Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.5% → 86.7%

Time: 26.5s

Alternatives: 18

Speedup: 1.3×

• 26.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt[3]{\sin k}\\ t_2 := 2 + \frac{\frac{k}{t}}{\frac{t}{k}}\\ \mathbf{if}\;t \leq -9.5 \cdot 10^{-58}:\\ \;\;\;\;\frac{\frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot t_1\right)}^{3} \cdot \tan k}}{t_2}\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{-45}:\\ \;\;\;\;\left(\frac{2}{{k}^{2}} \cdot \frac{\ell}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot {\left(\frac{t \cdot t_1}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}}{t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (cbrt (sin k))) (t_2 (+ 2.0 (/ (/ k t) (/ t k)))))
   (if (<= t -9.5e-58)
     (/ (/ 2.0 (* (pow (* (* t (pow (cbrt l) -2.0)) t_1) 3.0) (tan k))) t_2)
     (if (<= t 4.4e-45)
       (* (* (/ 2.0 (pow k 2.0)) (/ l t)) (/ (/ l (sin k)) (tan k)))
       (/
        (/ 2.0 (* (tan k) (pow (/ (* t t_1) (pow (cbrt l) 2.0)) 3.0)))
        t_2)))))

C

double code(double t, double l, double k) {
	double t_1 = cbrt(sin(k));
	double t_2 = 2.0 + ((k / t) / (t / k));
	double tmp;
	if (t <= -9.5e-58) {
		tmp = (2.0 / (pow(((t * pow(cbrt(l), -2.0)) * t_1), 3.0) * tan(k))) / t_2;
	} else if (t <= 4.4e-45) {
		tmp = ((2.0 / pow(k, 2.0)) * (l / t)) * ((l / sin(k)) / tan(k));
	} else {
		tmp = (2.0 / (tan(k) * pow(((t * t_1) / pow(cbrt(l), 2.0)), 3.0))) / t_2;
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.cbrt(Math.sin(k));
	double t_2 = 2.0 + ((k / t) / (t / k));
	double tmp;
	if (t <= -9.5e-58) {
		tmp = (2.0 / (Math.pow(((t * Math.pow(Math.cbrt(l), -2.0)) * t_1), 3.0) * Math.tan(k))) / t_2;
	} else if (t <= 4.4e-45) {
		tmp = ((2.0 / Math.pow(k, 2.0)) * (l / t)) * ((l / Math.sin(k)) / Math.tan(k));
	} else {
		tmp = (2.0 / (Math.tan(k) * Math.pow(((t * t_1) / Math.pow(Math.cbrt(l), 2.0)), 3.0))) / t_2;
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cbrt(sin(k))
	t_2 = Float64(2.0 + Float64(Float64(k / t) / Float64(t / k)))
	tmp = 0.0
	if (t <= -9.5e-58)
		tmp = Float64(Float64(2.0 / Float64((Float64(Float64(t * (cbrt(l) ^ -2.0)) * t_1) ^ 3.0) * tan(k))) / t_2);
	elseif (t <= 4.4e-45)
		tmp = Float64(Float64(Float64(2.0 / (k ^ 2.0)) * Float64(l / t)) * Float64(Float64(l / sin(k)) / tan(k)));
	else
		tmp = Float64(Float64(2.0 / Float64(tan(k) * (Float64(Float64(t * t_1) / (cbrt(l) ^ 2.0)) ^ 3.0))) / t_2);
	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[(2.0 + N[(N[(k / t), $MachinePrecision] / N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -9.5e-58], N[(N[(2.0 / N[(N[Power[N[(N[(t * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], 3.0], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t, 4.4e-45], N[(N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[Power[N[(N[(t * t$95$1), $MachinePrecision] / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -9.4999999999999994e-58

   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 72.2%

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

      1. unpow2 72.2%

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

      2. clear-num 72.2%

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

      3. un-div-inv 72.2%

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

   5. Applied egg-rr 72.2%

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

      1. add-cube-cbrt 72.1%

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

      2. pow3 72.1%

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

      3. cbrt-prod 72.0%

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

      4. cbrt-div 74.6%

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

      5. cbrt-div 74.5%

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

      6. unpow3 74.5%

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

      7. add-cbrt-cube 94.9%

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

      8. associate-/l/ 95.0%

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

      9. pow2 95.0%

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

   7. Applied egg-rr 95.0%

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

      1. expm1-log1p-u 58.0%

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

      2. expm1-udef 39.5%

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

      3. div-inv 39.5%

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

      4. pow-flip 39.5%

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

      5. metadata-eval 39.5%

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

   9. Applied egg-rr 39.5%

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

       1. expm1-def 58.1%

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

       2. expm1-log1p 95.0%

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

   11. Simplified 95.0%

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

   if -9.4999999999999994e-58 < t < 4.39999999999999987e-45

   1. Initial program 35.0%

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

   3. Simplified 33.2%

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

      1. associate-*r* 40.1%

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

      2. *-un-lft-identity 40.1%

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

      3. times-frac 40.1%

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

   5. Applied egg-rr 40.1%

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

      1. /-rgt-identity 40.1%

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

      2. associate-*r/ 40.1%

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

      3. times-frac 41.0%

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

      4. *-commutative 41.0%

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

      5. associate-/r* 41.0%

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

   7. Simplified 41.0%

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

   8. Taylor expanded in t around 0 78.8%

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

      1. associate-*r/ 78.8%

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

      2. times-frac 80.9%

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

   10. Simplified 80.9%

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

   if 4.39999999999999987e-45 < t

   1. Initial program 66.6%

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

   3. Simplified 72.2%

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

      1. unpow2 72.2%

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

      2. clear-num 72.2%

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

      3. un-div-inv 72.2%

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

   5. Applied egg-rr 72.2%

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

      1. add-cube-cbrt 72.1%

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

      2. pow3 72.1%

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

      3. cbrt-prod 72.0%

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

      4. cbrt-div 73.3%

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

      5. cbrt-div 73.2%

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

      6. unpow3 73.2%

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

      7. add-cbrt-cube 92.2%

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

      8. associate-/l/ 92.2%

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

      9. pow2 92.2%

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

   7. Applied egg-rr 92.2%

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

      1. associate-*l/ 92.3%

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

   9. Applied egg-rr 92.3%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{-58}:\\ \;\;\;\;\frac{\frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \tan k}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{-45}:\\ \;\;\;\;\left(\frac{2}{{k}^{2}} \cdot \frac{\ell}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot {\left(\frac{t \cdot \sqrt[3]{\sin k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \end{array} \]

Alternative 2: 78.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)} \leq 5 \cdot 10^{+301}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot \frac{t \cdot \frac{{t}^{2}}{\ell}}{\ell}\right)}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{k}^{2}} \cdot \frac{\ell}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<=
      (/
       2.0
       (*
        (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))
        (* (tan k) (* (sin k) (/ (pow t 3.0) (* l l))))))
      5e+301)
   (/
    (/ 2.0 (* (tan k) (* (sin k) (/ (* t (/ (pow t 2.0) l)) l))))
    (+ 2.0 (/ (/ k t) (/ t k))))
   (* (* (/ 2.0 (pow k 2.0)) (/ l t)) (/ (/ l (sin k)) (tan k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((2.0 / ((1.0 + (1.0 + pow((k / t), 2.0))) * (tan(k) * (sin(k) * (pow(t, 3.0) / (l * l)))))) <= 5e+301) {
		tmp = (2.0 / (tan(k) * (sin(k) * ((t * (pow(t, 2.0) / l)) / l)))) / (2.0 + ((k / t) / (t / k)));
	} else {
		tmp = ((2.0 / pow(k, 2.0)) * (l / t)) * ((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 ((2.0d0 / ((1.0d0 + (1.0d0 + ((k / t) ** 2.0d0))) * (tan(k) * (sin(k) * ((t ** 3.0d0) / (l * l)))))) <= 5d+301) then
        tmp = (2.0d0 / (tan(k) * (sin(k) * ((t * ((t ** 2.0d0) / l)) / l)))) / (2.0d0 + ((k / t) / (t / k)))
    else
        tmp = ((2.0d0 / (k ** 2.0d0)) * (l / t)) * ((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 ((2.0 / ((1.0 + (1.0 + Math.pow((k / t), 2.0))) * (Math.tan(k) * (Math.sin(k) * (Math.pow(t, 3.0) / (l * l)))))) <= 5e+301) {
		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 = ((2.0 / Math.pow(k, 2.0)) * (l / t)) * ((l / Math.sin(k)) / Math.tan(k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (2.0 / ((1.0 + (1.0 + math.pow((k / t), 2.0))) * (math.tan(k) * (math.sin(k) * (math.pow(t, 3.0) / (l * l)))))) <= 5e+301:
		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 = ((2.0 / math.pow(k, 2.0)) * (l / t)) * ((l / math.sin(k)) / math.tan(k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0))) * Float64(tan(k) * Float64(sin(k) * Float64((t ^ 3.0) / Float64(l * l)))))) <= 5e+301)
		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(Float64(2.0 / (k ^ 2.0)) * Float64(l / t)) * Float64(Float64(l / sin(k)) / tan(k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((2.0 / ((1.0 + (1.0 + ((k / t) ^ 2.0))) * (tan(k) * (sin(k) * ((t ^ 3.0) / (l * l)))))) <= 5e+301)
		tmp = (2.0 / (tan(k) * (sin(k) * ((t * ((t ^ 2.0) / l)) / l)))) / (2.0 + ((k / t) / (t / k)));
	else
		tmp = ((2.0 / (k ^ 2.0)) * (l / t)) * ((l / sin(k)) / tan(k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[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[(N[Sin[k], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+301], 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[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 5.0000000000000004e301

   1. Initial program 80.6%

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

   3. Simplified 85.4%

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

      1. unpow2 85.4%

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

      2. clear-num 85.4%

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

      3. un-div-inv 85.4%

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

   5. Applied egg-rr 85.4%

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

      1. cube-mult 85.4%

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

      2. *-un-lft-identity 85.4%

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

      3. times-frac 88.2%

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

      4. pow2 88.2%

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

   7. Applied egg-rr 88.2%

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

   if 5.0000000000000004e301 < (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))

