Toniolo and Linder, Equation (10+)

Percentage Accurate: 38.4% → 54.6%

Time: 28.1s

Precision: binary64

Cost: 34388

• 63.2% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;t \leq -540000000000:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{\frac{2}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}}{\sin k \cdot {t}^{3}}\right)\\ \mathbf{elif}\;t \leq -1.6 \cdot 10^{-104}:\\ \;\;\;\;\frac{\frac{1}{k \cdot k} + -0.16666666666666666}{\frac{{t}^{3}}{\ell \cdot \ell}}\\ \mathbf{elif}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{t}}{\tan k \cdot \left(\sin k \cdot t_1\right)}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-228}:\\ \;\;\;\;\frac{\ell}{k \cdot {t}^{3}} \cdot \frac{\ell}{k}\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{-90}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(t_1 \cdot \left(\tan k \cdot \sin k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\mathsf{fma}\left(0.16666666666666666, \frac{\ell}{\frac{{t}^{3}}{k}}, \frac{\frac{\ell}{{t}^{3}}}{k}\right) - {k}^{3} \cdot \frac{\ell \cdot -0.019444444444444445}{{t}^{3}}\right)\\ \end{array} \]

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))))

↓

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* k (/ k l))))
   (if (<= t -540000000000.0)
     (*
      l
      (*
       l
       (/
        (/ (/ 2.0 (tan k)) (+ 2.0 (pow (/ k t) 2.0)))
        (* (sin k) (pow t 3.0)))))
     (if (<= t -1.6e-104)
       (/ (+ (/ 1.0 (* k k)) -0.16666666666666666) (/ (pow t 3.0) (* l l)))
       (if (<= t -5e-310)
         (/ (* l (/ 2.0 t)) (* (tan k) (* (sin k) t_1)))
         (if (<= t 2.3e-228)
           (* (/ l (* k (pow t 3.0))) (/ l k))
           (if (<= t 3.7e-90)
             (/ 2.0 (* (/ t l) (* t_1 (* (tan k) (sin k)))))
             (*
              (/ l k)
              (-
               (fma
                0.16666666666666666
                (/ l (/ (pow t 3.0) k))
                (/ (/ l (pow t 3.0)) k))
               (*
                (pow k 3.0)
                (/ (* l -0.019444444444444445) (pow t 3.0))))))))))))

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));
}

↓

double code(double t, double l, double k) {
	double t_1 = k * (k / l);
	double tmp;
	if (t <= -540000000000.0) {
		tmp = l * (l * (((2.0 / tan(k)) / (2.0 + pow((k / t), 2.0))) / (sin(k) * pow(t, 3.0))));
	} else if (t <= -1.6e-104) {
		tmp = ((1.0 / (k * k)) + -0.16666666666666666) / (pow(t, 3.0) / (l * l));
	} else if (t <= -5e-310) {
		tmp = (l * (2.0 / t)) / (tan(k) * (sin(k) * t_1));
	} else if (t <= 2.3e-228) {
		tmp = (l / (k * pow(t, 3.0))) * (l / k);
	} else if (t <= 3.7e-90) {
		tmp = 2.0 / ((t / l) * (t_1 * (tan(k) * sin(k))));
	} else {
		tmp = (l / k) * (fma(0.16666666666666666, (l / (pow(t, 3.0) / k)), ((l / pow(t, 3.0)) / k)) - (pow(k, 3.0) * ((l * -0.019444444444444445) / pow(t, 3.0))));
	}
	return tmp;
}

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

↓

function code(t, l, k)
	t_1 = Float64(k * Float64(k / l))
	tmp = 0.0
	if (t <= -540000000000.0)
		tmp = Float64(l * Float64(l * Float64(Float64(Float64(2.0 / tan(k)) / Float64(2.0 + (Float64(k / t) ^ 2.0))) / Float64(sin(k) * (t ^ 3.0)))));
	elseif (t <= -1.6e-104)
		tmp = Float64(Float64(Float64(1.0 / Float64(k * k)) + -0.16666666666666666) / Float64((t ^ 3.0) / Float64(l * l)));
	elseif (t <= -5e-310)
		tmp = Float64(Float64(l * Float64(2.0 / t)) / Float64(tan(k) * Float64(sin(k) * t_1)));
	elseif (t <= 2.3e-228)
		tmp = Float64(Float64(l / Float64(k * (t ^ 3.0))) * Float64(l / k));
	elseif (t <= 3.7e-90)
		tmp = Float64(2.0 / Float64(Float64(t / l) * Float64(t_1 * Float64(tan(k) * sin(k)))));
	else
		tmp = Float64(Float64(l / k) * Float64(fma(0.16666666666666666, Float64(l / Float64((t ^ 3.0) / k)), Float64(Float64(l / (t ^ 3.0)) / k)) - Float64((k ^ 3.0) * Float64(Float64(l * -0.019444444444444445) / (t ^ 3.0)))));
	end
	return tmp
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]

