Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.3% → 88.9%

Time: 28.2s

Alternatives: 22

Speedup: 3.8×

• 26.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) + 1.0d0))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0));
}

Python

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) + 1.0))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) + 1.0));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 55.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) + 1.0d0))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0));
}

Python

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) + 1.0))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) + 1.0));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 88.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt[3]{\sin k}\\ t_2 := \sqrt[3]{\tan k}\\ \mathbf{if}\;t \leq -2.9 \cdot 10^{-29}:\\ \;\;\;\;\frac{2}{{\left(\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot t_1\right) \cdot t_2\right)}^{3} \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-48}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{2}{t \cdot {k}^{2}} \cdot \frac{\ell}{\sin k}\right)}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t_2 \cdot \left(t_1 \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)\right)}^{3} \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (cbrt (sin k))) (t_2 (cbrt (tan k))))
   (if (<= t -2.9e-29)
     (/
      2.0
      (*
       (pow (* (* (* t (pow (cbrt l) -2.0)) t_1) t_2) 3.0)
       (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))))
     (if (<= t 4.2e-48)
       (/ (* l (* (/ 2.0 (* t (pow k 2.0))) (/ l (sin k)))) (tan k))
       (/
        2.0
        (*
         (pow (* t_2 (* t_1 (/ t (pow (cbrt l) 2.0)))) 3.0)
         (+ 1.0 (+ 1.0 (/ (/ k t) (/ t k))))))))))

C

double code(double t, double l, double k) {
	double t_1 = cbrt(sin(k));
	double t_2 = cbrt(tan(k));
	double tmp;
	if (t <= -2.9e-29) {
		tmp = 2.0 / (pow((((t * pow(cbrt(l), -2.0)) * t_1) * t_2), 3.0) * (1.0 + (1.0 + pow((k / t), 2.0))));
	} else if (t <= 4.2e-48) {
		tmp = (l * ((2.0 / (t * pow(k, 2.0))) * (l / sin(k)))) / tan(k);
	} else {
		tmp = 2.0 / (pow((t_2 * (t_1 * (t / pow(cbrt(l), 2.0)))), 3.0) * (1.0 + (1.0 + ((k / t) / (t / k)))));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.cbrt(Math.sin(k));
	double t_2 = Math.cbrt(Math.tan(k));
	double tmp;
	if (t <= -2.9e-29) {
		tmp = 2.0 / (Math.pow((((t * Math.pow(Math.cbrt(l), -2.0)) * t_1) * t_2), 3.0) * (1.0 + (1.0 + Math.pow((k / t), 2.0))));
	} else if (t <= 4.2e-48) {
		tmp = (l * ((2.0 / (t * Math.pow(k, 2.0))) * (l / Math.sin(k)))) / Math.tan(k);
	} else {
		tmp = 2.0 / (Math.pow((t_2 * (t_1 * (t / Math.pow(Math.cbrt(l), 2.0)))), 3.0) * (1.0 + (1.0 + ((k / t) / (t / k)))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cbrt(sin(k))
	t_2 = cbrt(tan(k))
	tmp = 0.0
	if (t <= -2.9e-29)
		tmp = Float64(2.0 / Float64((Float64(Float64(Float64(t * (cbrt(l) ^ -2.0)) * t_1) * t_2) ^ 3.0) * Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0)))));
	elseif (t <= 4.2e-48)
		tmp = Float64(Float64(l * Float64(Float64(2.0 / Float64(t * (k ^ 2.0))) * Float64(l / sin(k)))) / tan(k));
	else
		tmp = Float64(2.0 / Float64((Float64(t_2 * Float64(t_1 * Float64(t / (cbrt(l) ^ 2.0)))) ^ 3.0) * Float64(1.0 + Float64(1.0 + Float64(Float64(k / t) / Float64(t / k))))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[t, -2.9e-29], N[(2.0 / N[(N[Power[N[(N[(N[(t * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], 3.0], $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.2e-48], N[(N[(l * N[(N[(2.0 / N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(t$95$2 * N[(t$95$1 * N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(1.0 + N[(1.0 + N[(N[(k / t), $MachinePrecision] / N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.90000000000000024e-29

   1. Initial program 58.1%

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

      1. associate-/r* 64.7%

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

      2. add-cube-cbrt 64.7%

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

      3. *-un-lft-identity 64.7%

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

      4. times-frac 64.7%

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

      5. pow2 64.7%

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

      6. cbrt-div 64.7%

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

      7. rem-cbrt-cube 64.7%

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

      8. cbrt-div 64.7%

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

      9. rem-cbrt-cube 79.2%

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

   3. Applied egg-rr 79.2%

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

      1. add-cube-cbrt 79.0%

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

      2. pow3 79.0%

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

      3. cbrt-prod 79.0%

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

      4. frac-times 73.3%

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

      5. unpow2 73.3%

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

      6. *-un-lft-identity 73.3%

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

      7. cbrt-div 74.7%

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

      8. add-cbrt-cube 87.7%

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

   5. Applied egg-rr 87.7%

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

      1. add-cube-cbrt 87.6%

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

      2. pow3 87.7%

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

      3. cbrt-prod 87.6%

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

      4. unpow3 87.6%

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

      5. add-cbrt-cube 91.6%

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

      6. associate-/l/ 91.5%

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

      7. pow2 91.5%

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

   7. Applied egg-rr 91.5%

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

      1. expm1-log1p-u 61.1%

         \[\leadsto \frac{2}{{\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)\right)} \cdot \sqrt[3]{\tan k}\right)}^{3} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. expm1-udef 49.2%

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

      3. div-inv 49.2%

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

      4. pow-flip 49.2%

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

      5. metadata-eval 49.2%

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

   9. Applied egg-rr 49.2%

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

       1. expm1-def 61.1%

          \[\leadsto \frac{2}{{\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k}\right)\right)} \cdot \sqrt[3]{\tan k}\right)}^{3} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

       2. expm1-log1p 91.6%

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

   11. Simplified 91.6%

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

   if -2.90000000000000024e-29 < t < 4.19999999999999977e-48

   1. Initial program 36.3%

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

   3. Simplified 35.4%

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

      1. associate-*r* 41.8%

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

      2. associate-*r* 41.8%

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

      3. times-frac 43.5%

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

      4. div-inv 43.5%

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

      5. pow-flip 43.5%

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

      6. metadata-eval 43.5%

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

   5. Applied egg-rr 43.5%

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

      1. times-frac 43.6%

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

   7. Simplified 43.6%

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

   8. Taylor expanded in t around 0 80.2%

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

      1. associate-*r/ 80.2%

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

      2. associate-*r* 80.2%

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

   10. Simplified 80.2%

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

       1. associate-*r/ 80.3%

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

       2. times-frac 82.5%

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

       3. *-commutative 82.5%

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

   12. Applied egg-rr 82.5%

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

   if 4.19999999999999977e-48 < t

   1. Initial program 67.1%

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

      1. associate-/r* 72.7%

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

      2. add-cube-cbrt 72.6%

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

      3. *-un-lft-identity 72.6%

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

      4. times-frac 72.6%

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

      5. pow2 72.6%

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

      6. cbrt-div 72.6%

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

      7. rem-cbrt-cube 72.6%

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

      8. cbrt-div 72.6%

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

      9. rem-cbrt-cube 85.0%

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

   3. Applied egg-rr 85.0%

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

      1. add-cube-cbrt 84.9%

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

      2. pow3 84.8%

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

      3. cbrt-prod 84.8%

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

      4. frac-times 78.6%

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

      5. unpow2 78.6%

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

      6. *-un-lft-identity 78.6%

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

      7. cbrt-div 78.6%

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

      8. add-cbrt-cube 89.5%

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

   5. Applied egg-rr 89.5%

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

      1. add-cube-cbrt 89.5%

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

      2. pow3 89.5%

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

      3. cbrt-prod 89.2%

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

      4. unpow3 89.3%

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

      5. add-cbrt-cube 95.2%

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

      6. associate-/l/ 95.3%

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

      7. pow2 95.3%

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

   7. Applied egg-rr 95.3%

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

      1. unpow2 95.3%

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

      2. clear-num 95.3%

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

      3. un-div-inv 95.3%

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

   9. Applied egg-rr 95.3%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{-29}:\\ \;\;\;\;\frac{2}{{\left(\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\tan k}\right)}^{3} \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-48}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{2}{t \cdot {k}^{2}} \cdot \frac{\ell}{\sin k}\right)}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\tan k} \cdot \left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)\right)}^{3} \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)}\\ \end{array} \]

Alternative 2: 78.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 80.8%

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

   3. Simplified 74.4%

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

      1. associate-*r* 75.6%

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

      2. associate-*r* 75.6%

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

      3. times-frac 87.4%

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

      4. div-inv 87.4%

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

      5. pow-flip 87.9%

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

      6. metadata-eval 87.9%

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

   5. Applied egg-rr 87.9%

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

      1. times-frac 86.2%

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

   7. Simplified 86.2%

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

      1. associate-*r/ 84.0%

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

   9. Applied egg-rr 84.0%

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

       1. associate-/l* 86.3%

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

       2. associate-*l/ 89.2%

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

       3. associate-*r/ 87.9%

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

       4. associate-/r* 87.9%

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

       5. times-frac 88.7%

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

   11. Simplified 88.7%

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

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

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

      1. associate-*r* 14.7%

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

      2. associate-*r* 14.7%

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

      3. times-frac 14.7%

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

      4. div-inv 14.7%

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

      5. pow-flip 16.8%

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

      6. metadata-eval 16.8%

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

   5. Applied egg-rr 16.8%

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

      1. times-frac 16.9%

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

   7. Simplified 16.9%

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

   8. Taylor expanded in t around 0 59.6%

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

      1. associate-*r/ 59.6%

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

      2. associate-*r* 59.6%

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

   10. Simplified 59.6%

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

   11. Taylor expanded in l around 0 59.6%

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

       1. *-commutative 59.6%

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

       2. *-commutative 59.6%

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

       3. associate-*r* 59.6%

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

       4. associate-/r* 61.3%

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

       5. *-commutative 61.3%

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

   13. Simplified 61.3%

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

4. Final simplification 78.6%

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

Alternative 3: 88.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.12 \cdot 10^{-30} \lor \neg \left(t \leq 3.9 \cdot 10^{-48}\right):\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\tan k} \cdot \left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)\right)}^{3} \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{2}{t \cdot {k}^{2}} \cdot \frac{\ell}{\sin k}\right)}{\tan k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -1.12e-30) (not (<= t 3.9e-48)))
   (/
    2.0
    (*
     (pow (* (cbrt (tan k)) (* (cbrt (sin k)) (/ t (pow (cbrt l) 2.0)))) 3.0)
     (+ 1.0 (+ 1.0 (/ (/ k t) (/ t k))))))
   (/ (* l (* (/ 2.0 (* t (pow k 2.0))) (/ l (sin k)))) (tan k))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.12e-30) || !(t <= 3.9e-48)) {
		tmp = 2.0 / (pow((cbrt(tan(k)) * (cbrt(sin(k)) * (t / pow(cbrt(l), 2.0)))), 3.0) * (1.0 + (1.0 + ((k / t) / (t / k)))));
	} else {
		tmp = (l * ((2.0 / (t * pow(k, 2.0))) * (l / sin(k)))) / tan(k);
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.12e-30) || !(t <= 3.9e-48)) {
		tmp = 2.0 / (Math.pow((Math.cbrt(Math.tan(k)) * (Math.cbrt(Math.sin(k)) * (t / Math.pow(Math.cbrt(l), 2.0)))), 3.0) * (1.0 + (1.0 + ((k / t) / (t / k)))));
	} else {
		tmp = (l * ((2.0 / (t * Math.pow(k, 2.0))) * (l / Math.sin(k)))) / Math.tan(k);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -1.12e-30) || !(t <= 3.9e-48))
		tmp = Float64(2.0 / Float64((Float64(cbrt(tan(k)) * Float64(cbrt(sin(k)) * Float64(t / (cbrt(l) ^ 2.0)))) ^ 3.0) * Float64(1.0 + Float64(1.0 + Float64(Float64(k / t) / Float64(t / k))))));
	else
		tmp = Float64(Float64(l * Float64(Float64(2.0 / Float64(t * (k ^ 2.0))) * Float64(l / sin(k)))) / tan(k));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -1.12e-30], N[Not[LessEqual[t, 3.9e-48]], $MachinePrecision]], N[(2.0 / N[(N[Power[N[(N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(1.0 + N[(1.0 + N[(N[(k / t), $MachinePrecision] / N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[(N[(2.0 / N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.12e-30 or 3.9e-48 < t

   1. Initial program 63.0%

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

      1. associate-/r* 69.1%

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

      2. add-cube-cbrt 69.0%

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

      3. *-un-lft-identity 69.0%

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

      4. times-frac 69.0%

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

      5. pow2 69.0%

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

      6. cbrt-div 69.0%

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

      7. rem-cbrt-cube 69.0%

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

      8. cbrt-div 69.0%

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

      9. rem-cbrt-cube 82.3%

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

   3. Applied egg-rr 82.3%

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

      1. add-cube-cbrt 82.2%

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

      2. pow3 82.2%

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

      3. cbrt-prod 82.1%

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

      4. frac-times 76.2%

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

      5. unpow2 76.2%

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

      6. *-un-lft-identity 76.2%

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

      7. cbrt-div 76.8%

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

      8. add-cbrt-cube 88.6%

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

   5. Applied egg-rr 88.6%

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

      1. add-cube-cbrt 88.6%

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

      2. pow3 88.7%

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

      3. cbrt-prod 88.5%

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

      4. unpow3 88.5%

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

      5. add-cbrt-cube 93.5%

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

      6. associate-/l/ 93.6%

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

      7. pow2 93.6%

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

   7. Applied egg-rr 93.6%

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

      1. unpow2 93.6%

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

      2. clear-num 93.6%

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

      3. un-div-inv 93.6%

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

   9. Applied egg-rr 93.6%

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

   if -1.12e-30 < t < 3.9e-48

   1. Initial program 36.3%

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

   3. Simplified 35.4%

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

      1. associate-*r* 41.8%

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

      2. associate-*r* 41.8%

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

      3. times-frac 43.5%

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

      4. div-inv 43.5%

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

      5. pow-flip 43.5%

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

      6. metadata-eval 43.5%

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

   5. Applied egg-rr 43.5%

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

      1. times-frac 43.6%

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

   7. Simplified 43.6%

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

   8. Taylor expanded in t around 0 80.2%

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

      1. associate-*r/ 80.2%

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

      2. associate-*r* 80.2%

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

   10. Simplified 80.2%

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

       1. associate-*r/ 80.3%

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

       2. times-frac 82.5%

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

       3. *-commutative 82.5%

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

   12. Applied egg-rr 82.5%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.12 \cdot 10^{-30} \lor \neg \left(t \leq 3.9 \cdot 10^{-48}\right):\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\tan k} \cdot \left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)\right)}^{3} \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{2}{t \cdot {k}^{2}} \cdot \frac{\ell}{\sin k}\right)}{\tan k}\\ \end{array} \]