   1. Initial program 19.1%

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

   3. Simplified 19.1%

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

      1. associate-*r* 29.2%

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

      2. *-un-lft-identity 29.2%

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

      3. times-frac 29.8%

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

   5. Applied egg-rr 29.8%

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

      1. /-rgt-identity 29.8%

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

      2. associate-*r/ 29.2%

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

      3. times-frac 29.8%

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

      4. *-commutative 29.8%

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

      5. associate-/r* 29.8%

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

   7. Simplified 29.8%

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

   8. Taylor expanded in t around 0 63.8%

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

      1. associate-*r/ 63.8%

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

      2. times-frac 64.0%

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

   10. Simplified 64.0%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)} \leq 5 \cdot 10^{+301}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot \frac{t \cdot \frac{{t}^{2}}{\ell}}{\ell}\right)}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{k}^{2}} \cdot \frac{\ell}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \end{array} \]

Alternative 3: 86.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{-58} \lor \neg \left(t \leq 1.35 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{\frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \tan k}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{k}^{2}} \cdot \frac{\ell}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -3.3e-58) (not (<= t 1.35e-46)))
   (/
    (/ 2.0 (* (pow (* (* t (pow (cbrt l) -2.0)) (cbrt (sin k))) 3.0) (tan k)))
    (+ 2.0 (/ (/ k t) (/ t k))))
   (* (* (/ 2.0 (pow k 2.0)) (/ l t)) (/ (/ l (sin k)) (tan k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -3.3e-58) || !(t <= 1.35e-46)) {
		tmp = (2.0 / (pow(((t * pow(cbrt(l), -2.0)) * cbrt(sin(k))), 3.0) * tan(k))) / (2.0 + ((k / t) / (t / k)));
	} else {
		tmp = ((2.0 / pow(k, 2.0)) * (l / t)) * ((l / sin(k)) / tan(k));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -3.3e-58) || !(t <= 1.35e-46)) {
		tmp = (2.0 / (Math.pow(((t * Math.pow(Math.cbrt(l), -2.0)) * Math.cbrt(Math.sin(k))), 3.0) * Math.tan(k))) / (2.0 + ((k / t) / (t / k)));
	} else {
		tmp = ((2.0 / Math.pow(k, 2.0)) * (l / t)) * ((l / Math.sin(k)) / Math.tan(k));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -3.3e-58) || !(t <= 1.35e-46))
		tmp = Float64(Float64(2.0 / Float64((Float64(Float64(t * (cbrt(l) ^ -2.0)) * cbrt(sin(k))) ^ 3.0) * tan(k))) / Float64(2.0 + Float64(Float64(k / t) / Float64(t / k))));
	else
		tmp = Float64(Float64(Float64(2.0 / (k ^ 2.0)) * Float64(l / t)) * Float64(Float64(l / sin(k)) / tan(k)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -3.3e-58], N[Not[LessEqual[t, 1.35e-46]], $MachinePrecision]], N[(N[(2.0 / N[(N[Power[N[(N[(t * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(k / t), $MachinePrecision] / N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.30000000000000026e-58 or 1.35e-46 < t

   1. Initial program 64.7%

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

   3. Simplified 72.2%

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

      1. unpow2 72.2%

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

      2. clear-num 72.2%

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

      3. un-div-inv 72.2%

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

   5. Applied egg-rr 72.2%

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

      1. add-cube-cbrt 72.1%

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

      2. pow3 72.1%

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

      3. cbrt-prod 72.0%

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

      4. cbrt-div 73.9%

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

      5. cbrt-div 73.8%

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

      6. unpow3 73.8%

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

      7. add-cbrt-cube 93.5%

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

      8. associate-/l/ 93.5%

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

      9. pow2 93.5%

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

   7. Applied egg-rr 93.5%

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

      1. expm1-log1p-u 56.2%

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

      2. expm1-udef 40.7%

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

      3. div-inv 40.7%

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

      4. pow-flip 40.7%

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

      5. metadata-eval 40.7%

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

   9. Applied egg-rr 40.7%

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

       1. expm1-def 56.2%

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

       2. expm1-log1p 93.5%

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

   11. Simplified 93.5%

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

   if -3.30000000000000026e-58 < t < 1.35e-46

   1. Initial program 35.0%

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

   3. Simplified 33.2%

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

      1. associate-*r* 40.1%

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

      2. *-un-lft-identity 40.1%

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

      3. times-frac 40.1%

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

   5. Applied egg-rr 40.1%

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

      1. /-rgt-identity 40.1%

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

      2. associate-*r/ 40.1%

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

      3. times-frac 41.0%

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

      4. *-commutative 41.0%

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

      5. associate-/r* 41.0%

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

   7. Simplified 41.0%

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

   8. Taylor expanded in t around 0 78.8%

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

      1. associate-*r/ 78.8%

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

      2. times-frac 80.9%

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

   10. Simplified 80.9%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{-58} \lor \neg \left(t \leq 1.35 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{\frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \tan k}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{k}^{2}} \cdot \frac{\ell}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \end{array} \]

Alternative 4: 86.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt[3]{\sin k}\\ t_2 := 2 + \frac{\frac{k}{t}}{\frac{t}{k}}\\ \mathbf{if}\;t \leq -6.2 \cdot 10^{-58}:\\ \;\;\;\;\frac{\frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot t_1\right)}^{3} \cdot \tan k}}{t_2}\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{-47}:\\ \;\;\;\;\left(\frac{2}{{k}^{2}} \cdot \frac{\ell}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot {\left(t_1 \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}}{t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (cbrt (sin k))) (t_2 (+ 2.0 (/ (/ k t) (/ t k)))))
   (if (<= t -6.2e-58)
     (/ (/ 2.0 (* (pow (* (* t (pow (cbrt l) -2.0)) t_1) 3.0) (tan k))) t_2)
     (if (<= t 4.3e-47)
       (* (* (/ 2.0 (pow k 2.0)) (/ l t)) (/ (/ l (sin k)) (tan k)))
       (/
        (/ 2.0 (* (tan k) (pow (* t_1 (/ t (pow (cbrt l) 2.0))) 3.0)))
        t_2)))))

C

double code(double t, double l, double k) {
	double t_1 = cbrt(sin(k));
	double t_2 = 2.0 + ((k / t) / (t / k));
	double tmp;
	if (t <= -6.2e-58) {
		tmp = (2.0 / (pow(((t * pow(cbrt(l), -2.0)) * t_1), 3.0) * tan(k))) / t_2;
	} else if (t <= 4.3e-47) {
		tmp = ((2.0 / pow(k, 2.0)) * (l / t)) * ((l / sin(k)) / tan(k));
	} else {
		tmp = (2.0 / (tan(k) * pow((t_1 * (t / pow(cbrt(l), 2.0))), 3.0))) / t_2;
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.cbrt(Math.sin(k));
	double t_2 = 2.0 + ((k / t) / (t / k));
	double tmp;
	if (t <= -6.2e-58) {
		tmp = (2.0 / (Math.pow(((t * Math.pow(Math.cbrt(l), -2.0)) * t_1), 3.0) * Math.tan(k))) / t_2;
	} else if (t <= 4.3e-47) {
		tmp = ((2.0 / Math.pow(k, 2.0)) * (l / t)) * ((l / Math.sin(k)) / Math.tan(k));
	} else {
		tmp = (2.0 / (Math.tan(k) * Math.pow((t_1 * (t / Math.pow(Math.cbrt(l), 2.0))), 3.0))) / t_2;
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cbrt(sin(k))
	t_2 = Float64(2.0 + Float64(Float64(k / t) / Float64(t / k)))
	tmp = 0.0
	if (t <= -6.2e-58)
		tmp = Float64(Float64(2.0 / Float64((Float64(Float64(t * (cbrt(l) ^ -2.0)) * t_1) ^ 3.0) * tan(k))) / t_2);
	elseif (t <= 4.3e-47)
		tmp = Float64(Float64(Float64(2.0 / (k ^ 2.0)) * Float64(l / t)) * Float64(Float64(l / sin(k)) / tan(k)));
	else
		tmp = Float64(Float64(2.0 / Float64(tan(k) * (Float64(t_1 * Float64(t / (cbrt(l) ^ 2.0))) ^ 3.0))) / t_2);
	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[(2.0 + N[(N[(k / t), $MachinePrecision] / N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -6.2e-58], N[(N[(2.0 / N[(N[Power[N[(N[(t * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], 3.0], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t, 4.3e-47], N[(N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / 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] / t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -6.1999999999999998e-58