↓

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -540000000000.0], N[(l * N[(l * N[(N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.6e-104], N[(N[(N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] / N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -5e-310], N[(N[(l * N[(2.0 / t), $MachinePrecision]), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.3e-228], N[(N[(l / N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.7e-90], N[(2.0 / N[(N[(t / l), $MachinePrecision] * N[(t$95$1 * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k), $MachinePrecision] * N[(N[(0.16666666666666666 * N[(l / N[(N[Power[t, 3.0], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] + N[(N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] - N[(N[Power[k, 3.0], $MachinePrecision] * N[(N[(l * -0.019444444444444445), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -5.4e11

   1. Initial program 39.0%

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

   2. Simplified 40.8%

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

      [Start] 39.0%	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l/ [<=] 39.0%	\[ \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]
      associate-*l/ [=>] 40.8%	\[ \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]
      associate-*l/ [=>] 40.8%	\[ \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]
      associate-/r/ [=>] 40.8%	\[ \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]
      *-commutative [=>] 40.8%	\[ \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]
      associate-/l/ [=>] 40.8%	\[ \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]
      associate-*r* [<=] 40.8%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      *-commutative [=>] 40.8%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]
      associate-*r* [=>] 40.8%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]
      *-commutative [=>] 40.8%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Applied egg-rr 26.0%

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

      [Start] 40.8%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)} \]
      expm1-log1p-u [=>] 31.4%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\right)} \]
      expm1-udef [=>] 26.0%	\[ \color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)} - 1} \]

   4. Simplified 46.4%

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

      [Start] 26.0%	\[ e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)} - 1 \]
      expm1-def [=>] 31.4%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\right)} \]
      expm1-log1p [=>] 40.8%	\[ \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]
      associate-*l* [=>] 46.5%	\[ \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)} \]
      associate-/r* [=>] 46.4%	\[ \ell \cdot \left(\ell \cdot \color{blue}{\frac{\frac{2}{\tan k}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)}}\right) \]
      associate-/r* [=>] 46.4%	\[ \ell \cdot \left(\ell \cdot \color{blue}{\frac{\frac{\frac{2}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}}{{t}^{3} \cdot \sin k}}\right) \]
      *-commutative [=>] 46.4%	\[ \ell \cdot \left(\ell \cdot \frac{\frac{\frac{2}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}}{\color{blue}{\sin k \cdot {t}^{3}}}\right) \]

   if -5.4e11 < t < -1.59999999999999994e-104

   1. Initial program 43.4%

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

   2. Simplified 43.5%

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

      [Start] 43.4%	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 43.4%	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      associate-*l/ [=>] 43.3%	\[ \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k\right)} \]
      associate-*l/ [=>] 43.3%	\[ \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]
      associate-*r/ [=>] 43.5%	\[ \frac{2}{\color{blue}{\frac{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right)}{\ell \cdot \ell}}} \]
      associate-/r/ [=>] 43.5%	\[ \color{blue}{\frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)} \]

   3. Taylor expanded in k around 0 57.6%

      \[\leadsto \color{blue}{\left(\frac{1}{{k}^{2} \cdot {t}^{3}} + -0.5 \cdot \frac{2 \cdot \left(0.3333333333333333 \cdot {t}^{3} + -0.16666666666666666 \cdot {t}^{3}\right) + t}{{t}^{6}}\right)} \cdot \left(\ell \cdot \ell\right) \]

   4. Simplified 57.6%

      \[\leadsto \color{blue}{\left(\frac{1}{\left(k \cdot k\right) \cdot {t}^{3}} + -0.5 \cdot \frac{\mathsf{fma}\left(2, {t}^{3} \cdot 0.16666666666666666, t\right)}{{t}^{6}}\right)} \cdot \left(\ell \cdot \ell\right) \]
      Step-by-step derivation

