Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.5% → 89.7%

Time: 23.7s

Alternatives: 18

Speedup: 3.8×

• 27.3% 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: 54.5% 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: 89.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ \mathbf{if}\;t \leq -8.8 \cdot 10^{+97}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{t}{t_2}\right)}^{3}\right) \cdot \left(\tan k \cdot \left(2 + t_1\right)\right)}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-24}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{k \cdot \tan k}\right) \cdot \frac{\frac{\ell}{k}}{t \cdot \sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t \cdot \sqrt[3]{\sin k}}{t_2}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(t_1 + 1\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)) (t_2 (pow (cbrt l) 2.0)))
   (if (<= t -8.8e+97)
     (/ 2.0 (* (* (sin k) (pow (/ t t_2) 3.0)) (* (tan k) (+ 2.0 t_1))))
     (if (<= t 5e-24)
       (* (* 2.0 (/ l (* k (tan k)))) (/ (/ l k) (* t (sin k))))
       (/
        2.0
        (*
         (pow (/ (* t (cbrt (sin k))) t_2) 3.0)
         (* (tan k) (+ 1.0 (+ t_1 1.0)))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double t_2 = pow(cbrt(l), 2.0);
	double tmp;
	if (t <= -8.8e+97) {
		tmp = 2.0 / ((sin(k) * pow((t / t_2), 3.0)) * (tan(k) * (2.0 + t_1)));
	} else if (t <= 5e-24) {
		tmp = (2.0 * (l / (k * tan(k)))) * ((l / k) / (t * sin(k)));
	} else {
		tmp = 2.0 / (pow(((t * cbrt(sin(k))) / t_2), 3.0) * (tan(k) * (1.0 + (t_1 + 1.0))));
	}
	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(Math.cbrt(l), 2.0);
	double tmp;
	if (t <= -8.8e+97) {
		tmp = 2.0 / ((Math.sin(k) * Math.pow((t / t_2), 3.0)) * (Math.tan(k) * (2.0 + t_1)));
	} else if (t <= 5e-24) {
		tmp = (2.0 * (l / (k * Math.tan(k)))) * ((l / k) / (t * Math.sin(k)));
	} else {
		tmp = 2.0 / (Math.pow(((t * Math.cbrt(Math.sin(k))) / t_2), 3.0) * (Math.tan(k) * (1.0 + (t_1 + 1.0))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	t_2 = cbrt(l) ^ 2.0
	tmp = 0.0
	if (t <= -8.8e+97)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * (Float64(t / t_2) ^ 3.0)) * Float64(tan(k) * Float64(2.0 + t_1))));
	elseif (t <= 5e-24)
		tmp = Float64(Float64(2.0 * Float64(l / Float64(k * tan(k)))) * Float64(Float64(l / k) / Float64(t * sin(k))));
	else
		tmp = Float64(2.0 / Float64((Float64(Float64(t * cbrt(sin(k))) / t_2) ^ 3.0) * Float64(tan(k) * Float64(1.0 + Float64(t_1 + 1.0)))));
	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[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t, -8.8e+97], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Power[N[(t / t$95$2), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5e-24], N[(N[(2.0 * N[(l / N[(k * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[(t * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -8.8000000000000003e97

   1. Initial program 58.5%

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

      1. associate-*l* 58.5%

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

      2. +-commutative 58.5%

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

   3. Simplified 58.5%

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

      1. add-cube-cbrt 58.5%

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

      2. pow3 58.5%

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

      3. cbrt-prod 58.5%

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

      4. cbrt-div 58.5%

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

      5. rem-cbrt-cube 66.0%

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

      6. cbrt-prod 89.1%

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

      7. pow2 89.1%

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

   5. Applied egg-rr 89.1%

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

      1. pow1 89.1%

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

      2. associate-+r+ 89.1%

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

      3. metadata-eval 89.1%

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

   7. Applied egg-rr 89.1%

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

      1. unpow1 89.1%

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

      2. *-commutative 89.1%

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

      3. cube-prod 89.1%

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

      4. rem-cube-cbrt 89.2%

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

   9. Simplified 89.2%

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

   if -8.8000000000000003e97 < t < 4.9999999999999998e-24

   1. Initial program 44.2%

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

      1. associate-/l/ 44.1%

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

      2. associate-*l/ 44.1%

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

      3. associate-*l/ 44.1%

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

      4. associate-/r/ 44.2%

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

      5. *-commutative 44.2%

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

      6. associate-/l/ 44.2%

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

      7. associate-*r* 44.5%

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

      8. *-commutative 44.5%

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

      9. associate-*r* 44.5%

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

      10. *-commutative 44.5%

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

   3. Simplified 44.5%

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

   4. Taylor expanded in k around inf 78.2%

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

      1. unpow2 78.2%

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

      2. *-commutative 78.2%

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

   6. Simplified 78.2%

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

      1. associate-*r/ 78.2%

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

   8. Applied egg-rr 78.2%

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

      1. unpow2 78.2%

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

      2. associate-*r/ 78.2%

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

      3. unpow2 78.2%

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

      4. associate-*l* 82.3%

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

      5. associate-*l* 88.8%

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

      6. *-commutative 88.8%

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

   10. Simplified 88.8%

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

       1. associate-*r/ 88.8%

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

   12. Applied egg-rr 88.8%

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

       1. associate-*r* 88.8%

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

       2. associate-/r* 92.5%

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

       3. *-commutative 92.5%

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

   14. Simplified 92.5%

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

       1. associate-*r/ 89.8%

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

       2. times-frac 89.8%

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

   16. Applied egg-rr 89.8%

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

       1. associate-*r* 89.8%

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

       2. times-frac 96.5%

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

       3. associate-*r/ 96.5%

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

       4. *-commutative 96.5%

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

       5. associate-/r* 96.5%

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

       6. *-rgt-identity 96.5%

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

       7. *-commutative 96.5%

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

       8. associate-*r/ 96.6%

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

       9. associate-*l* 96.6%

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

       10. associate-*r/ 96.5%

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

       11. *-rgt-identity 96.5%

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

   18. Simplified 96.5%

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

   if 4.9999999999999998e-24 < t

   1. Initial program 66.9%

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

      1. associate-*l* 66.9%

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

      2. +-commutative 66.9%

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

   3. Simplified 66.9%

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

      1. add-cube-cbrt 66.8%

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

      2. pow3 66.8%

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

      3. cbrt-prod 66.7%

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

      4. cbrt-div 66.7%

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

      5. rem-cbrt-cube 74.9%

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

      6. cbrt-prod 93.2%

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

      7. pow2 93.2%

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

   5. Applied egg-rr 93.2%

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

      1. associate-*l/ 93.3%

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

   7. Applied egg-rr 93.3%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.8 \cdot 10^{+97}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-24}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{k \cdot \tan k}\right) \cdot \frac{\frac{\ell}{k}}{t \cdot \sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t \cdot \sqrt[3]{\sin k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right)}\\ \end{array} \]

Alternative 2: 85.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / t) ^ 2.0;
	tmp = 0.0;
	if ((2.0 / ((1.0 + (t_1 + 1.0)) * (tan(k) * (sin(k) * ((t ^ 3.0) / (l * l)))))) <= Inf)
		tmp = ((2.0 * l) / (2.0 + t_1)) * ((l / sin(k)) * ((t ^ -3.0) / tan(k)));
	else
		tmp = (2.0 * (l / (k * tan(k)))) * ((l / k) / (t * sin(k)));
	end
	tmp_2 = 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[(1.0 + N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(2.0 * l), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t, -3.0], $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(l / N[(k * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[(t * N[Sin[k], $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 86.7%

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

      1. associate-*l* 87.0%

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

      2. associate-/l/ 87.0%

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

      3. *-commutative 87.0%

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

      4. associate-*r/ 87.0%

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

      5. associate-/l* 87.0%

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

      6. associate-/r/ 81.7%

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

   3. Simplified 84.7%

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

      1. expm1-log1p-u 68.4%

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

      2. expm1-udef 61.7%

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

      3. associate-*l/ 61.7%

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

      4. *-commutative 61.7%

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

      5. div-inv 61.7%

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

      6. pow-flip 61.7%

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

      7. metadata-eval 61.7%

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

   5. Applied egg-rr 61.7%

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

      1. expm1-def 68.4%

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

      2. expm1-log1p 84.7%

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

      3. associate-*r* 84.7%

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

      4. times-frac 85.4%

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

   7. Simplified 85.4%

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

      1. *-un-lft-identity 85.4%

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

      2. associate-/l* 84.8%

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

   9. Applied egg-rr 84.8%

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

       1. *-lft-identity 84.8%

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

       2. associate-/r/ 85.5%

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

       3. associate-*l/ 85.4%

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

       4. times-frac 92.4%

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

   11. Simplified 92.4%

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

   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. associate-/l/ 0.0%

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

      2. associate-*l/ 0.0%

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

      3. associate-*l/ 0.0%

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

      4. associate-/r/ 0.1%

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

      5. *-commutative 0.1%

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

      6. associate-/l/ 0.1%

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

      7. associate-*r* 0.1%

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

      8. *-commutative 0.1%

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

      9. associate-*r* 0.1%

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

      10. *-commutative 0.1%

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

   3. Simplified 0.1%

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

   4. Taylor expanded in k around inf 54.1%

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

      1. unpow2 54.1%

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

      2. *-commutative 54.1%

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

   6. Simplified 54.1%

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

      1. associate-*r/ 54.1%

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

   8. Applied egg-rr 54.1%

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

      1. unpow2 54.1%

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

      2. associate-*r/ 54.1%

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

      3. unpow2 54.1%

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

      4. associate-*l* 63.1%

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

      5. associate-*l* 72.5%

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

      6. *-commutative 72.5%

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

   10. Simplified 72.5%

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

       1. associate-*r/ 72.5%

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

   12. Applied egg-rr 72.5%

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

       1. associate-*r* 72.6%

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

       2. associate-/r* 77.2%

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

       3. *-commutative 77.2%

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

   14. Simplified 77.2%

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

       1. associate-*r/ 72.9%

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

       2. times-frac 72.9%

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

   16. Applied egg-rr 72.9%

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

       1. associate-*r* 72.9%

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

       2. times-frac 83.1%

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

       3. associate-*r/ 83.1%

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

       4. *-commutative 83.1%

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

       5. associate-/r* 83.0%

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

       6. *-rgt-identity 83.0%

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

       7. *-commutative 83.0%

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

       8. associate-*r/ 83.1%

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

       9. associate-*l* 83.1%

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

       10. associate-*r/ 83.0%

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

       11. *-rgt-identity 83.0%

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

   18. Simplified 83.0%

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

4. Final simplification 88.7%

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

Alternative 3: 89.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ \mathbf{if}\;t \leq -2.15 \cdot 10^{+98}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {t_2}^{3}\right) \cdot \left(\tan k \cdot \left(2 + t_1\right)\right)}\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-25}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{k \cdot \tan k}\right) \cdot \frac{\frac{\ell}{k}}{t \cdot \sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(t_1 + 1\right)\right)\right) \cdot {\left(t_2 \cdot \sqrt[3]{\sin k}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)) (t_2 (/ t (pow (cbrt l) 2.0))))
   (if (<= t -2.15e+98)
     (/ 2.0 (* (* (sin k) (pow t_2 3.0)) (* (tan k) (+ 2.0 t_1))))
     (if (<= t 6.8e-25)
       (* (* 2.0 (/ l (* k (tan k)))) (/ (/ l k) (* t (sin k))))
       (/
        2.0
        (*
         (* (tan k) (+ 1.0 (+ t_1 1.0)))
         (pow (* t_2 (cbrt (sin k))) 3.0)))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double t_2 = t / pow(cbrt(l), 2.0);
	double tmp;
	if (t <= -2.15e+98) {
		tmp = 2.0 / ((sin(k) * pow(t_2, 3.0)) * (tan(k) * (2.0 + t_1)));
	} else if (t <= 6.8e-25) {
		tmp = (2.0 * (l / (k * tan(k)))) * ((l / k) / (t * sin(k)));
	} else {
		tmp = 2.0 / ((tan(k) * (1.0 + (t_1 + 1.0))) * pow((t_2 * cbrt(sin(k))), 3.0));
	}
	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 = t / Math.pow(Math.cbrt(l), 2.0);
	double tmp;
	if (t <= -2.15e+98) {
		tmp = 2.0 / ((Math.sin(k) * Math.pow(t_2, 3.0)) * (Math.tan(k) * (2.0 + t_1)));
	} else if (t <= 6.8e-25) {
		tmp = (2.0 * (l / (k * Math.tan(k)))) * ((l / k) / (t * Math.sin(k)));
	} else {
		tmp = 2.0 / ((Math.tan(k) * (1.0 + (t_1 + 1.0))) * Math.pow((t_2 * Math.cbrt(Math.sin(k))), 3.0));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	t_2 = Float64(t / (cbrt(l) ^ 2.0))
	tmp = 0.0
	if (t <= -2.15e+98)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * (t_2 ^ 3.0)) * Float64(tan(k) * Float64(2.0 + t_1))));
	elseif (t <= 6.8e-25)
		tmp = Float64(Float64(2.0 * Float64(l / Float64(k * tan(k)))) * Float64(Float64(l / k) / Float64(t * sin(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(1.0 + Float64(t_1 + 1.0))) * (Float64(t_2 * cbrt(sin(k))) ^ 3.0)));
	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[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.15e+98], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$2, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.8e-25], N[(N[(2.0 * N[(l / N[(k * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(t$95$2 * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.1500000000000001e98