Alternative 4: 84.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{+68} \lor \neg \left(t \leq 2.4 \cdot 10^{-48}\right):\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot {\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{2}{t \cdot {k}^{2}} \cdot \frac{\ell}{\sin k}\right)}{\tan k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -5.5e+68) (not (<= t 2.4e-48)))
   (/
    2.0
    (*
     (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))
     (* (tan k) (pow (* (cbrt (sin k)) (/ t (pow (cbrt l) 2.0))) 3.0))))
   (/ (* l (* (/ 2.0 (* t (pow k 2.0))) (/ l (sin k)))) (tan k))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -5.5e+68) || !(t <= 2.4e-48)) {
		tmp = 2.0 / ((1.0 + (1.0 + pow((k / t), 2.0))) * (tan(k) * pow((cbrt(sin(k)) * (t / pow(cbrt(l), 2.0))), 3.0)));
	} else {
		tmp = (l * ((2.0 / (t * pow(k, 2.0))) * (l / sin(k)))) / tan(k);
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -5.5e+68) || !(t <= 2.4e-48)) {
		tmp = 2.0 / ((1.0 + (1.0 + Math.pow((k / t), 2.0))) * (Math.tan(k) * Math.pow((Math.cbrt(Math.sin(k)) * (t / Math.pow(Math.cbrt(l), 2.0))), 3.0)));
	} else {
		tmp = (l * ((2.0 / (t * Math.pow(k, 2.0))) * (l / Math.sin(k)))) / Math.tan(k);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -5.5e+68) || !(t <= 2.4e-48))
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0))) * Float64(tan(k) * (Float64(cbrt(sin(k)) * Float64(t / (cbrt(l) ^ 2.0))) ^ 3.0))));
	else
		tmp = Float64(Float64(l * Float64(Float64(2.0 / Float64(t * (k ^ 2.0))) * Float64(l / sin(k)))) / tan(k));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -5.5e+68], N[Not[LessEqual[t, 2.4e-48]], $MachinePrecision]], N[(2.0 / N[(N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[(N[(2.0 / N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.5000000000000004e68 or 2.4e-48 < t

   1. Initial program 61.0%

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

      1. associate-/r* 68.4%

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

      2. add-cube-cbrt 68.3%

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

      3. *-un-lft-identity 68.3%

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

      4. times-frac 68.3%

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

      5. pow2 68.3%

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

      6. cbrt-div 68.3%

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

      7. rem-cbrt-cube 68.3%

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

      8. cbrt-div 68.3%

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

      9. rem-cbrt-cube 84.4%

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

   3. Applied egg-rr 84.4%

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

      1. add-cube-cbrt 84.2%

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

      2. pow3 84.2%

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

      3. cbrt-prod 84.2%

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

      4. frac-times 77.1%

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

      5. unpow2 77.1%

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

      6. *-un-lft-identity 77.1%

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

      7. cbrt-div 77.0%

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

      8. add-cbrt-cube 91.2%

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

   5. Applied egg-rr 91.2%

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

      1. expm1-log1p-u 66.0%

         \[\leadsto \frac{2}{\left({\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}} \cdot \sqrt[3]{\sin k}\right)\right)\right)}}^{3} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. expm1-udef 49.9%

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

      3. associate-/l/ 49.9%

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

      4. pow2 49.9%

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

   7. Applied egg-rr 49.9%

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

      1. expm1-def 66.1%

         \[\leadsto \frac{2}{\left({\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)\right)\right)}}^{3} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. expm1-log1p 91.2%

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

   9. Simplified 91.2%

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

   if -5.5000000000000004e68 < t < 2.4e-48

   1. Initial program 42.6%

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

   3. Simplified 41.1%

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

      1. associate-*r* 46.4%

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

      2. associate-*r* 46.4%

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

      3. times-frac 51.9%

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

      4. div-inv 51.9%

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

      5. pow-flip 51.9%

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

      6. metadata-eval 51.9%

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

   5. Applied egg-rr 51.9%

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

      1. times-frac 51.9%

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

   7. Simplified 51.9%

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

   8. Taylor expanded in t around 0 80.1%

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

      1. associate-*r/ 80.1%

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

      2. associate-*r* 80.2%

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

   10. Simplified 80.2%

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

       1. associate-*r/ 80.2%

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

       2. times-frac 82.0%

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

       3. *-commutative 82.0%

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

   12. Applied egg-rr 82.0%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{+68} \lor \neg \left(t \leq 2.4 \cdot 10^{-48}\right):\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot {\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{2}{t \cdot {k}^{2}} \cdot \frac{\ell}{\sin k}\right)}{\tan k}\\ \end{array} \]

Alternative 5: 84.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt[3]{\sin k}\\ t_2 := 1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\\ \mathbf{if}\;t \leq -5.5 \cdot 10^{+68}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\tan k \cdot {\left(t_1 \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}\right)}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-48}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{2}{t \cdot {k}^{2}} \cdot \frac{\ell}{\sin k}\right)}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\tan k \cdot {\left(t_1 \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (cbrt (sin k))) (t_2 (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))))
   (if (<= t -5.5e+68)
     (/ 2.0 (* t_2 (* (tan k) (pow (* t_1 (/ (/ t (cbrt l)) (cbrt l))) 3.0))))
     (if (<= t 3.8e-48)
       (/ (* l (* (/ 2.0 (* t (pow k 2.0))) (/ l (sin k)))) (tan k))
       (/
        2.0
        (* t_2 (* (tan k) (pow (* t_1 (/ t (pow (cbrt l) 2.0))) 3.0))))))))

C

double code(double t, double l, double k) {
	double t_1 = cbrt(sin(k));
	double t_2 = 1.0 + (1.0 + pow((k / t), 2.0));
	double tmp;
	if (t <= -5.5e+68) {
		tmp = 2.0 / (t_2 * (tan(k) * pow((t_1 * ((t / cbrt(l)) / cbrt(l))), 3.0)));
	} else if (t <= 3.8e-48) {
		tmp = (l * ((2.0 / (t * pow(k, 2.0))) * (l / sin(k)))) / tan(k);
	} else {
		tmp = 2.0 / (t_2 * (tan(k) * pow((t_1 * (t / pow(cbrt(l), 2.0))), 3.0)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.cbrt(Math.sin(k));
	double t_2 = 1.0 + (1.0 + Math.pow((k / t), 2.0));
	double tmp;
	if (t <= -5.5e+68) {
		tmp = 2.0 / (t_2 * (Math.tan(k) * Math.pow((t_1 * ((t / Math.cbrt(l)) / Math.cbrt(l))), 3.0)));
	} else if (t <= 3.8e-48) {
		tmp = (l * ((2.0 / (t * Math.pow(k, 2.0))) * (l / Math.sin(k)))) / Math.tan(k);
	} else {
		tmp = 2.0 / (t_2 * (Math.tan(k) * Math.pow((t_1 * (t / Math.pow(Math.cbrt(l), 2.0))), 3.0)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cbrt(sin(k))
	t_2 = Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0)))
	tmp = 0.0
	if (t <= -5.5e+68)
		tmp = Float64(2.0 / Float64(t_2 * Float64(tan(k) * (Float64(t_1 * Float64(Float64(t / cbrt(l)) / cbrt(l))) ^ 3.0))));
	elseif (t <= 3.8e-48)
		tmp = Float64(Float64(l * Float64(Float64(2.0 / Float64(t * (k ^ 2.0))) * Float64(l / sin(k)))) / tan(k));
	else
		tmp = Float64(2.0 / Float64(t_2 * Float64(tan(k) * (Float64(t_1 * Float64(t / (cbrt(l) ^ 2.0))) ^ 3.0))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5.5e+68], N[(2.0 / N[(t$95$2 * N[(N[Tan[k], $MachinePrecision] * N[Power[N[(t$95$1 * N[(N[(t / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.8e-48], N[(N[(l * N[(N[(2.0 / N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$2 * N[(N[Tan[k], $MachinePrecision] * N[Power[N[(t$95$1 * N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.5000000000000004e68

   1. Initial program 49.7%

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

      1. associate-/r* 60.2%

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

      2. add-cube-cbrt 60.2%

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

      3. *-un-lft-identity 60.2%

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

      4. times-frac 60.2%

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

      5. pow2 60.2%

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

      6. cbrt-div 60.2%

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

      7. rem-cbrt-cube 60.2%

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

      8. cbrt-div 60.2%

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

      9. rem-cbrt-cube 83.2%

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

   3. Applied egg-rr 83.2%

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

      1. add-cube-cbrt 83.1%

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

      2. pow3 83.1%

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

      3. cbrt-prod 83.1%

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

      4. frac-times 74.1%

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

      5. unpow2 74.1%

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

      6. *-un-lft-identity 74.1%

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

      7. cbrt-div 74.0%

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

      8. add-cbrt-cube 94.5%

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

   5. Applied egg-rr 94.5%

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

   if -5.5000000000000004e68 < t < 3.80000000000000002e-48

   1. Initial program 42.6%

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

   3. Simplified 41.1%

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

      1. associate-*r* 46.4%

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

      2. associate-*r* 46.4%

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

      3. times-frac 51.9%

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

      4. div-inv 51.9%

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

      5. pow-flip 51.9%

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

      6. metadata-eval 51.9%

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

   5. Applied egg-rr 51.9%

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

      1. times-frac 51.9%

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

   7. Simplified 51.9%

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

   8. Taylor expanded in t around 0 80.1%

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

      1. associate-*r/ 80.1%

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

      2. associate-*r* 80.2%

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

   10. Simplified 80.2%

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

       1. associate-*r/ 80.2%

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

       2. times-frac 82.0%

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

       3. *-commutative 82.0%

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

   12. Applied egg-rr 82.0%

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

   if 3.80000000000000002e-48 < t

   1. Initial program 67.1%

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

      1. associate-/r* 72.7%

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

      2. add-cube-cbrt 72.6%

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

      3. *-un-lft-identity 72.6%

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

      4. times-frac 72.6%

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

      5. pow2 72.6%

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

      6. cbrt-div 72.6%

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

      7. rem-cbrt-cube 72.6%

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

      8. cbrt-div 72.6%

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

      9. rem-cbrt-cube 85.0%

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

   3. Applied egg-rr 85.0%

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

      1. add-cube-cbrt 84.9%

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

      2. pow3 84.8%

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

      3. cbrt-prod 84.8%

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

      4. frac-times 78.6%

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

      5. unpow2 78.6%

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

      6. *-un-lft-identity 78.6%

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

      7. cbrt-div 78.6%

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

      8. add-cbrt-cube 89.5%

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

   5. Applied egg-rr 89.5%

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

      1. expm1-log1p-u 72.4%

         \[\leadsto \frac{2}{\left({\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}} \cdot \sqrt[3]{\sin k}\right)\right)\right)}}^{3} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. expm1-udef 52.4%

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

      3. associate-/l/ 52.4%

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

      4. pow2 52.4%

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

   7. Applied egg-rr 52.4%

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

      1. expm1-def 72.4%

         \[\leadsto \frac{2}{\left({\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)\right)\right)}}^{3} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. expm1-log1p 89.5%

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

   9. Simplified 89.5%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{+68}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot {\left(\sqrt[3]{\sin k} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}\right)}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-48}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{2}{t \cdot {k}^{2}} \cdot \frac{\ell}{\sin k}\right)}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot {\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \end{array} \]

Alternative 6: 71.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \frac{2}{\left(1 + \left(1 + t_1\right)\right) \cdot \left(\tan k \cdot {\left(\sqrt[3]{\sin k} \cdot \left(t \cdot {\ell}^{-0.6666666666666666}\right)\right)}^{3}\right)}\\ t_3 := \frac{2 \cdot {t}^{-3}}{\sin k} \cdot \frac{\ell}{2 + t_1}\\ \mathbf{if}\;t \leq -6.7 \cdot 10^{+103}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -3 \cdot 10^{-30}:\\ \;\;\;\;\frac{t_3}{\frac{\tan k}{\ell}}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-48}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{2}{t \cdot {k}^{2}} \cdot \frac{\ell}{\sin k}\right)}{\tan k}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{+104}:\\ \;\;\;\;t_3 \cdot \frac{\ell}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0))
        (t_2
         (/
          2.0
          (*
           (+ 1.0 (+ 1.0 t_1))
           (*
            (tan k)
            (pow (* (cbrt (sin k)) (* t (pow l -0.6666666666666666))) 3.0)))))
        (t_3 (* (/ (* 2.0 (pow t -3.0)) (sin k)) (/ l (+ 2.0 t_1)))))
   (if (<= t -6.7e+103)
     t_2
     (if (<= t -3e-30)
       (/ t_3 (/ (tan k) l))
       (if (<= t 2.1e-48)
         (/ (* l (* (/ 2.0 (* t (pow k 2.0))) (/ l (sin k)))) (tan k))
         (if (<= t 2.6e+104) (* t_3 (/ l (tan k))) t_2))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double t_2 = 2.0 / ((1.0 + (1.0 + t_1)) * (tan(k) * pow((cbrt(sin(k)) * (t * pow(l, -0.6666666666666666))), 3.0)));
	double t_3 = ((2.0 * pow(t, -3.0)) / sin(k)) * (l / (2.0 + t_1));
	double tmp;
	if (t <= -6.7e+103) {
		tmp = t_2;
	} else if (t <= -3e-30) {
		tmp = t_3 / (tan(k) / l);
	} else if (t <= 2.1e-48) {
		tmp = (l * ((2.0 / (t * pow(k, 2.0))) * (l / sin(k)))) / tan(k);
	} else if (t <= 2.6e+104) {
		tmp = t_3 * (l / tan(k));
	} else {
		tmp = t_2;
	}
	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 = 2.0 / ((1.0 + (1.0 + t_1)) * (Math.tan(k) * Math.pow((Math.cbrt(Math.sin(k)) * (t * Math.pow(l, -0.6666666666666666))), 3.0)));
	double t_3 = ((2.0 * Math.pow(t, -3.0)) / Math.sin(k)) * (l / (2.0 + t_1));
	double tmp;
	if (t <= -6.7e+103) {
		tmp = t_2;
	} else if (t <= -3e-30) {
		tmp = t_3 / (Math.tan(k) / l);
	} else if (t <= 2.1e-48) {
		tmp = (l * ((2.0 / (t * Math.pow(k, 2.0))) * (l / Math.sin(k)))) / Math.tan(k);
	} else if (t <= 2.6e+104) {
		tmp = t_3 * (l / Math.tan(k));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	t_2 = Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + t_1)) * Float64(tan(k) * (Float64(cbrt(sin(k)) * Float64(t * (l ^ -0.6666666666666666))) ^ 3.0))))
	t_3 = Float64(Float64(Float64(2.0 * (t ^ -3.0)) / sin(k)) * Float64(l / Float64(2.0 + t_1)))
	tmp = 0.0
	if (t <= -6.7e+103)
		tmp = t_2;
	elseif (t <= -3e-30)
		tmp = Float64(t_3 / Float64(tan(k) / l));
	elseif (t <= 2.1e-48)
		tmp = Float64(Float64(l * Float64(Float64(2.0 / Float64(t * (k ^ 2.0))) * Float64(l / sin(k)))) / tan(k));
	elseif (t <= 2.6e+104)
		tmp = Float64(t_3 * Float64(l / tan(k)));
	else
		tmp = t_2;
	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[(2.0 / N[(N[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t * N[Power[l, -0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(2.0 * N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -6.7e+103], t$95$2, If[LessEqual[t, -3e-30], N[(t$95$3 / N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.1e-48], N[(N[(l * N[(N[(2.0 / N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.6e+104], N[(t$95$3 * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -6.70000000000000033e103 or 2.6e104 < t