      [Start] 57.6%	\[ \left(\frac{1}{{k}^{2} \cdot {t}^{3}} + -0.5 \cdot \frac{2 \cdot \left(0.3333333333333333 \cdot {t}^{3} + -0.16666666666666666 \cdot {t}^{3}\right) + t}{{t}^{6}}\right) \cdot \left(\ell \cdot \ell\right) \]
      unpow2 [=>] 57.6%	\[ \left(\frac{1}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} + -0.5 \cdot \frac{2 \cdot \left(0.3333333333333333 \cdot {t}^{3} + -0.16666666666666666 \cdot {t}^{3}\right) + t}{{t}^{6}}\right) \cdot \left(\ell \cdot \ell\right) \]
      fma-def [=>] 57.6%	\[ \left(\frac{1}{\left(k \cdot k\right) \cdot {t}^{3}} + -0.5 \cdot \frac{\color{blue}{\mathsf{fma}\left(2, 0.3333333333333333 \cdot {t}^{3} + -0.16666666666666666 \cdot {t}^{3}, t\right)}}{{t}^{6}}\right) \cdot \left(\ell \cdot \ell\right) \]
      distribute-rgt-out [=>] 57.6%	\[ \left(\frac{1}{\left(k \cdot k\right) \cdot {t}^{3}} + -0.5 \cdot \frac{\mathsf{fma}\left(2, \color{blue}{{t}^{3} \cdot \left(0.3333333333333333 + -0.16666666666666666\right)}, t\right)}{{t}^{6}}\right) \cdot \left(\ell \cdot \ell\right) \]
      metadata-eval [=>] 57.6%	\[ \left(\frac{1}{\left(k \cdot k\right) \cdot {t}^{3}} + -0.5 \cdot \frac{\mathsf{fma}\left(2, {t}^{3} \cdot \color{blue}{0.16666666666666666}, t\right)}{{t}^{6}}\right) \cdot \left(\ell \cdot \ell\right) \]

   5. Taylor expanded in t around inf 68.8%

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{{k}^{2}} - 0.16666666666666666\right) \cdot {\ell}^{2}}{{t}^{3}}} \]

   6. Simplified 68.8%

      \[\leadsto \color{blue}{\frac{\frac{1}{k \cdot k} + -0.16666666666666666}{\frac{{t}^{3}}{\ell \cdot \ell}}} \]
      Step-by-step derivation

      [Start] 68.8%	\[ \frac{\left(\frac{1}{{k}^{2}} - 0.16666666666666666\right) \cdot {\ell}^{2}}{{t}^{3}} \]
      associate-/l* [=>] 68.8%	\[ \color{blue}{\frac{\frac{1}{{k}^{2}} - 0.16666666666666666}{\frac{{t}^{3}}{{\ell}^{2}}}} \]
      sub-neg [=>] 68.8%	\[ \frac{\color{blue}{\frac{1}{{k}^{2}} + \left(-0.16666666666666666\right)}}{\frac{{t}^{3}}{{\ell}^{2}}} \]
      unpow2 [=>] 68.8%	\[ \frac{\frac{1}{\color{blue}{k \cdot k}} + \left(-0.16666666666666666\right)}{\frac{{t}^{3}}{{\ell}^{2}}} \]
      metadata-eval [=>] 68.8%	\[ \frac{\frac{1}{k \cdot k} + \color{blue}{-0.16666666666666666}}{\frac{{t}^{3}}{{\ell}^{2}}} \]
      unpow2 [=>] 68.8%	\[ \frac{\frac{1}{k \cdot k} + -0.16666666666666666}{\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

   if -1.59999999999999994e-104 < t < -4.999999999999985e-310

   1. Initial program 26.0%

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

   2. Simplified 26.0%

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

      [Start] 26.0%	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 26.0%	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      associate-*l* [=>] 26.0%	\[ \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]
      associate-*r* [=>] 26.0%	\[ \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
      +-commutative [=>] 26.0%	\[ \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]
      associate-+r+ [=>] 26.0%	\[ \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]
      metadata-eval [=>] 26.0%	\[ \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Taylor expanded in k around inf 51.0%

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

   4. Simplified 67.2%

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

      [Start] 51.0%	\[ \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]
      *-commutative [=>] 51.0%	\[ \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]
      unpow2 [=>] 51.0%	\[ \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]
      times-frac [=>] 67.2%	\[ \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]
      unpow2 [=>] 67.2%	\[ \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   5. Applied egg-rr 72.4%

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

      [Start] 67.2%	\[ \frac{2}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]
      div-inv [=>] 67.2%	\[ \color{blue}{2 \cdot \frac{1}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
      associate-*l* [=>] 67.2%	\[ 2 \cdot \frac{1}{\color{blue}{\frac{t}{\ell} \cdot \left(\frac{k \cdot k}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]
      associate-/l* [=>] 72.4%	\[ 2 \cdot \frac{1}{\frac{t}{\ell} \cdot \left(\color{blue}{\frac{k}{\frac{\ell}{k}}} \cdot \left(\sin k \cdot \tan k\right)\right)} \]

