Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.7% → 87.5%

Time: 36.1s

Alternatives: 18

Speedup: 3.8×

• 27.4% 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: 54.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{-18} \lor \neg \left(t \leq 8 \cdot 10^{-101}\right):\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{\sqrt[3]{\ell}}{\frac{\sqrt[3]{\sin k}}{\sqrt[3]{\ell}}}}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \left(\ell \cdot \cos k\right)}{{\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -9.5e-18) (not (<= t 8e-101)))
   (/
    (pow
     (/ (cbrt (/ 2.0 (tan k))) (/ t (/ (cbrt l) (/ (cbrt (sin k)) (cbrt l)))))
     3.0)
    (+ 2.0 (pow (/ k t) 2.0)))
   (* (/ 2.0 (* k (* t k))) (/ (* l (* l (cos k))) (pow (sin k) 2.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -9.5e-18) || !(t <= 8e-101)) {
		tmp = pow((cbrt((2.0 / tan(k))) / (t / (cbrt(l) / (cbrt(sin(k)) / cbrt(l))))), 3.0) / (2.0 + pow((k / t), 2.0));
	} else {
		tmp = (2.0 / (k * (t * k))) * ((l * (l * cos(k))) / pow(sin(k), 2.0));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -9.5e-18) || !(t <= 8e-101)) {
		tmp = Math.pow((Math.cbrt((2.0 / Math.tan(k))) / (t / (Math.cbrt(l) / (Math.cbrt(Math.sin(k)) / Math.cbrt(l))))), 3.0) / (2.0 + Math.pow((k / t), 2.0));
	} else {
		tmp = (2.0 / (k * (t * k))) * ((l * (l * Math.cos(k))) / Math.pow(Math.sin(k), 2.0));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -9.5e-18) || !(t <= 8e-101))
		tmp = Float64((Float64(cbrt(Float64(2.0 / tan(k))) / Float64(t / Float64(cbrt(l) / Float64(cbrt(sin(k)) / cbrt(l))))) ^ 3.0) / Float64(2.0 + (Float64(k / t) ^ 2.0)));
	else
		tmp = Float64(Float64(2.0 / Float64(k * Float64(t * k))) * Float64(Float64(l * Float64(l * cos(k))) / (sin(k) ^ 2.0)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -9.5e-18], N[Not[LessEqual[t, 8e-101]], $MachinePrecision]], N[(N[Power[N[(N[Power[N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[(t / N[(N[Power[l, 1/3], $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -9.5000000000000003e-18 or 8.00000000000000041e-101 < t

   1. Initial program 62.7%

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

      1. associate-/r* 62.8%

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

      2. associate-*l* 58.5%

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

      3. sqr-neg 58.5%

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

      4. associate-*l* 62.8%

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

      5. *-commutative 62.8%

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

      6. sqr-neg 62.8%

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

      7. associate-/r* 62.8%

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

   3. Simplified 62.7%

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

      1. associate-/l/ 62.7%

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

      2. associate-/r/ 62.8%

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

      3. add-cube-cbrt 62.8%

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

      4. pow3 62.8%

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

   5. Applied egg-rr 77.1%

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

      1. cbrt-div 92.4%

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

   7. Applied egg-rr 92.4%

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

      1. cbrt-div 95.0%

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

   9. Applied egg-rr 95.0%

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

   if -9.5000000000000003e-18 < t < 8.00000000000000041e-101

   1. Initial program 35.5%

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

      1. associate-/r* 35.5%

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

      2. associate-*l* 35.5%

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

      3. sqr-neg 35.5%

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

      4. associate-*l* 35.5%

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

      5. *-commutative 35.5%

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

      6. sqr-neg 35.5%

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

      7. associate-/r* 35.5%

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

   3. Simplified 35.5%

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

      1. associate-/l/ 35.5%

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

      2. associate-/r/ 35.5%

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

      3. add-cube-cbrt 35.4%

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

      4. pow3 35.4%

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

   5. Applied egg-rr 58.6%

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

      1. cbrt-div 54.6%

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

   7. Applied egg-rr 54.6%

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

   8. Taylor expanded in k around inf 76.7%

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

      1. associate-*r/ 76.7%

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

      2. associate-*r* 76.7%

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

      3. times-frac 77.0%

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

      4. unpow2 77.0%

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

      5. associate-*l* 81.9%

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

      6. unpow2 81.9%

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

      7. associate-*l* 81.9%

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

   10. Simplified 81.9%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{-18} \lor \neg \left(t \leq 8 \cdot 10^{-101}\right):\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{\sqrt[3]{\ell}}{\frac{\sqrt[3]{\sin k}}{\sqrt[3]{\ell}}}}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \left(\ell \cdot \cos k\right)}{{\sin k}^{2}}\\ \end{array} \]