   1. Initial program 52.7%

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

      1. add-cube-cbrt 52.7%

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

      2. pow3 52.7%

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

      3. cbrt-prod 52.7%

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

      4. div-inv 52.7%

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

      5. cbrt-prod 52.7%

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

      6. rem-cbrt-cube 67.2%

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

      7. pow2 67.2%

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

      8. pow-flip 68.5%

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

      9. metadata-eval 68.5%

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

   3. Applied egg-rr 68.5%

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

      1. pow1/3 67.8%

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

      2. pow-pow 37.8%

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

      3. metadata-eval 37.8%

         \[\leadsto \frac{2}{\left({\left(\left(t \cdot {\ell}^{\color{blue}{-0.6666666666666666}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Applied egg-rr 37.8%

      \[\leadsto \frac{2}{\left({\left(\left(t \cdot \color{blue}{{\ell}^{-0.6666666666666666}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   if -6.70000000000000033e103 < t < -2.9999999999999999e-30

   1. Initial program 67.0%

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

   3. Simplified 59.9%

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

      1. associate-*r* 63.2%

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

      2. associate-*r* 63.2%

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

      3. times-frac 88.7%

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

      4. div-inv 88.7%

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

      5. pow-flip 92.1%

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

      6. metadata-eval 92.1%

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

   5. Applied egg-rr 92.1%

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

      1. times-frac 89.6%

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

   7. Simplified 89.6%

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

      1. associate-*r/ 86.5%

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

   9. Applied egg-rr 86.5%

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

       1. associate-/l* 89.7%

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

       2. associate-*l/ 96.3%

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

       3. associate-*r/ 91.9%

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

       4. associate-/r* 92.2%

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

       5. times-frac 96.3%

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

   11. Simplified 96.3%

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

   if -2.9999999999999999e-30 < t < 2.09999999999999989e-48

   1. Initial program 36.3%

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

   3. Simplified 35.4%

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

      1. associate-*r* 41.8%

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

      2. associate-*r* 41.8%

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

      3. times-frac 43.5%

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

      4. div-inv 43.5%

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

      5. pow-flip 43.5%

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

      6. metadata-eval 43.5%

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

   5. Applied egg-rr 43.5%

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

      1. times-frac 43.6%

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

   7. Simplified 43.6%

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

   8. Taylor expanded in t around 0 80.2%

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

      1. associate-*r/ 80.2%

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

      2. associate-*r* 80.2%

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

   10. Simplified 80.2%

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

       1. associate-*r/ 80.3%

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

       2. times-frac 82.5%

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

       3. *-commutative 82.5%

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

   12. Applied egg-rr 82.5%

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

   if 2.09999999999999989e-48 < t < 2.6e104

   1. Initial program 81.7%

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

   3. Simplified 81.7%

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

      1. associate-*r* 84.4%

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

      2. associate-*r* 84.4%

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

      3. times-frac 91.9%

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

      4. div-inv 91.9%

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

      5. pow-flip 91.9%

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

      6. metadata-eval 91.9%

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

   5. Applied egg-rr 91.9%

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

      1. times-frac 86.5%

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

   7. Simplified 86.5%

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

      1. expm1-log1p-u 75.1%

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

      2. expm1-udef 62.3%

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

   9. Applied egg-rr 62.3%

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

       1. expm1-def 75.1%

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

       2. expm1-log1p 86.5%

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

       3. associate-*l/ 94.3%

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

       4. associate-*r/ 91.9%

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

       5. associate-/r* 91.9%

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

       6. times-frac 94.4%

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

   11. Simplified 94.4%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.7 \cdot 10^{+103}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot {\left(\sqrt[3]{\sin k} \cdot \left(t \cdot {\ell}^{-0.6666666666666666}\right)\right)}^{3}\right)}\\ \mathbf{elif}\;t \leq -3 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{2 \cdot {t}^{-3}}{\sin k} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}}{\frac{\tan k}{\ell}}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-48}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{2}{t \cdot {k}^{2}} \cdot \frac{\ell}{\sin k}\right)}{\tan k}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{+104}:\\ \;\;\;\;\left(\frac{2 \cdot {t}^{-3}}{\sin k} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\right) \cdot \frac{\ell}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot {\left(\sqrt[3]{\sin k} \cdot \left(t \cdot {\ell}^{-0.6666666666666666}\right)\right)}^{3}\right)}\\ \end{array} \]

Alternative 7: 84.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \frac{2}{t_1 \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)\right)}\\ \mathbf{if}\;t \leq -6.8 \cdot 10^{+103}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{2 \cdot {t}^{-3}}{\sin k} \cdot \frac{\ell}{t_1}}{\frac{\tan k}{\ell}}\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{-48}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{2}{t \cdot {k}^{2}} \cdot \frac{\ell}{\sin k}\right)}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (+ 2.0 (pow (/ k t) 2.0)))
        (t_2
         (/
          2.0
          (* t_1 (* (sin k) (* (tan k) (pow (/ t (pow (cbrt l) 2.0)) 3.0)))))))
   (if (<= t -6.8e+103)
     t_2
     (if (<= t -1e-30)
       (/ (* (/ (* 2.0 (pow t -3.0)) (sin k)) (/ l t_1)) (/ (tan k) l))
       (if (<= t 3.7e-48)
         (/ (* l (* (/ 2.0 (* t (pow k 2.0))) (/ l (sin k)))) (tan k))
         t_2)))))

C

double code(double t, double l, double k) {
	double t_1 = 2.0 + pow((k / t), 2.0);
	double t_2 = 2.0 / (t_1 * (sin(k) * (tan(k) * pow((t / pow(cbrt(l), 2.0)), 3.0))));
	double tmp;
	if (t <= -6.8e+103) {
		tmp = t_2;
	} else if (t <= -1e-30) {
		tmp = (((2.0 * pow(t, -3.0)) / sin(k)) * (l / t_1)) / (tan(k) / l);
	} else if (t <= 3.7e-48) {
		tmp = (l * ((2.0 / (t * pow(k, 2.0))) * (l / sin(k)))) / tan(k);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = 2.0 + Math.pow((k / t), 2.0);
	double t_2 = 2.0 / (t_1 * (Math.sin(k) * (Math.tan(k) * Math.pow((t / Math.pow(Math.cbrt(l), 2.0)), 3.0))));
	double tmp;
	if (t <= -6.8e+103) {
		tmp = t_2;
	} else if (t <= -1e-30) {
		tmp = (((2.0 * Math.pow(t, -3.0)) / Math.sin(k)) * (l / t_1)) / (Math.tan(k) / l);
	} else if (t <= 3.7e-48) {
		tmp = (l * ((2.0 / (t * Math.pow(k, 2.0))) * (l / Math.sin(k)))) / Math.tan(k);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(2.0 + (Float64(k / t) ^ 2.0))
	t_2 = Float64(2.0 / Float64(t_1 * Float64(sin(k) * Float64(tan(k) * (Float64(t / (cbrt(l) ^ 2.0)) ^ 3.0)))))
	tmp = 0.0
	if (t <= -6.8e+103)
		tmp = t_2;
	elseif (t <= -1e-30)
		tmp = Float64(Float64(Float64(Float64(2.0 * (t ^ -3.0)) / sin(k)) * Float64(l / t_1)) / Float64(tan(k) / l));
	elseif (t <= 3.7e-48)
		tmp = Float64(Float64(l * Float64(Float64(2.0 / Float64(t * (k ^ 2.0))) * Float64(l / sin(k)))) / tan(k));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 / N[(t$95$1 * N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Power[N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -6.8e+103], t$95$2, If[LessEqual[t, -1e-30], N[(N[(N[(N[(2.0 * N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.7e-48], N[(N[(l * N[(N[(2.0 / N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -6.7999999999999997e103 or 3.6999999999999998e-48 < t

   1. Initial program 61.9%

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

      1. associate-/r* 68.8%

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

      2. add-cube-cbrt 68.7%

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

      3. *-un-lft-identity 68.7%

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

      4. times-frac 68.7%

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

      5. pow2 68.7%

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

      6. cbrt-div 68.7%

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

      7. rem-cbrt-cube 68.7%

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

      8. cbrt-div 68.7%

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

      9. rem-cbrt-cube 84.6%

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

   3. Applied egg-rr 84.6%

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

      1. add-cube-cbrt 84.5%

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

      2. pow3 84.5%

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

      3. cbrt-prod 84.4%

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

      4. frac-times 77.0%

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

      5. unpow2 77.0%

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

      6. *-un-lft-identity 77.0%

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

      7. cbrt-div 76.9%

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

      8. add-cbrt-cube 90.9%

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

   5. Applied egg-rr 90.9%

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

      1. distribute-lft-in 90.9%

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

   7. Applied egg-rr 87.7%

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

      1. distribute-lft-out 87.7%

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

      2. associate-*l* 87.7%

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

      3. +-commutative 87.7%

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

      4. associate-+l+ 87.7%

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

      5. metadata-eval 87.7%

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

      6. +-commutative 87.7%

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

   9. Simplified 87.7%

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

   if -6.7999999999999997e103 < t < -1e-30

   1. Initial program 67.0%

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

   3. Simplified 59.9%

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

      1. associate-*r* 63.2%

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

      2. associate-*r* 63.2%

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

      3. times-frac 88.7%

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

      4. div-inv 88.7%

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

      5. pow-flip 92.1%

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

      6. metadata-eval 92.1%

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

   5. Applied egg-rr 92.1%

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

      1. times-frac 89.6%

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

   7. Simplified 89.6%

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

      1. associate-*r/ 86.5%

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

   9. Applied egg-rr 86.5%

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

       1. associate-/l* 89.7%

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

       2. associate-*l/ 96.3%

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

       3. associate-*r/ 91.9%

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

       4. associate-/r* 92.2%

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

       5. times-frac 96.3%

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

   11. Simplified 96.3%

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

   if -1e-30 < t < 3.6999999999999998e-48

   1. Initial program 36.3%

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

   3. Simplified 35.4%

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

      1. associate-*r* 41.8%

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

      2. associate-*r* 41.8%

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

      3. times-frac 43.5%

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

      4. div-inv 43.5%

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

      5. pow-flip 43.5%

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

      6. metadata-eval 43.5%

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

   5. Applied egg-rr 43.5%

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

      1. times-frac 43.6%

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

   7. Simplified 43.6%

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

   8. Taylor expanded in t around 0 80.2%

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

      1. associate-*r/ 80.2%

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

      2. associate-*r* 80.2%

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

   10. Simplified 80.2%

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

       1. associate-*r/ 80.3%

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

       2. times-frac 82.5%

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

       3. *-commutative 82.5%

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

   12. Applied egg-rr 82.5%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.8 \cdot 10^{+103}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)\right)}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{2 \cdot {t}^{-3}}{\sin k} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}}{\frac{\tan k}{\ell}}\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{-48}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{2}{t \cdot {k}^{2}} \cdot \frac{\ell}{\sin k}\right)}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)\right)}\\ \end{array} \]

Alternative 8: 81.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := 1 + \left(1 + t_1\right)\\ \mathbf{if}\;t \leq -6.4 \cdot 10^{+103}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{elif}\;t \leq -7 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{2 \cdot {t}^{-3}}{\sin k} \cdot \frac{\ell}{2 + t_1}}{\frac{\tan k}{\ell}}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-48}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{2}{t \cdot {k}^{2}} \cdot \frac{\ell}{\sin k}\right)}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)) (t_2 (+ 1.0 (+ 1.0 t_1))))
   (if (<= t -6.4e+103)
     (/ 2.0 (* t_2 (* (tan k) (* (sin k) (* (/ (pow t 2.0) l) (/ t l))))))
     (if (<= t -7e-32)
       (/ (* (/ (* 2.0 (pow t -3.0)) (sin k)) (/ l (+ 2.0 t_1))) (/ (tan k) l))
       (if (<= t 1.9e-48)
         (/ (* l (* (/ 2.0 (* t (pow k 2.0))) (/ l (sin k)))) (tan k))
         (/
          2.0
          (* t_2 (* (tan k) (* (sin k) (pow (/ (pow t 1.5) l) 2.0))))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double t_2 = 1.0 + (1.0 + t_1);
	double tmp;
	if (t <= -6.4e+103) {
		tmp = 2.0 / (t_2 * (tan(k) * (sin(k) * ((pow(t, 2.0) / l) * (t / l)))));
	} else if (t <= -7e-32) {
		tmp = (((2.0 * pow(t, -3.0)) / sin(k)) * (l / (2.0 + t_1))) / (tan(k) / l);
	} else if (t <= 1.9e-48) {
		tmp = (l * ((2.0 / (t * pow(k, 2.0))) * (l / sin(k)))) / tan(k);
	} else {
		tmp = 2.0 / (t_2 * (tan(k) * (sin(k) * pow((pow(t, 1.5) / l), 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (k / t) ** 2.0d0
    t_2 = 1.0d0 + (1.0d0 + t_1)
    if (t <= (-6.4d+103)) then
        tmp = 2.0d0 / (t_2 * (tan(k) * (sin(k) * (((t ** 2.0d0) / l) * (t / l)))))
    else if (t <= (-7d-32)) then
        tmp = (((2.0d0 * (t ** (-3.0d0))) / sin(k)) * (l / (2.0d0 + t_1))) / (tan(k) / l)
    else if (t <= 1.9d-48) then
        tmp = (l * ((2.0d0 / (t * (k ** 2.0d0))) * (l / sin(k)))) / tan(k)
    else
        tmp = 2.0d0 / (t_2 * (tan(k) * (sin(k) * (((t ** 1.5d0) / l) ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double t_2 = 1.0 + (1.0 + t_1);
	double tmp;
	if (t <= -6.4e+103) {
		tmp = 2.0 / (t_2 * (Math.tan(k) * (Math.sin(k) * ((Math.pow(t, 2.0) / l) * (t / l)))));
	} else if (t <= -7e-32) {
		tmp = (((2.0 * Math.pow(t, -3.0)) / Math.sin(k)) * (l / (2.0 + t_1))) / (Math.tan(k) / l);
	} else if (t <= 1.9e-48) {
		tmp = (l * ((2.0 / (t * Math.pow(k, 2.0))) * (l / Math.sin(k)))) / Math.tan(k);
	} else {
		tmp = 2.0 / (t_2 * (Math.tan(k) * (Math.sin(k) * Math.pow((Math.pow(t, 1.5) / l), 2.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow((k / t), 2.0)
	t_2 = 1.0 + (1.0 + t_1)
	tmp = 0
	if t <= -6.4e+103:
		tmp = 2.0 / (t_2 * (math.tan(k) * (math.sin(k) * ((math.pow(t, 2.0) / l) * (t / l)))))
	elif t <= -7e-32:
		tmp = (((2.0 * math.pow(t, -3.0)) / math.sin(k)) * (l / (2.0 + t_1))) / (math.tan(k) / l)
	elif t <= 1.9e-48:
		tmp = (l * ((2.0 / (t * math.pow(k, 2.0))) * (l / math.sin(k)))) / math.tan(k)
	else:
		tmp = 2.0 / (t_2 * (math.tan(k) * (math.sin(k) * math.pow((math.pow(t, 1.5) / l), 2.0))))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	t_2 = Float64(1.0 + Float64(1.0 + t_1))
	tmp = 0.0
	if (t <= -6.4e+103)
		tmp = Float64(2.0 / Float64(t_2 * Float64(tan(k) * Float64(sin(k) * Float64(Float64((t ^ 2.0) / l) * Float64(t / l))))));
	elseif (t <= -7e-32)
		tmp = Float64(Float64(Float64(Float64(2.0 * (t ^ -3.0)) / sin(k)) * Float64(l / Float64(2.0 + t_1))) / Float64(tan(k) / l));
	elseif (t <= 1.9e-48)
		tmp = Float64(Float64(l * Float64(Float64(2.0 / Float64(t * (k ^ 2.0))) * Float64(l / sin(k)))) / tan(k));
	else
		tmp = Float64(2.0 / Float64(t_2 * Float64(tan(k) * Float64(sin(k) * (Float64((t ^ 1.5) / l) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / t) ^ 2.0;
	t_2 = 1.0 + (1.0 + t_1);
	tmp = 0.0;
	if (t <= -6.4e+103)
		tmp = 2.0 / (t_2 * (tan(k) * (sin(k) * (((t ^ 2.0) / l) * (t / l)))));
	elseif (t <= -7e-32)
		tmp = (((2.0 * (t ^ -3.0)) / sin(k)) * (l / (2.0 + t_1))) / (tan(k) / l);
	elseif (t <= 1.9e-48)
		tmp = (l * ((2.0 / (t * (k ^ 2.0))) * (l / sin(k)))) / tan(k);
	else
		tmp = 2.0 / (t_2 * (tan(k) * (sin(k) * (((t ^ 1.5) / l) ^ 2.0))));
	end
	tmp_2 = 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[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -6.4e+103], N[(2.0 / N[(t$95$2 * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -7e-32], N[(N[(N[(N[(2.0 * N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.9e-48], N[(N[(l * N[(N[(2.0 / N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$2 * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -6.39999999999999985e103