   6. Simplified 74.8%

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

      [Start] 72.4%	\[ 2 \cdot \frac{1}{\frac{t}{\ell} \cdot \left(\frac{k}{\frac{\ell}{k}} \cdot \left(\sin k \cdot \tan k\right)\right)} \]
      associate-*r/ [=>] 72.4%	\[ \color{blue}{\frac{2 \cdot 1}{\frac{t}{\ell} \cdot \left(\frac{k}{\frac{\ell}{k}} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]
      metadata-eval [=>] 72.4%	\[ \frac{\color{blue}{2}}{\frac{t}{\ell} \cdot \left(\frac{k}{\frac{\ell}{k}} \cdot \left(\sin k \cdot \tan k\right)\right)} \]
      associate-/r* [=>] 74.7%	\[ \color{blue}{\frac{\frac{2}{\frac{t}{\ell}}}{\frac{k}{\frac{\ell}{k}} \cdot \left(\sin k \cdot \tan k\right)}} \]
      associate-/r/ [=>] 74.8%	\[ \frac{\color{blue}{\frac{2}{t} \cdot \ell}}{\frac{k}{\frac{\ell}{k}} \cdot \left(\sin k \cdot \tan k\right)} \]
      *-commutative [=>] 74.8%	\[ \frac{\frac{2}{t} \cdot \ell}{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \frac{k}{\frac{\ell}{k}}}} \]
      *-commutative [=>] 74.8%	\[ \frac{\frac{2}{t} \cdot \ell}{\color{blue}{\left(\tan k \cdot \sin k\right)} \cdot \frac{k}{\frac{\ell}{k}}} \]
      associate-*l* [=>] 74.8%	\[ \frac{\frac{2}{t} \cdot \ell}{\color{blue}{\tan k \cdot \left(\sin k \cdot \frac{k}{\frac{\ell}{k}}\right)}} \]
      associate-/r/ [=>] 74.8%	\[ \frac{\frac{2}{t} \cdot \ell}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left(\frac{k}{\ell} \cdot k\right)}\right)} \]
      *-commutative [=>] 74.8%	\[ \frac{\frac{2}{t} \cdot \ell}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left(k \cdot \frac{k}{\ell}\right)}\right)} \]

   if -4.999999999999985e-310 < t < 2.2999999999999999e-228

   1. Initial program 22.2%

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

   2. Simplified 22.2%

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

      [Start] 22.2%	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 22.2%	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      +-commutative [=>] 22.2%	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Taylor expanded in k around 0 66.7%

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

   4. Simplified 66.7%

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

      [Start] 66.7%	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(2 \cdot k\right)} \]
      *-commutative [=>] 66.7%	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   5. Taylor expanded in t around 0 66.7%

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

   6. Simplified 66.7%

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

      [Start] 66.7%	\[ \frac{{\ell}^{2}}{k \cdot \left(\sin k \cdot {t}^{3}\right)} \]
      associate-/r* [=>] 66.7%	\[ \color{blue}{\frac{\frac{{\ell}^{2}}{k}}{\sin k \cdot {t}^{3}}} \]
      unpow2 [=>] 66.7%	\[ \frac{\frac{\color{blue}{\ell \cdot \ell}}{k}}{\sin k \cdot {t}^{3}} \]

   7. Taylor expanded in l around 0 66.7%

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

   8. Simplified 72.5%

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

      [Start] 66.7%	\[ \frac{{\ell}^{2}}{k \cdot \left(\sin k \cdot {t}^{3}\right)} \]
      unpow2 [=>] 66.7%	\[ \frac{\color{blue}{\ell \cdot \ell}}{k \cdot \left(\sin k \cdot {t}^{3}\right)} \]
      *-commutative [=>] 66.7%	\[ \frac{\ell \cdot \ell}{\color{blue}{\left(\sin k \cdot {t}^{3}\right) \cdot k}} \]
      *-commutative [=>] 66.7%	\[ \frac{\ell \cdot \ell}{\color{blue}{\left({t}^{3} \cdot \sin k\right)} \cdot k} \]
      times-frac [=>] 72.5%	\[ \color{blue}{\frac{\ell}{{t}^{3} \cdot \sin k} \cdot \frac{\ell}{k}} \]
      *-commutative [<=] 72.5%	\[ \frac{\ell}{\color{blue}{\sin k \cdot {t}^{3}}} \cdot \frac{\ell}{k} \]

   9. Taylor expanded in k around 0 94.7%

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

   if 2.2999999999999999e-228 < t < 3.70000000000000018e-90

   1. Initial program 38.6%

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

   2. Simplified 38.6%

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

      [Start] 38.6%	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 38.6%	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      associate-*l* [=>] 38.6%	\[ \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]
      associate-*r* [=>] 38.6%	\[ \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
      +-commutative [=>] 38.6%	\[ \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]
      associate-+r+ [=>] 38.6%	\[ \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]
      metadata-eval [=>] 38.6%	\[ \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Taylor expanded in k around inf 54.5%

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

   4. Simplified 73.6%

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

      [Start] 54.5%	\[ \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]
      *-commutative [=>] 54.5%	\[ \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]
      unpow2 [=>] 54.5%	\[ \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]
      times-frac [=>] 73.6%	\[ \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]
      unpow2 [=>] 73.6%	\[ \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   5. Applied egg-rr 76.3%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t}{\ell} \cdot \left(\frac{k}{\frac{\ell}{k}} \cdot \left(\sin k \cdot \tan k\right)\right)\right)\right)}} \]
      Step-by-step derivation