   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 72.2%

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

      1. unpow2 72.2%

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

      2. clear-num 72.2%

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

      3. un-div-inv 72.2%

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

   5. Applied egg-rr 72.2%

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

      1. add-cube-cbrt 72.1%

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

      2. pow3 72.1%

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

      3. cbrt-prod 72.0%

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

      4. cbrt-div 74.6%

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

      5. cbrt-div 74.5%

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

      6. unpow3 74.5%

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

      7. add-cbrt-cube 94.9%

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

      8. associate-/l/ 95.0%

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

      9. pow2 95.0%

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

   7. Applied egg-rr 95.0%

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

      1. expm1-log1p-u 58.0%

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

      2. expm1-udef 39.5%

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

      3. div-inv 39.5%

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

      4. pow-flip 39.5%

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

      5. metadata-eval 39.5%

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

   9. Applied egg-rr 39.5%

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

       1. expm1-def 58.1%

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

       2. expm1-log1p 95.0%

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

   11. Simplified 95.0%

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

   if -6.1999999999999998e-58 < t < 4.2999999999999998e-47

   1. Initial program 35.0%

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

   3. Simplified 33.2%

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

      1. associate-*r* 40.1%

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

      2. *-un-lft-identity 40.1%

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

      3. times-frac 40.1%

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

   5. Applied egg-rr 40.1%

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

      1. /-rgt-identity 40.1%

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

      2. associate-*r/ 40.1%

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

      3. times-frac 41.0%

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

      4. *-commutative 41.0%

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

      5. associate-/r* 41.0%

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

   7. Simplified 41.0%

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

   8. Taylor expanded in t around 0 78.8%

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

      1. associate-*r/ 78.8%

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

      2. times-frac 80.9%

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

   10. Simplified 80.9%

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

   if 4.2999999999999998e-47 < t

   1. Initial program 66.6%

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

   3. Simplified 72.2%

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

      1. unpow2 72.2%

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

      2. clear-num 72.2%

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

      3. un-div-inv 72.2%

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

   5. Applied egg-rr 72.2%

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

      1. add-cube-cbrt 72.1%

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

      2. pow3 72.1%

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

      3. cbrt-prod 72.0%

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

      4. cbrt-div 73.3%

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

      5. cbrt-div 73.2%

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

      6. unpow3 73.2%

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

      7. add-cbrt-cube 92.2%

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

      8. associate-/l/ 92.2%

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

      9. pow2 92.2%

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

   7. Applied egg-rr 92.2%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.2 \cdot 10^{-58}:\\ \;\;\;\;\frac{\frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \tan k}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{-47}:\\ \;\;\;\;\left(\frac{2}{{k}^{2}} \cdot \frac{\ell}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot {\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \end{array} \]

Alternative 5: 83.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{-55}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;t \leq 1.28 \cdot 10^{-45}:\\ \;\;\;\;\left(\frac{2}{{k}^{2}} \cdot \frac{\ell}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -3.5e-55)
   (/
    2.0
    (*
     (* (tan k) (* (sin k) (pow (/ (/ t (cbrt l)) (cbrt l)) 3.0)))
     (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))))
   (if (<= t 1.28e-45)
     (* (* (/ 2.0 (pow k 2.0)) (/ l t)) (/ (/ l (sin k)) (tan k)))
     (/
      (/ 2.0 (* (tan k) (* (sin k) (pow (/ t (pow (cbrt l) 2.0)) 3.0))))
      (+ 2.0 (/ (/ k t) (/ t k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -3.5e-55) {
		tmp = 2.0 / ((tan(k) * (sin(k) * pow(((t / cbrt(l)) / cbrt(l)), 3.0))) * (1.0 + (1.0 + pow((k / t), 2.0))));
	} else if (t <= 1.28e-45) {
		tmp = ((2.0 / pow(k, 2.0)) * (l / t)) * ((l / sin(k)) / tan(k));
	} else {
		tmp = (2.0 / (tan(k) * (sin(k) * pow((t / pow(cbrt(l), 2.0)), 3.0)))) / (2.0 + ((k / t) / (t / k)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -3.5e-55) {
		tmp = 2.0 / ((Math.tan(k) * (Math.sin(k) * Math.pow(((t / Math.cbrt(l)) / Math.cbrt(l)), 3.0))) * (1.0 + (1.0 + Math.pow((k / t), 2.0))));
	} else if (t <= 1.28e-45) {
		tmp = ((2.0 / Math.pow(k, 2.0)) * (l / t)) * ((l / Math.sin(k)) / Math.tan(k));
	} else {
		tmp = (2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow((t / Math.pow(Math.cbrt(l), 2.0)), 3.0)))) / (2.0 + ((k / t) / (t / k)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -3.5e-55)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) * (Float64(Float64(t / cbrt(l)) / cbrt(l)) ^ 3.0))) * Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0)))));
	elseif (t <= 1.28e-45)
		tmp = Float64(Float64(Float64(2.0 / (k ^ 2.0)) * Float64(l / t)) * Float64(Float64(l / sin(k)) / tan(k)));
	else
		tmp = Float64(Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (Float64(t / (cbrt(l) ^ 2.0)) ^ 3.0)))) / Float64(2.0 + Float64(Float64(k / t) / Float64(t / k))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -3.5e-55], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[(t / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.28e-45], N[(N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $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 < -3.50000000000000025e-55

   1. Initial program 64.3%

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

      1. associate-/r* 74.2%

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

      2. add-cube-cbrt 74.2%

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

      3. *-un-lft-identity 74.2%

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

      4. times-frac 74.2%

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

      5. pow2 74.2%

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

      6. cbrt-div 74.1%

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

      7. rem-cbrt-cube 74.1%

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

      8. cbrt-div 74.1%

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

      9. rem-cbrt-cube 88.8%

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

   3. Applied egg-rr 88.8%

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

      1. add-cube-cbrt 88.6%

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

      2. pow3 88.7%

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

      3. frac-times 81.9%

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

      4. unpow2 81.9%

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

      5. *-un-lft-identity 81.9%

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

      6. cbrt-div 81.9%

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

      7. add-cbrt-cube 91.3%

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

   5. Applied egg-rr 91.3%

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

   if -3.50000000000000025e-55 < t < 1.28e-45

   1. Initial program 34.3%

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

   3. Simplified 32.5%

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

      1. associate-*r* 39.3%

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

      2. *-un-lft-identity 39.3%

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

      3. times-frac 39.3%

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

   5. Applied egg-rr 39.3%

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

      1. /-rgt-identity 39.3%

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

      2. associate-*r/ 39.3%

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

      3. times-frac 40.3%

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

      4. *-commutative 40.3%

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

      5. associate-/r* 41.2%

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

   7. Simplified 41.2%

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

   8. Taylor expanded in t around 0 77.4%

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

      1. associate-*r/ 77.4%

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

      2. times-frac 79.5%

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

   10. Simplified 79.5%

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

   if 1.28e-45 < t

   1. Initial program 66.6%

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

   3. Simplified 72.2%

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

      1. unpow2 72.2%

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

      2. clear-num 72.2%

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

      3. un-div-inv 72.2%

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

   5. Applied egg-rr 72.2%

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

      1. add-cube-cbrt 72.1%

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

      2. pow3 72.0%

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

      3. cbrt-div 72.1%

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

      4. cbrt-div 72.0%

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

      5. unpow3 72.0%

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

      6. add-cbrt-cube 87.5%

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

      7. associate-/l/ 87.5%

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

      8. pow2 87.5%

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

   7. Applied egg-rr 87.5%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{-55}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;t \leq 1.28 \cdot 10^{-45}:\\ \;\;\;\;\left(\frac{2}{{k}^{2}} \cdot \frac{\ell}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \end{array} \]

Alternative 6: 83.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-51} \lor \neg \left(t \leq 2 \cdot 10^{-43}\right):\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{k}^{2}} \cdot \frac{\ell}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -7.5e-51) (not (<= t 2e-43)))
   (/
    (/ 2.0 (* (tan k) (* (sin k) (pow (/ t (pow (cbrt l) 2.0)) 3.0))))
    (+ 2.0 (/ (/ k t) (/ t k))))
   (* (* (/ 2.0 (pow k 2.0)) (/ l t)) (/ (/ l (sin k)) (tan k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -7.5e-51) || !(t <= 2e-43)) {
		tmp = (2.0 / (tan(k) * (sin(k) * pow((t / pow(cbrt(l), 2.0)), 3.0)))) / (2.0 + ((k / t) / (t / k)));
	} else {
		tmp = ((2.0 / pow(k, 2.0)) * (l / t)) * ((l / sin(k)) / tan(k));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -7.5e-51) || !(t <= 2e-43)) {
		tmp = (2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow((t / Math.pow(Math.cbrt(l), 2.0)), 3.0)))) / (2.0 + ((k / t) / (t / k)));
	} else {
		tmp = ((2.0 / Math.pow(k, 2.0)) * (l / t)) * ((l / Math.sin(k)) / Math.tan(k));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -7.5e-51) || !(t <= 2e-43))
		tmp = Float64(Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (Float64(t / (cbrt(l) ^ 2.0)) ^ 3.0)))) / Float64(2.0 + Float64(Float64(k / t) / Float64(t / k))));
	else
		tmp = Float64(Float64(Float64(2.0 / (k ^ 2.0)) * Float64(l / t)) * Float64(Float64(l / sin(k)) / tan(k)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -7.5e-51], N[Not[LessEqual[t, 2e-43]], $MachinePrecision]], N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(k / t), $MachinePrecision] / N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -7.49999999999999976e-51 or 2.00000000000000015e-43 < t

   1. Initial program 65.5%

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

   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 73.1%

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

      2. clear-num 73.1%

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

      3. un-div-inv 73.1%

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

   5. Applied egg-rr 73.1%

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

      1. add-cube-cbrt 73.0%

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

      2. pow3 73.0%

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

      3. cbrt-div 73.0%

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

      4. cbrt-div 72.9%

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

      5. unpow3 72.9%

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

      6. add-cbrt-cube 89.3%

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

      7. associate-/l/ 89.3%

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

      8. pow2 89.3%

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

   7. Applied egg-rr 89.3%

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

   if -7.49999999999999976e-51 < t < 2.00000000000000015e-43

   1. Initial program 34.3%

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

   3. Simplified 32.5%

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

      1. associate-*r* 39.3%

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

      2. *-un-lft-identity 39.3%

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

      3. times-frac 39.3%

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

   5. Applied egg-rr 39.3%

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

      1. /-rgt-identity 39.3%

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

      2. associate-*r/ 39.3%

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

      3. times-frac 40.3%

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

      4. *-commutative 40.3%

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

      5. associate-/r* 41.2%

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

   7. Simplified 41.2%

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

   8. Taylor expanded in t around 0 77.4%

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

      1. associate-*r/ 77.4%

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

      2. times-frac 79.5%

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

   10. Simplified 79.5%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-51} \lor \neg \left(t \leq 2 \cdot 10^{-43}\right):\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{k}^{2}} \cdot \frac{\ell}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \end{array} \]