Alternative 2: 75.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;\frac{2}{\left(1 + \left(t_1 + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)} \leq 2 \cdot 10^{+258}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t}\right)}^{3}}{\frac{2 + t_1}{\frac{\ell}{\frac{\sin k}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \left(\ell \cdot \cos k\right)}{{\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<=
        (/
         2.0
         (*
          (+ 1.0 (+ t_1 1.0))
          (* (tan k) (* (sin k) (/ (pow t 3.0) (* l l))))))
        2e+258)
     (/
      (pow (/ (cbrt (/ 2.0 (tan k))) t) 3.0)
      (/ (+ 2.0 t_1) (/ l (/ (sin k) l))))
     (* (/ 2.0 (* k (* t k))) (/ (* l (* l (cos k))) (pow (sin k) 2.0))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((1.0 + (t_1 + 1.0)) * (tan(k) * (sin(k) * (pow(t, 3.0) / (l * l)))))) <= 2e+258) {
		tmp = pow((cbrt((2.0 / tan(k))) / t), 3.0) / ((2.0 + t_1) / (l / (sin(k) / l)));
	} else {
		tmp = (2.0 / (k * (t * k))) * ((l * (l * cos(k))) / pow(sin(k), 2.0));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((1.0 + (t_1 + 1.0)) * (Math.tan(k) * (Math.sin(k) * (Math.pow(t, 3.0) / (l * l)))))) <= 2e+258) {
		tmp = Math.pow((Math.cbrt((2.0 / Math.tan(k))) / t), 3.0) / ((2.0 + t_1) / (l / (Math.sin(k) / l)));
	} else {
		tmp = (2.0 / (k * (t * k))) * ((l * (l * Math.cos(k))) / Math.pow(Math.sin(k), 2.0));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(1.0 + Float64(t_1 + 1.0)) * Float64(tan(k) * Float64(sin(k) * Float64((t ^ 3.0) / Float64(l * l)))))) <= 2e+258)
		tmp = Float64((Float64(cbrt(Float64(2.0 / tan(k))) / t) ^ 3.0) / Float64(Float64(2.0 + t_1) / Float64(l / Float64(sin(k) / l))));
	else
		tmp = Float64(Float64(2.0 / Float64(k * Float64(t * k))) * Float64(Float64(l * Float64(l * cos(k))) / (sin(k) ^ 2.0)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(2.0 / N[(N[(1.0 + N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+258], N[(N[Power[N[(N[Power[N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision] / N[(N[(2.0 + t$95$1), $MachinePrecision] / N[(l / N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 2.00000000000000011e258

   1. Initial program 76.3%

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

      1. associate-/r* 76.4%

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

      2. associate-*l* 71.5%

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

      3. sqr-neg 71.5%

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

      4. associate-*l* 76.4%

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

      5. *-commutative 76.4%

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

      6. sqr-neg 76.4%

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

      7. associate-/r* 76.4%

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

   3. Simplified 76.2%

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

      1. associate-/l/ 76.3%

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

      2. associate-/r/ 76.4%

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

      3. add-cube-cbrt 76.3%

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

      4. pow3 76.3%

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

   5. Applied egg-rr 88.8%

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

      1. div-inv 88.8%

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

      2. associate-/r/ 88.9%

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

      3. associate-/r/ 88.9%

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

   7. Applied egg-rr 88.9%

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

      1. associate-*r/ 88.9%

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

      2. associate-*l/ 88.9%

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

      3. *-rgt-identity 88.9%

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

      4. cube-prod 84.7%

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

      5. rem-cube-cbrt 84.8%

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

      6. associate-/l* 84.7%

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

      7. associate-*l/ 79.6%

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

      8. associate-/l* 84.7%

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

   9. Simplified 84.7%

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

   if 2.00000000000000011e258 < (/.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 24.4%