   1. Initial program 58.5%

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

      1. associate-*l* 58.5%

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

      2. +-commutative 58.5%

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

   3. Simplified 58.5%

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

      1. add-cube-cbrt 58.5%

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

      2. pow3 58.5%

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

      3. cbrt-prod 58.5%

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

      4. cbrt-div 58.5%

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

      5. rem-cbrt-cube 66.0%

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

      6. cbrt-prod 89.1%

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

      7. pow2 89.1%

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

   5. Applied egg-rr 89.1%

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

      1. pow1 89.1%

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

      2. associate-+r+ 89.1%

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

      3. metadata-eval 89.1%

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

   7. Applied egg-rr 89.1%

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

      1. unpow1 89.1%

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

      2. *-commutative 89.1%

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

      3. cube-prod 89.1%

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

      4. rem-cube-cbrt 89.2%

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

   9. Simplified 89.2%

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

   if -2.1500000000000001e98 < t < 6.80000000000000003e-25

   1. Initial program 44.2%

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

      1. associate-/l/ 44.1%

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

      2. associate-*l/ 44.1%

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

      3. associate-*l/ 44.1%

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

      4. associate-/r/ 44.2%

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

      5. *-commutative 44.2%

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

      6. associate-/l/ 44.2%

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

      7. associate-*r* 44.5%

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

      8. *-commutative 44.5%

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

      9. associate-*r* 44.5%

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

      10. *-commutative 44.5%

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

   3. Simplified 44.5%

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

   4. Taylor expanded in k around inf 78.2%

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

      1. unpow2 78.2%

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

      2. *-commutative 78.2%

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

   6. Simplified 78.2%

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

      1. associate-*r/ 78.2%

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

   8. Applied egg-rr 78.2%

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

      1. unpow2 78.2%

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

      2. associate-*r/ 78.2%

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

      3. unpow2 78.2%

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

      4. associate-*l* 82.3%

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

      5. associate-*l* 88.8%

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

      6. *-commutative 88.8%

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

   10. Simplified 88.8%

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

       1. associate-*r/ 88.8%

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

   12. Applied egg-rr 88.8%

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

       1. associate-*r* 88.8%

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

       2. associate-/r* 92.5%

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

       3. *-commutative 92.5%

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

   14. Simplified 92.5%

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

       1. associate-*r/ 89.8%

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

       2. times-frac 89.8%

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

   16. Applied egg-rr 89.8%

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

       1. associate-*r* 89.8%

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

       2. times-frac 96.5%

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

       3. associate-*r/ 96.5%

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

       4. *-commutative 96.5%

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

       5. associate-/r* 96.5%

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

       6. *-rgt-identity 96.5%

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

       7. *-commutative 96.5%

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

       8. associate-*r/ 96.6%

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

       9. associate-*l* 96.6%

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

       10. associate-*r/ 96.5%

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

       11. *-rgt-identity 96.5%

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

   18. Simplified 96.5%

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

   if 6.80000000000000003e-25 < t

   1. Initial program 66.9%

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

      1. associate-*l* 66.9%

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

      2. +-commutative 66.9%

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

   3. Simplified 66.9%

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

      1. add-cube-cbrt 66.8%

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

      2. pow3 66.8%

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

      3. cbrt-prod 66.7%

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

      4. cbrt-div 66.7%

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

      5. rem-cbrt-cube 74.9%

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

      6. cbrt-prod 93.2%

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

      7. pow2 93.2%

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

   5. Applied egg-rr 93.2%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.15 \cdot 10^{+98}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-25}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{k \cdot \tan k}\right) \cdot \frac{\frac{\ell}{k}}{t \cdot \sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3}}\\ \end{array} \]

Alternative 4: 88.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -8.8 \cdot 10^{+97} \lor \neg \left(t \leq 2.7 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{k \cdot \tan k}\right) \cdot \frac{\frac{\ell}{k}}{t \cdot \sin k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -8.8e+97) (not (<= t 2.7e-24)))
   (/
    2.0
    (*
     (* (sin k) (pow (/ t (pow (cbrt l) 2.0)) 3.0))
     (* (tan k) (+ 2.0 (pow (/ k t) 2.0)))))
   (* (* 2.0 (/ l (* k (tan k)))) (/ (/ l k) (* t (sin k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -8.8e+97) || !(t <= 2.7e-24)) {
		tmp = 2.0 / ((sin(k) * pow((t / pow(cbrt(l), 2.0)), 3.0)) * (tan(k) * (2.0 + pow((k / t), 2.0))));
	} else {
		tmp = (2.0 * (l / (k * tan(k)))) * ((l / k) / (t * sin(k)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -8.8e+97) || !(t <= 2.7e-24)) {
		tmp = 2.0 / ((Math.sin(k) * Math.pow((t / Math.pow(Math.cbrt(l), 2.0)), 3.0)) * (Math.tan(k) * (2.0 + Math.pow((k / t), 2.0))));
	} else {
		tmp = (2.0 * (l / (k * Math.tan(k)))) * ((l / k) / (t * Math.sin(k)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -8.8e+97) || !(t <= 2.7e-24))
		tmp = Float64(2.0 / Float64(Float64(sin(k) * (Float64(t / (cbrt(l) ^ 2.0)) ^ 3.0)) * Float64(tan(k) * Float64(2.0 + (Float64(k / t) ^ 2.0)))));
	else
		tmp = Float64(Float64(2.0 * Float64(l / Float64(k * tan(k)))) * Float64(Float64(l / k) / Float64(t * sin(k))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -8.8e+97], N[Not[LessEqual[t, 2.7e-24]], $MachinePrecision]], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Power[N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(l / N[(k * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -8.8000000000000003e97 or 2.70000000000000007e-24 < t

   1. Initial program 63.5%

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

      1. associate-*l* 63.5%

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

      2. +-commutative 63.5%

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

   3. Simplified 63.5%

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

      1. add-cube-cbrt 63.4%

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

      2. pow3 63.4%

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

      3. cbrt-prod 63.4%

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

      4. cbrt-div 63.4%

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

      5. rem-cbrt-cube 71.2%

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

      6. cbrt-prod 91.6%

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

      7. pow2 91.6%

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

   5. Applied egg-rr 91.6%

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

      1. pow1 91.6%

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

      2. associate-+r+ 91.6%

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

      3. metadata-eval 91.6%

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

   7. Applied egg-rr 91.6%

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

      1. unpow1 91.6%

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

      2. *-commutative 91.6%

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

      3. cube-prod 88.2%

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

      4. rem-cube-cbrt 88.3%

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

   9. Simplified 88.3%

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

   if -8.8000000000000003e97 < t < 2.70000000000000007e-24

   1. Initial program 44.2%

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

      1. associate-/l/ 44.1%

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

      2. associate-*l/ 44.1%

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

      3. associate-*l/ 44.1%

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

      4. associate-/r/ 44.2%

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

      5. *-commutative 44.2%

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

      6. associate-/l/ 44.2%

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

      7. associate-*r* 44.5%

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

      8. *-commutative 44.5%

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

      9. associate-*r* 44.5%

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

      10. *-commutative 44.5%

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

   3. Simplified 44.5%

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

   4. Taylor expanded in k around inf 78.2%

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

      1. unpow2 78.2%

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

      2. *-commutative 78.2%

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

   6. Simplified 78.2%

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

      1. associate-*r/ 78.2%

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

   8. Applied egg-rr 78.2%

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

      1. unpow2 78.2%

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

      2. associate-*r/ 78.2%

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

      3. unpow2 78.2%

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

      4. associate-*l* 82.3%

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

      5. associate-*l* 88.8%

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

      6. *-commutative 88.8%

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

   10. Simplified 88.8%

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

       1. associate-*r/ 88.8%

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

   12. Applied egg-rr 88.8%

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

       1. associate-*r* 88.8%

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

       2. associate-/r* 92.5%

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

       3. *-commutative 92.5%

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

   14. Simplified 92.5%

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

       1. associate-*r/ 89.8%

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

       2. times-frac 89.8%

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

   16. Applied egg-rr 89.8%

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

       1. associate-*r* 89.8%

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

       2. times-frac 96.5%

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

       3. associate-*r/ 96.5%

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

       4. *-commutative 96.5%

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

       5. associate-/r* 96.5%

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

       6. *-rgt-identity 96.5%

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

       7. *-commutative 96.5%

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

       8. associate-*r/ 96.6%

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

       9. associate-*l* 96.6%

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

       10. associate-*r/ 96.5%

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

       11. *-rgt-identity 96.5%

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

   18. Simplified 96.5%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.8 \cdot 10^{+97} \lor \neg \left(t \leq 2.7 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{k \cdot \tan k}\right) \cdot \frac{\frac{\ell}{k}}{t \cdot \sin k}\\ \end{array} \]

Alternative 5: 87.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{if}\;t \leq -4.2 \cdot 10^{+100}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-22}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{k \cdot \tan k}\right) \cdot \frac{\frac{\ell}{k}}{t \cdot \sin k}\\ \mathbf{elif}\;t \leq 1.16 \cdot 10^{+106}:\\ \;\;\;\;\frac{2 \cdot \ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{{t}^{-3}}{\tan k}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1
         (/
          2.0
          (*
           (pow (* (/ t (pow (cbrt l) 2.0)) (cbrt (sin k))) 3.0)
           (* 2.0 k)))))
   (if (<= t -4.2e+100)
     t_1
     (if (<= t 1.45e-22)
       (* (* 2.0 (/ l (* k (tan k)))) (/ (/ l k) (* t (sin k))))
       (if (<= t 1.16e+106)
         (*
          (/ (* 2.0 l) (+ 2.0 (pow (/ k t) 2.0)))
          (* (/ l (sin k)) (/ (pow t -3.0) (tan k))))
         t_1)))))

C

double code(double t, double l, double k) {
	double t_1 = 2.0 / (pow(((t / pow(cbrt(l), 2.0)) * cbrt(sin(k))), 3.0) * (2.0 * k));
	double tmp;
	if (t <= -4.2e+100) {
		tmp = t_1;
	} else if (t <= 1.45e-22) {
		tmp = (2.0 * (l / (k * tan(k)))) * ((l / k) / (t * sin(k)));
	} else if (t <= 1.16e+106) {
		tmp = ((2.0 * l) / (2.0 + pow((k / t), 2.0))) * ((l / sin(k)) * (pow(t, -3.0) / tan(k)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = 2.0 / (Math.pow(((t / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(Math.sin(k))), 3.0) * (2.0 * k));
	double tmp;
	if (t <= -4.2e+100) {
		tmp = t_1;
	} else if (t <= 1.45e-22) {
		tmp = (2.0 * (l / (k * Math.tan(k)))) * ((l / k) / (t * Math.sin(k)));
	} else if (t <= 1.16e+106) {
		tmp = ((2.0 * l) / (2.0 + Math.pow((k / t), 2.0))) * ((l / Math.sin(k)) * (Math.pow(t, -3.0) / Math.tan(k)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(2.0 / Float64((Float64(Float64(t / (cbrt(l) ^ 2.0)) * cbrt(sin(k))) ^ 3.0) * Float64(2.0 * k)))
	tmp = 0.0
	if (t <= -4.2e+100)
		tmp = t_1;
	elseif (t <= 1.45e-22)
		tmp = Float64(Float64(2.0 * Float64(l / Float64(k * tan(k)))) * Float64(Float64(l / k) / Float64(t * sin(k))));
	elseif (t <= 1.16e+106)
		tmp = Float64(Float64(Float64(2.0 * l) / Float64(2.0 + (Float64(k / t) ^ 2.0))) * Float64(Float64(l / sin(k)) * Float64((t ^ -3.0) / tan(k))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(2.0 / N[(N[Power[N[(N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4.2e+100], t$95$1, If[LessEqual[t, 1.45e-22], N[(N[(2.0 * N[(l / N[(k * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.16e+106], N[(N[(N[(2.0 * l), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t, -3.0], $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.1999999999999997e100 or 1.16000000000000004e106 < t

   1. Initial program 62.8%

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

      1. associate-*l* 62.8%

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

      2. +-commutative 62.8%

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

   3. Simplified 62.8%

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

      1. add-cube-cbrt 62.8%

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

      2. pow3 62.8%

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

      3. cbrt-prod 62.8%

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

      4. cbrt-div 62.8%

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

      5. rem-cbrt-cube 73.0%

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

      6. cbrt-prod 93.8%

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

      7. pow2 93.8%

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

   5. Applied egg-rr 93.8%

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

   6. Taylor expanded in k around 0 86.1%

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

   if -4.1999999999999997e100 < t < 1.4500000000000001e-22

   1. Initial program 43.9%

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

      1. associate-/l/ 43.9%

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

      2. associate-*l/ 43.8%

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

      3. associate-*l/ 43.8%

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

      4. associate-/r/ 43.9%

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

      5. *-commutative 43.9%

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

      6. associate-/l/ 43.9%

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

      7. associate-*r* 44.2%

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

      8. *-commutative 44.2%

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

      9. associate-*r* 44.2%

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

      10. *-commutative 44.2%

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

   3. Simplified 44.2%

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

   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. unpow2 77.7%

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

      2. *-commutative 77.7%

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

   6. Simplified 77.7%

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

      1. associate-*r/ 77.7%

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

   8. Applied egg-rr 77.7%

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

      1. unpow2 77.7%

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

      2. associate-*r/ 77.7%

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

      3. unpow2 77.7%

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

      4. associate-*l* 81.8%

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

      5. associate-*l* 88.2%

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

      6. *-commutative 88.2%

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

   10. Simplified 88.2%

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

       1. associate-*r/ 88.2%

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

   12. Applied egg-rr 88.2%

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

       1. associate-*r* 88.3%

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

       2. associate-/r* 92.0%

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

       3. *-commutative 92.0%

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

   14. Simplified 92.0%

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

       1. associate-*r/ 89.2%

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

       2. times-frac 89.2%

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

   16. Applied egg-rr 89.2%

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

       1. associate-*r* 89.2%

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

       2. times-frac 95.9%

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

       3. associate-*r/ 95.9%

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

       4. *-commutative 95.9%

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

       5. associate-/r* 95.9%

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

       6. *-rgt-identity 95.9%

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

       7. *-commutative 95.9%

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

       8. associate-*r/ 96.0%

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

       9. associate-*l* 96.0%

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

       10. associate-*r/ 95.9%

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

       11. *-rgt-identity 95.9%

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

   18. Simplified 95.9%

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

   if 1.4500000000000001e-22 < t < 1.16000000000000004e106

   1. Initial program 68.6%

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

      1. associate-*l* 68.5%

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

      2. associate-/l/ 68.5%

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

      3. *-commutative 68.5%

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

      4. associate-*r/ 68.6%

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

      5. associate-/l* 68.4%

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

      6. associate-/r/ 68.1%

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

   3. Simplified 75.2%

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

      1. expm1-log1p-u 74.9%

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

      2. expm1-udef 47.4%

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

      3. associate-*l/ 47.4%

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

      4. *-commutative 47.4%

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

      5. div-inv 47.4%

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

      6. pow-flip 51.1%

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

      7. metadata-eval 51.1%

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

   5. Applied egg-rr 51.1%

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

      1. expm1-def 78.8%

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

      2. expm1-log1p 79.3%

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

      3. associate-*r* 79.3%

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

      4. times-frac 87.3%

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

   7. Simplified 87.3%

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

      1. *-un-lft-identity 87.3%

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

      2. associate-/l* 83.4%

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

   9. Applied egg-rr 83.4%

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

       1. *-lft-identity 83.4%

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

       2. associate-/r/ 87.3%

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

       3. associate-*l/ 87.3%

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

       4. times-frac 91.5%

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

   11. Simplified 91.5%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{+100}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-22}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{k \cdot \tan k}\right) \cdot \frac{\frac{\ell}{k}}{t \cdot \sin k}\\ \mathbf{elif}\;t \leq 1.16 \cdot 10^{+106}:\\ \;\;\;\;\frac{2 \cdot \ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{{t}^{-3}}{\tan k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \]