   1. Initial program 50.9%

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

      1. unpow3 50.9%

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

      2. times-frac 73.5%

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

      3. pow2 73.5%

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

   3. Applied egg-rr 73.5%

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

   if -6.39999999999999985e103 < t < -6.9999999999999997e-32

   1. Initial program 67.0%

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

   3. Simplified 59.9%

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

      1. associate-*r* 63.2%

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

      2. associate-*r* 63.2%

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

      3. times-frac 88.7%

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

      4. div-inv 88.7%

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

      5. pow-flip 92.1%

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

      6. metadata-eval 92.1%

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

   5. Applied egg-rr 92.1%

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

      1. times-frac 89.6%

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

   7. Simplified 89.6%

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

      1. associate-*r/ 86.5%

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

   9. Applied egg-rr 86.5%

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

       1. associate-/l* 89.7%

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

       2. associate-*l/ 96.3%

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

       3. associate-*r/ 91.9%

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

       4. associate-/r* 92.2%

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

       5. times-frac 96.3%

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

   11. Simplified 96.3%

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

   if -6.9999999999999997e-32 < t < 1.90000000000000001e-48

   1. Initial program 36.3%

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

   3. Simplified 35.4%

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

      1. associate-*r* 41.8%

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

      2. associate-*r* 41.8%

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

      3. times-frac 43.5%

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

      4. div-inv 43.5%

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

      5. pow-flip 43.5%

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

      6. metadata-eval 43.5%

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

   5. Applied egg-rr 43.5%

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

      1. times-frac 43.6%

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

   7. Simplified 43.6%

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

   8. Taylor expanded in t around 0 80.2%

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

      1. associate-*r/ 80.2%

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

      2. associate-*r* 80.2%

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

   10. Simplified 80.2%

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

       1. associate-*r/ 80.3%

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

       2. times-frac 82.5%

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

       3. *-commutative 82.5%

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

   12. Applied egg-rr 82.5%

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

   if 1.90000000000000001e-48 < t

   1. Initial program 67.1%

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

      1. add-sqr-sqrt 67.0%

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

      2. pow2 67.0%

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

      3. sqrt-div 67.1%

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

      4. sqrt-pow1 71.0%

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

      5. metadata-eval 71.0%

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

      6. sqrt-prod 36.9%

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

      7. add-sqr-sqrt 82.6%

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

   3. Applied egg-rr 82.6%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.4 \cdot 10^{+103}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{elif}\;t \leq -7 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{2 \cdot {t}^{-3}}{\sin k} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}}{\frac{\tan k}{\ell}}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-48}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{2}{t \cdot {k}^{2}} \cdot \frac{\ell}{\sin k}\right)}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right)}\\ \end{array} \]

Alternative 9: 78.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{-30} \lor \neg \left(t \leq 3.2 \cdot 10^{-48}\right):\\ \;\;\;\;\left(\frac{2 \cdot {t}^{-3}}{\sin k} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\right) \cdot \frac{\ell}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{2}{t \cdot {k}^{2}} \cdot \frac{\ell}{\sin k}\right)}{\tan k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -9e-30) (not (<= t 3.2e-48)))
   (*
    (* (/ (* 2.0 (pow t -3.0)) (sin k)) (/ l (+ 2.0 (pow (/ k t) 2.0))))
    (/ l (tan k)))
   (/ (* l (* (/ 2.0 (* t (pow k 2.0))) (/ l (sin k)))) (tan k))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -9e-30) || !(t <= 3.2e-48)) {
		tmp = (((2.0 * pow(t, -3.0)) / sin(k)) * (l / (2.0 + pow((k / t), 2.0)))) * (l / tan(k));
	} else {
		tmp = (l * ((2.0 / (t * pow(k, 2.0))) * (l / sin(k)))) / tan(k);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-9d-30)) .or. (.not. (t <= 3.2d-48))) then
        tmp = (((2.0d0 * (t ** (-3.0d0))) / sin(k)) * (l / (2.0d0 + ((k / t) ** 2.0d0)))) * (l / tan(k))
    else
        tmp = (l * ((2.0d0 / (t * (k ** 2.0d0))) * (l / sin(k)))) / tan(k)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -9e-30) || !(t <= 3.2e-48)) {
		tmp = (((2.0 * Math.pow(t, -3.0)) / Math.sin(k)) * (l / (2.0 + Math.pow((k / t), 2.0)))) * (l / Math.tan(k));
	} else {
		tmp = (l * ((2.0 / (t * Math.pow(k, 2.0))) * (l / Math.sin(k)))) / Math.tan(k);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -9e-30) or not (t <= 3.2e-48):
		tmp = (((2.0 * math.pow(t, -3.0)) / math.sin(k)) * (l / (2.0 + math.pow((k / t), 2.0)))) * (l / math.tan(k))
	else:
		tmp = (l * ((2.0 / (t * math.pow(k, 2.0))) * (l / math.sin(k)))) / math.tan(k)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -9e-30) || !(t <= 3.2e-48))
		tmp = Float64(Float64(Float64(Float64(2.0 * (t ^ -3.0)) / sin(k)) * Float64(l / Float64(2.0 + (Float64(k / t) ^ 2.0)))) * Float64(l / tan(k)));
	else
		tmp = Float64(Float64(l * Float64(Float64(2.0 / Float64(t * (k ^ 2.0))) * Float64(l / sin(k)))) / tan(k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -9e-30) || ~((t <= 3.2e-48)))
		tmp = (((2.0 * (t ^ -3.0)) / sin(k)) * (l / (2.0 + ((k / t) ^ 2.0)))) * (l / tan(k));
	else
		tmp = (l * ((2.0 / (t * (k ^ 2.0))) * (l / sin(k)))) / tan(k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -9e-30], N[Not[LessEqual[t, 3.2e-48]], $MachinePrecision]], N[(N[(N[(N[(2.0 * N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[(N[(2.0 / N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -8.99999999999999935e-30 or 3.1999999999999998e-48 < t

   1. Initial program 63.0%

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

   3. Simplified 56.4%

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

      1. associate-*r* 62.4%

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

      2. associate-*r* 62.4%

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

      3. times-frac 74.5%

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

      4. div-inv 74.5%

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

      5. pow-flip 76.5%

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

      6. metadata-eval 76.5%

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

   5. Applied egg-rr 76.5%

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

      1. times-frac 74.6%

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

   7. Simplified 74.6%

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

      1. expm1-log1p-u 65.3%

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

      2. expm1-udef 57.3%

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

   9. Applied egg-rr 57.3%

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

       1. expm1-def 65.3%

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

       2. expm1-log1p 74.6%

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

       3. associate-*l/ 77.9%

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

       4. associate-*r/ 76.5%

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

       5. associate-/r* 76.5%

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

       6. times-frac 78.0%

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

   11. Simplified 78.0%

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

   if -8.99999999999999935e-30 < t < 3.1999999999999998e-48

   1. Initial program 36.3%

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

   3. Simplified 35.4%

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

      1. associate-*r* 41.8%

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

      2. associate-*r* 41.8%

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

      3. times-frac 43.5%

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

      4. div-inv 43.5%

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

      5. pow-flip 43.5%

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

      6. metadata-eval 43.5%

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

   5. Applied egg-rr 43.5%

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

      1. times-frac 43.6%

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

   7. Simplified 43.6%

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

   8. Taylor expanded in t around 0 80.2%

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

      1. associate-*r/ 80.2%

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

      2. associate-*r* 80.2%

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

   10. Simplified 80.2%

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

       1. associate-*r/ 80.3%

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

       2. times-frac 82.5%

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

       3. *-commutative 82.5%

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

   12. Applied egg-rr 82.5%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{-30} \lor \neg \left(t \leq 3.2 \cdot 10^{-48}\right):\\ \;\;\;\;\left(\frac{2 \cdot {t}^{-3}}{\sin k} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\right) \cdot \frac{\ell}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{2}{t \cdot {k}^{2}} \cdot \frac{\ell}{\sin k}\right)}{\tan k}\\ \end{array} \]

Alternative 10: 77.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{+68} \lor \neg \left(t \leq 2 \cdot 10^{-49}\right):\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot \frac{t \cdot \frac{{t}^{2}}{\ell}}{\ell}\right)}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{2}{t \cdot {k}^{2}} \cdot \frac{\ell}{\sin k}\right)}{\tan k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -5.5e+68) (not (<= t 2e-49)))
   (/
    (/ 2.0 (* (tan k) (* (sin k) (/ (* t (/ (pow t 2.0) l)) l))))
    (+ 2.0 (/ (/ k t) (/ t k))))
   (/ (* l (* (/ 2.0 (* t (pow k 2.0))) (/ l (sin k)))) (tan k))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -5.5e+68) || !(t <= 2e-49)) {
		tmp = (2.0 / (tan(k) * (sin(k) * ((t * (pow(t, 2.0) / l)) / l)))) / (2.0 + ((k / t) / (t / k)));
	} else {
		tmp = (l * ((2.0 / (t * pow(k, 2.0))) * (l / sin(k)))) / tan(k);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-5.5d+68)) .or. (.not. (t <= 2d-49))) then
        tmp = (2.0d0 / (tan(k) * (sin(k) * ((t * ((t ** 2.0d0) / l)) / l)))) / (2.0d0 + ((k / t) / (t / k)))
    else
        tmp = (l * ((2.0d0 / (t * (k ** 2.0d0))) * (l / sin(k)))) / tan(k)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -5.5e+68) || !(t <= 2e-49)) {
		tmp = (2.0 / (Math.tan(k) * (Math.sin(k) * ((t * (Math.pow(t, 2.0) / l)) / l)))) / (2.0 + ((k / t) / (t / k)));
	} else {
		tmp = (l * ((2.0 / (t * Math.pow(k, 2.0))) * (l / Math.sin(k)))) / Math.tan(k);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -5.5e+68) or not (t <= 2e-49):
		tmp = (2.0 / (math.tan(k) * (math.sin(k) * ((t * (math.pow(t, 2.0) / l)) / l)))) / (2.0 + ((k / t) / (t / k)))
	else:
		tmp = (l * ((2.0 / (t * math.pow(k, 2.0))) * (l / math.sin(k)))) / math.tan(k)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -5.5e+68) || !(t <= 2e-49))
		tmp = Float64(Float64(2.0 / Float64(tan(k) * Float64(sin(k) * Float64(Float64(t * Float64((t ^ 2.0) / l)) / l)))) / Float64(2.0 + Float64(Float64(k / t) / Float64(t / k))));
	else
		tmp = Float64(Float64(l * Float64(Float64(2.0 / Float64(t * (k ^ 2.0))) * Float64(l / sin(k)))) / tan(k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -5.5e+68) || ~((t <= 2e-49)))
		tmp = (2.0 / (tan(k) * (sin(k) * ((t * ((t ^ 2.0) / l)) / l)))) / (2.0 + ((k / t) / (t / k)));
	else
		tmp = (l * ((2.0 / (t * (k ^ 2.0))) * (l / sin(k)))) / tan(k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -5.5e+68], N[Not[LessEqual[t, 2e-49]], $MachinePrecision]], N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(t * N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(k / t), $MachinePrecision] / N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[(N[(2.0 / N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.5000000000000004e68 or 1.99999999999999987e-49 < t

   1. Initial program 61.4%

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

   3. Simplified 68.7%

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

      1. unpow2 96.4%

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

      2. clear-num 96.4%

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

      3. un-div-inv 96.4%

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

   5. Applied egg-rr 68.7%

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

      1. cube-mult 68.7%

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

      2. *-un-lft-identity 68.7%

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

      3. times-frac 76.0%

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

      4. pow2 76.0%

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

   7. Applied egg-rr 76.0%

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

   if -5.5000000000000004e68 < t < 1.99999999999999987e-49

   1. Initial program 42.2%

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

   3. Simplified 40.7%

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

      1. associate-*r* 46.0%

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

      2. associate-*r* 46.0%

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

      3. times-frac 51.6%

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

      4. div-inv 51.6%

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

      5. pow-flip 51.6%

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

      6. metadata-eval 51.6%

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

   5. Applied egg-rr 51.6%

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

      1. times-frac 51.6%

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

   7. Simplified 51.6%

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

   8. Taylor expanded in t around 0 80.0%

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

      1. associate-*r/ 80.0%

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

      2. associate-*r* 80.0%

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

   10. Simplified 80.0%

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

       1. associate-*r/ 80.1%

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

       2. times-frac 81.8%

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

       3. *-commutative 81.8%

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

   12. Applied egg-rr 81.8%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{+68} \lor \neg \left(t \leq 2 \cdot 10^{-49}\right):\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot \frac{t \cdot \frac{{t}^{2}}{\ell}}{\ell}\right)}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{2}{t \cdot {k}^{2}} \cdot \frac{\ell}{\sin k}\right)}{\tan k}\\ \end{array} \]