Alternative 7: 59.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.15 \cdot 10^{-36} \lor \neg \left(t \leq 5 \cdot 10^{-45}\right):\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(t \cdot {\ell}^{-0.6666666666666666}\right)}^{3}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{k}^{2}} \cdot \frac{\ell}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -2.15e-36) (not (<= t 5e-45)))
   (/
    2.0
    (*
     (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))
     (* (tan k) (* (sin k) (pow (* t (pow l -0.6666666666666666)) 3.0)))))
   (* (* (/ 2.0 (pow k 2.0)) (/ l t)) (/ (/ l (sin k)) (tan k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -2.15e-36) || !(t <= 5e-45)) {
		tmp = 2.0 / ((1.0 + (1.0 + pow((k / t), 2.0))) * (tan(k) * (sin(k) * pow((t * pow(l, -0.6666666666666666)), 3.0))));
	} else {
		tmp = ((2.0 / pow(k, 2.0)) * (l / t)) * ((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 <= (-2.15d-36)) .or. (.not. (t <= 5d-45))) then
        tmp = 2.0d0 / ((1.0d0 + (1.0d0 + ((k / t) ** 2.0d0))) * (tan(k) * (sin(k) * ((t * (l ** (-0.6666666666666666d0))) ** 3.0d0))))
    else
        tmp = ((2.0d0 / (k ** 2.0d0)) * (l / t)) * ((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 <= -2.15e-36) || !(t <= 5e-45)) {
		tmp = 2.0 / ((1.0 + (1.0 + Math.pow((k / t), 2.0))) * (Math.tan(k) * (Math.sin(k) * Math.pow((t * Math.pow(l, -0.6666666666666666)), 3.0))));
	} else {
		tmp = ((2.0 / Math.pow(k, 2.0)) * (l / t)) * ((l / Math.sin(k)) / Math.tan(k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -2.15e-36) or not (t <= 5e-45):
		tmp = 2.0 / ((1.0 + (1.0 + math.pow((k / t), 2.0))) * (math.tan(k) * (math.sin(k) * math.pow((t * math.pow(l, -0.6666666666666666)), 3.0))))
	else:
		tmp = ((2.0 / math.pow(k, 2.0)) * (l / t)) * ((l / math.sin(k)) / math.tan(k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -2.15e-36) || !(t <= 5e-45))
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0))) * Float64(tan(k) * Float64(sin(k) * (Float64(t * (l ^ -0.6666666666666666)) ^ 3.0)))));
	else
		tmp = Float64(Float64(Float64(2.0 / (k ^ 2.0)) * Float64(l / t)) * Float64(Float64(l / sin(k)) / tan(k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -2.15e-36) || ~((t <= 5e-45)))
		tmp = 2.0 / ((1.0 + (1.0 + ((k / t) ^ 2.0))) * (tan(k) * (sin(k) * ((t * (l ^ -0.6666666666666666)) ^ 3.0))));
	else
		tmp = ((2.0 / (k ^ 2.0)) * (l / t)) * ((l / sin(k)) / tan(k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -2.15e-36], N[Not[LessEqual[t, 5e-45]], $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[(N[Sin[k], $MachinePrecision] * N[Power[N[(t * N[Power[l, -0.6666666666666666], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.1500000000000001e-36 or 4.99999999999999976e-45 < t

   1. Initial program 65.4%

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

      1. add-cube-cbrt 65.4%

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

      2. pow3 65.4%

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

      3. div-inv 65.4%

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

      4. cbrt-prod 65.3%

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

      5. rem-cbrt-cube 72.4%

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

      6. pow2 72.4%

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

      7. pow-flip 72.6%

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

      8. metadata-eval 72.6%

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

   3. Applied egg-rr 72.6%

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

      1. pow1/3 71.7%

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

      2. pow-pow 40.4%

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

      3. metadata-eval 40.4%

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

   5. Applied egg-rr 40.4%

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

   if -2.1500000000000001e-36 < t < 4.99999999999999976e-45

   1. Initial program 35.3%

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

   3. Simplified 33.5%

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

      1. associate-*r* 41.0%

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

      2. *-un-lft-identity 41.0%

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

      3. times-frac 41.0%

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

   5. Applied egg-rr 41.0%

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

      1. /-rgt-identity 41.0%

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

      2. associate-*r/ 41.0%

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

      3. times-frac 41.9%

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

      4. *-commutative 41.9%

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

      5. associate-/r* 42.8%

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

   7. Simplified 42.8%

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

   8. Taylor expanded in t around 0 78.0%

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

      1. associate-*r/ 78.0%

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

      2. times-frac 80.0%

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

   10. Simplified 80.0%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.15 \cdot 10^{-36} \lor \neg \left(t \leq 5 \cdot 10^{-45}\right):\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(t \cdot {\ell}^{-0.6666666666666666}\right)}^{3}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{k}^{2}} \cdot \frac{\ell}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \end{array} \]

Alternative 8: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 + \frac{\frac{k}{t}}{\frac{t}{k}}\\ \mathbf{if}\;t \leq -2.1 \cdot 10^{+169}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \frac{k}{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3}}}}{t_1}\\ \mathbf{elif}\;t \leq -1.4 \cdot 10^{-49}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-45}:\\ \;\;\;\;\left(\frac{2}{{k}^{2}} \cdot \frac{\ell}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot \frac{{t}^{1.5} \cdot \frac{{t}^{1.5}}{\ell}}{\ell}\right)}}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (+ 2.0 (/ (/ k t) (/ t k)))))
   (if (<= t -2.1e+169)
     (/ (/ 2.0 (* (tan k) (/ k (pow (/ (pow (cbrt l) 2.0) t) 3.0)))) t_1)
     (if (<= t -1.4e-49)
       (/
        2.0
        (*
         (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))
         (* (tan k) (* (sin k) (* (/ (pow t 2.0) l) (/ t l))))))
       (if (<= t 1.3e-45)
         (* (* (/ 2.0 (pow k 2.0)) (/ l t)) (/ (/ l (sin k)) (tan k)))
         (/
          (/
           2.0
           (* (tan k) (* (sin k) (/ (* (pow t 1.5) (/ (pow t 1.5) l)) l))))
          t_1))))))

C

double code(double t, double l, double k) {
	double t_1 = 2.0 + ((k / t) / (t / k));
	double tmp;
	if (t <= -2.1e+169) {
		tmp = (2.0 / (tan(k) * (k / pow((pow(cbrt(l), 2.0) / t), 3.0)))) / t_1;
	} else if (t <= -1.4e-49) {
		tmp = 2.0 / ((1.0 + (1.0 + pow((k / t), 2.0))) * (tan(k) * (sin(k) * ((pow(t, 2.0) / l) * (t / l)))));
	} else if (t <= 1.3e-45) {
		tmp = ((2.0 / pow(k, 2.0)) * (l / t)) * ((l / sin(k)) / tan(k));
	} else {
		tmp = (2.0 / (tan(k) * (sin(k) * ((pow(t, 1.5) * (pow(t, 1.5) / l)) / l)))) / t_1;
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = 2.0 + ((k / t) / (t / k));
	double tmp;
	if (t <= -2.1e+169) {
		tmp = (2.0 / (Math.tan(k) * (k / Math.pow((Math.pow(Math.cbrt(l), 2.0) / t), 3.0)))) / t_1;
	} else if (t <= -1.4e-49) {
		tmp = 2.0 / ((1.0 + (1.0 + Math.pow((k / t), 2.0))) * (Math.tan(k) * (Math.sin(k) * ((Math.pow(t, 2.0) / l) * (t / l)))));
	} else if (t <= 1.3e-45) {
		tmp = ((2.0 / Math.pow(k, 2.0)) * (l / t)) * ((l / Math.sin(k)) / Math.tan(k));
	} else {
		tmp = (2.0 / (Math.tan(k) * (Math.sin(k) * ((Math.pow(t, 1.5) * (Math.pow(t, 1.5) / l)) / l)))) / t_1;
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(2.0 + Float64(Float64(k / t) / Float64(t / k)))
	tmp = 0.0
	if (t <= -2.1e+169)
		tmp = Float64(Float64(2.0 / Float64(tan(k) * Float64(k / (Float64((cbrt(l) ^ 2.0) / t) ^ 3.0)))) / t_1);
	elseif (t <= -1.4e-49)
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0))) * Float64(tan(k) * Float64(sin(k) * Float64(Float64((t ^ 2.0) / l) * Float64(t / l))))));
	elseif (t <= 1.3e-45)
		tmp = Float64(Float64(Float64(2.0 / (k ^ 2.0)) * Float64(l / t)) * Float64(Float64(l / sin(k)) / tan(k)));
	else
		tmp = Float64(Float64(2.0 / Float64(tan(k) * Float64(sin(k) * Float64(Float64((t ^ 1.5) * Float64((t ^ 1.5) / l)) / l)))) / t_1);
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(2.0 + N[(N[(k / t), $MachinePrecision] / N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.1e+169], N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(k / N[Power[N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t, -1.4e-49], 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[(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, 1.3e-45], N[(N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t, 1.5], $MachinePrecision] * N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -2.1000000000000001e169