Alternative 6: 82.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\ell \cdot \ell\right) \cdot \frac{2}{{\left(\left(t \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{2 \cdot k}\right)}^{3}}\\ \mathbf{if}\;t \leq -3 \cdot 10^{+99}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-24}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{k \cdot \tan k}\right) \cdot \frac{\frac{\ell}{k}}{t \cdot \sin k}\\ \mathbf{elif}\;t \leq 2550000000:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+103}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\cos k}{\frac{k}{\ell} \cdot \frac{k}{\ell}}}{{\sin k}^{2}}}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1
         (*
          (* l l)
          (/ 2.0 (pow (* (* t (cbrt (sin k))) (cbrt (* 2.0 k))) 3.0)))))
   (if (<= t -3e+99)
     t_1
     (if (<= t 5e-24)
       (* (* 2.0 (/ l (* k (tan k)))) (/ (/ l k) (* t (sin k))))
       (if (<= t 2550000000.0)
         (* (/ l k) (/ l (* k (pow t 3.0))))
         (if (<= t 1.1e+103)
           (* 2.0 (/ (/ (/ (cos k) (* (/ k l) (/ k l))) (pow (sin k) 2.0)) t))
           t_1))))))

C

double code(double t, double l, double k) {
	double t_1 = (l * l) * (2.0 / pow(((t * cbrt(sin(k))) * cbrt((2.0 * k))), 3.0));
	double tmp;
	if (t <= -3e+99) {
		tmp = t_1;
	} else if (t <= 5e-24) {
		tmp = (2.0 * (l / (k * tan(k)))) * ((l / k) / (t * sin(k)));
	} else if (t <= 2550000000.0) {
		tmp = (l / k) * (l / (k * pow(t, 3.0)));
	} else if (t <= 1.1e+103) {
		tmp = 2.0 * (((cos(k) / ((k / l) * (k / l))) / pow(sin(k), 2.0)) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = (l * l) * (2.0 / Math.pow(((t * Math.cbrt(Math.sin(k))) * Math.cbrt((2.0 * k))), 3.0));
	double tmp;
	if (t <= -3e+99) {
		tmp = t_1;
	} else if (t <= 5e-24) {
		tmp = (2.0 * (l / (k * Math.tan(k)))) * ((l / k) / (t * Math.sin(k)));
	} else if (t <= 2550000000.0) {
		tmp = (l / k) * (l / (k * Math.pow(t, 3.0)));
	} else if (t <= 1.1e+103) {
		tmp = 2.0 * (((Math.cos(k) / ((k / l) * (k / l))) / Math.pow(Math.sin(k), 2.0)) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(Float64(l * l) * Float64(2.0 / (Float64(Float64(t * cbrt(sin(k))) * cbrt(Float64(2.0 * k))) ^ 3.0)))
	tmp = 0.0
	if (t <= -3e+99)
		tmp = t_1;
	elseif (t <= 5e-24)
		tmp = Float64(Float64(2.0 * Float64(l / Float64(k * tan(k)))) * Float64(Float64(l / k) / Float64(t * sin(k))));
	elseif (t <= 2550000000.0)
		tmp = Float64(Float64(l / k) * Float64(l / Float64(k * (t ^ 3.0))));
	elseif (t <= 1.1e+103)
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k) / Float64(Float64(k / l) * Float64(k / l))) / (sin(k) ^ 2.0)) / t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = 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, -3e+99], t$95$1, If[LessEqual[t, 5e-24], N[(N[(2.0 * N[(l / N[(k * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2550000000.0], N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.1e+103], N[(2.0 * N[(N[(N[(N[Cos[k], $MachinePrecision] / N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -3.00000000000000014e99 or 1.09999999999999996e103 < t

   1. Initial program 62.8%

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

      1. associate-/l/ 62.8%

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

      2. associate-*l/ 62.8%

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

      3. associate-*l/ 62.8%

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

      4. associate-/r/ 62.8%

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

      5. *-commutative 62.8%

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

      6. associate-/l/ 62.8%

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

      7. associate-*r* 62.8%

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

      8. *-commutative 62.8%

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

      9. associate-*r* 62.8%

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

      10. *-commutative 62.8%

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

   3. Simplified 62.8%

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

      1. add-cube-cbrt 62.8%

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

      2. pow3 62.8%

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

   5. Applied egg-rr 77.8%

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

   6. Taylor expanded in k around 0 77.8%

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

      1. *-commutative 77.8%

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

   8. Simplified 77.8%

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

   if -3.00000000000000014e99 < t < 4.9999999999999998e-24

   1. Initial program 43.9%

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

      1. associate-/l/ 43.9%

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

      2. associate-*l/ 43.8%

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

      3. associate-*l/ 43.8%

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

      4. associate-/r/ 43.9%

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

      5. *-commutative 43.9%

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

      6. associate-/l/ 43.9%

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

      7. associate-*r* 44.2%

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

      8. *-commutative 44.2%

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

      9. associate-*r* 44.2%

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

      10. *-commutative 44.2%

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

   3. Simplified 44.2%

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

   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. unpow2 77.7%

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

      2. *-commutative 77.7%

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

   6. Simplified 77.7%

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

      1. associate-*r/ 77.7%

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

   8. Applied egg-rr 77.7%

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

      1. unpow2 77.7%

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

      2. associate-*r/ 77.7%

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

      3. unpow2 77.7%

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

      4. associate-*l* 81.8%

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

      5. associate-*l* 88.2%

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

      6. *-commutative 88.2%

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

   10. Simplified 88.2%

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

       1. associate-*r/ 88.2%

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

   12. Applied egg-rr 88.2%

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

       1. associate-*r* 88.3%

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

       2. associate-/r* 92.0%

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

       3. *-commutative 92.0%

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

   14. Simplified 92.0%

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

       1. associate-*r/ 89.2%

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

       2. times-frac 89.2%

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

   16. Applied egg-rr 89.2%

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

       1. associate-*r* 89.2%

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

       2. times-frac 95.9%

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

       3. associate-*r/ 95.9%

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

       4. *-commutative 95.9%

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

       5. associate-/r* 95.9%

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

       6. *-rgt-identity 95.9%

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

       7. *-commutative 95.9%

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

       8. associate-*r/ 96.0%

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

       9. associate-*l* 96.0%

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

       10. associate-*r/ 95.9%

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

       11. *-rgt-identity 95.9%

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

   18. Simplified 95.9%

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

   if 4.9999999999999998e-24 < t < 2.55e9

   1. Initial program 54.6%

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

      1. associate-/l/ 54.6%

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

      2. associate-*l/ 54.8%

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

      3. associate-*l/ 54.0%

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

      4. associate-/r/ 54.4%

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

      5. *-commutative 54.4%

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

      6. associate-/l/ 54.4%

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

      7. associate-*r* 54.4%

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

      8. *-commutative 54.4%

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

      9. associate-*r* 54.4%

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

      10. *-commutative 54.4%

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

   3. Simplified 54.4%

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

   4. Taylor expanded in k around 0 54.2%

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

      1. unpow2 54.2%

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

      2. *-commutative 54.2%

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

      3. times-frac 75.0%

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

      4. unpow2 75.0%

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

   6. Simplified 75.0%

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

   7. Taylor expanded in l around 0 54.2%

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

      1. unpow2 54.2%

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

      2. *-commutative 54.2%

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

      3. unpow2 54.2%

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

      4. associate-*r* 54.2%

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

      5. times-frac 88.0%

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

   9. Simplified 88.0%

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

   if 2.55e9 < t < 1.09999999999999996e103

   1. Initial program 75.5%

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

      1. associate-/l/ 75.3%

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

      2. associate-*l/ 75.2%

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

      3. associate-*l/ 75.1%

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

      4. associate-/r/ 74.9%

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

      5. *-commutative 74.9%

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

      6. associate-/l/ 75.0%

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

      7. associate-*r* 75.1%

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

      8. *-commutative 75.1%

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

      9. associate-*r* 75.0%

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

      10. *-commutative 75.0%

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

   3. Simplified 75.0%

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

   4. Taylor expanded in k around inf 68.3%

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

      1. unpow2 68.3%

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

      2. *-commutative 68.3%

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

   6. Simplified 68.3%

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

      1. associate-*r/ 68.3%

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

   8. Applied egg-rr 68.3%

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

      1. unpow2 68.3%

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

      2. associate-*r/ 68.3%

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

      3. unpow2 68.3%

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

      4. associate-*l* 74.6%

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

      5. associate-*l* 74.6%

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

      6. *-commutative 74.6%

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

   10. Simplified 74.6%

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

   11. Taylor expanded in l around 0 68.4%

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

       1. associate-/r* 68.5%

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

       2. associate-/r* 68.5%

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

       3. unpow2 68.5%

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

       4. times-frac 68.7%

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

       5. unpow2 68.7%

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

   13. Simplified 68.7%

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

   14. Taylor expanded in k around inf 68.5%

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

       1. associate-/l* 68.5%

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

       2. unpow2 68.5%

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

       3. unpow2 68.5%

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

       4. times-frac 86.8%

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

   16. Simplified 86.8%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{+99}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{\left(\left(t \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{2 \cdot k}\right)}^{3}}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-24}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{k \cdot \tan k}\right) \cdot \frac{\frac{\ell}{k}}{t \cdot \sin k}\\ \mathbf{elif}\;t \leq 2550000000:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+103}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\cos k}{\frac{k}{\ell} \cdot \frac{k}{\ell}}}{{\sin k}^{2}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{\left(\left(t \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{2 \cdot k}\right)}^{3}}\\ \end{array} \]

Alternative 7: 79.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{+99}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{3} \cdot \frac{\frac{\ell}{k}}{k}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-22}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{k \cdot \tan k}\right) \cdot \frac{\frac{\ell}{k}}{t \cdot \sin k}\\ \mathbf{elif}\;t \leq 29500 \lor \neg \left(t \leq 6 \cdot 10^{+120}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\cos k}{\frac{k}{\ell} \cdot \frac{k}{\ell}}}{{\sin k}^{2}}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -3e+99)
   (* (pow (/ (cbrt l) t) 3.0) (/ (/ l k) k))
   (if (<= t 6e-22)
     (* (* 2.0 (/ l (* k (tan k)))) (/ (/ l k) (* t (sin k))))
     (if (or (<= t 29500.0) (not (<= t 6e+120)))
       (* (/ l k) (/ l (* k (pow t 3.0))))
       (* 2.0 (/ (/ (/ (cos k) (* (/ k l) (/ k l))) (pow (sin k) 2.0)) t))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -3e+99) {
		tmp = pow((cbrt(l) / t), 3.0) * ((l / k) / k);
	} else if (t <= 6e-22) {
		tmp = (2.0 * (l / (k * tan(k)))) * ((l / k) / (t * sin(k)));
	} else if ((t <= 29500.0) || !(t <= 6e+120)) {
		tmp = (l / k) * (l / (k * pow(t, 3.0)));
	} else {
		tmp = 2.0 * (((cos(k) / ((k / l) * (k / l))) / pow(sin(k), 2.0)) / t);
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -3e+99) {
		tmp = Math.pow((Math.cbrt(l) / t), 3.0) * ((l / k) / k);
	} else if (t <= 6e-22) {
		tmp = (2.0 * (l / (k * Math.tan(k)))) * ((l / k) / (t * Math.sin(k)));
	} else if ((t <= 29500.0) || !(t <= 6e+120)) {
		tmp = (l / k) * (l / (k * Math.pow(t, 3.0)));
	} else {
		tmp = 2.0 * (((Math.cos(k) / ((k / l) * (k / l))) / Math.pow(Math.sin(k), 2.0)) / t);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -3e+99)
		tmp = Float64((Float64(cbrt(l) / t) ^ 3.0) * Float64(Float64(l / k) / k));
	elseif (t <= 6e-22)
		tmp = Float64(Float64(2.0 * Float64(l / Float64(k * tan(k)))) * Float64(Float64(l / k) / Float64(t * sin(k))));
	elseif ((t <= 29500.0) || !(t <= 6e+120))
		tmp = Float64(Float64(l / k) * Float64(l / Float64(k * (t ^ 3.0))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k) / Float64(Float64(k / l) * Float64(k / l))) / (sin(k) ^ 2.0)) / t));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -3e+99], N[(N[Power[N[(N[Power[l, 1/3], $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6e-22], N[(N[(2.0 * N[(l / N[(k * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t, 29500.0], N[Not[LessEqual[t, 6e+120]], $MachinePrecision]], N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[Cos[k], $MachinePrecision] / N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -3.00000000000000014e99