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

      1. associate-/r* 24.4%

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

      2. associate-*l* 24.4%

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

      3. sqr-neg 24.4%

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

      4. associate-*l* 24.4%

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

      5. *-commutative 24.4%

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

      6. sqr-neg 24.4%

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

      7. associate-/r* 24.4%

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

   3. Simplified 24.4%

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

      1. associate-/l/ 24.4%

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

      2. associate-/r/ 24.4%

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

      3. add-cube-cbrt 24.4%

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

      4. pow3 24.4%

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

   5. Applied egg-rr 48.3%

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

      1. cbrt-div 58.3%

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

   7. Applied egg-rr 58.3%

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

   8. Taylor expanded in k around inf 62.4%

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

      1. associate-*r/ 62.4%

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

      2. associate-*r* 62.4%

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

      3. times-frac 62.7%

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

      4. unpow2 62.7%

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

      5. associate-*l* 66.8%

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

      6. unpow2 66.8%

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

      7. associate-*l* 66.8%

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

   10. Simplified 66.8%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)} \leq 2 \cdot 10^{+258}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t}\right)}^{3}}{\frac{2 + {\left(\frac{k}{t}\right)}^{2}}{\frac{\ell}{\frac{\sin k}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \left(\ell \cdot \cos k\right)}{{\sin k}^{2}}\\ \end{array} \]

Alternative 3: 74.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \ell \cdot \left(\ell \cdot \cos k\right)\\ \mathbf{if}\;\frac{2}{\left(1 + \left(t_1 + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)} \leq 5 \cdot 10^{+197}:\\ \;\;\;\;\frac{2}{t \cdot {\left(t \cdot \sin k\right)}^{2}} \cdot \frac{t_2}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(t \cdot k\right)} \cdot \frac{t_2}{{\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)) (t_2 (* l (* l (cos k)))))
   (if (<=
        (/
         2.0
         (*
          (+ 1.0 (+ t_1 1.0))
          (* (tan k) (* (sin k) (/ (pow t 3.0) (* l l))))))
        5e+197)
     (* (/ 2.0 (* t (pow (* t (sin k)) 2.0))) (/ t_2 (+ 2.0 t_1)))
     (* (/ 2.0 (* k (* t k))) (/ t_2 (pow (sin k) 2.0))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double t_2 = l * (l * cos(k));
	double tmp;
	if ((2.0 / ((1.0 + (t_1 + 1.0)) * (tan(k) * (sin(k) * (pow(t, 3.0) / (l * l)))))) <= 5e+197) {
		tmp = (2.0 / (t * pow((t * sin(k)), 2.0))) * (t_2 / (2.0 + t_1));
	} else {
		tmp = (2.0 / (k * (t * k))) * (t_2 / pow(sin(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) :: t_2
    real(8) :: tmp
    t_1 = (k / t) ** 2.0d0
    t_2 = l * (l * cos(k))
    if ((2.0d0 / ((1.0d0 + (t_1 + 1.0d0)) * (tan(k) * (sin(k) * ((t ** 3.0d0) / (l * l)))))) <= 5d+197) then
        tmp = (2.0d0 / (t * ((t * sin(k)) ** 2.0d0))) * (t_2 / (2.0d0 + t_1))
    else
        tmp = (2.0d0 / (k * (t * k))) * (t_2 / (sin(k) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double t_2 = l * (l * Math.cos(k));
	double tmp;
	if ((2.0 / ((1.0 + (t_1 + 1.0)) * (Math.tan(k) * (Math.sin(k) * (Math.pow(t, 3.0) / (l * l)))))) <= 5e+197) {
		tmp = (2.0 / (t * Math.pow((t * Math.sin(k)), 2.0))) * (t_2 / (2.0 + t_1));
	} else {
		tmp = (2.0 / (k * (t * k))) * (t_2 / Math.pow(Math.sin(k), 2.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow((k / t), 2.0)
	t_2 = l * (l * math.cos(k))
	tmp = 0
	if (2.0 / ((1.0 + (t_1 + 1.0)) * (math.tan(k) * (math.sin(k) * (math.pow(t, 3.0) / (l * l)))))) <= 5e+197:
		tmp = (2.0 / (t * math.pow((t * math.sin(k)), 2.0))) * (t_2 / (2.0 + t_1))
	else:
		tmp = (2.0 / (k * (t * k))) * (t_2 / math.pow(math.sin(k), 2.0))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	t_2 = Float64(l * Float64(l * cos(k)))
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(1.0 + Float64(t_1 + 1.0)) * Float64(tan(k) * Float64(sin(k) * Float64((t ^ 3.0) / Float64(l * l)))))) <= 5e+197)
		tmp = Float64(Float64(2.0 / Float64(t * (Float64(t * sin(k)) ^ 2.0))) * Float64(t_2 / Float64(2.0 + t_1)));
	else
		tmp = Float64(Float64(2.0 / Float64(k * Float64(t * k))) * Float64(t_2 / (sin(k) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / t) ^ 2.0;
	t_2 = l * (l * cos(k));
	tmp = 0.0;
	if ((2.0 / ((1.0 + (t_1 + 1.0)) * (tan(k) * (sin(k) * ((t ^ 3.0) / (l * l)))))) <= 5e+197)
		tmp = (2.0 / (t * ((t * sin(k)) ^ 2.0))) * (t_2 / (2.0 + t_1));
	else
		tmp = (2.0 / (k * (t * k))) * (t_2 / (sin(k) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(l * N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(2.0 / N[(N[(1.0 + N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+197], N[(N[(2.0 / N[(t * N[Power[N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 / N[Power[N[Sin[k], $MachinePrecision], 2.0], $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))) < 5.00000000000000009e197