   1. Initial program 62.3%

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

   3. Simplified 68.8%

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

      1. unpow2 68.8%

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

      2. clear-num 68.8%

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

      3. un-div-inv 68.8%

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

   5. Applied egg-rr 68.8%

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

   6. Taylor expanded in k around 0 62.3%

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

      1. associate-/l* 62.3%

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

   8. Simplified 62.3%

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

      1. add-cube-cbrt 62.3%

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

      2. pow2 62.3%

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

      3. cbrt-div 62.3%

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

      4. unpow2 62.3%

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

      5. cbrt-prod 62.3%

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

      6. unpow2 62.3%

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

      7. unpow3 62.3%

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

      8. add-cbrt-cube 62.3%

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

      9. cbrt-div 62.3%

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

      10. unpow2 62.3%

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

      11. cbrt-prod 68.8%

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

      12. unpow2 68.8%

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

      13. unpow3 68.8%

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

      14. add-cbrt-cube 82.6%

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

   10. Applied egg-rr 82.6%

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

       1. pow-plus 82.6%

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

       2. metadata-eval 82.6%

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

   12. Simplified 82.6%

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

   if -2.1000000000000001e169 < t < -1.39999999999999999e-49

   1. Initial program 66.1%

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

      1. unpow3 66.1%

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

      2. times-frac 94.4%

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

      3. pow2 94.4%

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

   3. Applied egg-rr 94.4%

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

   if -1.39999999999999999e-49 < t < 1.29999999999999993e-45

   1. Initial program 34.3%

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

   3. Simplified 32.5%

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

      1. associate-*r* 39.3%

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

      2. *-un-lft-identity 39.3%

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

      3. times-frac 39.3%

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

   5. Applied egg-rr 39.3%

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

      1. /-rgt-identity 39.3%

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

      2. associate-*r/ 39.3%

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

      3. times-frac 40.3%

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

      4. *-commutative 40.3%

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

      5. associate-/r* 41.2%

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

   7. Simplified 41.2%

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

   8. Taylor expanded in t around 0 77.4%

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

      1. associate-*r/ 77.4%

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

      2. times-frac 79.5%

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

   10. Simplified 79.5%

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

   if 1.29999999999999993e-45 < t

   1. Initial program 66.6%

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

   3. Simplified 72.2%

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

      1. unpow2 72.2%

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

      2. clear-num 72.2%

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

      3. un-div-inv 72.2%

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

   5. Applied egg-rr 72.2%

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

      1. add-sqr-sqrt 72.2%

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

      2. *-un-lft-identity 72.2%

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

      3. times-frac 72.2%

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

      4. sqrt-pow1 72.2%

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

      5. metadata-eval 72.2%

         \[\leadsto \frac{\frac{2}{\left(\frac{\frac{{t}^{\color{blue}{1.5}}}{1} \cdot \frac{\sqrt{{t}^{3}}}{\ell}}{\ell} \cdot \sin k\right) \cdot \tan k}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}} \]

      6. sqrt-pow1 81.9%

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

      7. metadata-eval 81.9%

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

   7. Applied egg-rr 81.9%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.1 \cdot 10^{+169}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \frac{k}{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3}}}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \mathbf{elif}\;t \leq -1.4 \cdot 10^{-49}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-45}:\\ \;\;\;\;\left(\frac{2}{{k}^{2}} \cdot \frac{\ell}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot \frac{{t}^{1.5} \cdot \frac{{t}^{1.5}}{\ell}}{\ell}\right)}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \end{array} \]

Alternative 9: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 + \frac{\frac{k}{t}}{\frac{t}{k}}\\ \mathbf{if}\;t \leq -5 \cdot 10^{+169}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \frac{k}{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3}}}}{t_1}\\ \mathbf{elif}\;t \leq -1.8 \cdot 10^{-49}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-46}:\\ \;\;\;\;\left(\frac{2}{{k}^{2}} \cdot \frac{\ell}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}\right)}}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (+ 2.0 (/ (/ k t) (/ t k)))))
   (if (<= t -5e+169)
     (/ (/ 2.0 (* (tan k) (/ k (pow (/ (pow (cbrt l) 2.0) t) 3.0)))) t_1)
     (if (<= t -1.8e-49)
       (/
        2.0
        (*
         (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))
         (* (tan k) (* (sin k) (* (/ (pow t 2.0) l) (/ t l))))))
       (if (<= t 5.5e-46)
         (* (* (/ 2.0 (pow k 2.0)) (/ l t)) (/ (/ l (sin k)) (tan k)))
         (/
          (/ 2.0 (* (tan k) (* (sin k) (/ (pow (/ t (cbrt l)) 3.0) l))))
          t_1))))))

C

double code(double t, double l, double k) {
	double t_1 = 2.0 + ((k / t) / (t / k));
	double tmp;
	if (t <= -5e+169) {
		tmp = (2.0 / (tan(k) * (k / pow((pow(cbrt(l), 2.0) / t), 3.0)))) / t_1;
	} else if (t <= -1.8e-49) {
		tmp = 2.0 / ((1.0 + (1.0 + pow((k / t), 2.0))) * (tan(k) * (sin(k) * ((pow(t, 2.0) / l) * (t / l)))));
	} else if (t <= 5.5e-46) {
		tmp = ((2.0 / pow(k, 2.0)) * (l / t)) * ((l / sin(k)) / tan(k));
	} else {
		tmp = (2.0 / (tan(k) * (sin(k) * (pow((t / cbrt(l)), 3.0) / l)))) / t_1;
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = 2.0 + ((k / t) / (t / k));
	double tmp;
	if (t <= -5e+169) {
		tmp = (2.0 / (Math.tan(k) * (k / Math.pow((Math.pow(Math.cbrt(l), 2.0) / t), 3.0)))) / t_1;
	} else if (t <= -1.8e-49) {
		tmp = 2.0 / ((1.0 + (1.0 + Math.pow((k / t), 2.0))) * (Math.tan(k) * (Math.sin(k) * ((Math.pow(t, 2.0) / l) * (t / l)))));
	} else if (t <= 5.5e-46) {
		tmp = ((2.0 / Math.pow(k, 2.0)) * (l / t)) * ((l / Math.sin(k)) / Math.tan(k));
	} else {
		tmp = (2.0 / (Math.tan(k) * (Math.sin(k) * (Math.pow((t / Math.cbrt(l)), 3.0) / l)))) / t_1;
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(2.0 + Float64(Float64(k / t) / Float64(t / k)))
	tmp = 0.0
	if (t <= -5e+169)
		tmp = Float64(Float64(2.0 / Float64(tan(k) * Float64(k / (Float64((cbrt(l) ^ 2.0) / t) ^ 3.0)))) / t_1);
	elseif (t <= -1.8e-49)
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0))) * Float64(tan(k) * Float64(sin(k) * Float64(Float64((t ^ 2.0) / l) * Float64(t / l))))));
	elseif (t <= 5.5e-46)
		tmp = Float64(Float64(Float64(2.0 / (k ^ 2.0)) * Float64(l / t)) * Float64(Float64(l / sin(k)) / tan(k)));
	else
		tmp = Float64(Float64(2.0 / Float64(tan(k) * Float64(sin(k) * Float64((Float64(t / cbrt(l)) ^ 3.0) / l)))) / t_1);
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(2.0 + N[(N[(k / t), $MachinePrecision] / N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5e+169], N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(k / N[Power[N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t, -1.8e-49], 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[(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, 5.5e-46], N[(N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[N[(t / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -5.00000000000000017e169

   1. Initial program 62.3%

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

   3. Simplified 68.8%

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

      1. unpow2 68.8%

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

      2. clear-num 68.8%

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

      3. un-div-inv 68.8%

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

   5. Applied egg-rr 68.8%

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

   6. Taylor expanded in k around 0 62.3%

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

      1. associate-/l* 62.3%

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

   8. Simplified 62.3%

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

      1. add-cube-cbrt 62.3%

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

      2. pow2 62.3%

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

      3. cbrt-div 62.3%

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

      4. unpow2 62.3%

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

      5. cbrt-prod 62.3%

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

      6. unpow2 62.3%

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

      7. unpow3 62.3%

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

      8. add-cbrt-cube 62.3%

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

      9. cbrt-div 62.3%

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

      10. unpow2 62.3%

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

      11. cbrt-prod 68.8%

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

      12. unpow2 68.8%

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

      13. unpow3 68.8%

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

      14. add-cbrt-cube 82.6%

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

   10. Applied egg-rr 82.6%

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

       1. pow-plus 82.6%

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

       2. metadata-eval 82.6%

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

   12. Simplified 82.6%

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

   if -5.00000000000000017e169 < t < -1.79999999999999985e-49

   1. Initial program 66.1%

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

      1. unpow3 66.1%

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

      2. times-frac 94.4%

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

      3. pow2 94.4%

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

   3. Applied egg-rr 94.4%

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

   if -1.79999999999999985e-49 < t < 5.49999999999999983e-46

   1. Initial program 34.3%

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

   3. Simplified 32.5%

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

      1. associate-*r* 39.3%

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

      2. *-un-lft-identity 39.3%

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

      3. times-frac 39.3%

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

   5. Applied egg-rr 39.3%

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

      1. /-rgt-identity 39.3%

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

      2. associate-*r/ 39.3%

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

      3. times-frac 40.3%

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

      4. *-commutative 40.3%

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

      5. associate-/r* 41.2%

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

   7. Simplified 41.2%

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

   8. Taylor expanded in t around 0 77.4%

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

      1. associate-*r/ 77.4%

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

      2. times-frac 79.5%

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

   10. Simplified 79.5%

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

   if 5.49999999999999983e-46 < t

   1. Initial program 66.6%

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

   3. Simplified 72.2%

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

      1. unpow2 72.2%

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

      2. clear-num 72.2%

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

      3. un-div-inv 72.2%

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

   5. Applied egg-rr 72.2%

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

      1. add-cube-cbrt 72.1%

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

      2. pow3 72.1%

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

      3. cbrt-div 72.0%

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

      4. rem-cbrt-cube 81.7%

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

   7. Applied egg-rr 81.7%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+169}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \frac{k}{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3}}}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \mathbf{elif}\;t \leq -1.8 \cdot 10^{-49}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-46}:\\ \;\;\;\;\left(\frac{2}{{k}^{2}} \cdot \frac{\ell}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}\right)}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \end{array} \]