      [Start] 73.6%	\[ \frac{2}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]
      expm1-log1p-u [=>] 73.0%	\[ \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)\right)\right)}} \]
      associate-*l* [=>] 76.3%	\[ \frac{2}{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{t}{\ell} \cdot \left(\frac{k \cdot k}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right)}\right)\right)} \]
      associate-/l* [=>] 76.3%	\[ \frac{2}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t}{\ell} \cdot \left(\color{blue}{\frac{k}{\frac{\ell}{k}}} \cdot \left(\sin k \cdot \tan k\right)\right)\right)\right)} \]

   6. Applied egg-rr 77.3%

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

      [Start] 76.3%	\[ \frac{2}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t}{\ell} \cdot \left(\frac{k}{\frac{\ell}{k}} \cdot \left(\sin k \cdot \tan k\right)\right)\right)\right)} \]
      expm1-log1p-u [<=] 77.3%	\[ \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\frac{k}{\frac{\ell}{k}} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]
      associate-/r/ [=>] 77.3%	\[ \frac{2}{\frac{t}{\ell} \cdot \left(\color{blue}{\left(\frac{k}{\ell} \cdot k\right)} \cdot \left(\sin k \cdot \tan k\right)\right)} \]

   if 3.70000000000000018e-90 < t

   1. Initial program 45.2%

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

   2. Simplified 45.2%

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

      [Start] 45.2%	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 45.2%	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      +-commutative [=>] 45.2%	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Taylor expanded in k around 0 46.6%

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

   4. Simplified 46.6%

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

      [Start] 46.6%	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(2 \cdot k\right)} \]
      *-commutative [=>] 46.6%	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   5. Taylor expanded in t around 0 41.3%

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

   6. Simplified 44.5%

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

      [Start] 41.3%	\[ \frac{{\ell}^{2}}{k \cdot \left(\sin k \cdot {t}^{3}\right)} \]
      associate-/r* [=>] 44.5%	\[ \color{blue}{\frac{\frac{{\ell}^{2}}{k}}{\sin k \cdot {t}^{3}}} \]
      unpow2 [=>] 44.5%	\[ \frac{\frac{\color{blue}{\ell \cdot \ell}}{k}}{\sin k \cdot {t}^{3}} \]

   7. Taylor expanded in l around 0 41.3%

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

   8. Simplified 44.7%

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

      [Start] 41.3%	\[ \frac{{\ell}^{2}}{k \cdot \left(\sin k \cdot {t}^{3}\right)} \]
      unpow2 [=>] 41.3%	\[ \frac{\color{blue}{\ell \cdot \ell}}{k \cdot \left(\sin k \cdot {t}^{3}\right)} \]
      *-commutative [=>] 41.3%	\[ \frac{\ell \cdot \ell}{\color{blue}{\left(\sin k \cdot {t}^{3}\right) \cdot k}} \]
      *-commutative [=>] 41.3%	\[ \frac{\ell \cdot \ell}{\color{blue}{\left({t}^{3} \cdot \sin k\right)} \cdot k} \]
      times-frac [=>] 44.7%	\[ \color{blue}{\frac{\ell}{{t}^{3} \cdot \sin k} \cdot \frac{\ell}{k}} \]
      *-commutative [<=] 44.7%	\[ \frac{\ell}{\color{blue}{\sin k \cdot {t}^{3}}} \cdot \frac{\ell}{k} \]

   9. Taylor expanded in k around 0 54.7%

      \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot \frac{k \cdot \ell}{{t}^{3}} + \left(\frac{\ell}{k \cdot {t}^{3}} + -1 \cdot \left({k}^{3} \cdot \left(0.008333333333333333 \cdot \frac{\ell}{{t}^{3}} + -0.027777777777777776 \cdot \frac{\ell}{{t}^{3}}\right)\right)\right)\right)} \cdot \frac{\ell}{k} \]

   10. Simplified 59.0%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.16666666666666666, \frac{\ell}{\frac{{t}^{3}}{k}}, \frac{\frac{\ell}{{t}^{3}}}{k}\right) - {k}^{3} \cdot \frac{\ell \cdot -0.019444444444444445}{{t}^{3}}\right)} \cdot \frac{\ell}{k} \]
       Step-by-step derivation