Alternative 11: 76.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.95 \cdot 10^{+77}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\ell}{{\left(t \cdot \sqrt[3]{k}\right)}^{3}}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-48}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{2}{t \cdot {k}^{2}} \cdot \frac{\ell}{\sin k}\right)}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}\right)}}{2 + \frac{k}{t} \cdot \frac{k}{t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -1.95e+77)
   (* (/ l (tan k)) (/ l (pow (* t (cbrt k)) 3.0)))
   (if (<= t 2.55e-48)
     (/ (* l (* (/ 2.0 (* t (pow k 2.0))) (/ l (sin k)))) (tan k))
     (/
      (/ 2.0 (* (tan k) (* (sin k) (/ (/ (pow t 3.0) l) l))))
      (+ 2.0 (* (/ k t) (/ k t)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.95e+77) {
		tmp = (l / tan(k)) * (l / pow((t * cbrt(k)), 3.0));
	} else if (t <= 2.55e-48) {
		tmp = (l * ((2.0 / (t * pow(k, 2.0))) * (l / sin(k)))) / tan(k);
	} else {
		tmp = (2.0 / (tan(k) * (sin(k) * ((pow(t, 3.0) / l) / l)))) / (2.0 + ((k / t) * (k / t)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.95e+77) {
		tmp = (l / Math.tan(k)) * (l / Math.pow((t * Math.cbrt(k)), 3.0));
	} else if (t <= 2.55e-48) {
		tmp = (l * ((2.0 / (t * Math.pow(k, 2.0))) * (l / Math.sin(k)))) / Math.tan(k);
	} else {
		tmp = (2.0 / (Math.tan(k) * (Math.sin(k) * ((Math.pow(t, 3.0) / l) / l)))) / (2.0 + ((k / t) * (k / t)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -1.95e+77)
		tmp = Float64(Float64(l / tan(k)) * Float64(l / (Float64(t * cbrt(k)) ^ 3.0)));
	elseif (t <= 2.55e-48)
		tmp = Float64(Float64(l * Float64(Float64(2.0 / Float64(t * (k ^ 2.0))) * Float64(l / sin(k)))) / tan(k));
	else
		tmp = Float64(Float64(2.0 / Float64(tan(k) * Float64(sin(k) * Float64(Float64((t ^ 3.0) / l) / l)))) / Float64(2.0 + Float64(Float64(k / t) * Float64(k / t))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -1.95e+77], N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[N[(t * N[Power[k, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.55e-48], N[(N[(l * N[(N[(2.0 / N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(k / t), $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.9499999999999999e77

   1. Initial program 50.9%

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

   3. Simplified 35.5%

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

      1. associate-*r* 43.8%

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

      2. associate-*r* 43.8%

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

      3. times-frac 60.7%

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

      4. div-inv 60.7%

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

      5. pow-flip 65.2%

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

      6. metadata-eval 65.2%

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

   5. Applied egg-rr 65.2%

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

      1. times-frac 65.8%

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

   7. Simplified 65.8%

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

   8. Taylor expanded in k around 0 60.4%

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

      1. add-cube-cbrt 60.4%

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

      2. pow3 60.4%

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

      3. *-commutative 60.4%

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

      4. cbrt-prod 60.4%

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

      5. unpow3 60.4%

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

      6. add-cbrt-cube 70.2%

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

   10. Applied egg-rr 70.2%

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

   if -1.9499999999999999e77 < t < 2.55000000000000006e-48

   1. Initial program 42.3%

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

   3. Simplified 40.9%

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

      1. associate-*r* 46.8%

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

      2. associate-*r* 46.8%

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

      3. times-frac 52.3%

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

      4. div-inv 52.3%

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

      5. pow-flip 52.2%

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

      6. metadata-eval 52.2%

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

   5. Applied egg-rr 52.2%

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

      1. times-frac 51.6%

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

   7. Simplified 51.6%

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

   8. Taylor expanded in t around 0 79.6%

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

      1. associate-*r/ 79.6%

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

      2. associate-*r* 79.6%

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

   10. Simplified 79.6%

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

       1. associate-*r/ 79.6%

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

       2. times-frac 81.4%

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

       3. *-commutative 81.4%

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

   12. Applied egg-rr 81.4%

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

   if 2.55000000000000006e-48 < t

   1. Initial program 67.1%

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

   3. Simplified 72.8%

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

      1. unpow2 72.8%

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

   5. Applied egg-rr 72.8%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.95 \cdot 10^{+77}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\ell}{{\left(t \cdot \sqrt[3]{k}\right)}^{3}}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-48}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{2}{t \cdot {k}^{2}} \cdot \frac{\ell}{\sin k}\right)}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}\right)}}{2 + \frac{k}{t} \cdot \frac{k}{t}}\\ \end{array} \]

Alternative 12: 76.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.95 \cdot 10^{+77}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\ell}{{\left(t \cdot \sqrt[3]{k}\right)}^{3}}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-48}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{2}{t \cdot {k}^{2}} \cdot \frac{\ell}{\sin k}\right)}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}\right)}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -1.95e+77)
   (* (/ l (tan k)) (/ l (pow (* t (cbrt k)) 3.0)))
   (if (<= t 1.4e-48)
     (/ (* l (* (/ 2.0 (* t (pow k 2.0))) (/ l (sin k)))) (tan k))
     (/
      (/ 2.0 (* (tan k) (* (sin k) (/ (/ (pow t 3.0) l) l))))
      (+ 2.0 (/ (/ k t) (/ t k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.95e+77) {
		tmp = (l / tan(k)) * (l / pow((t * cbrt(k)), 3.0));
	} else if (t <= 1.4e-48) {
		tmp = (l * ((2.0 / (t * pow(k, 2.0))) * (l / sin(k)))) / tan(k);
	} else {
		tmp = (2.0 / (tan(k) * (sin(k) * ((pow(t, 3.0) / l) / l)))) / (2.0 + ((k / t) / (t / k)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.95e+77) {
		tmp = (l / Math.tan(k)) * (l / Math.pow((t * Math.cbrt(k)), 3.0));
	} else if (t <= 1.4e-48) {
		tmp = (l * ((2.0 / (t * Math.pow(k, 2.0))) * (l / Math.sin(k)))) / Math.tan(k);
	} else {
		tmp = (2.0 / (Math.tan(k) * (Math.sin(k) * ((Math.pow(t, 3.0) / l) / l)))) / (2.0 + ((k / t) / (t / k)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -1.95e+77)
		tmp = Float64(Float64(l / tan(k)) * Float64(l / (Float64(t * cbrt(k)) ^ 3.0)));
	elseif (t <= 1.4e-48)
		tmp = Float64(Float64(l * Float64(Float64(2.0 / Float64(t * (k ^ 2.0))) * Float64(l / sin(k)))) / tan(k));
	else
		tmp = Float64(Float64(2.0 / Float64(tan(k) * Float64(sin(k) * Float64(Float64((t ^ 3.0) / l) / l)))) / Float64(2.0 + Float64(Float64(k / t) / Float64(t / k))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -1.95e+77], N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[N[(t * N[Power[k, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.4e-48], N[(N[(l * N[(N[(2.0 / N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(k / t), $MachinePrecision] / N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.9499999999999999e77

   1. Initial program 50.9%

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

   3. Simplified 35.5%

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

      1. associate-*r* 43.8%

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

      2. associate-*r* 43.8%

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

      3. times-frac 60.7%

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

      4. div-inv 60.7%

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

      5. pow-flip 65.2%

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

      6. metadata-eval 65.2%

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

   5. Applied egg-rr 65.2%

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

      1. times-frac 65.8%

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

   7. Simplified 65.8%

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

   8. Taylor expanded in k around 0 60.4%

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

      1. add-cube-cbrt 60.4%

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

      2. pow3 60.4%

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

      3. *-commutative 60.4%

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

      4. cbrt-prod 60.4%

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

      5. unpow3 60.4%

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

      6. add-cbrt-cube 70.2%

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

   10. Applied egg-rr 70.2%

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

   if -1.9499999999999999e77 < t < 1.40000000000000002e-48

   1. Initial program 41.9%

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

   3. Simplified 40.4%

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

      1. associate-*r* 46.4%

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

      2. associate-*r* 46.4%

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

      3. times-frac 51.9%

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

      4. div-inv 51.9%

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

      5. pow-flip 51.9%

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

      6. metadata-eval 51.9%

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

   5. Applied egg-rr 51.9%

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

      1. times-frac 51.3%

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

   7. Simplified 51.3%

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

   8. Taylor expanded in t around 0 79.4%

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

      1. associate-*r/ 79.4%

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

      2. associate-*r* 79.5%

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

   10. Simplified 79.5%

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

       1. associate-*r/ 79.5%

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

       2. times-frac 81.3%

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

       3. *-commutative 81.3%

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

   12. Applied egg-rr 81.3%

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

   if 1.40000000000000002e-48 < t

   1. Initial program 67.5%

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

   3. Simplified 73.1%

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

      1. unpow2 95.3%

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

      2. clear-num 95.4%

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

      3. un-div-inv 95.4%

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

   5. Applied egg-rr 73.1%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.95 \cdot 10^{+77}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\ell}{{\left(t \cdot \sqrt[3]{k}\right)}^{3}}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-48}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{2}{t \cdot {k}^{2}} \cdot \frac{\ell}{\sin k}\right)}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}\right)}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \end{array} \]

Alternative 13: 75.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell}{\tan k}\\ t_2 := \frac{\ell}{{\left(t \cdot \sqrt[3]{k}\right)}^{3}}\\ \mathbf{if}\;t \leq -1.95 \cdot 10^{+77}:\\ \;\;\;\;t_1 \cdot t_2\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{-48}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \frac{\ell}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ l (tan k))) (t_2 (/ l (pow (* t (cbrt k)) 3.0))))
   (if (<= t -1.95e+77)
     (* t_1 t_2)
     (if (<= t 4.1e-48)
       (* t_1 (* 2.0 (/ l (* (pow k 2.0) (* t (sin k))))))
       (* t_2 (/ l k))))))

C

double code(double t, double l, double k) {
	double t_1 = l / tan(k);
	double t_2 = l / pow((t * cbrt(k)), 3.0);
	double tmp;
	if (t <= -1.95e+77) {
		tmp = t_1 * t_2;
	} else if (t <= 4.1e-48) {
		tmp = t_1 * (2.0 * (l / (pow(k, 2.0) * (t * sin(k)))));
	} else {
		tmp = t_2 * (l / k);
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = l / Math.tan(k);
	double t_2 = l / Math.pow((t * Math.cbrt(k)), 3.0);
	double tmp;
	if (t <= -1.95e+77) {
		tmp = t_1 * t_2;
	} else if (t <= 4.1e-48) {
		tmp = t_1 * (2.0 * (l / (Math.pow(k, 2.0) * (t * Math.sin(k)))));
	} else {
		tmp = t_2 * (l / k);
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(l / tan(k))
	t_2 = Float64(l / (Float64(t * cbrt(k)) ^ 3.0))
	tmp = 0.0
	if (t <= -1.95e+77)
		tmp = Float64(t_1 * t_2);
	elseif (t <= 4.1e-48)
		tmp = Float64(t_1 * Float64(2.0 * Float64(l / Float64((k ^ 2.0) * Float64(t * sin(k))))));
	else
		tmp = Float64(t_2 * Float64(l / k));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(l / N[Power[N[(t * N[Power[k, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.95e+77], N[(t$95$1 * t$95$2), $MachinePrecision], If[LessEqual[t, 4.1e-48], N[(t$95$1 * N[(2.0 * N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(l / k), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.9499999999999999e77

   1. Initial program 50.9%

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

   3. Simplified 35.5%

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

      1. associate-*r* 43.8%

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

      2. associate-*r* 43.8%

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

      3. times-frac 60.7%

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

      4. div-inv 60.7%

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

      5. pow-flip 65.2%

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

      6. metadata-eval 65.2%

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

   5. Applied egg-rr 65.2%

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

      1. times-frac 65.8%

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

   7. Simplified 65.8%

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

   8. Taylor expanded in k around 0 60.4%

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

      1. add-cube-cbrt 60.4%

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

      2. pow3 60.4%

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

      3. *-commutative 60.4%

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

      4. cbrt-prod 60.4%

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

      5. unpow3 60.4%

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

      6. add-cbrt-cube 70.2%

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

   10. Applied egg-rr 70.2%

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

   if -1.9499999999999999e77 < t < 4.10000000000000014e-48

   1. Initial program 42.3%

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

   3. Simplified 40.9%

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

      1. associate-*r* 46.8%

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

      2. associate-*r* 46.8%

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

      3. times-frac 52.3%

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

      4. div-inv 52.3%

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

      5. pow-flip 52.2%

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

      6. metadata-eval 52.2%

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

   5. Applied egg-rr 52.2%

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

      1. times-frac 51.6%

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

   7. Simplified 51.6%

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

   8. Taylor expanded in t around 0 79.6%

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

   if 4.10000000000000014e-48 < t

   1. Initial program 67.1%

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

   3. Simplified 64.3%

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

      1. associate-*r* 69.9%

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

      2. associate-*r* 69.9%

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

      3. times-frac 76.0%

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

      4. div-inv 76.0%

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

      5. pow-flip 77.3%

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

      6. metadata-eval 77.3%

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

   5. Applied egg-rr 77.3%

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

      1. times-frac 74.8%

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

   7. Simplified 74.8%

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

   8. Taylor expanded in k around 0 65.9%

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

   9. Taylor expanded in k around 0 69.9%

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

       1. add-cube-cbrt 69.8%

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

       2. pow3 69.8%

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

       3. *-commutative 69.8%

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

       4. cbrt-prod 69.8%

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

       5. unpow3 69.8%

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

       6. add-cbrt-cube 72.4%

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

   11. Applied egg-rr 72.4%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.95 \cdot 10^{+77}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\ell}{{\left(t \cdot \sqrt[3]{k}\right)}^{3}}\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{-48}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(t \cdot \sqrt[3]{k}\right)}^{3}} \cdot \frac{\ell}{k}\\ \end{array} \]

Alternative 14: 76.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell}{\tan k}\\ t_2 := \frac{\ell}{{\left(t \cdot \sqrt[3]{k}\right)}^{3}}\\ \mathbf{if}\;t \leq -1.95 \cdot 10^{+77}:\\ \;\;\;\;t_1 \cdot t_2\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{-48}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \frac{\frac{\ell}{\sin k}}{t \cdot {k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \frac{\ell}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ l (tan k))) (t_2 (/ l (pow (* t (cbrt k)) 3.0))))
   (if (<= t -1.95e+77)
     (* t_1 t_2)
     (if (<= t 4.4e-48)
       (* t_1 (* 2.0 (/ (/ l (sin k)) (* t (pow k 2.0)))))
       (* t_2 (/ l k))))))