Alternative 10: 70.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 7.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \frac{k}{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3}}}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{\tan k} \cdot \left(2 \cdot \frac{\ell}{t \cdot {k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 7.2e-12)
   (/
    (/ 2.0 (* (tan k) (/ k (pow (/ (pow (cbrt l) 2.0) t) 3.0))))
    (+ 2.0 (/ (/ k t) (/ t k))))
   (* (/ (/ l (sin k)) (tan k)) (* 2.0 (/ l (* t (pow k 2.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 7.2e-12) {
		tmp = (2.0 / (tan(k) * (k / pow((pow(cbrt(l), 2.0) / t), 3.0)))) / (2.0 + ((k / t) / (t / k)));
	} else {
		tmp = ((l / sin(k)) / tan(k)) * (2.0 * (l / (t * pow(k, 2.0))));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 7.2e-12) {
		tmp = (2.0 / (Math.tan(k) * (k / Math.pow((Math.pow(Math.cbrt(l), 2.0) / t), 3.0)))) / (2.0 + ((k / t) / (t / k)));
	} else {
		tmp = ((l / Math.sin(k)) / Math.tan(k)) * (2.0 * (l / (t * Math.pow(k, 2.0))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 7.2e-12)
		tmp = Float64(Float64(2.0 / Float64(tan(k) * Float64(k / (Float64((cbrt(l) ^ 2.0) / t) ^ 3.0)))) / Float64(2.0 + Float64(Float64(k / t) / Float64(t / k))));
	else
		tmp = Float64(Float64(Float64(l / sin(k)) / tan(k)) * Float64(2.0 * Float64(l / Float64(t * (k ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 7.2e-12], N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(k / N[Power[N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(k / t), $MachinePrecision] / N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(l / N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.2e-12

   1. Initial program 54.2%

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

   3. Simplified 61.3%

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

      1. unpow2 61.3%

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

      2. clear-num 61.4%

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

      3. un-div-inv 61.4%

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

   5. Applied egg-rr 61.4%

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

   6. Taylor expanded in k around 0 52.8%

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

      1. associate-/l* 52.6%

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

   8. Simplified 52.6%

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

      1. add-cube-cbrt 52.6%

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

      2. pow2 52.6%

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

      3. cbrt-div 52.6%

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

      4. unpow2 52.6%

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

      5. cbrt-prod 52.6%

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

      6. unpow2 52.6%

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

      7. unpow3 52.5%

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

      8. add-cbrt-cube 52.6%

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

      9. cbrt-div 52.6%

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

      10. unpow2 52.6%

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

      11. cbrt-prod 59.7%

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

      12. unpow2 59.7%

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

      13. unpow3 59.7%

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

      14. add-cbrt-cube 72.1%

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

   10. Applied egg-rr 72.1%

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

       1. pow-plus 72.1%

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

       2. metadata-eval 72.1%

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

   12. Simplified 72.1%

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

   if 7.2e-12 < k

   1. Initial program 48.7%

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

   3. Simplified 48.6%

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

      1. associate-*r* 56.1%

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

      2. *-un-lft-identity 56.1%

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

      3. times-frac 56.2%

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

   5. Applied egg-rr 56.2%

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

      1. /-rgt-identity 56.2%

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

      2. associate-*r/ 56.1%

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

      3. times-frac 56.1%

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

      4. *-commutative 56.1%

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

      5. associate-/r* 56.2%

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

   7. Simplified 56.2%

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

   8. Taylor expanded in t around 0 73.1%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \frac{k}{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3}}}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{\tan k} \cdot \left(2 \cdot \frac{\ell}{t \cdot {k}^{2}}\right)\\ \end{array} \]

Alternative 11: 76.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{-49} \lor \neg \left(t \leq 1.1 \cdot 10^{-44}\right):\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}\right)}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{k}^{2}} \cdot \frac{\ell}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -5.8e-49) (not (<= t 1.1e-44)))
   (/
    (/ 2.0 (* (tan k) (* (sin k) (/ (/ (pow t 3.0) l) l))))
    (+ 2.0 (/ (/ k t) (/ t k))))
   (* (* (/ 2.0 (pow k 2.0)) (/ l t)) (/ (/ l (sin k)) (tan k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -5.8e-49) || !(t <= 1.1e-44)) {
		tmp = (2.0 / (tan(k) * (sin(k) * ((pow(t, 3.0) / l) / l)))) / (2.0 + ((k / t) / (t / k)));
	} else {
		tmp = ((2.0 / pow(k, 2.0)) * (l / t)) * ((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 <= (-5.8d-49)) .or. (.not. (t <= 1.1d-44))) then
        tmp = (2.0d0 / (tan(k) * (sin(k) * (((t ** 3.0d0) / l) / l)))) / (2.0d0 + ((k / t) / (t / k)))
    else
        tmp = ((2.0d0 / (k ** 2.0d0)) * (l / t)) * ((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 <= -5.8e-49) || !(t <= 1.1e-44)) {
		tmp = (2.0 / (Math.tan(k) * (Math.sin(k) * ((Math.pow(t, 3.0) / l) / l)))) / (2.0 + ((k / t) / (t / k)));
	} else {
		tmp = ((2.0 / Math.pow(k, 2.0)) * (l / t)) * ((l / Math.sin(k)) / Math.tan(k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -5.8e-49) or not (t <= 1.1e-44):
		tmp = (2.0 / (math.tan(k) * (math.sin(k) * ((math.pow(t, 3.0) / l) / l)))) / (2.0 + ((k / t) / (t / k)))
	else:
		tmp = ((2.0 / math.pow(k, 2.0)) * (l / t)) * ((l / math.sin(k)) / math.tan(k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -5.8e-49) || !(t <= 1.1e-44))
		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))));
	else
		tmp = Float64(Float64(Float64(2.0 / (k ^ 2.0)) * Float64(l / t)) * Float64(Float64(l / sin(k)) / tan(k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -5.8e-49) || ~((t <= 1.1e-44)))
		tmp = (2.0 / (tan(k) * (sin(k) * (((t ^ 3.0) / l) / l)))) / (2.0 + ((k / t) / (t / k)));
	else
		tmp = ((2.0 / (k ^ 2.0)) * (l / t)) * ((l / sin(k)) / tan(k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -5.8e-49], N[Not[LessEqual[t, 1.1e-44]], $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], N[(N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.8e-49 or 1.10000000000000006e-44 < t

   1. Initial program 65.5%

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

   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 73.1%

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

      2. clear-num 73.1%

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

      3. un-div-inv 73.1%

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

   5. Applied egg-rr 73.1%

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

   if -5.8e-49 < t < 1.10000000000000006e-44

   1. Initial program 34.3%

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

   3. Simplified 32.5%

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

      1. associate-*r* 39.3%

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

      2. *-un-lft-identity 39.3%

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

      3. times-frac 39.3%

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

   5. Applied egg-rr 39.3%

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

      1. /-rgt-identity 39.3%

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

      2. associate-*r/ 39.3%

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

      3. times-frac 40.3%

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

      4. *-commutative 40.3%

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

      5. associate-/r* 41.2%

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

   7. Simplified 41.2%

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

   8. Taylor expanded in t around 0 77.4%

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

      1. associate-*r/ 77.4%

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

      2. times-frac 79.5%

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

   10. Simplified 79.5%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{-49} \lor \neg \left(t \leq 1.1 \cdot 10^{-44}\right):\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}\right)}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{k}^{2}} \cdot \frac{\ell}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \end{array} \]

Alternative 12: 77.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{-57} \lor \neg \left(t \leq 7.5 \cdot 10^{-45}\right):\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \frac{\sin k \cdot \frac{{t}^{3}}{\ell}}{\ell}}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{k}^{2}} \cdot \frac{\ell}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -7e-57) (not (<= t 7.5e-45)))
   (/
    (/ 2.0 (* (tan k) (/ (* (sin k) (/ (pow t 3.0) l)) l)))
    (+ 2.0 (/ (/ k t) (/ t k))))
   (* (* (/ 2.0 (pow k 2.0)) (/ l t)) (/ (/ l (sin k)) (tan k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -7e-57) || !(t <= 7.5e-45)) {
		tmp = (2.0 / (tan(k) * ((sin(k) * (pow(t, 3.0) / l)) / l))) / (2.0 + ((k / t) / (t / k)));
	} else {
		tmp = ((2.0 / pow(k, 2.0)) * (l / t)) * ((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 <= (-7d-57)) .or. (.not. (t <= 7.5d-45))) then
        tmp = (2.0d0 / (tan(k) * ((sin(k) * ((t ** 3.0d0) / l)) / l))) / (2.0d0 + ((k / t) / (t / k)))
    else
        tmp = ((2.0d0 / (k ** 2.0d0)) * (l / t)) * ((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 <= -7e-57) || !(t <= 7.5e-45)) {
		tmp = (2.0 / (Math.tan(k) * ((Math.sin(k) * (Math.pow(t, 3.0) / l)) / l))) / (2.0 + ((k / t) / (t / k)));
	} else {
		tmp = ((2.0 / Math.pow(k, 2.0)) * (l / t)) * ((l / Math.sin(k)) / Math.tan(k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -7e-57) or not (t <= 7.5e-45):
		tmp = (2.0 / (math.tan(k) * ((math.sin(k) * (math.pow(t, 3.0) / l)) / l))) / (2.0 + ((k / t) / (t / k)))
	else:
		tmp = ((2.0 / math.pow(k, 2.0)) * (l / t)) * ((l / math.sin(k)) / math.tan(k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -7e-57) || !(t <= 7.5e-45))
		tmp = Float64(Float64(2.0 / Float64(tan(k) * Float64(Float64(sin(k) * Float64((t ^ 3.0) / l)) / l))) / Float64(2.0 + Float64(Float64(k / t) / Float64(t / k))));
	else
		tmp = Float64(Float64(Float64(2.0 / (k ^ 2.0)) * Float64(l / t)) * Float64(Float64(l / sin(k)) / tan(k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -7e-57) || ~((t <= 7.5e-45)))
		tmp = (2.0 / (tan(k) * ((sin(k) * ((t ^ 3.0) / l)) / l))) / (2.0 + ((k / t) / (t / k)));
	else
		tmp = ((2.0 / (k ^ 2.0)) * (l / t)) * ((l / sin(k)) / tan(k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -7e-57], N[Not[LessEqual[t, 7.5e-45]], $MachinePrecision]], N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(k / t), $MachinePrecision] / N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -6.99999999999999983e-57 or 7.5000000000000006e-45 < t