       [Start] 54.7%	\[ \left(0.16666666666666666 \cdot \frac{k \cdot \ell}{{t}^{3}} + \left(\frac{\ell}{k \cdot {t}^{3}} + -1 \cdot \left({k}^{3} \cdot \left(0.008333333333333333 \cdot \frac{\ell}{{t}^{3}} + -0.027777777777777776 \cdot \frac{\ell}{{t}^{3}}\right)\right)\right)\right) \cdot \frac{\ell}{k} \]
       associate-+r+ [=>] 54.7%	\[ \color{blue}{\left(\left(0.16666666666666666 \cdot \frac{k \cdot \ell}{{t}^{3}} + \frac{\ell}{k \cdot {t}^{3}}\right) + -1 \cdot \left({k}^{3} \cdot \left(0.008333333333333333 \cdot \frac{\ell}{{t}^{3}} + -0.027777777777777776 \cdot \frac{\ell}{{t}^{3}}\right)\right)\right)} \cdot \frac{\ell}{k} \]
       mul-1-neg [=>] 54.7%	\[ \left(\left(0.16666666666666666 \cdot \frac{k \cdot \ell}{{t}^{3}} + \frac{\ell}{k \cdot {t}^{3}}\right) + \color{blue}{\left(-{k}^{3} \cdot \left(0.008333333333333333 \cdot \frac{\ell}{{t}^{3}} + -0.027777777777777776 \cdot \frac{\ell}{{t}^{3}}\right)\right)}\right) \cdot \frac{\ell}{k} \]
       unsub-neg [=>] 54.7%	\[ \color{blue}{\left(\left(0.16666666666666666 \cdot \frac{k \cdot \ell}{{t}^{3}} + \frac{\ell}{k \cdot {t}^{3}}\right) - {k}^{3} \cdot \left(0.008333333333333333 \cdot \frac{\ell}{{t}^{3}} + -0.027777777777777776 \cdot \frac{\ell}{{t}^{3}}\right)\right)} \cdot \frac{\ell}{k} \]
       fma-def [=>] 54.7%	\[ \left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, \frac{k \cdot \ell}{{t}^{3}}, \frac{\ell}{k \cdot {t}^{3}}\right)} - {k}^{3} \cdot \left(0.008333333333333333 \cdot \frac{\ell}{{t}^{3}} + -0.027777777777777776 \cdot \frac{\ell}{{t}^{3}}\right)\right) \cdot \frac{\ell}{k} \]
       *-commutative [=>] 54.7%	\[ \left(\mathsf{fma}\left(0.16666666666666666, \frac{\color{blue}{\ell \cdot k}}{{t}^{3}}, \frac{\ell}{k \cdot {t}^{3}}\right) - {k}^{3} \cdot \left(0.008333333333333333 \cdot \frac{\ell}{{t}^{3}} + -0.027777777777777776 \cdot \frac{\ell}{{t}^{3}}\right)\right) \cdot \frac{\ell}{k} \]
       associate-/l* [=>] 54.7%	\[ \left(\mathsf{fma}\left(0.16666666666666666, \color{blue}{\frac{\ell}{\frac{{t}^{3}}{k}}}, \frac{\ell}{k \cdot {t}^{3}}\right) - {k}^{3} \cdot \left(0.008333333333333333 \cdot \frac{\ell}{{t}^{3}} + -0.027777777777777776 \cdot \frac{\ell}{{t}^{3}}\right)\right) \cdot \frac{\ell}{k} \]
       *-commutative [=>] 54.7%	\[ \left(\mathsf{fma}\left(0.16666666666666666, \frac{\ell}{\frac{{t}^{3}}{k}}, \frac{\ell}{\color{blue}{{t}^{3} \cdot k}}\right) - {k}^{3} \cdot \left(0.008333333333333333 \cdot \frac{\ell}{{t}^{3}} + -0.027777777777777776 \cdot \frac{\ell}{{t}^{3}}\right)\right) \cdot \frac{\ell}{k} \]
       associate-/r* [=>] 55.8%	\[ \left(\mathsf{fma}\left(0.16666666666666666, \frac{\ell}{\frac{{t}^{3}}{k}}, \color{blue}{\frac{\frac{\ell}{{t}^{3}}}{k}}\right) - {k}^{3} \cdot \left(0.008333333333333333 \cdot \frac{\ell}{{t}^{3}} + -0.027777777777777776 \cdot \frac{\ell}{{t}^{3}}\right)\right) \cdot \frac{\ell}{k} \]
       +-commutative [=>] 55.8%	\[ \left(\mathsf{fma}\left(0.16666666666666666, \frac{\ell}{\frac{{t}^{3}}{k}}, \frac{\frac{\ell}{{t}^{3}}}{k}\right) - {k}^{3} \cdot \color{blue}{\left(-0.027777777777777776 \cdot \frac{\ell}{{t}^{3}} + 0.008333333333333333 \cdot \frac{\ell}{{t}^{3}}\right)}\right) \cdot \frac{\ell}{k} \]
       distribute-rgt-out [=>] 59.0%	\[ \left(\mathsf{fma}\left(0.16666666666666666, \frac{\ell}{\frac{{t}^{3}}{k}}, \frac{\frac{\ell}{{t}^{3}}}{k}\right) - {k}^{3} \cdot \color{blue}{\left(\frac{\ell}{{t}^{3}} \cdot \left(-0.027777777777777776 + 0.008333333333333333\right)\right)}\right) \cdot \frac{\ell}{k} \]
       associate-*l/ [=>] 59.0%	\[ \left(\mathsf{fma}\left(0.16666666666666666, \frac{\ell}{\frac{{t}^{3}}{k}}, \frac{\frac{\ell}{{t}^{3}}}{k}\right) - {k}^{3} \cdot \color{blue}{\frac{\ell \cdot \left(-0.027777777777777776 + 0.008333333333333333\right)}{{t}^{3}}}\right) \cdot \frac{\ell}{k} \]
       metadata-eval [=>] 59.0%	\[ \left(\mathsf{fma}\left(0.16666666666666666, \frac{\ell}{\frac{{t}^{3}}{k}}, \frac{\frac{\ell}{{t}^{3}}}{k}\right) - {k}^{3} \cdot \frac{\ell \cdot \color{blue}{-0.019444444444444445}}{{t}^{3}}\right) \cdot \frac{\ell}{k} \]