C

double code(double t, double l, double k) {
	double t_1 = l / tan(k);
	double t_2 = l / pow((t * cbrt(k)), 3.0);
	double tmp;
	if (t <= -1.95e+77) {
		tmp = t_1 * t_2;
	} else if (t <= 4.4e-48) {
		tmp = t_1 * (2.0 * ((l / sin(k)) / (t * pow(k, 2.0))));
	} else {
		tmp = t_2 * (l / k);
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = l / Math.tan(k);
	double t_2 = l / Math.pow((t * Math.cbrt(k)), 3.0);
	double tmp;
	if (t <= -1.95e+77) {
		tmp = t_1 * t_2;
	} else if (t <= 4.4e-48) {
		tmp = t_1 * (2.0 * ((l / Math.sin(k)) / (t * Math.pow(k, 2.0))));
	} else {
		tmp = t_2 * (l / k);
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(l / tan(k))
	t_2 = Float64(l / (Float64(t * cbrt(k)) ^ 3.0))
	tmp = 0.0
	if (t <= -1.95e+77)
		tmp = Float64(t_1 * t_2);
	elseif (t <= 4.4e-48)
		tmp = Float64(t_1 * Float64(2.0 * Float64(Float64(l / sin(k)) / Float64(t * (k ^ 2.0)))));
	else
		tmp = Float64(t_2 * Float64(l / k));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(l / N[Power[N[(t * N[Power[k, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.95e+77], N[(t$95$1 * t$95$2), $MachinePrecision], If[LessEqual[t, 4.4e-48], N[(t$95$1 * N[(2.0 * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(l / k), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.9499999999999999e77

   1. Initial program 50.9%

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

   3. Simplified 35.5%

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

      1. associate-*r* 43.8%

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

      2. associate-*r* 43.8%

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

      3. times-frac 60.7%

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

      4. div-inv 60.7%

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

      5. pow-flip 65.2%

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

      6. metadata-eval 65.2%

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

   5. Applied egg-rr 65.2%

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

      1. times-frac 65.8%

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

   7. Simplified 65.8%

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

   8. Taylor expanded in k around 0 60.4%

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

      1. add-cube-cbrt 60.4%

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

      2. pow3 60.4%

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

      3. *-commutative 60.4%

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

      4. cbrt-prod 60.4%

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

      5. unpow3 60.4%

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

      6. add-cbrt-cube 70.2%

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

   10. Applied egg-rr 70.2%

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

   if -1.9499999999999999e77 < t < 4.40000000000000025e-48

   1. Initial program 42.3%

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

   3. Simplified 40.9%

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

      1. associate-*r* 46.8%

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

      2. associate-*r* 46.8%

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

      3. times-frac 52.3%

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

      4. div-inv 52.3%

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

      5. pow-flip 52.2%

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

      6. metadata-eval 52.2%

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

   5. Applied egg-rr 52.2%

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

      1. times-frac 51.6%

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

   7. Simplified 51.6%

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

   8. Taylor expanded in t around 0 79.6%

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

      1. associate-*r/ 79.6%

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

      2. associate-*r* 79.6%

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

   10. Simplified 79.6%

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

   11. Taylor expanded in l around 0 79.6%

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

       1. *-commutative 79.6%

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

       2. *-commutative 79.6%

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

       3. associate-*r* 79.6%

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

       4. associate-/r* 81.4%

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

       5. *-commutative 81.4%

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

   13. Simplified 81.4%

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

   if 4.40000000000000025e-48 < t

   1. Initial program 67.1%

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

   3. Simplified 64.3%

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

      1. associate-*r* 69.9%

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

      2. associate-*r* 69.9%

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

      3. times-frac 76.0%

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

      4. div-inv 76.0%

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

      5. pow-flip 77.3%

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

      6. metadata-eval 77.3%

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

   5. Applied egg-rr 77.3%

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

      1. times-frac 74.8%

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

   7. Simplified 74.8%

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

   8. Taylor expanded in k around 0 65.9%

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

   9. Taylor expanded in k around 0 69.9%

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

       1. add-cube-cbrt 69.8%

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

       2. pow3 69.8%

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

       3. *-commutative 69.8%

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

       4. cbrt-prod 69.8%

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

       5. unpow3 69.8%

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

       6. add-cbrt-cube 72.4%

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

   11. Applied egg-rr 72.4%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.95 \cdot 10^{+77}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\ell}{{\left(t \cdot \sqrt[3]{k}\right)}^{3}}\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{-48}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \left(2 \cdot \frac{\frac{\ell}{\sin k}}{t \cdot {k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(t \cdot \sqrt[3]{k}\right)}^{3}} \cdot \frac{\ell}{k}\\ \end{array} \]

Alternative 15: 76.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell}{{\left(t \cdot \sqrt[3]{k}\right)}^{3}}\\ \mathbf{if}\;t \leq -1.95 \cdot 10^{+77}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot t_1\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{-48}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{2}{t \cdot {k}^{2}} \cdot \frac{\ell}{\sin k}\right)}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{\ell}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ l (pow (* t (cbrt k)) 3.0))))
   (if (<= t -1.95e+77)
     (* (/ l (tan k)) t_1)
     (if (<= t 4.4e-48)
       (/ (* l (* (/ 2.0 (* t (pow k 2.0))) (/ l (sin k)))) (tan k))
       (* t_1 (/ l k))))))

C

double code(double t, double l, double k) {
	double t_1 = l / pow((t * cbrt(k)), 3.0);
	double tmp;
	if (t <= -1.95e+77) {
		tmp = (l / tan(k)) * t_1;
	} else if (t <= 4.4e-48) {
		tmp = (l * ((2.0 / (t * pow(k, 2.0))) * (l / sin(k)))) / tan(k);
	} else {
		tmp = t_1 * (l / k);
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = l / Math.pow((t * Math.cbrt(k)), 3.0);
	double tmp;
	if (t <= -1.95e+77) {
		tmp = (l / Math.tan(k)) * t_1;
	} else if (t <= 4.4e-48) {
		tmp = (l * ((2.0 / (t * Math.pow(k, 2.0))) * (l / Math.sin(k)))) / Math.tan(k);
	} else {
		tmp = t_1 * (l / k);
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(l / (Float64(t * cbrt(k)) ^ 3.0))
	tmp = 0.0
	if (t <= -1.95e+77)
		tmp = Float64(Float64(l / tan(k)) * t_1);
	elseif (t <= 4.4e-48)
		tmp = Float64(Float64(l * Float64(Float64(2.0 / Float64(t * (k ^ 2.0))) * Float64(l / sin(k)))) / tan(k));
	else
		tmp = Float64(t_1 * Float64(l / k));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(l / N[Power[N[(t * N[Power[k, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.95e+77], N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t, 4.4e-48], N[(N[(l * N[(N[(2.0 / N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(l / k), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.9499999999999999e77

   1. Initial program 50.9%

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

   3. Simplified 35.5%

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

      1. associate-*r* 43.8%

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

      2. associate-*r* 43.8%

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

      3. times-frac 60.7%

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

      4. div-inv 60.7%

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

      5. pow-flip 65.2%

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

      6. metadata-eval 65.2%

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

   5. Applied egg-rr 65.2%

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

      1. times-frac 65.8%

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

   7. Simplified 65.8%

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

   8. Taylor expanded in k around 0 60.4%

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

      1. add-cube-cbrt 60.4%

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

      2. pow3 60.4%

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

      3. *-commutative 60.4%

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

      4. cbrt-prod 60.4%

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

      5. unpow3 60.4%

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

      6. add-cbrt-cube 70.2%

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

   10. Applied egg-rr 70.2%

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

   if -1.9499999999999999e77 < t < 4.40000000000000025e-48

   1. Initial program 42.3%

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

   3. Simplified 40.9%

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

      1. associate-*r* 46.8%

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

      2. associate-*r* 46.8%

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

      3. times-frac 52.3%

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

      4. div-inv 52.3%

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

      5. pow-flip 52.2%

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

      6. metadata-eval 52.2%

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

   5. Applied egg-rr 52.2%

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

      1. times-frac 51.6%

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

   7. Simplified 51.6%

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

   8. Taylor expanded in t around 0 79.6%

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

      1. associate-*r/ 79.6%

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

      2. associate-*r* 79.6%

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

   10. Simplified 79.6%

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

       1. associate-*r/ 79.6%

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

       2. times-frac 81.4%

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

       3. *-commutative 81.4%

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

   12. Applied egg-rr 81.4%

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

   if 4.40000000000000025e-48 < t

   1. Initial program 67.1%

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

   3. Simplified 64.3%

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

      1. associate-*r* 69.9%

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

      2. associate-*r* 69.9%

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

      3. times-frac 76.0%

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

      4. div-inv 76.0%

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{{t}^{3}}\right)} \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k} \]

      5. pow-flip 77.3%

         \[\leadsto \frac{\left(2 \cdot \color{blue}{{t}^{\left(-3\right)}}\right) \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k} \]

      6. metadata-eval 77.3%

         \[\leadsto \frac{\left(2 \cdot {t}^{\color{blue}{-3}}\right) \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k} \]

   5. Applied egg-rr 77.3%

      \[\leadsto \color{blue}{\frac{\left(2 \cdot {t}^{-3}\right) \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k}} \]
   6. Step-by-step derivation

      1. times-frac 74.8%

         \[\leadsto \color{blue}{\left(\frac{2 \cdot {t}^{-3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\sin k}\right)} \cdot \frac{\ell}{\tan k} \]

   7. Simplified 74.8%

      \[\leadsto \color{blue}{\left(\frac{2 \cdot {t}^{-3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\ell}{\tan k}} \]

   8. Taylor expanded in k around 0 65.9%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot {t}^{3}}} \cdot \frac{\ell}{\tan k} \]

   9. Taylor expanded in k around 0 69.9%

      \[\leadsto \frac{\ell}{k \cdot {t}^{3}} \cdot \color{blue}{\frac{\ell}{k}} \]
   10. Step-by-step derivation

       1. add-cube-cbrt 69.8%

          \[\leadsto \frac{\ell}{\color{blue}{\left(\sqrt[3]{k \cdot {t}^{3}} \cdot \sqrt[3]{k \cdot {t}^{3}}\right) \cdot \sqrt[3]{k \cdot {t}^{3}}}} \cdot \frac{\ell}{k} \]

       2. pow3 69.8%

          \[\leadsto \frac{\ell}{\color{blue}{{\left(\sqrt[3]{k \cdot {t}^{3}}\right)}^{3}}} \cdot \frac{\ell}{k} \]

       3. *-commutative 69.8%

          \[\leadsto \frac{\ell}{{\left(\sqrt[3]{\color{blue}{{t}^{3} \cdot k}}\right)}^{3}} \cdot \frac{\ell}{k} \]

       4. cbrt-prod 69.8%

          \[\leadsto \frac{\ell}{{\color{blue}{\left(\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{k}\right)}}^{3}} \cdot \frac{\ell}{k} \]

       5. unpow3 69.8%

          \[\leadsto \frac{\ell}{{\left(\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}} \cdot \sqrt[3]{k}\right)}^{3}} \cdot \frac{\ell}{k} \]

       6. add-cbrt-cube 72.4%

          \[\leadsto \frac{\ell}{{\left(\color{blue}{t} \cdot \sqrt[3]{k}\right)}^{3}} \cdot \frac{\ell}{k} \]

   11. Applied egg-rr 72.4%

       \[\leadsto \frac{\ell}{\color{blue}{{\left(t \cdot \sqrt[3]{k}\right)}^{3}}} \cdot \frac{\ell}{k} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.95 \cdot 10^{+77}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\ell}{{\left(t \cdot \sqrt[3]{k}\right)}^{3}}\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{-48}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{2}{t \cdot {k}^{2}} \cdot \frac{\ell}{\sin k}\right)}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(t \cdot \sqrt[3]{k}\right)}^{3}} \cdot \frac{\ell}{k}\\ \end{array} \]

Alternative 16: 69.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{-56} \lor \neg \left(t \leq 2.55 \cdot 10^{-48}\right):\\ \;\;\;\;\frac{\ell}{{\left(t \cdot \sqrt[3]{k}\right)}^{3}} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \left(2 \cdot \frac{\ell}{t \cdot {k}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -1.5e-56) (not (<= t 2.55e-48)))
   (* (/ l (pow (* t (cbrt k)) 3.0)) (/ l k))
   (* (/ l (tan k)) (* 2.0 (/ l (* t (pow k 3.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.5e-56) || !(t <= 2.55e-48)) {
		tmp = (l / pow((t * cbrt(k)), 3.0)) * (l / k);
	} else {
		tmp = (l / tan(k)) * (2.0 * (l / (t * pow(k, 3.0))));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.5e-56) || !(t <= 2.55e-48)) {
		tmp = (l / Math.pow((t * Math.cbrt(k)), 3.0)) * (l / k);
	} else {
		tmp = (l / Math.tan(k)) * (2.0 * (l / (t * Math.pow(k, 3.0))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -1.5e-56) || !(t <= 2.55e-48))
		tmp = Float64(Float64(l / (Float64(t * cbrt(k)) ^ 3.0)) * Float64(l / k));
	else
		tmp = Float64(Float64(l / tan(k)) * Float64(2.0 * Float64(l / Float64(t * (k ^ 3.0)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -1.5e-56], N[Not[LessEqual[t, 2.55e-48]], $MachinePrecision]], N[(N[(l / N[Power[N[(t * N[Power[k, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(l / N[(t * N[Power[k, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.49999999999999995e-56 or 2.55000000000000006e-48 < t

   1. Initial program 62.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 56.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* 62.1%

         \[\leadsto \frac{\color{blue}{\left(\frac{2}{{t}^{3}} \cdot \ell\right) \cdot \ell}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      2. associate-*r* 62.1%

         \[\leadsto \frac{\left(\frac{2}{{t}^{3}} \cdot \ell\right) \cdot \ell}{\color{blue}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right) \cdot \tan k}} \]

      3. times-frac 73.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}} \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k}} \]

      4. div-inv 73.9%

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{{t}^{3}}\right)} \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k} \]

      5. pow-flip 75.8%

         \[\leadsto \frac{\left(2 \cdot \color{blue}{{t}^{\left(-3\right)}}\right) \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k} \]

      6. metadata-eval 75.8%

         \[\leadsto \frac{\left(2 \cdot {t}^{\color{blue}{-3}}\right) \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k} \]

   5. Applied egg-rr 75.8%

      \[\leadsto \color{blue}{\frac{\left(2 \cdot {t}^{-3}\right) \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k}} \]
   6. Step-by-step derivation

      1. times-frac 74.0%

         \[\leadsto \color{blue}{\left(\frac{2 \cdot {t}^{-3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\sin k}\right)} \cdot \frac{\ell}{\tan k} \]

   7. Simplified 74.0%

      \[\leadsto \color{blue}{\left(\frac{2 \cdot {t}^{-3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\ell}{\tan k}} \]

   8. Taylor expanded in k around 0 62.5%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot {t}^{3}}} \cdot \frac{\ell}{\tan k} \]

   9. Taylor expanded in k around 0 65.4%

      \[\leadsto \frac{\ell}{k \cdot {t}^{3}} \cdot \color{blue}{\frac{\ell}{k}} \]
   10. Step-by-step derivation