   1. Initial program 59.8%

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

      1. associate-/l/ 59.8%

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

      2. associate-*l/ 59.8%

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

      3. associate-*l/ 59.8%

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

      4. associate-/r/ 59.8%

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

      5. *-commutative 59.8%

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

      6. associate-/l/ 59.8%

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

      7. associate-*r* 59.8%

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

      8. *-commutative 59.8%

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

      9. associate-*r* 59.8%

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

      10. *-commutative 59.8%

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

   3. Simplified 59.8%

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

   4. Taylor expanded in k around 0 50.1%

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

      1. unpow2 50.1%

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

      2. *-commutative 50.1%

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

      3. times-frac 57.3%

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

      4. unpow2 57.3%

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

   6. Simplified 57.3%

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

      1. add-cube-cbrt 57.3%

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

      2. pow2 57.3%

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

      3. cbrt-div 57.3%

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

      4. rem-cbrt-cube 57.3%

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

      5. cbrt-div 57.3%

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

      6. rem-cbrt-cube 66.3%

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

   8. Applied egg-rr 66.3%

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

      1. pow-plus 66.3%

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

      2. metadata-eval 66.3%

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

   10. Simplified 66.3%

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

   11. Taylor expanded in l around 0 66.3%

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

       1. unpow2 66.3%

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

       2. associate-/r* 71.0%

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

   13. Simplified 71.0%

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

   if -3.00000000000000014e99 < t < 5.9999999999999998e-22

   1. Initial program 43.9%

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

      1. associate-/l/ 43.9%

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

      2. associate-*l/ 43.8%

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

      3. associate-*l/ 43.8%

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

      4. associate-/r/ 43.9%

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

      5. *-commutative 43.9%

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

      6. associate-/l/ 43.9%

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

      7. associate-*r* 44.2%

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

      8. *-commutative 44.2%

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

      9. associate-*r* 44.2%

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

      10. *-commutative 44.2%

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

   3. Simplified 44.2%

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

   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. unpow2 77.7%

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

      2. *-commutative 77.7%

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

   6. Simplified 77.7%

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

      1. associate-*r/ 77.7%

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

   8. Applied egg-rr 77.7%

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

      1. unpow2 77.7%

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

      2. associate-*r/ 77.7%

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

      3. unpow2 77.7%

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

      4. associate-*l* 81.8%

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

      5. associate-*l* 88.2%

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

      6. *-commutative 88.2%

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

   10. Simplified 88.2%

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

       1. associate-*r/ 88.2%

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

   12. Applied egg-rr 88.2%

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

       1. associate-*r* 88.3%

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

       2. associate-/r* 92.0%

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

       3. *-commutative 92.0%

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

   14. Simplified 92.0%

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

       1. associate-*r/ 89.2%

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

       2. times-frac 89.2%

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

   16. Applied egg-rr 89.2%

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

       1. associate-*r* 89.2%

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

       2. times-frac 95.9%

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

       3. associate-*r/ 95.9%

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

       4. *-commutative 95.9%

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

       5. associate-/r* 95.9%

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

       6. *-rgt-identity 95.9%

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

       7. *-commutative 95.9%

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

       8. associate-*r/ 96.0%

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

       9. associate-*l* 96.0%

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

       10. associate-*r/ 95.9%

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

       11. *-rgt-identity 95.9%

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

   18. Simplified 95.9%

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

   if 5.9999999999999998e-22 < t < 29500 or 6e120 < t

   1. Initial program 64.6%

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

      1. associate-/l/ 64.6%

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

      2. associate-*l/ 64.6%

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

      3. associate-*l/ 64.5%

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

      4. associate-/r/ 64.5%

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

      5. *-commutative 64.5%

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

      6. associate-/l/ 64.5%

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

      7. associate-*r* 64.5%

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

      8. *-commutative 64.5%

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

      9. associate-*r* 64.5%

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

      10. *-commutative 64.5%

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

   3. Simplified 64.5%

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

   4. Taylor expanded in k around 0 55.3%

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

      1. unpow2 55.3%

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

      2. *-commutative 55.3%

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

      3. times-frac 61.7%

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

      4. unpow2 61.7%

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

   6. Simplified 61.7%

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

   7. Taylor expanded in l around 0 55.3%

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

      1. unpow2 55.3%

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

      2. *-commutative 55.3%

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

      3. unpow2 55.3%

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

      4. associate-*r* 64.5%

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

      5. times-frac 73.2%

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

   9. Simplified 73.2%

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

   if 29500 < t < 6e120

   1. Initial program 71.8%

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

      1. associate-/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/ 71.6%

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

      3. associate-*l/ 71.5%

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

      4. associate-/r/ 71.4%

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

      5. *-commutative 71.4%

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

      6. associate-/l/ 71.5%

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

      7. associate-*r* 71.5%

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

      8. *-commutative 71.5%

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

      9. associate-*r* 71.5%

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

      10. *-commutative 71.5%

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

   3. Simplified 71.5%

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

   4. Taylor expanded in k around inf 75.9%

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

      1. unpow2 75.9%

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

      2. *-commutative 75.9%

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

   6. Simplified 75.9%

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

      1. associate-*r/ 75.8%

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

   8. Applied egg-rr 75.8%

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

      1. unpow2 75.8%

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

      2. associate-*r/ 75.9%

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

      3. unpow2 75.9%

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

      4. associate-*l* 80.7%

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

      5. associate-*l* 80.7%

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

      6. *-commutative 80.7%

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

   10. Simplified 80.7%

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

   11. Taylor expanded in l around 0 75.9%

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

       1. associate-/r* 76.0%

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

       2. associate-/r* 76.0%

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

       3. unpow2 76.0%

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

       4. times-frac 76.2%

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

       5. unpow2 76.2%

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

   13. Simplified 76.2%

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

   14. Taylor expanded in k around inf 76.0%

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

       1. associate-/l* 76.0%

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

       2. unpow2 76.0%

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

       3. unpow2 76.0%

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

       4. times-frac 89.9%

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

   16. Simplified 89.9%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{+99}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{3} \cdot \frac{\frac{\ell}{k}}{k}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-22}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{k \cdot \tan k}\right) \cdot \frac{\frac{\ell}{k}}{t \cdot \sin k}\\ \mathbf{elif}\;t \leq 29500 \lor \neg \left(t \leq 6 \cdot 10^{+120}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\cos k}{\frac{k}{\ell} \cdot \frac{k}{\ell}}}{{\sin k}^{2}}}{t}\\ \end{array} \]

Alternative 8: 79.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{+100}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{3} \cdot \frac{\frac{\ell}{k}}{k}\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-22} \lor \neg \left(t \leq 800000000\right) \land t \leq 5.1 \cdot 10^{+120}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{k \cdot \tan k}\right) \cdot \frac{\frac{\ell}{k}}{t \cdot \sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -7e+100)
   (* (pow (/ (cbrt l) t) 3.0) (/ (/ l k) k))
   (if (or (<= t 6.8e-22) (and (not (<= t 800000000.0)) (<= t 5.1e+120)))
     (* (* 2.0 (/ l (* k (tan k)))) (/ (/ l k) (* t (sin k))))
     (* (/ l k) (/ l (* k (pow t 3.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -7e+100) {
		tmp = pow((cbrt(l) / t), 3.0) * ((l / k) / k);
	} else if ((t <= 6.8e-22) || (!(t <= 800000000.0) && (t <= 5.1e+120))) {
		tmp = (2.0 * (l / (k * tan(k)))) * ((l / k) / (t * sin(k)));
	} else {
		tmp = (l / k) * (l / (k * pow(t, 3.0)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -7e+100) {
		tmp = Math.pow((Math.cbrt(l) / t), 3.0) * ((l / k) / k);
	} else if ((t <= 6.8e-22) || (!(t <= 800000000.0) && (t <= 5.1e+120))) {
		tmp = (2.0 * (l / (k * Math.tan(k)))) * ((l / k) / (t * Math.sin(k)));
	} else {
		tmp = (l / k) * (l / (k * Math.pow(t, 3.0)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -7e+100)
		tmp = Float64((Float64(cbrt(l) / t) ^ 3.0) * Float64(Float64(l / k) / k));
	elseif ((t <= 6.8e-22) || (!(t <= 800000000.0) && (t <= 5.1e+120)))
		tmp = Float64(Float64(2.0 * Float64(l / Float64(k * tan(k)))) * Float64(Float64(l / k) / Float64(t * sin(k))));
	else
		tmp = Float64(Float64(l / k) * Float64(l / Float64(k * (t ^ 3.0))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -7e+100], N[(N[Power[N[(N[Power[l, 1/3], $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t, 6.8e-22], And[N[Not[LessEqual[t, 800000000.0]], $MachinePrecision], LessEqual[t, 5.1e+120]]], N[(N[(2.0 * N[(l / N[(k * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -6.99999999999999953e100

   1. Initial program 59.8%

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

      1. associate-/l/ 59.8%

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

      2. associate-*l/ 59.8%

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

      3. associate-*l/ 59.8%

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

      4. associate-/r/ 59.8%

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

      5. *-commutative 59.8%

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

      6. associate-/l/ 59.8%

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

      7. associate-*r* 59.8%

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

      8. *-commutative 59.8%

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

      9. associate-*r* 59.8%

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

      10. *-commutative 59.8%

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

   3. Simplified 59.8%

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

   4. Taylor expanded in k around 0 50.1%

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

      1. unpow2 50.1%

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

      2. *-commutative 50.1%

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

      3. times-frac 57.3%

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

      4. unpow2 57.3%

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

   6. Simplified 57.3%

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

      1. add-cube-cbrt 57.3%

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

      2. pow2 57.3%

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

      3. cbrt-div 57.3%

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

      4. rem-cbrt-cube 57.3%

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

      5. cbrt-div 57.3%

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

      6. rem-cbrt-cube 66.3%

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

   8. Applied egg-rr 66.3%

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

      1. pow-plus 66.3%

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

      2. metadata-eval 66.3%

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

   10. Simplified 66.3%

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

   11. Taylor expanded in l around 0 66.3%

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

       1. unpow2 66.3%

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

       2. associate-/r* 71.0%

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

   13. Simplified 71.0%

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

   if -6.99999999999999953e100 < t < 6.7999999999999997e-22 or 8e8 < t < 5.10000000000000027e120

   1. Initial program 47.4%

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

      1. associate-/l/ 47.3%

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

      2. associate-*l/ 47.3%

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

      3. associate-*l/ 47.3%

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

      4. associate-/r/ 47.3%

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

      5. *-commutative 47.3%

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

      6. associate-/l/ 47.4%

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

      7. associate-*r* 47.6%

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

      8. *-commutative 47.6%

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

      9. associate-*r* 47.6%

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

      10. *-commutative 47.6%

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

   3. Simplified 47.6%

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

   4. Taylor expanded in k around inf 77.5%

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

      1. unpow2 77.5%

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

      2. *-commutative 77.5%

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

   6. Simplified 77.5%

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

      1. associate-*r/ 77.5%

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

   8. Applied egg-rr 77.5%

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

      1. unpow2 77.5%

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

      2. associate-*r/ 77.5%

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

      3. unpow2 77.5%

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

      4. associate-*l* 81.6%

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

      5. associate-*l* 87.3%

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

      6. *-commutative 87.3%

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

   10. Simplified 87.3%

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

       1. associate-*r/ 87.3%

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

   12. Applied egg-rr 87.3%

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

       1. associate-*r* 87.3%

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

       2. associate-/r* 91.1%

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

       3. *-commutative 91.1%

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

   14. Simplified 91.1%

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

       1. associate-*r/ 88.1%

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

       2. times-frac 88.2%

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

   16. Applied egg-rr 88.2%

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

       1. associate-*r* 88.2%

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

       2. times-frac 95.2%

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

       3. associate-*r/ 95.2%

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

       4. *-commutative 95.2%

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

       5. associate-/r* 95.2%

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

       6. *-rgt-identity 95.2%

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

       7. *-commutative 95.2%

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

       8. associate-*r/ 95.2%

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

       9. associate-*l* 95.2%

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

       10. associate-*r/ 95.2%

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

       11. *-rgt-identity 95.2%

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

   18. Simplified 95.2%

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

   if 6.7999999999999997e-22 < t < 8e8 or 5.10000000000000027e120 < t

   1. Initial program 64.6%

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

      1. associate-/l/ 64.6%

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

      2. associate-*l/ 64.6%

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

      3. associate-*l/ 64.5%

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

      4. associate-/r/ 64.5%

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

      5. *-commutative 64.5%

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

      6. associate-/l/ 64.5%

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

      7. associate-*r* 64.5%

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

      8. *-commutative 64.5%

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

      9. associate-*r* 64.5%

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

      10. *-commutative 64.5%

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

   3. Simplified 64.5%

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

   4. Taylor expanded in k around 0 55.3%

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

      1. unpow2 55.3%

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

      2. *-commutative 55.3%

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

      3. times-frac 61.7%

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

      4. unpow2 61.7%

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

   6. Simplified 61.7%

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

   7. Taylor expanded in l around 0 55.3%

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

      1. unpow2 55.3%

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

      2. *-commutative 55.3%

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

      3. unpow2 55.3%

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

      4. associate-*r* 64.5%

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

      5. times-frac 73.2%

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

   9. Simplified 73.2%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{+100}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{3} \cdot \frac{\frac{\ell}{k}}{k}\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-22} \lor \neg \left(t \leq 800000000\right) \land t \leq 5.1 \cdot 10^{+120}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{k \cdot \tan k}\right) \cdot \frac{\frac{\ell}{k}}{t \cdot \sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \end{array} \]

Alternative 9: 76.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -1.2 \cdot 10^{-18} \lor \neg \left(k \leq 5.2 \cdot 10^{-16}\right):\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{3} \cdot \frac{\frac{\ell}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= k -1.2e-18) (not (<= k 5.2e-16)))
   (* l (* l (/ 2.0 (* (tan k) (* k (* k (* t (sin k))))))))
   (* (pow (/ (cbrt l) t) 3.0) (/ (/ l k) k))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((k <= -1.2e-18) || !(k <= 5.2e-16)) {
		tmp = l * (l * (2.0 / (tan(k) * (k * (k * (t * sin(k)))))));
	} else {
		tmp = pow((cbrt(l) / t), 3.0) * ((l / k) / k);
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((k <= -1.2e-18) || !(k <= 5.2e-16)) {
		tmp = l * (l * (2.0 / (Math.tan(k) * (k * (k * (t * Math.sin(k)))))));
	} else {
		tmp = Math.pow((Math.cbrt(l) / t), 3.0) * ((l / k) / k);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if ((k <= -1.2e-18) || !(k <= 5.2e-16))
		tmp = Float64(l * Float64(l * Float64(2.0 / Float64(tan(k) * Float64(k * Float64(k * Float64(t * sin(k))))))));
	else
		tmp = Float64((Float64(cbrt(l) / t) ^ 3.0) * Float64(Float64(l / k) / k));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[k, -1.2e-18], N[Not[LessEqual[k, 5.2e-16]], $MachinePrecision]], N[(l * N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k * N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[Power[l, 1/3], $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < -1.19999999999999997e-18 or 5.1999999999999997e-16 < k