3. Recombined 6 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -540000000000:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{\frac{2}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}}{\sin k \cdot {t}^{3}}\right)\\ \mathbf{elif}\;t \leq -1.6 \cdot 10^{-104}:\\ \;\;\;\;\frac{\frac{1}{k \cdot k} + -0.16666666666666666}{\frac{{t}^{3}}{\ell \cdot \ell}}\\ \mathbf{elif}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{t}}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-228}:\\ \;\;\;\;\frac{\ell}{k \cdot {t}^{3}} \cdot \frac{\ell}{k}\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{-90}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\left(k \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\mathsf{fma}\left(0.16666666666666666, \frac{\ell}{\frac{{t}^{3}}{k}}, \frac{\frac{\ell}{{t}^{3}}}{k}\right) - {k}^{3} \cdot \frac{\ell \cdot -0.019444444444444445}{{t}^{3}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	54.6%
Cost	34388

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;t \leq -540000000000:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{\frac{2}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}}{\sin k \cdot {t}^{3}}\right)\\ \mathbf{elif}\;t \leq -1.6 \cdot 10^{-104}:\\ \;\;\;\;\frac{\frac{1}{k \cdot k} + -0.16666666666666666}{\frac{{t}^{3}}{\ell \cdot \ell}}\\ \mathbf{elif}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{t}}{\tan k \cdot \left(\sin k \cdot t_1\right)}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-228}:\\ \;\;\;\;\frac{\ell}{k \cdot {t}^{3}} \cdot \frac{\ell}{k}\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{-90}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(t_1 \cdot \left(\tan k \cdot \sin k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\mathsf{fma}\left(0.16666666666666666, \frac{\ell}{\frac{{t}^{3}}{k}}, \frac{\frac{\ell}{{t}^{3}}}{k}\right) - {k}^{3} \cdot \frac{\ell \cdot -0.019444444444444445}{{t}^{3}}\right)\\ \end{array} \]

Alternative 2
Accuracy	54.4%
Cost	28180

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot {t}^{3}}\\ t_2 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;t \leq -17000000000000:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{\frac{2}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}}{\sin k \cdot {t}^{3}}\right)\\ \mathbf{elif}\;t \leq -2.3 \cdot 10^{-104}:\\ \;\;\;\;\frac{\frac{1}{k \cdot k} + -0.16666666666666666}{\frac{{t}^{3}}{\ell \cdot \ell}}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{t}}{\tan k \cdot \left(\sin k \cdot t_2\right)}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-220}:\\ \;\;\;\;t_1 \cdot \frac{\ell}{k}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(t_2 \cdot \left(\tan k \cdot \sin k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\ell}{k}, t_1, \frac{\left(\ell \cdot \ell\right) \cdot 0.16666666666666666}{{t}^{3}} - k \cdot \left(k \cdot \left(-0.019444444444444445 \cdot \frac{\ell \cdot \ell}{{t}^{3}}\right)\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	56.1%
Cost	26892

\[\begin{array}{l} t_1 := \frac{t}{\ell} \cdot \left(k \cdot \frac{k}{\ell}\right)\\ \mathbf{if}\;k \leq -4.2 \cdot 10^{+84}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-0.5}{{t}^{5}}\right)\right)\right)\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-46}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \sin k\right) \cdot \sqrt[3]{t_1 \cdot \left(t_1 \cdot t_1\right)}}\\ \mathbf{elif}\;k \leq 9.5 \cdot 10^{+82}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\log \left(e^{\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \left(2 \cdot k\right)}\\ \end{array} \]

