \left(t \cdot k\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{k \cdot \left(t \cdot k\right)} - \frac{0.16666666666666666}{t}\right)\right)\\ \end{array} \]

Alternative 21: 55.2% accurate, 24.8× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{k \cdot \left(t \cdot k\right)}\right) \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* 2.0 (* (* (/ l k) (/ l k)) (/ 1.0 (* k (* t k))))))

C

double code(double t, double l, double k) {
	return 2.0 * (((l / k) * (l / k)) * (1.0 / (k * (t * k))));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * (((l / k) * (l / k)) * (1.0d0 / (k * (t * k))))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * (((l / k) * (l / k)) * (1.0 / (k * (t * k))));
}

Python

def code(t, l, k):
	return 2.0 * (((l / k) * (l / k)) * (1.0 / (k * (t * k))))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(1.0 / Float64(k * Float64(t * k)))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * (((l / k) * (l / k)) * (1.0 / (k * (t * k))));
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/l/ 54.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   2. associate-*l/ 54.3%

      \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

   3. associate-*l/ 53.9%

      \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

   4. associate-/r/ 54.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

   5. *-commutative 54.3%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

   6. associate-/l/ 54.3%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

   7. associate-*r* 54.7%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   8. *-commutative 54.7%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

   9. associate-*r* 54.7%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

   10. *-commutative 54.7%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

3. Simplified 54.7%

   \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]
4. Step-by-step derivation

   1. add-cube-cbrt 54.5%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)} \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right) \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}} \]

   2. pow3 54.5%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)}^{3}}} \]

5. Applied egg-rr 64.4%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]

6. Taylor expanded in k around inf 56.1%

   \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
7. Step-by-step derivation

   1. *-commutative 56.1%

      \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

   2. times-frac 54.0%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

   3. unpow2 54.0%

      \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   4. unpow2 54.0%

      \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   5. times-frac 62.0%

      \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   6. *-commutative 62.0%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   7. associate-/r* 62.0%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{\sin k}^{2}}}\right) \]

8. Simplified 62.0%

   \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)} \]

9. Taylor expanded in k around 0 54.2%

   \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\frac{1}{{k}^{2} \cdot t}}\right) \]
10. Step-by-step derivation

    1. unpow2 54.2%

       \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{\color{blue}{\left(k \cdot k\right)} \cdot t}\right) \]

    2. associate-*r* 54.3%

       \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{\color{blue}{k \cdot \left(k \cdot t\right)}}\right) \]

11. Simplified 54.3%

    \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\frac{1}{k \cdot \left(k \cdot t\right)}}\right) \]

12. Final simplification 54.3%

    \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{k \cdot \left(t \cdot k\right)}\right) \]

Alternative 22: 55.9% accurate, 24.8× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\frac{\ell \cdot \frac{\ell}{k}}{k} \cdot \frac{1}{k \cdot \left(t \cdot k\right)}\right) \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* 2.0 (* (/ (* l (/ l k)) k) (/ 1.0 (* k (* t k))))))

C

double code(double t, double l, double k) {
	return 2.0 * (((l * (l / k)) / k) * (1.0 / (k * (t * k))));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * (((l * (l / k)) / k) * (1.0d0 / (k * (t * k))))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * (((l * (l / k)) / k) * (1.0 / (k * (t * k))));
}

Python

def code(t, l, k):
	return 2.0 * (((l * (l / k)) / k) * (1.0 / (k * (t * k))))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64(Float64(l * Float64(l / k)) / k) * Float64(1.0 / Float64(k * Float64(t * k)))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * (((l * (l / k)) / k) * (1.0 / (k * (t * k))));
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(N[(l * N[(l / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(1.0 / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/l/ 54.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   2. associate-*l/ 54.3%

      \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

   3. associate-*l/ 53.9%

      \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

   4. associate-/r/ 54.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

   5. *-commutative 54.3%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

   6. associate-/l/ 54.3%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

   7. associate-*r* 54.7%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   8. *-commutative 54.7%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

   9. associate-*r* 54.7%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

   10. *-commutative 54.7%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

3. Simplified 54.7%

   \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]
4. Step-by-step derivation

   1. add-cube-cbrt 54.5%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)} \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right) \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}} \]

   2. pow3 54.5%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)}^{3}}} \]

5. Applied egg-rr 64.4%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]

6. Taylor expanded in k around inf 56.1%

   \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
7. Step-by-step derivation

   1. *-commutative 56.1%

      \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

   2. times-frac 54.0%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

   3. unpow2 54.0%

      \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   4. unpow2 54.0%

      \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   5. times-frac 62.0%

      \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   6. *-commutative 62.0%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   7. associate-/r* 62.0%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{\sin k}^{2}}}\right) \]

8. Simplified 62.0%

   \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)} \]

9. Taylor expanded in k around 0 54.2%

   \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\frac{1}{{k}^{2} \cdot t}}\right) \]
10. Step-by-step derivation

    1. unpow2 54.2%

       \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{\color{blue}{\left(k \cdot k\right)} \cdot t}\right) \]

    2. associate-*r* 54.3%

       \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{\color{blue}{k \cdot \left(k \cdot t\right)}}\right) \]

11. Simplified 54.3%

    \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\frac{1}{k \cdot \left(k \cdot t\right)}}\right) \]
12. Step-by-step derivation

    1. associate-*r/ 54.9%

       \[\leadsto 2 \cdot \left(\color{blue}{\frac{\frac{\ell}{k} \cdot \ell}{k}} \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\right) \]

13. Applied egg-rr 54.9%

    \[\leadsto 2 \cdot \left(\color{blue}{\frac{\frac{\ell}{k} \cdot \ell}{k}} \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\right) \]

14. Final simplification 54.9%

    \[\leadsto 2 \cdot \left(\frac{\ell \cdot \frac{\ell}{k}}{k} \cdot \frac{1}{k \cdot \left(t \cdot k\right)}\right) \]

Reproduce

herbie shell --seed 2023230 
(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))))