   1. Initial program 49.2%

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

      1. associate-/l/ 49.2%

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

      2. associate-*l/ 49.2%

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

      3. associate-*l/ 49.2%

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

      4. associate-/r/ 49.3%

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

      5. *-commutative 49.3%

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

      6. associate-/l/ 49.3%

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

      7. associate-*r* 49.3%

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

      8. *-commutative 49.3%

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

      9. associate-*r* 49.3%

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

      10. *-commutative 49.3%

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

   3. Simplified 49.3%

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

   4. Taylor expanded in k around inf 76.3%

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

      1. unpow2 76.3%

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

      2. *-commutative 76.3%

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

   6. Simplified 76.3%

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

      1. associate-*r/ 76.3%

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

   8. Applied egg-rr 76.3%

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

      1. unpow2 76.3%

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

      2. associate-*r/ 76.3%

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

      3. unpow2 76.3%

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

      4. associate-*l* 80.6%

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

      5. associate-*l* 86.8%

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

      6. *-commutative 86.8%

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

   10. Simplified 86.8%

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

   if -1.19999999999999997e-18 < k < 5.1999999999999997e-16

   1. Initial program 57.4%

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

      1. associate-/l/ 57.4%

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

      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/ 57.3%

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

      4. associate-/r/ 57.3%

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

      5. *-commutative 57.3%

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

      6. associate-/l/ 57.3%

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

      7. associate-*r* 57.8%

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

      8. *-commutative 57.8%

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

      9. associate-*r* 57.8%

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

      10. *-commutative 57.8%

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

   3. Simplified 57.8%

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

   4. Taylor expanded in k around 0 53.1%

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

      1. unpow2 53.1%

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

      2. *-commutative 53.1%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac 59.2%

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. unpow2 59.2%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   6. Simplified 59.2%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 59.2%

         \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\frac{\ell}{{t}^{3}}} \cdot \sqrt[3]{\frac{\ell}{{t}^{3}}}\right) \cdot \sqrt[3]{\frac{\ell}{{t}^{3}}}\right)} \cdot \frac{\ell}{k \cdot k} \]

      2. pow2 59.2%

         \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{\frac{\ell}{{t}^{3}}}\right)}^{2}} \cdot \sqrt[3]{\frac{\ell}{{t}^{3}}}\right) \cdot \frac{\ell}{k \cdot k} \]

      3. cbrt-div 59.2%

         \[\leadsto \left({\color{blue}{\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{{t}^{3}}}\right)}}^{2} \cdot \sqrt[3]{\frac{\ell}{{t}^{3}}}\right) \cdot \frac{\ell}{k \cdot k} \]

      4. rem-cbrt-cube 59.1%

         \[\leadsto \left({\left(\frac{\sqrt[3]{\ell}}{\color{blue}{t}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{{t}^{3}}}\right) \cdot \frac{\ell}{k \cdot k} \]

      5. cbrt-div 59.1%

         \[\leadsto \left({\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{2} \cdot \color{blue}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{{t}^{3}}}}\right) \cdot \frac{\ell}{k \cdot k} \]

      6. rem-cbrt-cube 66.0%

         \[\leadsto \left({\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{2} \cdot \frac{\sqrt[3]{\ell}}{\color{blue}{t}}\right) \cdot \frac{\ell}{k \cdot k} \]

   8. Applied egg-rr 66.0%

      \[\leadsto \color{blue}{\left({\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{2} \cdot \frac{\sqrt[3]{\ell}}{t}\right)} \cdot \frac{\ell}{k \cdot k} \]
   9. Step-by-step derivation

      1. pow-plus 66.0%

         \[\leadsto \color{blue}{{\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{\left(2 + 1\right)}} \cdot \frac{\ell}{k \cdot k} \]

      2. metadata-eval 66.0%

         \[\leadsto {\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{\color{blue}{3}} \cdot \frac{\ell}{k \cdot k} \]

   10. Simplified 66.0%

       \[\leadsto \color{blue}{{\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{3}} \cdot \frac{\ell}{k \cdot k} \]

   11. Taylor expanded in l around 0 66.0%

       \[\leadsto {\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{3} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]
   12. Step-by-step derivation

       1. unpow2 66.0%

          \[\leadsto {\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{3} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

       2. associate-/r* 73.8%

          \[\leadsto {\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{3} \cdot \color{blue}{\frac{\frac{\ell}{k}}{k}} \]

   13. Simplified 73.8%

       \[\leadsto {\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{3} \cdot \color{blue}{\frac{\frac{\ell}{k}}{k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.2 \cdot 10^{-18} \lor \neg \left(k \leq 5.2 \cdot 10^{-16}\right):\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{3} \cdot \frac{\frac{\ell}{k}}{k}\\ \end{array} \]

Alternative 10: 79.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -1.95 \cdot 10^{-18} \lor \neg \left(k \leq 4.1 \cdot 10^{-16}\right):\\ \;\;\;\;\ell \cdot \left(\frac{\ell}{k} \cdot \frac{2}{\left(k \cdot \tan k\right) \cdot \left(t \cdot \sin k\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{3} \cdot \frac{\frac{\ell}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= k -1.95e-18) (not (<= k 4.1e-16)))
   (* l (* (/ l k) (/ 2.0 (* (* k (tan k)) (* t (sin k))))))
   (* (pow (/ (cbrt l) t) 3.0) (/ (/ l k) k))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((k <= -1.95e-18) || !(k <= 4.1e-16)) {
		tmp = l * ((l / k) * (2.0 / ((k * tan(k)) * (t * sin(k)))));
	} else {
		tmp = pow((cbrt(l) / t), 3.0) * ((l / k) / k);
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((k <= -1.95e-18) || !(k <= 4.1e-16)) {
		tmp = l * ((l / k) * (2.0 / ((k * Math.tan(k)) * (t * Math.sin(k)))));
	} else {
		tmp = Math.pow((Math.cbrt(l) / t), 3.0) * ((l / k) / k);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if ((k <= -1.95e-18) || !(k <= 4.1e-16))
		tmp = Float64(l * Float64(Float64(l / k) * Float64(2.0 / Float64(Float64(k * tan(k)) * Float64(t * sin(k))))));
	else
		tmp = Float64((Float64(cbrt(l) / t) ^ 3.0) * Float64(Float64(l / k) / k));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[k, -1.95e-18], N[Not[LessEqual[k, 4.1e-16]], $MachinePrecision]], N[(l * N[(N[(l / k), $MachinePrecision] * N[(2.0 / N[(N[(k * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[Power[l, 1/3], $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < -1.95000000000000002e-18 or 4.10000000000000006e-16 < k

   1. Initial program 49.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 49.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 49.2%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 49.2%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 49.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 49.3%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 49.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 49.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 49.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 49.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 49.3%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 49.3%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 76.3%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. unpow2 76.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(\sin k \cdot t\right)\right)} \]

      2. *-commutative 76.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \sin k\right)}\right)} \]

   6. Simplified 76.3%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]
   7. Step-by-step derivation

      1. associate-*r/ 76.3%

         \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]

   8. Applied egg-rr 76.3%

      \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]
   9. Step-by-step derivation

      1. unpow2 76.3%

         \[\leadsto \frac{\color{blue}{{\ell}^{2}} \cdot 2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)} \]

      2. associate-*r/ 76.3%

         \[\leadsto \color{blue}{{\ell}^{2} \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]

      3. unpow2 76.3%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)} \]

      4. associate-*l* 80.6%

         \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\right)} \]

      5. associate-*l* 86.8%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \color{blue}{\left(k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)\right)}}\right) \]

      6. *-commutative 86.8%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(k \cdot \left(k \cdot \color{blue}{\left(\sin k \cdot t\right)}\right)\right)}\right) \]

   10. Simplified 86.8%

       \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(k \cdot \left(k \cdot \left(\sin k \cdot t\right)\right)\right)}\right)} \]
   11. Step-by-step derivation

       1. associate-*r/ 86.8%

          \[\leadsto \ell \cdot \color{blue}{\frac{\ell \cdot 2}{\tan k \cdot \left(k \cdot \left(k \cdot \left(\sin k \cdot t\right)\right)\right)}} \]

   12. Applied egg-rr 86.8%

       \[\leadsto \ell \cdot \color{blue}{\frac{\ell \cdot 2}{\tan k \cdot \left(k \cdot \left(k \cdot \left(\sin k \cdot t\right)\right)\right)}} \]
   13. Step-by-step derivation

       1. associate-*r* 86.8%

          \[\leadsto \ell \cdot \frac{\ell \cdot 2}{\color{blue}{\left(\tan k \cdot k\right) \cdot \left(k \cdot \left(\sin k \cdot t\right)\right)}} \]

       2. associate-/r* 89.3%

          \[\leadsto \ell \cdot \color{blue}{\frac{\frac{\ell \cdot 2}{\tan k \cdot k}}{k \cdot \left(\sin k \cdot t\right)}} \]

       3. *-commutative 89.3%

          \[\leadsto \ell \cdot \frac{\frac{\color{blue}{2 \cdot \ell}}{\tan k \cdot k}}{k \cdot \left(\sin k \cdot t\right)} \]

   14. Simplified 89.3%

       \[\leadsto \ell \cdot \color{blue}{\frac{\frac{2 \cdot \ell}{\tan k \cdot k}}{k \cdot \left(\sin k \cdot t\right)}} \]
   15. Step-by-step derivation

       1. div-inv 89.3%

          \[\leadsto \ell \cdot \color{blue}{\left(\frac{2 \cdot \ell}{\tan k \cdot k} \cdot \frac{1}{k \cdot \left(\sin k \cdot t\right)}\right)} \]

       2. times-frac 89.2%

          \[\leadsto \ell \cdot \left(\color{blue}{\left(\frac{2}{\tan k} \cdot \frac{\ell}{k}\right)} \cdot \frac{1}{k \cdot \left(\sin k \cdot t\right)}\right) \]

   16. Applied egg-rr 89.2%

       \[\leadsto \ell \cdot \color{blue}{\left(\left(\frac{2}{\tan k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{k \cdot \left(\sin k \cdot t\right)}\right)} \]
   17. Step-by-step derivation

       1. *-commutative 89.2%

          \[\leadsto \ell \cdot \color{blue}{\left(\frac{1}{k \cdot \left(\sin k \cdot t\right)} \cdot \left(\frac{2}{\tan k} \cdot \frac{\ell}{k}\right)\right)} \]

       2. associate-*l/ 89.3%

          \[\leadsto \ell \cdot \left(\frac{1}{k \cdot \left(\sin k \cdot t\right)} \cdot \color{blue}{\frac{2 \cdot \frac{\ell}{k}}{\tan k}}\right) \]

       3. associate-*r/ 89.3%

          \[\leadsto \ell \cdot \left(\frac{1}{k \cdot \left(\sin k \cdot t\right)} \cdot \frac{\color{blue}{\frac{2 \cdot \ell}{k}}}{\tan k}\right) \]

       4. associate-/r* 89.3%

          \[\leadsto \ell \cdot \left(\frac{1}{k \cdot \left(\sin k \cdot t\right)} \cdot \color{blue}{\frac{2 \cdot \ell}{k \cdot \tan k}}\right) \]

       5. times-frac 86.8%

          \[\leadsto \ell \cdot \color{blue}{\frac{1 \cdot \left(2 \cdot \ell\right)}{\left(k \cdot \left(\sin k \cdot t\right)\right) \cdot \left(k \cdot \tan k\right)}} \]

       6. *-lft-identity 86.8%

          \[\leadsto \ell \cdot \frac{\color{blue}{2 \cdot \ell}}{\left(k \cdot \left(\sin k \cdot t\right)\right) \cdot \left(k \cdot \tan k\right)} \]

       7. associate-*l* 86.9%

          \[\leadsto \ell \cdot \frac{2 \cdot \ell}{\color{blue}{k \cdot \left(\left(\sin k \cdot t\right) \cdot \left(k \cdot \tan k\right)\right)}} \]

       8. *-commutative 86.9%

          \[\leadsto \ell \cdot \frac{\color{blue}{\ell \cdot 2}}{k \cdot \left(\left(\sin k \cdot t\right) \cdot \left(k \cdot \tan k\right)\right)} \]

       9. times-frac 89.3%

          \[\leadsto \ell \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{2}{\left(\sin k \cdot t\right) \cdot \left(k \cdot \tan k\right)}\right)} \]

   18. Simplified 89.3%

       \[\leadsto \ell \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{2}{\left(\sin k \cdot t\right) \cdot \left(k \cdot \tan k\right)}\right)} \]

   if -1.95000000000000002e-18 < k < 4.10000000000000006e-16

   1. Initial program 57.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 57.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      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/ 57.3%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 57.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 57.3%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 57.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 57.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 57.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 57.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 57.8%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 57.8%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around 0 53.1%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 53.1%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 53.1%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac 59.2%

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. unpow2 59.2%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   6. Simplified 59.2%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 59.2%

         \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\frac{\ell}{{t}^{3}}} \cdot \sqrt[3]{\frac{\ell}{{t}^{3}}}\right) \cdot \sqrt[3]{\frac{\ell}{{t}^{3}}}\right)} \cdot \frac{\ell}{k \cdot k} \]

      2. pow2 59.2%

         \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{\frac{\ell}{{t}^{3}}}\right)}^{2}} \cdot \sqrt[3]{\frac{\ell}{{t}^{3}}}\right) \cdot \frac{\ell}{k \cdot k} \]

      3. cbrt-div 59.2%

         \[\leadsto \left({\color{blue}{\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{{t}^{3}}}\right)}}^{2} \cdot \sqrt[3]{\frac{\ell}{{t}^{3}}}\right) \cdot \frac{\ell}{k \cdot k} \]

      4. rem-cbrt-cube 59.1%

         \[\leadsto \left({\left(\frac{\sqrt[3]{\ell}}{\color{blue}{t}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{{t}^{3}}}\right) \cdot \frac{\ell}{k \cdot k} \]

      5. cbrt-div 59.1%

         \[\leadsto \left({\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{2} \cdot \color{blue}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{{t}^{3}}}}\right) \cdot \frac{\ell}{k \cdot k} \]

      6. rem-cbrt-cube 66.0%

         \[\leadsto \left({\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{2} \cdot \frac{\sqrt[3]{\ell}}{\color{blue}{t}}\right) \cdot \frac{\ell}{k \cdot k} \]

   8. Applied egg-rr 66.0%

      \[\leadsto \color{blue}{\left({\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{2} \cdot \frac{\sqrt[3]{\ell}}{t}\right)} \cdot \frac{\ell}{k \cdot k} \]
   9. Step-by-step derivation

      1. pow-plus 66.0%

         \[\leadsto \color{blue}{{\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{\left(2 + 1\right)}} \cdot \frac{\ell}{k \cdot k} \]