Alternative 4
Accuracy	52.8%
Cost	21704

\[\begin{array}{l} t_1 := \ell \cdot \left(\ell \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-0.5}{{t}^{5}}\right)\right)\right)\\ t_2 := \frac{t}{\ell} \cdot \left(k \cdot \frac{k}{\ell}\right)\\ \mathbf{if}\;k \leq -5.5 \cdot 10^{+84}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -8.5 \cdot 10^{-46}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \sin k\right) \cdot \sqrt[3]{t_2 \cdot \left(t_2 \cdot t_2\right)}}\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{+17}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{+229}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \frac{k}{\frac{\ell \cdot \ell}{{t}^{3}}}}\\ \end{array} \]

Alternative 5
Accuracy	53.2%
Cost	20240

\[\begin{array}{l} t_1 := \ell \cdot \left(\ell \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-0.5}{{t}^{5}}\right)\right)\right)\\ \mathbf{if}\;k \leq -5.5 \cdot 10^{+84}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -1.16 \cdot 10^{-45}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \sin k\right) \cdot \left(k \cdot \frac{t}{\frac{\ell}{\frac{k}{\ell}}}\right)}\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{+17}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}\\ \mathbf{elif}\;k \leq 1.65 \cdot 10^{+229}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \frac{k}{\frac{\ell \cdot \ell}{{t}^{3}}}}\\ \end{array} \]

Alternative 6
Accuracy	51.5%
Cost	14020

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+232}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \sin k\right) \cdot \left(k \cdot \frac{t}{\frac{\ell}{\frac{k}{\ell}}}\right)}\\ \end{array} \]

Alternative 7
Accuracy	50.2%
Cost	13828

\[\begin{array}{l} \mathbf{if}\;\ell \leq 4.3 \cdot 10^{+120}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \frac{k}{\frac{\ell \cdot \ell}{{t}^{3}}}}\\ \end{array} \]

Alternative 8
Accuracy	48.4%
Cost	13700

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+134}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{\sin k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \left(-\ell\right)}{\frac{{t}^{3}}{0.16666666666666666 + \frac{\frac{-1}{k}}{k}}}\\ \end{array} \]

Alternative 9
Accuracy	50.7%
Cost	13572

\[\begin{array}{l} \mathbf{if}\;k \leq -1.6 \cdot 10^{+170}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{\sin k \cdot {t}^{3}}}{k}\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{+17}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{-0.5}{{t}^{5}}\right)\\ \end{array} \]

Alternative 10
Accuracy	48.4%
Cost	7620

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+134}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \left(-\ell\right)}{\frac{{t}^{3}}{0.16666666666666666 + \frac{\frac{-1}{k}}{k}}}\\ \end{array} \]

Alternative 11
Accuracy	48.5%
Cost	7556

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \frac{k}{\frac{\ell \cdot \ell}{{t}^{3}}}}\\ \end{array} \]

Alternative 12
Accuracy	48.4%
Cost	7556

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+134}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{k \cdot k} + -0.16666666666666666}{\frac{{t}^{3}}{\ell \cdot \ell}}\\ \end{array} \]

Alternative 13
Accuracy	50.6%
Cost	7304

\[\begin{array}{l} \mathbf{if}\;k \leq -1.45 \cdot 10^{+197}:\\ \;\;\;\;-0.16666666666666666 \cdot \frac{\ell}{\frac{{t}^{3}}{\ell}}\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{+17}:\\ \;\;\;\;\frac{\ell}{k \cdot {t}^{3}} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{-0.5}{{t}^{5}}\right)\\ \end{array} \]

Alternative 14
Accuracy	50.5%
Cost	7304

\[\begin{array}{l} \mathbf{if}\;k \leq -9.2 \cdot 10^{+195}:\\ \;\;\;\;-0.16666666666666666 \cdot \frac{\ell}{\frac{{t}^{3}}{\ell}}\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{+17}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{-0.5}{{t}^{5}}\right)\\ \end{array} \]

Alternative 15
Accuracy	46.0%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;k \leq -5500000000 \lor \neg \left(k \leq 1.26 \cdot 10^{+79}\right):\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{-0.5}{{t}^{5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot \frac{k}{\frac{\ell}{k}}\right)}\\ \end{array} \]

Alternative 16
Accuracy	45.7%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;k \leq -5500000000:\\ \;\;\;\;-0.16666666666666666 \cdot \frac{\ell}{\frac{{t}^{3}}{\ell}}\\ \mathbf{elif}\;k \leq 1.05 \cdot 10^{+79}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot \frac{k}{\frac{\ell}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{-0.5}{{t}^{5}}\right)\\ \end{array} \]

Alternative 17
Accuracy	38.6%
Cost	960

\[\frac{2}{\left(k \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot \frac{k}{\frac{\ell}{k}}\right)} \]

Reproduce

herbie shell --seed 2023255 
(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))))