   1. Initial program 64.7%

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

   3. Simplified 72.2%

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

      1. unpow2 72.2%

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

      2. clear-num 72.2%

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

      3. un-div-inv 72.2%

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

   5. Applied egg-rr 72.2%

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

      1. associate-*l/ 74.1%

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

   7. Applied egg-rr 74.1%

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

   if -6.99999999999999983e-57 < t < 7.5000000000000006e-45

   1. Initial program 35.0%

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

   3. Simplified 33.2%

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

      1. associate-*r* 40.1%

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

      2. *-un-lft-identity 40.1%

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

      3. times-frac 40.1%

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

   5. Applied egg-rr 40.1%

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

      1. /-rgt-identity 40.1%

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

      2. associate-*r/ 40.1%

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

      3. times-frac 41.0%

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

      4. *-commutative 41.0%

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

      5. associate-/r* 41.0%

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

   7. Simplified 41.0%

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

   8. Taylor expanded in t around 0 78.8%

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

      1. associate-*r/ 78.8%

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

      2. times-frac 80.9%

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

   10. Simplified 80.9%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{-57} \lor \neg \left(t \leq 7.5 \cdot 10^{-45}\right):\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \frac{\sin k \cdot \frac{{t}^{3}}{\ell}}{\ell}}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{k}^{2}} \cdot \frac{\ell}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \end{array} \]

Alternative 13: 69.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{if}\;t \leq -8 \cdot 10^{-34} \lor \neg \left(t \leq 7.2 \cdot 10^{+133}\right):\\ \;\;\;\;t_1 \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \frac{\ell}{t \cdot {k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (/ l (sin k)) (tan k))))
   (if (or (<= t -8e-34) (not (<= t 7.2e+133)))
     (* t_1 (/ l (pow t 3.0)))
     (* t_1 (* 2.0 (/ l (* t (pow k 2.0))))))))

C

double code(double t, double l, double k) {
	double t_1 = (l / sin(k)) / tan(k);
	double tmp;
	if ((t <= -8e-34) || !(t <= 7.2e+133)) {
		tmp = t_1 * (l / pow(t, 3.0));
	} else {
		tmp = t_1 * (2.0 * (l / (t * pow(k, 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) :: tmp
    t_1 = (l / sin(k)) / tan(k)
    if ((t <= (-8d-34)) .or. (.not. (t <= 7.2d+133))) then
        tmp = t_1 * (l / (t ** 3.0d0))
    else
        tmp = t_1 * (2.0d0 * (l / (t * (k ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = (l / Math.sin(k)) / Math.tan(k);
	double tmp;
	if ((t <= -8e-34) || !(t <= 7.2e+133)) {
		tmp = t_1 * (l / Math.pow(t, 3.0));
	} else {
		tmp = t_1 * (2.0 * (l / (t * Math.pow(k, 2.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (l / math.sin(k)) / math.tan(k)
	tmp = 0
	if (t <= -8e-34) or not (t <= 7.2e+133):
		tmp = t_1 * (l / math.pow(t, 3.0))
	else:
		tmp = t_1 * (2.0 * (l / (t * math.pow(k, 2.0))))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(l / sin(k)) / tan(k))
	tmp = 0.0
	if ((t <= -8e-34) || !(t <= 7.2e+133))
		tmp = Float64(t_1 * Float64(l / (t ^ 3.0)));
	else
		tmp = Float64(t_1 * Float64(2.0 * Float64(l / Float64(t * (k ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (l / sin(k)) / tan(k);
	tmp = 0.0;
	if ((t <= -8e-34) || ~((t <= 7.2e+133)))
		tmp = t_1 * (l / (t ^ 3.0));
	else
		tmp = t_1 * (2.0 * (l / (t * (k ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t, -8e-34], N[Not[LessEqual[t, 7.2e+133]], $MachinePrecision]], N[(t$95$1 * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(2.0 * N[(l / N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -7.99999999999999942e-34 or 7.19999999999999956e133 < 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 53.4%

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

      1. associate-*r* 61.5%

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

      2. *-un-lft-identity 61.5%

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

      3. times-frac 62.0%

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

   5. Applied egg-rr 62.0%

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

      1. /-rgt-identity 62.0%

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

      2. associate-*r/ 61.5%

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

      3. times-frac 62.0%

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

      4. *-commutative 62.0%

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

      5. associate-/r* 67.9%

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

   7. Simplified 67.9%

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

   8. Taylor expanded in t around inf 64.9%

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

   if -7.99999999999999942e-34 < t < 7.19999999999999956e133

   1. Initial program 45.5%

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

   3. Simplified 42.8%

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

      1. associate-*r* 48.6%

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

      2. *-un-lft-identity 48.6%

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

      3. times-frac 49.3%

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

   5. Applied egg-rr 49.3%

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

      1. /-rgt-identity 49.3%

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

      2. associate-*r/ 48.6%

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

      3. times-frac 50.0%

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

      4. *-commutative 50.0%

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

      5. associate-/r* 51.3%

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

   7. Simplified 51.3%

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

   8. Taylor expanded in t around 0 77.5%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{-34} \lor \neg \left(t \leq 7.2 \cdot 10^{+133}\right):\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{\tan k} \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{\tan k} \cdot \left(2 \cdot \frac{\ell}{t \cdot {k}^{2}}\right)\\ \end{array} \]

Alternative 14: 70.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{if}\;t \leq -2.5 \cdot 10^{-36} \lor \neg \left(t \leq 2.8 \cdot 10^{+144}\right):\\ \;\;\;\;t_1 \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{k}^{2}} \cdot \frac{\ell}{t}\right) \cdot t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (/ l (sin k)) (tan k))))
   (if (or (<= t -2.5e-36) (not (<= t 2.8e+144)))
     (* t_1 (/ l (pow t 3.0)))
     (* (* (/ 2.0 (pow k 2.0)) (/ l t)) t_1))))

C

double code(double t, double l, double k) {
	double t_1 = (l / sin(k)) / tan(k);
	double tmp;
	if ((t <= -2.5e-36) || !(t <= 2.8e+144)) {
		tmp = t_1 * (l / pow(t, 3.0));
	} else {
		tmp = ((2.0 / pow(k, 2.0)) * (l / t)) * 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 / sin(k)) / tan(k)
    if ((t <= (-2.5d-36)) .or. (.not. (t <= 2.8d+144))) then
        tmp = t_1 * (l / (t ** 3.0d0))
    else
        tmp = ((2.0d0 / (k ** 2.0d0)) * (l / t)) * t_1
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = (l / Math.sin(k)) / Math.tan(k);
	double tmp;
	if ((t <= -2.5e-36) || !(t <= 2.8e+144)) {
		tmp = t_1 * (l / Math.pow(t, 3.0));
	} else {
		tmp = ((2.0 / Math.pow(k, 2.0)) * (l / t)) * t_1;
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (l / math.sin(k)) / math.tan(k)
	tmp = 0
	if (t <= -2.5e-36) or not (t <= 2.8e+144):
		tmp = t_1 * (l / math.pow(t, 3.0))
	else:
		tmp = ((2.0 / math.pow(k, 2.0)) * (l / t)) * t_1
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(l / sin(k)) / tan(k))
	tmp = 0.0
	if ((t <= -2.5e-36) || !(t <= 2.8e+144))
		tmp = Float64(t_1 * Float64(l / (t ^ 3.0)));
	else
		tmp = Float64(Float64(Float64(2.0 / (k ^ 2.0)) * Float64(l / t)) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (l / sin(k)) / tan(k);
	tmp = 0.0;
	if ((t <= -2.5e-36) || ~((t <= 2.8e+144)))
		tmp = t_1 * (l / (t ^ 3.0));
	else
		tmp = ((2.0 / (k ^ 2.0)) * (l / t)) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t, -2.5e-36], N[Not[LessEqual[t, 2.8e+144]], $MachinePrecision]], N[(t$95$1 * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.50000000000000002e-36 or 2.80000000000000007e144 < t

   1. Initial program 61.6%

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

   3. Simplified 53.4%

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

      1. associate-*r* 61.7%

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

      2. *-un-lft-identity 61.7%

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

      3. times-frac 62.2%

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

   5. Applied egg-rr 62.2%

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

      1. /-rgt-identity 62.2%

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

      2. associate-*r/ 61.7%

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

      3. times-frac 62.2%

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

      4. *-commutative 62.2%

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

      5. associate-/r* 68.2%

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

   7. Simplified 68.2%

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

   8. Taylor expanded in t around inf 65.2%

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

   if -2.50000000000000002e-36 < t < 2.80000000000000007e144

   1. Initial program 45.5%

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

   3. Simplified 42.9%

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

      1. associate-*r* 48.6%

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

      2. *-un-lft-identity 48.6%

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

      3. times-frac 49.3%

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

   5. Applied egg-rr 49.3%

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

      1. /-rgt-identity 49.3%

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

      2. associate-*r/ 48.6%

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

      3. times-frac 50.0%

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

      4. *-commutative 50.0%

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

      5. associate-/r* 51.3%

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

   7. Simplified 51.3%

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

   8. Taylor expanded in t around 0 76.5%

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

      1. associate-*r/ 76.5%

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

      2. times-frac 78.7%

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

   10. Simplified 78.7%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{-36} \lor \neg \left(t \leq 2.8 \cdot 10^{+144}\right):\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{\tan k} \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{k}^{2}} \cdot \frac{\ell}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \end{array} \]

Alternative 15: 59.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 7.2 \cdot 10^{+189}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{\tan k} \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 7.2e+189)
   (* (/ (/ l (sin k)) (tan k)) (/ l (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 <= 7.2e+189) {
		tmp = ((l / sin(k)) / tan(k)) * (l / 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 <= 7.2d+189) then
        tmp = ((l / sin(k)) / tan(k)) * (l / (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 <= 7.2e+189) {
		tmp = ((l / Math.sin(k)) / Math.tan(k)) * (l / 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 <= 7.2e+189:
		tmp = ((l / math.sin(k)) / math.tan(k)) * (l / 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 <= 7.2e+189)
		tmp = Float64(Float64(Float64(l / sin(k)) / tan(k)) * Float64(l / (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 <= 7.2e+189)
		tmp = ((l / sin(k)) / tan(k)) * (l / (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, 7.2e+189], N[(N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[t, 3.0], $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 < 7.20000000000000017e189