      2. metadata-eval 66.0%

         \[\leadsto {\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{\color{blue}{3}} \cdot \frac{\ell}{k \cdot k} \]

   10. Simplified 66.0%

       \[\leadsto \color{blue}{{\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{3}} \cdot \frac{\ell}{k \cdot k} \]

   11. Taylor expanded in l around 0 66.0%

       \[\leadsto {\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{3} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]
   12. Step-by-step derivation

       1. unpow2 66.0%

          \[\leadsto {\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{3} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

       2. associate-/r* 73.8%

          \[\leadsto {\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{3} \cdot \color{blue}{\frac{\frac{\ell}{k}}{k}} \]

   13. Simplified 73.8%

       \[\leadsto {\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{3} \cdot \color{blue}{\frac{\frac{\ell}{k}}{k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.95 \cdot 10^{-18} \lor \neg \left(k \leq 4.1 \cdot 10^{-16}\right):\\ \;\;\;\;\ell \cdot \left(\frac{\ell}{k} \cdot \frac{2}{\left(k \cdot \tan k\right) \cdot \left(t \cdot \sin k\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{3} \cdot \frac{\frac{\ell}{k}}{k}\\ \end{array} \]

Alternative 11: 66.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.0062 \lor \neg \left(t \leq 1.35 \cdot 10^{-25}\right):\\ \;\;\;\;{\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{3} \cdot \frac{\frac{\ell}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{\sin k}^{2}}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -0.0062) (not (<= t 1.35e-25)))
   (* (pow (/ (cbrt l) t) 3.0) (/ (/ l k) k))
   (* 2.0 (/ (/ (* (/ l k) (/ l k)) (pow (sin k) 2.0)) t))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -0.0062) || !(t <= 1.35e-25)) {
		tmp = pow((cbrt(l) / t), 3.0) * ((l / k) / k);
	} else {
		tmp = 2.0 * ((((l / k) * (l / k)) / pow(sin(k), 2.0)) / t);
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -0.0062) || !(t <= 1.35e-25)) {
		tmp = Math.pow((Math.cbrt(l) / t), 3.0) * ((l / k) / k);
	} else {
		tmp = 2.0 * ((((l / k) * (l / k)) / Math.pow(Math.sin(k), 2.0)) / t);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -0.0062) || !(t <= 1.35e-25))
		tmp = Float64((Float64(cbrt(l) / t) ^ 3.0) * Float64(Float64(l / k) / k));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(l / k) * Float64(l / k)) / (sin(k) ^ 2.0)) / t));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -0.0062], N[Not[LessEqual[t, 1.35e-25]], $MachinePrecision]], N[(N[Power[N[(N[Power[l, 1/3], $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.00619999999999999978 or 1.35000000000000008e-25 < t

   1. Initial program 65.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 65.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 65.8%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 65.7%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 65.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 65.0%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 65.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 65.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 65.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 65.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 65.0%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 65.0%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around 0 55.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 55.3%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 55.3%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac 60.2%

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. unpow2 60.2%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   6. Simplified 60.2%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 60.2%

         \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\frac{\ell}{{t}^{3}}} \cdot \sqrt[3]{\frac{\ell}{{t}^{3}}}\right) \cdot \sqrt[3]{\frac{\ell}{{t}^{3}}}\right)} \cdot \frac{\ell}{k \cdot k} \]

      2. pow2 60.2%

         \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{\frac{\ell}{{t}^{3}}}\right)}^{2}} \cdot \sqrt[3]{\frac{\ell}{{t}^{3}}}\right) \cdot \frac{\ell}{k \cdot k} \]

      3. cbrt-div 60.2%

         \[\leadsto \left({\color{blue}{\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{{t}^{3}}}\right)}}^{2} \cdot \sqrt[3]{\frac{\ell}{{t}^{3}}}\right) \cdot \frac{\ell}{k \cdot k} \]

      4. rem-cbrt-cube 60.1%

         \[\leadsto \left({\left(\frac{\sqrt[3]{\ell}}{\color{blue}{t}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{{t}^{3}}}\right) \cdot \frac{\ell}{k \cdot k} \]

      5. cbrt-div 60.1%

         \[\leadsto \left({\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{2} \cdot \color{blue}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{{t}^{3}}}}\right) \cdot \frac{\ell}{k \cdot k} \]

      6. rem-cbrt-cube 65.6%

         \[\leadsto \left({\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{2} \cdot \frac{\sqrt[3]{\ell}}{\color{blue}{t}}\right) \cdot \frac{\ell}{k \cdot k} \]

   8. Applied egg-rr 65.6%

      \[\leadsto \color{blue}{\left({\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{2} \cdot \frac{\sqrt[3]{\ell}}{t}\right)} \cdot \frac{\ell}{k \cdot k} \]
   9. Step-by-step derivation

      1. pow-plus 65.6%

         \[\leadsto \color{blue}{{\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{\left(2 + 1\right)}} \cdot \frac{\ell}{k \cdot k} \]

      2. metadata-eval 65.6%

         \[\leadsto {\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{\color{blue}{3}} \cdot \frac{\ell}{k \cdot k} \]

   10. Simplified 65.6%

       \[\leadsto \color{blue}{{\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{3}} \cdot \frac{\ell}{k \cdot k} \]

   11. Taylor expanded in l around 0 65.6%

       \[\leadsto {\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{3} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]
   12. Step-by-step derivation

       1. unpow2 65.6%

          \[\leadsto {\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{3} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

       2. associate-/r* 71.3%

          \[\leadsto {\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{3} \cdot \color{blue}{\frac{\frac{\ell}{k}}{k}} \]

   13. Simplified 71.3%

       \[\leadsto {\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{3} \cdot \color{blue}{\frac{\frac{\ell}{k}}{k}} \]

   if -0.00619999999999999978 < t < 1.35000000000000008e-25

   1. Initial program 39.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 39.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 39.5%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 39.5%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 40.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 40.3%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 40.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 40.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 40.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 40.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 40.7%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 40.7%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 78.5%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. unpow2 78.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(\sin k \cdot t\right)\right)} \]

      2. *-commutative 78.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \sin k\right)}\right)} \]

   6. Simplified 78.5%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]
   7. Step-by-step derivation

      1. associate-*r/ 78.6%

         \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]

   8. Applied egg-rr 78.6%

      \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]
   9. Step-by-step derivation

      1. unpow2 78.6%

         \[\leadsto \frac{\color{blue}{{\ell}^{2}} \cdot 2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)} \]

      2. associate-*r/ 78.5%

         \[\leadsto \color{blue}{{\ell}^{2} \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]

      3. unpow2 78.5%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)} \]

      4. associate-*l* 83.0%

         \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\right)} \]

      5. associate-*l* 90.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \color{blue}{\left(k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)\right)}}\right) \]

      6. *-commutative 90.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(k \cdot \left(k \cdot \color{blue}{\left(\sin k \cdot t\right)}\right)\right)}\right) \]

   10. Simplified 90.4%

       \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(k \cdot \left(k \cdot \left(\sin k \cdot t\right)\right)\right)}\right)} \]

   11. Taylor expanded in l around 0 78.5%

       \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   12. Step-by-step derivation

       1. associate-/r* 76.3%

          \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{{\sin k}^{2} \cdot t}} \]

       2. associate-/r* 76.4%

          \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{{\sin k}^{2}}}{t}} \]

       3. unpow2 76.4%

          \[\leadsto 2 \cdot \frac{\frac{\frac{\cos k \cdot {\ell}^{2}}{\color{blue}{k \cdot k}}}{{\sin k}^{2}}}{t} \]

       4. times-frac 81.8%

          \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\frac{\cos k}{k} \cdot \frac{{\ell}^{2}}{k}}}{{\sin k}^{2}}}{t} \]

       5. unpow2 81.8%

          \[\leadsto 2 \cdot \frac{\frac{\frac{\cos k}{k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{k}}{{\sin k}^{2}}}{t} \]

   13. Simplified 81.8%

       \[\leadsto \color{blue}{2 \cdot \frac{\frac{\frac{\cos k}{k} \cdot \frac{\ell \cdot \ell}{k}}{{\sin k}^{2}}}{t}} \]

   14. Taylor expanded in k around 0 65.0%

       \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2}}}}{{\sin k}^{2}}}{t} \]
   15. Step-by-step derivation

       1. unpow2 65.0%

          \[\leadsto 2 \cdot \frac{\frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{{\sin k}^{2}}}{t} \]

       2. unpow2 65.0%

          \[\leadsto 2 \cdot \frac{\frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{{\sin k}^{2}}}{t} \]

       3. times-frac 69.9%

          \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{{\sin k}^{2}}}{t} \]

   16. Simplified 69.9%

       \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{{\sin k}^{2}}}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.0062 \lor \neg \left(t \leq 1.35 \cdot 10^{-25}\right):\\ \;\;\;\;{\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{3} \cdot \frac{\frac{\ell}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{\sin k}^{2}}}{t}\\ \end{array} \]

Alternative 12: 65.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.6 \cdot 10^{-40} \lor \neg \left(t \leq 5 \cdot 10^{-49}\right):\\ \;\;\;\;{\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{3} \cdot \frac{\frac{\ell}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2 \cdot \ell}{{k}^{4}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -7.6e-40) (not (<= t 5e-49)))
   (* (pow (/ (cbrt l) t) 3.0) (/ (/ l k) k))
   (* (/ l t) (/ (* 2.0 l) (pow k 4.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -7.6e-40) || !(t <= 5e-49)) {
		tmp = pow((cbrt(l) / t), 3.0) * ((l / k) / k);
	} else {
		tmp = (l / t) * ((2.0 * l) / pow(k, 4.0));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -7.6e-40) || !(t <= 5e-49)) {
		tmp = Math.pow((Math.cbrt(l) / t), 3.0) * ((l / k) / k);
	} else {
		tmp = (l / t) * ((2.0 * l) / Math.pow(k, 4.0));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -7.6e-40) || !(t <= 5e-49))
		tmp = Float64((Float64(cbrt(l) / t) ^ 3.0) * Float64(Float64(l / k) / k));
	else
		tmp = Float64(Float64(l / t) * Float64(Float64(2.0 * l) / (k ^ 4.0)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -7.6e-40], N[Not[LessEqual[t, 5e-49]], $MachinePrecision]], N[(N[Power[N[(N[Power[l, 1/3], $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(N[(l / t), $MachinePrecision] * N[(N[(2.0 * l), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -7.5999999999999998e-40 or 4.9999999999999999e-49 < t

   1. Initial program 67.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 67.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 67.8%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 67.8%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 67.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 67.1%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 67.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 67.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 67.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 67.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 67.2%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 67.2%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around 0 58.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 58.6%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 58.6%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac 63.0%

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. unpow2 63.0%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   6. Simplified 63.0%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 62.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\frac{\ell}{{t}^{3}}} \cdot \sqrt[3]{\frac{\ell}{{t}^{3}}}\right) \cdot \sqrt[3]{\frac{\ell}{{t}^{3}}}\right)} \cdot \frac{\ell}{k \cdot k} \]

      2. pow2 62.9%

         \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{\frac{\ell}{{t}^{3}}}\right)}^{2}} \cdot \sqrt[3]{\frac{\ell}{{t}^{3}}}\right) \cdot \frac{\ell}{k \cdot k} \]

      3. cbrt-div 63.0%

         \[\leadsto \left({\color{blue}{\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{{t}^{3}}}\right)}}^{2} \cdot \sqrt[3]{\frac{\ell}{{t}^{3}}}\right) \cdot \frac{\ell}{k \cdot k} \]

      4. rem-cbrt-cube 62.9%

         \[\leadsto \left({\left(\frac{\sqrt[3]{\ell}}{\color{blue}{t}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{{t}^{3}}}\right) \cdot \frac{\ell}{k \cdot k} \]

      5. cbrt-div 62.9%

         \[\leadsto \left({\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{2} \cdot \color{blue}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{{t}^{3}}}}\right) \cdot \frac{\ell}{k \cdot k} \]

      6. rem-cbrt-cube 67.8%

         \[\leadsto \left({\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{2} \cdot \frac{\sqrt[3]{\ell}}{\color{blue}{t}}\right) \cdot \frac{\ell}{k \cdot k} \]

   8. Applied egg-rr 67.8%

      \[\leadsto \color{blue}{\left({\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{2} \cdot \frac{\sqrt[3]{\ell}}{t}\right)} \cdot \frac{\ell}{k \cdot k} \]
   9. Step-by-step derivation

      1. pow-plus 67.8%

         \[\leadsto \color{blue}{{\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{\left(2 + 1\right)}} \cdot \frac{\ell}{k \cdot k} \]

      2. metadata-eval 67.8%

         \[\leadsto {\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{\color{blue}{3}} \cdot \frac{\ell}{k \cdot k} \]

   10. Simplified 67.8%

       \[\leadsto \color{blue}{{\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{3}} \cdot \frac{\ell}{k \cdot k} \]

   11. Taylor expanded in l around 0 67.8%

       \[\leadsto {\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{3} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]
   12. Step-by-step derivation

       1. unpow2 67.8%

          \[\leadsto {\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{3} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

       2. associate-/r* 72.7%

          \[\leadsto {\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{3} \cdot \color{blue}{\frac{\frac{\ell}{k}}{k}} \]

   13. Simplified 72.7%

       \[\leadsto {\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{3} \cdot \color{blue}{\frac{\frac{\ell}{k}}{k}} \]

   if -7.5999999999999998e-40 < t < 4.9999999999999999e-49

   1. Initial program 33.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 33.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 32.9%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 32.9%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 33.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 33.8%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 33.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 34.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 34.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 34.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 34.2%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 34.2%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 77.8%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. unpow2 77.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(\sin k \cdot t\right)\right)} \]

      2. *-commutative 77.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \sin k\right)}\right)} \]

   6. Simplified 77.8%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]
   7. Step-by-step derivation

      1. associate-*r/ 77.8%

         \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]

   8. Applied egg-rr 77.8%

      \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]
   9. Step-by-step derivation

      1. unpow2 77.8%

         \[\leadsto \frac{\color{blue}{{\ell}^{2}} \cdot 2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)} \]

      2. associate-*r/ 77.8%

         \[\leadsto \color{blue}{{\ell}^{2} \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]

      3. unpow2 77.8%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)} \]

      4. associate-*l* 83.0%

         \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\right)} \]