       1. add-cube-cbrt 65.3%

          \[\leadsto \frac{\ell}{\color{blue}{\left(\sqrt[3]{k \cdot {t}^{3}} \cdot \sqrt[3]{k \cdot {t}^{3}}\right) \cdot \sqrt[3]{k \cdot {t}^{3}}}} \cdot \frac{\ell}{k} \]

       2. pow3 65.3%

          \[\leadsto \frac{\ell}{\color{blue}{{\left(\sqrt[3]{k \cdot {t}^{3}}\right)}^{3}}} \cdot \frac{\ell}{k} \]

       3. *-commutative 65.3%

          \[\leadsto \frac{\ell}{{\left(\sqrt[3]{\color{blue}{{t}^{3} \cdot k}}\right)}^{3}} \cdot \frac{\ell}{k} \]

       4. cbrt-prod 65.4%

          \[\leadsto \frac{\ell}{{\color{blue}{\left(\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{k}\right)}}^{3}} \cdot \frac{\ell}{k} \]

       5. unpow3 65.4%

          \[\leadsto \frac{\ell}{{\left(\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}} \cdot \sqrt[3]{k}\right)}^{3}} \cdot \frac{\ell}{k} \]

       6. add-cbrt-cube 69.4%

          \[\leadsto \frac{\ell}{{\left(\color{blue}{t} \cdot \sqrt[3]{k}\right)}^{3}} \cdot \frac{\ell}{k} \]

   11. Applied egg-rr 69.4%

       \[\leadsto \frac{\ell}{\color{blue}{{\left(t \cdot \sqrt[3]{k}\right)}^{3}}} \cdot \frac{\ell}{k} \]

   if -1.49999999999999995e-56 < t < 2.55000000000000006e-48

   1. Initial program 35.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 34.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* 41.5%

         \[\leadsto \frac{\color{blue}{\left(\frac{2}{{t}^{3}} \cdot \ell\right) \cdot \ell}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      2. associate-*r* 41.5%

         \[\leadsto \frac{\left(\frac{2}{{t}^{3}} \cdot \ell\right) \cdot \ell}{\color{blue}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right) \cdot \tan k}} \]

      3. times-frac 43.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}} \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k}} \]

      4. div-inv 43.3%

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{{t}^{3}}\right)} \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k} \]

      5. pow-flip 43.3%

         \[\leadsto \frac{\left(2 \cdot \color{blue}{{t}^{\left(-3\right)}}\right) \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k} \]

      6. metadata-eval 43.3%

         \[\leadsto \frac{\left(2 \cdot {t}^{\color{blue}{-3}}\right) \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k} \]

   5. Applied egg-rr 43.3%

      \[\leadsto \color{blue}{\frac{\left(2 \cdot {t}^{-3}\right) \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k}} \]
   6. Step-by-step derivation

      1. times-frac 43.3%

         \[\leadsto \color{blue}{\left(\frac{2 \cdot {t}^{-3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\sin k}\right)} \cdot \frac{\ell}{\tan k} \]

   7. Simplified 43.3%

      \[\leadsto \color{blue}{\left(\frac{2 \cdot {t}^{-3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\ell}{\tan k}} \]

   8. Taylor expanded in t around 0 81.3%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\right)} \cdot \frac{\ell}{\tan k} \]
   9. Step-by-step derivation

      1. associate-*r/ 81.3%

         \[\leadsto \color{blue}{\frac{2 \cdot \ell}{{k}^{2} \cdot \left(t \cdot \sin k\right)}} \cdot \frac{\ell}{\tan k} \]

      2. associate-*r* 81.3%

         \[\leadsto \frac{2 \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot \sin k}} \cdot \frac{\ell}{\tan k} \]

   10. Simplified 81.3%

       \[\leadsto \color{blue}{\frac{2 \cdot \ell}{\left({k}^{2} \cdot t\right) \cdot \sin k}} \cdot \frac{\ell}{\tan k} \]

   11. Taylor expanded in k around 0 63.3%

       \[\leadsto \color{blue}{\left(2 \cdot \frac{\ell}{{k}^{3} \cdot t}\right)} \cdot \frac{\ell}{\tan k} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{-56} \lor \neg \left(t \leq 2.55 \cdot 10^{-48}\right):\\ \;\;\;\;\frac{\ell}{{\left(t \cdot \sqrt[3]{k}\right)}^{3}} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \left(2 \cdot \frac{\ell}{t \cdot {k}^{3}}\right)\\ \end{array} \]

Alternative 17: 61.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{+152}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.35e+152)
   (* (/ l (tan k)) (/ l (* k (pow t 3.0))))
   (* 2.0 (/ (pow l 2.0) (* t (pow k 4.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.35e+152) {
		tmp = (l / tan(k)) * (l / (k * pow(t, 3.0)));
	} else {
		tmp = 2.0 * (pow(l, 2.0) / (t * pow(k, 4.0)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.35d+152) then
        tmp = (l / tan(k)) * (l / (k * (t ** 3.0d0)))
    else
        tmp = 2.0d0 * ((l ** 2.0d0) / (t * (k ** 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.35e+152) {
		tmp = (l / Math.tan(k)) * (l / (k * Math.pow(t, 3.0)));
	} else {
		tmp = 2.0 * (Math.pow(l, 2.0) / (t * Math.pow(k, 4.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.35e+152:
		tmp = (l / math.tan(k)) * (l / (k * math.pow(t, 3.0)))
	else:
		tmp = 2.0 * (math.pow(l, 2.0) / (t * math.pow(k, 4.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.35e+152)
		tmp = Float64(Float64(l / tan(k)) * Float64(l / Float64(k * (t ^ 3.0))));
	else
		tmp = Float64(2.0 * Float64((l ^ 2.0) / Float64(t * (k ^ 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.35e+152)
		tmp = (l / tan(k)) * (l / (k * (t ^ 3.0)));
	else
		tmp = 2.0 * ((l ^ 2.0) / (t * (k ^ 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.35e+152], N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.35000000000000007e152

   1. Initial program 50.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 46.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* 52.8%

         \[\leadsto \frac{\color{blue}{\left(\frac{2}{{t}^{3}} \cdot \ell\right) \cdot \ell}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      2. associate-*r* 52.8%

         \[\leadsto \frac{\left(\frac{2}{{t}^{3}} \cdot \ell\right) \cdot \ell}{\color{blue}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right) \cdot \tan k}} \]

      3. times-frac 61.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}} \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k}} \]

      4. div-inv 61.4%

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{{t}^{3}}\right)} \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k} \]

      5. pow-flip 62.6%

         \[\leadsto \frac{\left(2 \cdot \color{blue}{{t}^{\left(-3\right)}}\right) \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k} \]

      6. metadata-eval 62.6%

         \[\leadsto \frac{\left(2 \cdot {t}^{\color{blue}{-3}}\right) \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k} \]

   5. Applied egg-rr 62.6%

      \[\leadsto \color{blue}{\frac{\left(2 \cdot {t}^{-3}\right) \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k}} \]
   6. Step-by-step derivation

      1. times-frac 61.4%

         \[\leadsto \color{blue}{\left(\frac{2 \cdot {t}^{-3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\sin k}\right)} \cdot \frac{\ell}{\tan k} \]

   7. Simplified 61.4%

      \[\leadsto \color{blue}{\left(\frac{2 \cdot {t}^{-3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\ell}{\tan k}} \]

   8. Taylor expanded in k around 0 56.2%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot {t}^{3}}} \cdot \frac{\ell}{\tan k} \]

   if 1.35000000000000007e152 < k

   1. Initial program 52.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 59.2%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. add-cube-cbrt 59.2%

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}}}{\ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. *-un-lft-identity 59.2%

         \[\leadsto \frac{2}{\left(\left(\frac{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}}{\color{blue}{1 \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. times-frac 59.2%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. pow2 59.2%

         \[\leadsto \frac{2}{\left(\left(\left(\frac{\color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}}\right)}^{2}}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. cbrt-div 59.2%

         \[\leadsto \frac{2}{\left(\left(\left(\frac{{\color{blue}{\left(\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell}}\right)}}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. rem-cbrt-cube 59.2%

         \[\leadsto \frac{2}{\left(\left(\left(\frac{{\left(\frac{\color{blue}{t}}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. cbrt-div 59.2%

         \[\leadsto \frac{2}{\left(\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\color{blue}{\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell}}}}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. rem-cbrt-cube 73.4%

         \[\leadsto \frac{2}{\left(\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{\color{blue}{t}}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied egg-rr 73.4%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Taylor expanded in t around 0 68.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]

   5. Taylor expanded in k around 0 68.8%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{+152}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}}\\ \end{array} \]

Alternative 18: 59.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.05 \cdot 10^{+76}:\\ \;\;\;\;\frac{\ell}{k \cdot {t}^{3}} \cdot \left(\frac{\ell}{k} + -0.3333333333333333 \cdot \left(\ell \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.05e+76)
   (* (/ l (* k (pow t 3.0))) (+ (/ l k) (* -0.3333333333333333 (* l k))))
   (* 2.0 (/ (pow l 2.0) (* t (pow k 4.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.05e+76) {
		tmp = (l / (k * pow(t, 3.0))) * ((l / k) + (-0.3333333333333333 * (l * k)));
	} else {
		tmp = 2.0 * (pow(l, 2.0) / (t * pow(k, 4.0)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.05d+76) then
        tmp = (l / (k * (t ** 3.0d0))) * ((l / k) + ((-0.3333333333333333d0) * (l * k)))
    else
        tmp = 2.0d0 * ((l ** 2.0d0) / (t * (k ** 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.05e+76) {
		tmp = (l / (k * Math.pow(t, 3.0))) * ((l / k) + (-0.3333333333333333 * (l * k)));
	} else {
		tmp = 2.0 * (Math.pow(l, 2.0) / (t * Math.pow(k, 4.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.05e+76:
		tmp = (l / (k * math.pow(t, 3.0))) * ((l / k) + (-0.3333333333333333 * (l * k)))
	else:
		tmp = 2.0 * (math.pow(l, 2.0) / (t * math.pow(k, 4.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.05e+76)
		tmp = Float64(Float64(l / Float64(k * (t ^ 3.0))) * Float64(Float64(l / k) + Float64(-0.3333333333333333 * Float64(l * k))));
	else
		tmp = Float64(2.0 * Float64((l ^ 2.0) / Float64(t * (k ^ 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.05e+76)
		tmp = (l / (k * (t ^ 3.0))) * ((l / k) + (-0.3333333333333333 * (l * k)));
	else
		tmp = 2.0 * ((l ^ 2.0) / (t * (k ^ 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.05e+76], N[(N[(l / N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] + N[(-0.3333333333333333 * N[(l * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.05000000000000003e76

   1. Initial program 52.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 47.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* 53.6%

         \[\leadsto \frac{\color{blue}{\left(\frac{2}{{t}^{3}} \cdot \ell\right) \cdot \ell}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      2. associate-*r* 53.6%

         \[\leadsto \frac{\left(\frac{2}{{t}^{3}} \cdot \ell\right) \cdot \ell}{\color{blue}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right) \cdot \tan k}} \]

      3. times-frac 61.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}} \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k}} \]

      4. div-inv 61.9%

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{{t}^{3}}\right)} \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k} \]

      5. pow-flip 63.3%

         \[\leadsto \frac{\left(2 \cdot \color{blue}{{t}^{\left(-3\right)}}\right) \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k} \]

      6. metadata-eval 63.3%

         \[\leadsto \frac{\left(2 \cdot {t}^{\color{blue}{-3}}\right) \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k} \]

   5. Applied egg-rr 63.3%

      \[\leadsto \color{blue}{\frac{\left(2 \cdot {t}^{-3}\right) \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k}} \]
   6. Step-by-step derivation

      1. times-frac 62.4%

         \[\leadsto \color{blue}{\left(\frac{2 \cdot {t}^{-3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\sin k}\right)} \cdot \frac{\ell}{\tan k} \]

   7. Simplified 62.4%

      \[\leadsto \color{blue}{\left(\frac{2 \cdot {t}^{-3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\ell}{\tan k}} \]

   8. Taylor expanded in k around 0 57.2%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot {t}^{3}}} \cdot \frac{\ell}{\tan k} \]

   9. Taylor expanded in k around 0 56.3%

      \[\leadsto \frac{\ell}{k \cdot {t}^{3}} \cdot \color{blue}{\left(-0.3333333333333333 \cdot \left(k \cdot \ell\right) + \frac{\ell}{k}\right)} \]

   if 1.05000000000000003e76 < k

   1. Initial program 47.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 53.9%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. add-cube-cbrt 53.9%

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}}}{\ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. *-un-lft-identity 53.9%

         \[\leadsto \frac{2}{\left(\left(\frac{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}}{\color{blue}{1 \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. times-frac 53.9%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. pow2 53.9%

         \[\leadsto \frac{2}{\left(\left(\left(\frac{\color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}}\right)}^{2}}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. cbrt-div 53.9%

         \[\leadsto \frac{2}{\left(\left(\left(\frac{{\color{blue}{\left(\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell}}\right)}}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. rem-cbrt-cube 53.9%

         \[\leadsto \frac{2}{\left(\left(\left(\frac{{\left(\frac{\color{blue}{t}}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. cbrt-div 53.9%

         \[\leadsto \frac{2}{\left(\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\color{blue}{\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell}}}}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. rem-cbrt-cube 65.6%

         \[\leadsto \frac{2}{\left(\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{\color{blue}{t}}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied egg-rr 65.6%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Taylor expanded in t around 0 66.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]

   5. Taylor expanded in k around 0 56.3%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.05 \cdot 10^{+76}:\\ \;\;\;\;\frac{\ell}{k \cdot {t}^{3}} \cdot \left(\frac{\ell}{k} + -0.3333333333333333 \cdot \left(\ell \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}}\\ \end{array} \]

Alternative 19: 63.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\ell}{{\left(t \cdot \sqrt[3]{k}\right)}^{3}} \cdot \frac{\ell}{k} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* (/ l (pow (* t (cbrt k)) 3.0)) (/ l k)))

C

double code(double t, double l, double k) {
	return (l / pow((t * cbrt(k)), 3.0)) * (l / k);
}

Java

public static double code(double t, double l, double k) {
	return (l / Math.pow((t * Math.cbrt(k)), 3.0)) * (l / k);
}

Julia

function code(t, l, k)
	return Float64(Float64(l / (Float64(t * cbrt(k)) ^ 3.0)) * Float64(l / k))
end

Wolfram

code[t_, l_, k_] := N[(N[(l / N[Power[N[(t * N[Power[k, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

3. Simplified 47.1%

   \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
4. Step-by-step derivation

   1. associate-*r* 53.2%

      \[\leadsto \frac{\color{blue}{\left(\frac{2}{{t}^{3}} \cdot \ell\right) \cdot \ell}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   2. associate-*r* 53.3%

      \[\leadsto \frac{\left(\frac{2}{{t}^{3}} \cdot \ell\right) \cdot \ell}{\color{blue}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right) \cdot \tan k}} \]

   3. times-frac 60.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}} \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k}} \]