   1. Initial program 76.1%

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

      1. associate-/r* 76.3%

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

      2. associate-*l* 72.0%

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

      3. sqr-neg 72.0%

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

      4. associate-*l* 76.3%

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

      5. *-commutative 76.3%

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

      6. sqr-neg 76.3%

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

      7. associate-/r* 76.3%

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

   3. Simplified 76.1%

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

      1. associate-/l/ 76.1%

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

      2. associate-/r/ 76.3%

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

      3. add-cube-cbrt 76.2%

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

      4. pow3 76.2%

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

   5. Applied egg-rr 88.8%

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

      1. cbrt-div 94.3%

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

   7. Applied egg-rr 94.3%

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

   8. Taylor expanded in l around 0 58.9%

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

      1. associate-*r/ 58.9%

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

      2. associate-*r* 58.9%

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

      3. times-frac 58.8%

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

      4. cube-mult 58.8%

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

      5. unpow2 58.8%

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

      6. associate-*l* 60.4%

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

      7. unpow2 60.4%

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

      8. unpow2 60.4%

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

      9. swap-sqr 70.0%

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

      10. unpow1 70.0%

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

      11. pow-plus 70.0%

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

      12. metadata-eval 70.0%

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

   10. Simplified 83.3%

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

   if 5.00000000000000009e197 < (/.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 25.0%

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

      1. associate-/r* 25.0%

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

      2. associate-*l* 24.2%

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

      3. sqr-neg 24.2%

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

      4. associate-*l* 25.0%

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

      5. *-commutative 25.0%

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

      6. sqr-neg 25.0%

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

      7. associate-/r* 25.0%

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

   3. Simplified 25.0%

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

      1. associate-/l/ 25.0%

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

      2. associate-/r/ 25.0%

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

      3. add-cube-cbrt 25.0%

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

      4. pow3 25.0%

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

   5. Applied egg-rr 48.7%

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

      1. cbrt-div 58.6%

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

   7. Applied egg-rr 58.6%

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

   8. Taylor expanded in k around inf 62.0%

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

      1. associate-*r/ 62.0%

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

      2. associate-*r* 62.0%

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

      3. times-frac 62.2%

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

      4. unpow2 62.2%

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

      5. associate-*l* 66.3%

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

      6. unpow2 66.3%

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

      7. associate-*l* 66.3%

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

   10. Simplified 66.3%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)} \leq 5 \cdot 10^{+197}:\\ \;\;\;\;\frac{2}{t \cdot {\left(t \cdot \sin k\right)}^{2}} \cdot \frac{\ell \cdot \left(\ell \cdot \cos k\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \left(\ell \cdot \cos k\right)}{{\sin k}^{2}}\\ \end{array} \]