      5. associate-*l* 91.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \color{blue}{\left(k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)\right)}}\right) \]

      6. *-commutative 91.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(k \cdot \left(k \cdot \color{blue}{\left(\sin k \cdot t\right)}\right)\right)}\right) \]

   10. Simplified 91.4%

       \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(k \cdot \left(k \cdot \left(\sin k \cdot t\right)\right)\right)}\right)} \]

   11. Taylor expanded in k around 0 59.0%

       \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   12. Step-by-step derivation

       1. associate-*r/ 59.0%

          \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

       2. *-commutative 59.0%

          \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot 2}}{{k}^{4} \cdot t} \]

       3. unpow2 59.0%

          \[\leadsto \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot 2}{{k}^{4} \cdot t} \]

       4. associate-*l* 59.0%

          \[\leadsto \frac{\color{blue}{\ell \cdot \left(\ell \cdot 2\right)}}{{k}^{4} \cdot t} \]

       5. *-commutative 59.0%

          \[\leadsto \frac{\ell \cdot \left(\ell \cdot 2\right)}{\color{blue}{t \cdot {k}^{4}}} \]

       6. times-frac 65.3%

          \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell \cdot 2}{{k}^{4}}} \]

       7. *-commutative 65.3%

          \[\leadsto \frac{\ell}{t} \cdot \frac{\color{blue}{2 \cdot \ell}}{{k}^{4}} \]

   13. Simplified 65.3%

       \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{2 \cdot \ell}{{k}^{4}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.6 \cdot 10^{-40} \lor \neg \left(t \leq 5 \cdot 10^{-49}\right):\\ \;\;\;\;{\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{3} \cdot \frac{\frac{\ell}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2 \cdot \ell}{{k}^{4}}\\ \end{array} \]

Alternative 13: 61.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{-35} \lor \neg \left(t \leq 4.8 \cdot 10^{-48}\right):\\ \;\;\;\;\left(\ell \cdot {t}^{-3}\right) \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2 \cdot \ell}{{k}^{4}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -3.8e-35) (not (<= t 4.8e-48)))
   (* (* l (pow t -3.0)) (/ l (* k k)))
   (* (/ l t) (/ (* 2.0 l) (pow k 4.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -3.8e-35) || !(t <= 4.8e-48)) {
		tmp = (l * pow(t, -3.0)) * (l / (k * k));
	} else {
		tmp = (l / t) * ((2.0 * 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 ((t <= (-3.8d-35)) .or. (.not. (t <= 4.8d-48))) then
        tmp = (l * (t ** (-3.0d0))) * (l / (k * k))
    else
        tmp = (l / t) * ((2.0d0 * l) / (k ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -3.8e-35) || !(t <= 4.8e-48)) {
		tmp = (l * Math.pow(t, -3.0)) * (l / (k * k));
	} else {
		tmp = (l / t) * ((2.0 * l) / Math.pow(k, 4.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -3.8e-35) or not (t <= 4.8e-48):
		tmp = (l * math.pow(t, -3.0)) * (l / (k * k))
	else:
		tmp = (l / t) * ((2.0 * l) / math.pow(k, 4.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -3.8e-35) || !(t <= 4.8e-48))
		tmp = Float64(Float64(l * (t ^ -3.0)) * Float64(l / Float64(k * k)));
	else
		tmp = Float64(Float64(l / t) * Float64(Float64(2.0 * l) / (k ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -3.8e-35) || ~((t <= 4.8e-48)))
		tmp = (l * (t ^ -3.0)) * (l / (k * k));
	else
		tmp = (l / t) * ((2.0 * l) / (k ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -3.8e-35], N[Not[LessEqual[t, 4.8e-48]], $MachinePrecision]], N[(N[(l * N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t), $MachinePrecision] * N[(N[(2.0 * l), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.8000000000000001e-35 or 4.8e-48 < t

   1. Initial program 67.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 67.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 67.8%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 67.8%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 67.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 67.1%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 67.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 67.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 67.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 67.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 67.2%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 67.2%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around 0 58.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 58.6%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 58.6%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac 63.0%

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. unpow2 63.0%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   6. Simplified 63.0%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 57.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell}{{t}^{3}}\right)\right)} \cdot \frac{\ell}{k \cdot k} \]

      2. expm1-udef 52.5%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\ell}{{t}^{3}}\right)} - 1\right)} \cdot \frac{\ell}{k \cdot k} \]

      3. div-inv 52.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\ell \cdot \frac{1}{{t}^{3}}}\right)} - 1\right) \cdot \frac{\ell}{k \cdot k} \]

      4. pow-flip 52.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\ell \cdot \color{blue}{{t}^{\left(-3\right)}}\right)} - 1\right) \cdot \frac{\ell}{k \cdot k} \]

      5. metadata-eval 52.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\ell \cdot {t}^{\color{blue}{-3}}\right)} - 1\right) \cdot \frac{\ell}{k \cdot k} \]

   8. Applied egg-rr 52.5%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\ell \cdot {t}^{-3}\right)} - 1\right)} \cdot \frac{\ell}{k \cdot k} \]
   9. Step-by-step derivation

      1. expm1-def 58.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot {t}^{-3}\right)\right)} \cdot \frac{\ell}{k \cdot k} \]

      2. expm1-log1p 63.5%

         \[\leadsto \color{blue}{\left(\ell \cdot {t}^{-3}\right)} \cdot \frac{\ell}{k \cdot k} \]

   10. Simplified 63.5%

       \[\leadsto \color{blue}{\left(\ell \cdot {t}^{-3}\right)} \cdot \frac{\ell}{k \cdot k} \]

   if -3.8000000000000001e-35 < t < 4.8e-48

   1. Initial program 33.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 33.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 32.9%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 32.9%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 33.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 33.8%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 33.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 34.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 34.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 34.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 34.2%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 34.2%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 77.8%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. unpow2 77.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(\sin k \cdot t\right)\right)} \]

      2. *-commutative 77.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \sin k\right)}\right)} \]

   6. Simplified 77.8%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]
   7. Step-by-step derivation

      1. associate-*r/ 77.8%

         \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]

   8. Applied egg-rr 77.8%

      \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]
   9. Step-by-step derivation

      1. unpow2 77.8%

         \[\leadsto \frac{\color{blue}{{\ell}^{2}} \cdot 2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)} \]

      2. associate-*r/ 77.8%

         \[\leadsto \color{blue}{{\ell}^{2} \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]

      3. unpow2 77.8%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)} \]

      4. associate-*l* 83.0%

         \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\right)} \]

      5. associate-*l* 91.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \color{blue}{\left(k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)\right)}}\right) \]

      6. *-commutative 91.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(k \cdot \left(k \cdot \color{blue}{\left(\sin k \cdot t\right)}\right)\right)}\right) \]

   10. Simplified 91.4%

       \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(k \cdot \left(k \cdot \left(\sin k \cdot t\right)\right)\right)}\right)} \]

   11. Taylor expanded in k around 0 59.0%

       \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   12. Step-by-step derivation

       1. associate-*r/ 59.0%

          \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

       2. *-commutative 59.0%

          \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot 2}}{{k}^{4} \cdot t} \]

       3. unpow2 59.0%

          \[\leadsto \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot 2}{{k}^{4} \cdot t} \]

       4. associate-*l* 59.0%

          \[\leadsto \frac{\color{blue}{\ell \cdot \left(\ell \cdot 2\right)}}{{k}^{4} \cdot t} \]

       5. *-commutative 59.0%

          \[\leadsto \frac{\ell \cdot \left(\ell \cdot 2\right)}{\color{blue}{t \cdot {k}^{4}}} \]

       6. times-frac 65.3%

          \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell \cdot 2}{{k}^{4}}} \]

       7. *-commutative 65.3%

          \[\leadsto \frac{\ell}{t} \cdot \frac{\color{blue}{2 \cdot \ell}}{{k}^{4}} \]

   13. Simplified 65.3%

       \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{2 \cdot \ell}{{k}^{4}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{-35} \lor \neg \left(t \leq 4.8 \cdot 10^{-48}\right):\\ \;\;\;\;\left(\ell \cdot {t}^{-3}\right) \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2 \cdot \ell}{{k}^{4}}\\ \end{array} \]

Alternative 14: 67.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6.6 \cdot 10^{-37} \lor \neg \left(t \leq 6.4 \cdot 10^{-41}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2 \cdot \ell}{{k}^{4}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -6.6e-37) (not (<= t 6.4e-41)))
   (* (/ l k) (/ l (* k (pow t 3.0))))
   (* (/ l t) (/ (* 2.0 l) (pow k 4.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -6.6e-37) || !(t <= 6.4e-41)) {
		tmp = (l / k) * (l / (k * pow(t, 3.0)));
	} else {
		tmp = (l / t) * ((2.0 * 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 ((t <= (-6.6d-37)) .or. (.not. (t <= 6.4d-41))) then
        tmp = (l / k) * (l / (k * (t ** 3.0d0)))
    else
        tmp = (l / t) * ((2.0d0 * l) / (k ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -6.6e-37) || !(t <= 6.4e-41)) {
		tmp = (l / k) * (l / (k * Math.pow(t, 3.0)));
	} else {
		tmp = (l / t) * ((2.0 * l) / Math.pow(k, 4.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -6.6e-37) or not (t <= 6.4e-41):
		tmp = (l / k) * (l / (k * math.pow(t, 3.0)))
	else:
		tmp = (l / t) * ((2.0 * l) / math.pow(k, 4.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -6.6e-37) || !(t <= 6.4e-41))
		tmp = Float64(Float64(l / k) * Float64(l / Float64(k * (t ^ 3.0))));
	else
		tmp = Float64(Float64(l / t) * Float64(Float64(2.0 * l) / (k ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -6.6e-37) || ~((t <= 6.4e-41)))
		tmp = (l / k) * (l / (k * (t ^ 3.0)));
	else
		tmp = (l / t) * ((2.0 * l) / (k ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -6.6e-37], N[Not[LessEqual[t, 6.4e-41]], $MachinePrecision]], N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t), $MachinePrecision] * N[(N[(2.0 * l), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -6.59999999999999964e-37 or 6.40000000000000024e-41 < t

   1. Initial program 68.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 68.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 68.1%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 68.0%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 67.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 67.4%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 67.4%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 67.4%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 67.4%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 67.4%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 67.4%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 67.4%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around 0 58.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 58.7%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 58.7%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac 63.1%

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. unpow2 63.1%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   6. Simplified 63.1%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]

   7. Taylor expanded in l around 0 58.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   8. Step-by-step derivation

      1. unpow2 58.7%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 58.7%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. unpow2 58.7%

         \[\leadsto \frac{\ell \cdot \ell}{{t}^{3} \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. associate-*r* 64.6%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({t}^{3} \cdot k\right) \cdot k}} \]

      5. times-frac 71.1%

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3} \cdot k} \cdot \frac{\ell}{k}} \]

   9. Simplified 71.1%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3} \cdot k} \cdot \frac{\ell}{k}} \]

   if -6.59999999999999964e-37 < t < 6.40000000000000024e-41

   1. Initial program 33.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 33.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 33.2%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 33.2%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 34.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 34.1%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 34.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 34.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 34.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 34.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 34.5%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 34.5%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 77.4%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. unpow2 77.4%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(\sin k \cdot t\right)\right)} \]

      2. *-commutative 77.4%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \sin k\right)}\right)} \]

   6. Simplified 77.4%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]
   7. Step-by-step derivation

      1. associate-*r/ 77.4%

         \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]

   8. Applied egg-rr 77.4%

      \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]
   9. Step-by-step derivation

      1. unpow2 77.4%

         \[\leadsto \frac{\color{blue}{{\ell}^{2}} \cdot 2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)} \]

      2. associate-*r/ 77.4%

         \[\leadsto \color{blue}{{\ell}^{2} \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]

      3. unpow2 77.4%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)} \]

      4. associate-*l* 82.4%

         \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\right)} \]

      5. associate-*l* 90.7%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \color{blue}{\left(k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)\right)}}\right) \]

      6. *-commutative 90.7%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(k \cdot \left(k \cdot \color{blue}{\left(\sin k \cdot t\right)}\right)\right)}\right) \]

   10. Simplified 90.7%

       \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(k \cdot \left(k \cdot \left(\sin k \cdot t\right)\right)\right)}\right)} \]

   11. Taylor expanded in k around 0 58.9%

       \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   12. Step-by-step derivation

       1. associate-*r/ 58.9%

          \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

       2. *-commutative 58.9%

          \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot 2}}{{k}^{4} \cdot t} \]

       3. unpow2 58.9%

          \[\leadsto \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot 2}{{k}^{4} \cdot t} \]

       4. associate-*l* 58.9%

          \[\leadsto \frac{\color{blue}{\ell \cdot \left(\ell \cdot 2\right)}}{{k}^{4} \cdot t} \]

       5. *-commutative 58.9%

          \[\leadsto \frac{\ell \cdot \left(\ell \cdot 2\right)}{\color{blue}{t \cdot {k}^{4}}} \]

       6. times-frac 65.1%

          \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell \cdot 2}{{k}^{4}}} \]

       7. *-commutative 65.1%

          \[\leadsto \frac{\ell}{t} \cdot \frac{\color{blue}{2 \cdot \ell}}{{k}^{4}} \]

   13. Simplified 65.1%

       \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{2 \cdot \ell}{{k}^{4}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.6 \cdot 10^{-37} \lor \neg \left(t \leq 6.4 \cdot 10^{-41}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2 \cdot \ell}{{k}^{4}}\\ \end{array} \]

Alternative 15: 58.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.6 \cdot 10^{-47}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\frac{\ell}{t}}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot {t}^{-3}\right) \cdot \frac{\ell}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 3.6e-47)
   (* l (* 2.0 (/ (/ l t) (pow k 4.0))))
   (* (* l (pow t -3.0)) (/ l (* k k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 3.6e-47) {
		tmp = l * (2.0 * ((l / t) / 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) :: tmp
    if (t <= 3.6d-47) then
        tmp = l * (2.0d0 * ((l / t) / (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 tmp;
	if (t <= 3.6e-47) {
		tmp = l * (2.0 * ((l / t) / Math.pow(k, 4.0)));
	} else {
		tmp = (l * Math.pow(t, -3.0)) * (l / (k * k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 3.6e-47:
		tmp = l * (2.0 * ((l / t) / math.pow(k, 4.0)))
	else:
		tmp = (l * math.pow(t, -3.0)) * (l / (k * k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 3.6e-47)
		tmp = Float64(l * Float64(2.0 * Float64(Float64(l / t) / (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)
	tmp = 0.0;
	if (t <= 3.6e-47)
		tmp = l * (2.0 * ((l / t) / (k ^ 4.0)));
	else
		tmp = (l * (t ^ -3.0)) * (l / (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 3.6e-47], N[(l * N[(2.0 * N[(N[(l / t), $MachinePrecision] / 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 2 regimes