   4. div-inv 60.7%

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{{t}^{3}}\right)} \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k} \]

   5. pow-flip 61.8%

      \[\leadsto \frac{\left(2 \cdot \color{blue}{{t}^{\left(-3\right)}}\right) \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k} \]

   6. metadata-eval 61.8%

      \[\leadsto \frac{\left(2 \cdot {t}^{\color{blue}{-3}}\right) \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k} \]

5. Applied egg-rr 61.8%

   \[\leadsto \color{blue}{\frac{\left(2 \cdot {t}^{-3}\right) \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k}} \]
6. Step-by-step derivation

   1. times-frac 60.8%

      \[\leadsto \color{blue}{\left(\frac{2 \cdot {t}^{-3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\sin k}\right)} \cdot \frac{\ell}{\tan k} \]

7. Simplified 60.8%

   \[\leadsto \color{blue}{\left(\frac{2 \cdot {t}^{-3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\ell}{\tan k}} \]

8. Taylor expanded in k around 0 56.7%

   \[\leadsto \color{blue}{\frac{\ell}{k \cdot {t}^{3}}} \cdot \frac{\ell}{\tan k} \]

9. Taylor expanded in k around 0 57.8%

   \[\leadsto \frac{\ell}{k \cdot {t}^{3}} \cdot \color{blue}{\frac{\ell}{k}} \]
10. Step-by-step derivation

    1. add-cube-cbrt 57.8%

       \[\leadsto \frac{\ell}{\color{blue}{\left(\sqrt[3]{k \cdot {t}^{3}} \cdot \sqrt[3]{k \cdot {t}^{3}}\right) \cdot \sqrt[3]{k \cdot {t}^{3}}}} \cdot \frac{\ell}{k} \]

    2. pow3 57.8%

       \[\leadsto \frac{\ell}{\color{blue}{{\left(\sqrt[3]{k \cdot {t}^{3}}\right)}^{3}}} \cdot \frac{\ell}{k} \]

    3. *-commutative 57.8%

       \[\leadsto \frac{\ell}{{\left(\sqrt[3]{\color{blue}{{t}^{3} \cdot k}}\right)}^{3}} \cdot \frac{\ell}{k} \]

    4. cbrt-prod 57.8%

       \[\leadsto \frac{\ell}{{\color{blue}{\left(\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{k}\right)}}^{3}} \cdot \frac{\ell}{k} \]

    5. unpow3 57.8%

       \[\leadsto \frac{\ell}{{\left(\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}} \cdot \sqrt[3]{k}\right)}^{3}} \cdot \frac{\ell}{k} \]

    6. add-cbrt-cube 60.9%

       \[\leadsto \frac{\ell}{{\left(\color{blue}{t} \cdot \sqrt[3]{k}\right)}^{3}} \cdot \frac{\ell}{k} \]

11. Applied egg-rr 60.9%

    \[\leadsto \frac{\ell}{\color{blue}{{\left(t \cdot \sqrt[3]{k}\right)}^{3}}} \cdot \frac{\ell}{k} \]

12. Final simplification 60.9%

    \[\leadsto \frac{\ell}{{\left(t \cdot \sqrt[3]{k}\right)}^{3}} \cdot \frac{\ell}{k} \]

Alternative 20: 57.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{if}\;k \leq 1.9 \cdot 10^{+100}:\\ \;\;\;\;t_1 \cdot \left(\frac{\ell}{k} + -0.3333333333333333 \cdot \left(\ell \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ l (* k (pow t 3.0)))))
   (if (<= k 1.9e+100)
     (* t_1 (+ (/ l k) (* -0.3333333333333333 (* l k))))
     (* (/ l k) t_1))))

C

double code(double t, double l, double k) {
	double t_1 = l / (k * pow(t, 3.0));
	double tmp;
	if (k <= 1.9e+100) {
		tmp = t_1 * ((l / k) + (-0.3333333333333333 * (l * k)));
	} else {
		tmp = (l / k) * t_1;
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = l / (k * (t ** 3.0d0))
    if (k <= 1.9d+100) then
        tmp = t_1 * ((l / k) + ((-0.3333333333333333d0) * (l * k)))
    else
        tmp = (l / k) * t_1
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = l / (k * Math.pow(t, 3.0));
	double tmp;
	if (k <= 1.9e+100) {
		tmp = t_1 * ((l / k) + (-0.3333333333333333 * (l * k)));
	} else {
		tmp = (l / k) * t_1;
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = l / (k * math.pow(t, 3.0))
	tmp = 0
	if k <= 1.9e+100:
		tmp = t_1 * ((l / k) + (-0.3333333333333333 * (l * k)))
	else:
		tmp = (l / k) * t_1
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(l / Float64(k * (t ^ 3.0)))
	tmp = 0.0
	if (k <= 1.9e+100)
		tmp = Float64(t_1 * Float64(Float64(l / k) + Float64(-0.3333333333333333 * Float64(l * k))));
	else
		tmp = Float64(Float64(l / k) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = l / (k * (t ^ 3.0));
	tmp = 0.0;
	if (k <= 1.9e+100)
		tmp = t_1 * ((l / k) + (-0.3333333333333333 * (l * k)));
	else
		tmp = (l / k) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(l / N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.9e+100], N[(t$95$1 * N[(N[(l / k), $MachinePrecision] + N[(-0.3333333333333333 * N[(l * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k), $MachinePrecision] * t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.89999999999999982e100

   1. Initial program 52.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 47.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* 53.5%

         \[\leadsto \frac{\color{blue}{\left(\frac{2}{{t}^{3}} \cdot \ell\right) \cdot \ell}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      2. associate-*r* 53.6%

         \[\leadsto \frac{\left(\frac{2}{{t}^{3}} \cdot \ell\right) \cdot \ell}{\color{blue}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right) \cdot \tan k}} \]

      3. times-frac 61.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}} \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k}} \]

      4. div-inv 61.8%

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{{t}^{3}}\right)} \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k} \]

      5. pow-flip 63.1%

         \[\leadsto \frac{\left(2 \cdot \color{blue}{{t}^{\left(-3\right)}}\right) \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k} \]

      6. metadata-eval 63.1%

         \[\leadsto \frac{\left(2 \cdot {t}^{\color{blue}{-3}}\right) \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k} \]

   5. Applied egg-rr 63.1%

      \[\leadsto \color{blue}{\frac{\left(2 \cdot {t}^{-3}\right) \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k}} \]
   6. Step-by-step derivation

      1. times-frac 62.3%

         \[\leadsto \color{blue}{\left(\frac{2 \cdot {t}^{-3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\sin k}\right)} \cdot \frac{\ell}{\tan k} \]

   7. Simplified 62.3%

      \[\leadsto \color{blue}{\left(\frac{2 \cdot {t}^{-3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\ell}{\tan k}} \]

   8. Taylor expanded in k around 0 57.1%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot {t}^{3}}} \cdot \frac{\ell}{\tan k} \]

   9. Taylor expanded in k around 0 56.3%

      \[\leadsto \frac{\ell}{k \cdot {t}^{3}} \cdot \color{blue}{\left(-0.3333333333333333 \cdot \left(k \cdot \ell\right) + \frac{\ell}{k}\right)} \]

   if 1.89999999999999982e100 < k

   1. Initial program 47.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 45.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* 51.9%

         \[\leadsto \frac{\color{blue}{\left(\frac{2}{{t}^{3}} \cdot \ell\right) \cdot \ell}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      2. associate-*r* 51.9%

         \[\leadsto \frac{\left(\frac{2}{{t}^{3}} \cdot \ell\right) \cdot \ell}{\color{blue}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right) \cdot \tan k}} \]

      3. times-frac 55.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}} \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k}} \]

      4. div-inv 55.9%

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{{t}^{3}}\right)} \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k} \]

      5. pow-flip 55.9%

         \[\leadsto \frac{\left(2 \cdot \color{blue}{{t}^{\left(-3\right)}}\right) \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k} \]

      6. metadata-eval 55.9%

         \[\leadsto \frac{\left(2 \cdot {t}^{\color{blue}{-3}}\right) \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k} \]

   5. Applied egg-rr 55.9%

      \[\leadsto \color{blue}{\frac{\left(2 \cdot {t}^{-3}\right) \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k}} \]
   6. Step-by-step derivation

      1. times-frac 54.0%

         \[\leadsto \color{blue}{\left(\frac{2 \cdot {t}^{-3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\sin k}\right)} \cdot \frac{\ell}{\tan k} \]

   7. Simplified 54.0%

      \[\leadsto \color{blue}{\left(\frac{2 \cdot {t}^{-3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\ell}{\tan k}} \]

   8. Taylor expanded in k around 0 54.7%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot {t}^{3}}} \cdot \frac{\ell}{\tan k} \]

   9. Taylor expanded in k around 0 56.5%

      \[\leadsto \frac{\ell}{k \cdot {t}^{3}} \cdot \color{blue}{\frac{\ell}{k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.9 \cdot 10^{+100}:\\ \;\;\;\;\frac{\ell}{k \cdot {t}^{3}} \cdot \left(\frac{\ell}{k} + -0.3333333333333333 \cdot \left(\ell \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \end{array} \]

Alternative 21: 60.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \frac{\ell}{k} \cdot \frac{\ell}{k \cdot \left(t \cdot {t}^{2}\right)} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* (/ l k) (/ l (* k (* t (pow t 2.0))))))

C

double code(double t, double l, double k) {
	return (l / k) * (l / (k * (t * pow(t, 2.0))));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (l / k) * (l / (k * (t * (t ** 2.0d0))))
end function

Java

public static double code(double t, double l, double k) {
	return (l / k) * (l / (k * (t * Math.pow(t, 2.0))));
}

Python

def code(t, l, k):
	return (l / k) * (l / (k * (t * math.pow(t, 2.0))))

Julia

function code(t, l, k)
	return Float64(Float64(l / k) * Float64(l / Float64(k * Float64(t * (t ^ 2.0)))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l / k) * (l / (k * (t * (t ^ 2.0))));
end

Wolfram

code[t_, l_, k_] := N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * N[(t * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

3. Simplified 47.1%

   \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
4. Step-by-step derivation

   1. associate-*r* 53.2%

      \[\leadsto \frac{\color{blue}{\left(\frac{2}{{t}^{3}} \cdot \ell\right) \cdot \ell}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   2. associate-*r* 53.3%

      \[\leadsto \frac{\left(\frac{2}{{t}^{3}} \cdot \ell\right) \cdot \ell}{\color{blue}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right) \cdot \tan k}} \]

   3. times-frac 60.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}} \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k}} \]

   4. div-inv 60.7%

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{{t}^{3}}\right)} \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k} \]

   5. pow-flip 61.8%

      \[\leadsto \frac{\left(2 \cdot \color{blue}{{t}^{\left(-3\right)}}\right) \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k} \]

   6. metadata-eval 61.8%

      \[\leadsto \frac{\left(2 \cdot {t}^{\color{blue}{-3}}\right) \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k} \]

5. Applied egg-rr 61.8%

   \[\leadsto \color{blue}{\frac{\left(2 \cdot {t}^{-3}\right) \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k}} \]
6. Step-by-step derivation

   1. times-frac 60.8%

      \[\leadsto \color{blue}{\left(\frac{2 \cdot {t}^{-3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\sin k}\right)} \cdot \frac{\ell}{\tan k} \]

7. Simplified 60.8%

   \[\leadsto \color{blue}{\left(\frac{2 \cdot {t}^{-3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\ell}{\tan k}} \]

8. Taylor expanded in k around 0 56.7%

   \[\leadsto \color{blue}{\frac{\ell}{k \cdot {t}^{3}}} \cdot \frac{\ell}{\tan k} \]

9. Taylor expanded in k around 0 57.8%

   \[\leadsto \frac{\ell}{k \cdot {t}^{3}} \cdot \color{blue}{\frac{\ell}{k}} \]
10. Step-by-step derivation

    1. unpow3 57.8%

       \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \cdot \frac{\ell}{k} \]

    2. pow2 57.8%

       \[\leadsto \frac{\ell}{k \cdot \left(\color{blue}{{t}^{2}} \cdot t\right)} \cdot \frac{\ell}{k} \]

11. Applied egg-rr 57.8%

    \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left({t}^{2} \cdot t\right)}} \cdot \frac{\ell}{k} \]

12. Final simplification 57.8%

    \[\leadsto \frac{\ell}{k} \cdot \frac{\ell}{k \cdot \left(t \cdot {t}^{2}\right)} \]

Alternative 22: 60.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* (/ l k) (/ l (* k (pow t 3.0)))))

C

double code(double t, double l, double k) {
	return (l / k) * (l / (k * pow(t, 3.0)));
}

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 / k) * (l / (k * (t ** 3.0d0)))
end function

Java

public static double code(double t, double l, double k) {
	return (l / k) * (l / (k * Math.pow(t, 3.0)));
}

Python

def code(t, l, k):
	return (l / k) * (l / (k * math.pow(t, 3.0)))

Julia

function code(t, l, k)
	return Float64(Float64(l / k) * Float64(l / Float64(k * (t ^ 3.0))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l / k) * (l / (k * (t ^ 3.0)));
end

Wolfram

code[t_, l_, k_] := N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

3. Simplified 47.1%

   \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
4. Step-by-step derivation

   1. associate-*r* 53.2%

      \[\leadsto \frac{\color{blue}{\left(\frac{2}{{t}^{3}} \cdot \ell\right) \cdot \ell}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   2. associate-*r* 53.3%

      \[\leadsto \frac{\left(\frac{2}{{t}^{3}} \cdot \ell\right) \cdot \ell}{\color{blue}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right) \cdot \tan k}} \]

   3. times-frac 60.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}} \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k}} \]

   4. div-inv 60.7%

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{{t}^{3}}\right)} \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k} \]

   5. pow-flip 61.8%

      \[\leadsto \frac{\left(2 \cdot \color{blue}{{t}^{\left(-3\right)}}\right) \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k} \]

   6. metadata-eval 61.8%

      \[\leadsto \frac{\left(2 \cdot {t}^{\color{blue}{-3}}\right) \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k} \]

5. Applied egg-rr 61.8%

   \[\leadsto \color{blue}{\frac{\left(2 \cdot {t}^{-3}\right) \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\ell}{\tan k}} \]
6. Step-by-step derivation

   1. times-frac 60.8%

      \[\leadsto \color{blue}{\left(\frac{2 \cdot {t}^{-3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\sin k}\right)} \cdot \frac{\ell}{\tan k} \]

7. Simplified 60.8%

   \[\leadsto \color{blue}{\left(\frac{2 \cdot {t}^{-3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\ell}{\tan k}} \]

8. Taylor expanded in k around 0 56.7%

   \[\leadsto \color{blue}{\frac{\ell}{k \cdot {t}^{3}}} \cdot \frac{\ell}{\tan k} \]

9. Taylor expanded in k around 0 57.8%

   \[\leadsto \frac{\ell}{k \cdot {t}^{3}} \cdot \color{blue}{\frac{\ell}{k}} \]

10. Final simplification 57.8%

    \[\leadsto \frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}} \]

Reproduce

herbie shell --seed 2023320 
(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))))