   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 46.2%

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

      1. associate-*r* 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. *-un-lft-identity 53.5%

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

      3. times-frac 54.2%

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

   5. Applied egg-rr 54.2%

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

      1. /-rgt-identity 54.2%

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

      2. associate-*r/ 53.5%

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

      3. times-frac 54.6%

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

      4. *-commutative 54.6%

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

      5. associate-/r* 58.5%

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

   7. Simplified 58.5%

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

   8. Taylor expanded in t around inf 59.0%

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

   if 7.20000000000000017e189 < k

   1. Initial program 58.6%

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

   3. Simplified 58.6%

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

   4. Taylor expanded in t around 0 72.7%

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

      1. times-frac 72.7%

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

   6. Simplified 72.7%

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

   7. Taylor expanded in k around 0 72.7%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.2 \cdot 10^{+189}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{\tan k} \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}}\\ \end{array} \]

Alternative 16: 54.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{+220} \lor \neg \left(t \leq 8.8 \cdot 10^{+136}\right):\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \frac{2}{{t}^{3}}}{2 \cdot {k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{\frac{{k}^{4}}{\frac{{\ell}^{2}}{t}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -3.3e+220) (not (<= t 8.8e+136)))
   (/ (* (* l l) (/ 2.0 (pow t 3.0))) (* 2.0 (pow k 2.0)))
   (* 2.0 (/ 1.0 (/ (pow k 4.0) (/ (pow l 2.0) t))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -3.3e+220) || !(t <= 8.8e+136)) {
		tmp = ((l * l) * (2.0 / pow(t, 3.0))) / (2.0 * pow(k, 2.0));
	} else {
		tmp = 2.0 * (1.0 / (pow(k, 4.0) / (pow(l, 2.0) / t)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-3.3d+220)) .or. (.not. (t <= 8.8d+136))) then
        tmp = ((l * l) * (2.0d0 / (t ** 3.0d0))) / (2.0d0 * (k ** 2.0d0))
    else
        tmp = 2.0d0 * (1.0d0 / ((k ** 4.0d0) / ((l ** 2.0d0) / t)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -3.3e+220) || !(t <= 8.8e+136)) {
		tmp = ((l * l) * (2.0 / Math.pow(t, 3.0))) / (2.0 * Math.pow(k, 2.0));
	} else {
		tmp = 2.0 * (1.0 / (Math.pow(k, 4.0) / (Math.pow(l, 2.0) / t)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -3.3e+220) or not (t <= 8.8e+136):
		tmp = ((l * l) * (2.0 / math.pow(t, 3.0))) / (2.0 * math.pow(k, 2.0))
	else:
		tmp = 2.0 * (1.0 / (math.pow(k, 4.0) / (math.pow(l, 2.0) / t)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -3.3e+220) || !(t <= 8.8e+136))
		tmp = Float64(Float64(Float64(l * l) * Float64(2.0 / (t ^ 3.0))) / Float64(2.0 * (k ^ 2.0)));
	else
		tmp = Float64(2.0 * Float64(1.0 / Float64((k ^ 4.0) / Float64((l ^ 2.0) / t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -3.3e+220) || ~((t <= 8.8e+136)))
		tmp = ((l * l) * (2.0 / (t ^ 3.0))) / (2.0 * (k ^ 2.0));
	else
		tmp = 2.0 * (1.0 / ((k ^ 4.0) / ((l ^ 2.0) / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -3.3e+220], N[Not[LessEqual[t, 8.8e+136]], $MachinePrecision]], N[(N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(1.0 / N[(N[Power[k, 4.0], $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.30000000000000021e220 or 8.7999999999999998e136 < t

   1. Initial program 64.6%

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

   3. Simplified 53.0%

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

   4. Taylor expanded in k around 0 53.0%

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

   if -3.30000000000000021e220 < t < 8.7999999999999998e136

   1. Initial program 48.3%

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

   3. Simplified 45.6%

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

   4. Taylor expanded in t around 0 64.6%

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

      1. times-frac 64.6%

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

   6. Simplified 64.6%

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

   7. Taylor expanded in k around 0 53.0%

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

      1. *-commutative 53.0%

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

      2. associate-/r* 54.8%

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

   9. Simplified 54.8%

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

       1. clear-num 54.8%

          \[\leadsto 2 \cdot \color{blue}{\frac{1}{\frac{{k}^{4}}{\frac{{\ell}^{2}}{t}}}} \]

       2. inv-pow 54.8%

          \[\leadsto 2 \cdot \color{blue}{{\left(\frac{{k}^{4}}{\frac{{\ell}^{2}}{t}}\right)}^{-1}} \]

   11. Applied egg-rr 54.8%

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

       1. unpow-1 54.8%

          \[\leadsto 2 \cdot \color{blue}{\frac{1}{\frac{{k}^{4}}{\frac{{\ell}^{2}}{t}}}} \]

   13. Simplified 54.8%

       \[\leadsto 2 \cdot \color{blue}{\frac{1}{\frac{{k}^{4}}{\frac{{\ell}^{2}}{t}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{+220} \lor \neg \left(t \leq 8.8 \cdot 10^{+136}\right):\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \frac{2}{{t}^{3}}}{2 \cdot {k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{\frac{{k}^{4}}{\frac{{\ell}^{2}}{t}}}\\ \end{array} \]

Alternative 17: 51.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \frac{1}{\frac{{k}^{4}}{\frac{{\ell}^{2}}{t}}} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* 2.0 (/ 1.0 (/ (pow k 4.0) (/ (pow l 2.0) t)))))

C

double code(double t, double l, double k) {
	return 2.0 * (1.0 / (pow(k, 4.0) / (pow(l, 2.0) / t)));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * (1.0d0 / ((k ** 4.0d0) / ((l ** 2.0d0) / t)))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * (1.0 / (Math.pow(k, 4.0) / (Math.pow(l, 2.0) / t)));
}

Python

def code(t, l, k):
	return 2.0 * (1.0 / (math.pow(k, 4.0) / (math.pow(l, 2.0) / t)))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(1.0 / Float64((k ^ 4.0) / Float64((l ^ 2.0) / t))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * (1.0 / ((k ^ 4.0) / ((l ^ 2.0) / t)));
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(1.0 / N[(N[Power[k, 4.0], $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.7%

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

3. Simplified 47.6%

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

4. Taylor expanded in t around 0 57.4%

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

   1. times-frac 58.2%

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

6. Simplified 58.2%

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

7. Taylor expanded in k around 0 48.6%

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

   1. *-commutative 48.6%

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

   2. associate-/r* 49.9%

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

9. Simplified 49.9%

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

    1. clear-num 49.9%

       \[\leadsto 2 \cdot \color{blue}{\frac{1}{\frac{{k}^{4}}{\frac{{\ell}^{2}}{t}}}} \]

    2. inv-pow 49.9%

       \[\leadsto 2 \cdot \color{blue}{{\left(\frac{{k}^{4}}{\frac{{\ell}^{2}}{t}}\right)}^{-1}} \]

11. Applied egg-rr 49.9%

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

    1. unpow-1 49.9%

       \[\leadsto 2 \cdot \color{blue}{\frac{1}{\frac{{k}^{4}}{\frac{{\ell}^{2}}{t}}}} \]

13. Simplified 49.9%

    \[\leadsto 2 \cdot \color{blue}{\frac{1}{\frac{{k}^{4}}{\frac{{\ell}^{2}}{t}}}} \]

14. Final simplification 49.9%

    \[\leadsto 2 \cdot \frac{1}{\frac{{k}^{4}}{\frac{{\ell}^{2}}{t}}} \]

Alternative 18: 51.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\frac{{\ell}^{2}}{t} \cdot {k}^{-4}\right) \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* 2.0 (* (/ (pow l 2.0) t) (pow k -4.0))))

C

double code(double t, double l, double k) {
	return 2.0 * ((pow(l, 2.0) / t) * pow(k, -4.0));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * (((l ** 2.0d0) / t) * (k ** (-4.0d0)))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * ((Math.pow(l, 2.0) / t) * Math.pow(k, -4.0));
}

Python

def code(t, l, k):
	return 2.0 * ((math.pow(l, 2.0) / t) * math.pow(k, -4.0))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64((l ^ 2.0) / t) * (k ^ -4.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * (((l ^ 2.0) / t) * (k ^ -4.0));
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / t), $MachinePrecision] * N[Power[k, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.7%

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

3. Simplified 47.6%

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

4. Taylor expanded in t around 0 57.4%

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

   1. times-frac 58.2%

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

6. Simplified 58.2%

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

7. Taylor expanded in k around 0 48.6%

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

   1. *-commutative 48.6%

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

   2. associate-/r* 49.9%

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

9. Simplified 49.9%

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

    1. expm1-log1p-u 37.6%

       \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}\right)\right)} \]

    2. expm1-udef 38.1%

       \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}\right)} - 1\right)} \]

    3. div-inv 38.1%

       \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{{\ell}^{2}}{t} \cdot \frac{1}{{k}^{4}}}\right)} - 1\right) \]

    4. pow-flip 38.1%

       \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1\right) \]

    5. metadata-eval 38.1%

       \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{\color{blue}{-4}}\right)} - 1\right) \]

11. Applied egg-rr 38.1%

    \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-4}\right)} - 1\right)} \]
12. Step-by-step derivation

    1. expm1-def 37.6%

       \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-4}\right)\right)} \]

    2. expm1-log1p 49.9%

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

13. Simplified 49.9%

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

14. Final simplification 49.9%

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

Reproduce

herbie shell --seed 2023299 
(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))))