2. if t < 3.59999999999999991e-47

   1. Initial program 46.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 46.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 46.6%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 46.6%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 46.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 46.6%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 46.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 46.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 46.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 46.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 46.9%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 46.9%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 71.0%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. unpow2 71.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(\sin k \cdot t\right)\right)} \]

      2. *-commutative 71.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \sin k\right)}\right)} \]

   6. Simplified 71.0%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]
   7. Step-by-step derivation

      1. associate-*r/ 71.0%

         \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]

   8. Applied egg-rr 71.0%

      \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]
   9. Step-by-step derivation

      1. unpow2 71.0%

         \[\leadsto \frac{\color{blue}{{\ell}^{2}} \cdot 2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)} \]

      2. associate-*r/ 71.0%

         \[\leadsto \color{blue}{{\ell}^{2} \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]

      3. unpow2 71.0%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)} \]

      4. associate-*l* 75.4%

         \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\right)} \]

      5. associate-*l* 80.6%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \color{blue}{\left(k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)\right)}}\right) \]

      6. *-commutative 80.6%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(k \cdot \left(k \cdot \color{blue}{\left(\sin k \cdot t\right)}\right)\right)}\right) \]

   10. Simplified 80.6%

       \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(k \cdot \left(k \cdot \left(\sin k \cdot t\right)\right)\right)}\right)} \]

   11. Taylor expanded in k around 0 61.6%

       \[\leadsto \ell \cdot \color{blue}{\left(2 \cdot \frac{\ell}{{k}^{4} \cdot t}\right)} \]
   12. Step-by-step derivation

       1. *-commutative 61.6%

          \[\leadsto \ell \cdot \left(2 \cdot \frac{\ell}{\color{blue}{t \cdot {k}^{4}}}\right) \]

       2. associate-/r* 63.0%

          \[\leadsto \ell \cdot \left(2 \cdot \color{blue}{\frac{\frac{\ell}{t}}{{k}^{4}}}\right) \]

   13. Simplified 63.0%

       \[\leadsto \ell \cdot \color{blue}{\left(2 \cdot \frac{\frac{\ell}{t}}{{k}^{4}}\right)} \]

   if 3.59999999999999991e-47 < t

   1. Initial program 67.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 67.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 67.3%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 67.2%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 67.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 67.3%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 67.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 67.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 67.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 67.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 67.3%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 67.3%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around 0 57.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 57.6%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 57.6%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac 61.8%

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. unpow2 61.8%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   6. Simplified 61.8%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 57.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell}{{t}^{3}}\right)\right)} \cdot \frac{\ell}{k \cdot k} \]

      2. expm1-udef 50.9%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\ell}{{t}^{3}}\right)} - 1\right)} \cdot \frac{\ell}{k \cdot k} \]

      3. div-inv 50.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\ell \cdot \frac{1}{{t}^{3}}}\right)} - 1\right) \cdot \frac{\ell}{k \cdot k} \]

      4. pow-flip 50.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\ell \cdot \color{blue}{{t}^{\left(-3\right)}}\right)} - 1\right) \cdot \frac{\ell}{k \cdot k} \]

      5. metadata-eval 50.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\ell \cdot {t}^{\color{blue}{-3}}\right)} - 1\right) \cdot \frac{\ell}{k \cdot k} \]

   8. Applied egg-rr 50.9%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\ell \cdot {t}^{-3}\right)} - 1\right)} \cdot \frac{\ell}{k \cdot k} \]
   9. Step-by-step derivation

      1. expm1-def 57.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot {t}^{-3}\right)\right)} \cdot \frac{\ell}{k \cdot k} \]

      2. expm1-log1p 61.9%

         \[\leadsto \color{blue}{\left(\ell \cdot {t}^{-3}\right)} \cdot \frac{\ell}{k \cdot k} \]

   10. Simplified 61.9%

       \[\leadsto \color{blue}{\left(\ell \cdot {t}^{-3}\right)} \cdot \frac{\ell}{k \cdot k} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.6 \cdot 10^{-47}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\frac{\ell}{t}}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot {t}^{-3}\right) \cdot \frac{\ell}{k \cdot k}\\ \end{array} \]

Alternative 16: 51.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \frac{\ell \cdot \ell}{t \cdot {k}^{4}} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* 2.0 (/ (* l l) (* t (pow k 4.0)))))

C

double code(double t, double l, double k) {
	return 2.0 * ((l * l) / (t * pow(k, 4.0)));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * ((l * l) / (t * (k ** 4.0d0)))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * ((l * l) / (t * Math.pow(k, 4.0)));
}

Python

def code(t, l, k):
	return 2.0 * ((l * l) / (t * math.pow(k, 4.0)))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64(l * l) / Float64(t * (k ^ 4.0))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * ((l * l) / (t * (k ^ 4.0)));
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(l * l), $MachinePrecision] / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/l/ 52.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   2. associate-*l/ 52.4%

      \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

   3. associate-*l/ 52.4%

      \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

   4. associate-/r/ 52.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

   5. *-commutative 52.4%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

   6. associate-/l/ 52.4%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

   7. associate-*r* 52.6%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   8. *-commutative 52.6%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

   9. associate-*r* 52.6%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

   10. *-commutative 52.6%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

3. Simplified 52.6%

   \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

4. Taylor expanded in k around inf 67.0%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}} \]
5. Step-by-step derivation

   1. unpow2 67.0%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(\sin k \cdot t\right)\right)} \]

   2. *-commutative 67.0%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \sin k\right)}\right)} \]

6. Simplified 67.0%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]

7. Taylor expanded in k around 0 56.8%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
8. Step-by-step derivation

   1. unpow2 56.8%

      \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

   2. *-commutative 56.8%

      \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

9. Simplified 56.8%

   \[\leadsto \color{blue}{2 \cdot \frac{\ell \cdot \ell}{t \cdot {k}^{4}}} \]

10. Final simplification 56.8%

    \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{t \cdot {k}^{4}} \]

Alternative 17: 55.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* 2.0 (/ (* l (/ l (pow k 4.0))) t)))

C

double code(double t, double l, double k) {
	return 2.0 * ((l * (l / pow(k, 4.0))) / t);
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * ((l * (l / (k ** 4.0d0))) / t)
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * ((l * (l / Math.pow(k, 4.0))) / t);
}

Python

def code(t, l, k):
	return 2.0 * ((l * (l / math.pow(k, 4.0))) / t)

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64(l * Float64(l / (k ^ 4.0))) / t))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * ((l * (l / (k ^ 4.0))) / t);
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(l * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/l/ 52.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   2. associate-*l/ 52.4%

      \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

   3. associate-*l/ 52.4%

      \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

   4. associate-/r/ 52.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

   5. *-commutative 52.4%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

   6. associate-/l/ 52.4%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

   7. associate-*r* 52.6%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   8. *-commutative 52.6%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

   9. associate-*r* 52.6%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

   10. *-commutative 52.6%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

3. Simplified 52.6%

   \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

4. Taylor expanded in k around inf 67.0%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}} \]
5. Step-by-step derivation

   1. unpow2 67.0%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(\sin k \cdot t\right)\right)} \]

   2. *-commutative 67.0%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \sin k\right)}\right)} \]

6. Simplified 67.0%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]
7. Step-by-step derivation

   1. associate-*r/ 67.0%

      \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]

8. Applied egg-rr 67.0%

   \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]
9. Step-by-step derivation

   1. unpow2 67.0%

      \[\leadsto \frac{\color{blue}{{\ell}^{2}} \cdot 2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)} \]

   2. associate-*r/ 67.0%

      \[\leadsto \color{blue}{{\ell}^{2} \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]

   3. unpow2 67.0%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)} \]

   4. associate-*l* 70.7%

      \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\right)} \]

   5. associate-*l* 74.4%

      \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \color{blue}{\left(k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)\right)}}\right) \]

   6. *-commutative 74.4%

      \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(k \cdot \left(k \cdot \color{blue}{\left(\sin k \cdot t\right)}\right)\right)}\right) \]

10. Simplified 74.4%

    \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(k \cdot \left(k \cdot \left(\sin k \cdot t\right)\right)\right)}\right)} \]
11. Step-by-step derivation

    1. associate-*r/ 74.4%

       \[\leadsto \ell \cdot \color{blue}{\frac{\ell \cdot 2}{\tan k \cdot \left(k \cdot \left(k \cdot \left(\sin k \cdot t\right)\right)\right)}} \]

12. Applied egg-rr 74.4%

    \[\leadsto \ell \cdot \color{blue}{\frac{\ell \cdot 2}{\tan k \cdot \left(k \cdot \left(k \cdot \left(\sin k \cdot t\right)\right)\right)}} \]
13. Step-by-step derivation

    1. associate-*r* 74.4%

       \[\leadsto \ell \cdot \frac{\ell \cdot 2}{\color{blue}{\left(\tan k \cdot k\right) \cdot \left(k \cdot \left(\sin k \cdot t\right)\right)}} \]

    2. associate-/r* 76.6%

       \[\leadsto \ell \cdot \color{blue}{\frac{\frac{\ell \cdot 2}{\tan k \cdot k}}{k \cdot \left(\sin k \cdot t\right)}} \]

    3. *-commutative 76.6%

       \[\leadsto \ell \cdot \frac{\frac{\color{blue}{2 \cdot \ell}}{\tan k \cdot k}}{k \cdot \left(\sin k \cdot t\right)} \]

14. Simplified 76.6%

    \[\leadsto \ell \cdot \color{blue}{\frac{\frac{2 \cdot \ell}{\tan k \cdot k}}{k \cdot \left(\sin k \cdot t\right)}} \]

15. Taylor expanded in k around 0 56.8%

    \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
16. Step-by-step derivation

    1. unpow2 56.8%

       \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

    2. *-commutative 56.8%

       \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

    3. associate-*r/ 59.6%

       \[\leadsto 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{t \cdot {k}^{4}}\right)} \]

    4. *-commutative 59.6%

       \[\leadsto 2 \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{{k}^{4} \cdot t}}\right) \]

    5. associate-/r* 60.4%

       \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\frac{\frac{\ell}{{k}^{4}}}{t}}\right) \]

    6. associate-*r/ 60.4%

       \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}} \]

17. Simplified 60.4%

    \[\leadsto \color{blue}{2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}} \]

18. Final simplification 60.4%

    \[\leadsto 2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t} \]

Alternative 18: 55.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \ell \cdot \left(2 \cdot \frac{\frac{\ell}{t}}{{k}^{4}}\right) \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* l (* 2.0 (/ (/ l t) (pow k 4.0)))))

C

double code(double t, double l, double k) {
	return l * (2.0 * ((l / t) / pow(k, 4.0)));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = l * (2.0d0 * ((l / t) / (k ** 4.0d0)))
end function

Java

public static double code(double t, double l, double k) {
	return l * (2.0 * ((l / t) / Math.pow(k, 4.0)));
}

Python

def code(t, l, k):
	return l * (2.0 * ((l / t) / math.pow(k, 4.0)))

Julia

function code(t, l, k)
	return Float64(l * Float64(2.0 * Float64(Float64(l / t) / (k ^ 4.0))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = l * (2.0 * ((l / t) / (k ^ 4.0)));
end

Wolfram

code[t_, l_, k_] := N[(l * N[(2.0 * N[(N[(l / t), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/l/ 52.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   2. associate-*l/ 52.4%

      \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

   3. associate-*l/ 52.4%

      \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

   4. associate-/r/ 52.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

   5. *-commutative 52.4%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

   6. associate-/l/ 52.4%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

   7. associate-*r* 52.6%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   8. *-commutative 52.6%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

   9. associate-*r* 52.6%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

   10. *-commutative 52.6%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

3. Simplified 52.6%

   \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

4. Taylor expanded in k around inf 67.0%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}} \]
5. Step-by-step derivation

   1. unpow2 67.0%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(\sin k \cdot t\right)\right)} \]

   2. *-commutative 67.0%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \sin k\right)}\right)} \]

6. Simplified 67.0%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]
7. Step-by-step derivation

   1. associate-*r/ 67.0%

      \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]

8. Applied egg-rr 67.0%

   \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]
9. Step-by-step derivation

   1. unpow2 67.0%

      \[\leadsto \frac{\color{blue}{{\ell}^{2}} \cdot 2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)} \]

   2. associate-*r/ 67.0%

      \[\leadsto \color{blue}{{\ell}^{2} \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]

   3. unpow2 67.0%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)} \]

   4. associate-*l* 70.7%

      \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\right)} \]

   5. associate-*l* 74.4%

      \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \color{blue}{\left(k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)\right)}}\right) \]

   6. *-commutative 74.4%

      \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(k \cdot \left(k \cdot \color{blue}{\left(\sin k \cdot t\right)}\right)\right)}\right) \]

10. Simplified 74.4%

    \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(k \cdot \left(k \cdot \left(\sin k \cdot t\right)\right)\right)}\right)} \]

11. Taylor expanded in k around 0 59.6%

    \[\leadsto \ell \cdot \color{blue}{\left(2 \cdot \frac{\ell}{{k}^{4} \cdot t}\right)} \]
12. Step-by-step derivation

    1. *-commutative 59.6%

       \[\leadsto \ell \cdot \left(2 \cdot \frac{\ell}{\color{blue}{t \cdot {k}^{4}}}\right) \]

    2. associate-/r* 60.6%

       \[\leadsto \ell \cdot \left(2 \cdot \color{blue}{\frac{\frac{\ell}{t}}{{k}^{4}}}\right) \]

13. Simplified 60.6%

    \[\leadsto \ell \cdot \color{blue}{\left(2 \cdot \frac{\frac{\ell}{t}}{{k}^{4}}\right)} \]

14. Final simplification 60.6%

    \[\leadsto \ell \cdot \left(2 \cdot \frac{\frac{\ell}{t}}{{k}^{4}}\right) \]

Reproduce

herbie shell --seed 2023182 
(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))))