Alternative 4: 85.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.7 \cdot 10^{-10} \lor \neg \left(t \leq 1.25 \cdot 10^{-95}\right):\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{\frac{\sin k}{\ell}}}}}\right)}^{3}}{2 + \frac{k}{t} \cdot \frac{k}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \left(\ell \cdot \cos k\right)}{{\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -2.7e-10) (not (<= t 1.25e-95)))
   (/
    (pow
     (/ (cbrt (/ 2.0 (tan k))) (/ t (/ (cbrt l) (cbrt (/ (sin k) l)))))
     3.0)
    (+ 2.0 (* (/ k t) (/ k t))))
   (* (/ 2.0 (* k (* t k))) (/ (* l (* l (cos k))) (pow (sin k) 2.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -2.7e-10) || !(t <= 1.25e-95)) {
		tmp = pow((cbrt((2.0 / tan(k))) / (t / (cbrt(l) / cbrt((sin(k) / l))))), 3.0) / (2.0 + ((k / t) * (k / t)));
	} else {
		tmp = (2.0 / (k * (t * k))) * ((l * (l * cos(k))) / pow(sin(k), 2.0));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -2.7e-10) || !(t <= 1.25e-95)) {
		tmp = Math.pow((Math.cbrt((2.0 / Math.tan(k))) / (t / (Math.cbrt(l) / Math.cbrt((Math.sin(k) / l))))), 3.0) / (2.0 + ((k / t) * (k / t)));
	} else {
		tmp = (2.0 / (k * (t * k))) * ((l * (l * Math.cos(k))) / Math.pow(Math.sin(k), 2.0));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -2.7e-10) || !(t <= 1.25e-95))
		tmp = Float64((Float64(cbrt(Float64(2.0 / tan(k))) / Float64(t / Float64(cbrt(l) / cbrt(Float64(sin(k) / l))))) ^ 3.0) / Float64(2.0 + Float64(Float64(k / t) * Float64(k / t))));
	else
		tmp = Float64(Float64(2.0 / Float64(k * Float64(t * k))) * Float64(Float64(l * Float64(l * cos(k))) / (sin(k) ^ 2.0)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -2.7e-10], N[Not[LessEqual[t, 1.25e-95]], $MachinePrecision]], N[(N[Power[N[(N[Power[N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[(t / N[(N[Power[l, 1/3], $MachinePrecision] / N[Power[N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[(2.0 + N[(N[(k / t), $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.7e-10 or 1.2499999999999999e-95 < t

   1. Initial program 62.7%

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

      1. associate-/r* 62.8%

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

      2. associate-*l* 58.5%

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

      3. sqr-neg 58.5%

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

      4. associate-*l* 62.8%

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

      5. *-commutative 62.8%

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

      6. sqr-neg 62.8%

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

      7. associate-/r* 62.8%

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

   3. Simplified 62.7%

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

      1. associate-/l/ 62.7%

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

      2. associate-/r/ 62.8%

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

      3. add-cube-cbrt 62.8%

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

      4. pow3 62.8%

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

   5. Applied egg-rr 77.1%

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

      1. cbrt-div 92.4%

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

   7. Applied egg-rr 92.4%

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

      1. unpow2 92.4%

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

   9. Applied egg-rr 92.4%

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

   if -2.7e-10 < t < 1.2499999999999999e-95

   1. Initial program 35.5%

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

      1. associate-/r* 35.5%

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

      2. associate-*l* 35.5%

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

      3. sqr-neg 35.5%

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

      4. associate-*l* 35.5%

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

      5. *-commutative 35.5%

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

      6. sqr-neg 35.5%

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

      7. associate-/r* 35.5%

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

   3. Simplified 35.5%

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

      1. associate-/l/ 35.5%

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

      2. associate-/r/ 35.5%

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

      3. add-cube-cbrt 35.4%

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

      4. pow3 35.4%

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

   5. Applied egg-rr 58.6%

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

      1. cbrt-div 54.6%

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

   7. Applied egg-rr 54.6%

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

   8. Taylor expanded in k around inf 76.7%

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

      1. associate-*r/ 76.7%

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

      2. associate-*r* 76.7%

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

      3. times-frac 77.0%

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

      4. unpow2 77.0%

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

      5. associate-*l* 81.9%

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

      6. unpow2 81.9%

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

      7. associate-*l* 81.9%

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

   10. Simplified 81.9%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.7 \cdot 10^{-10} \lor \neg \left(t \leq 1.25 \cdot 10^{-95}\right):\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{\frac{\sin k}{\ell}}}}}\right)}^{3}}{2 + \frac{k}{t} \cdot \frac{k}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \left(\ell \cdot \cos k\right)}{{\sin k}^{2}}\\ \end{array} \]

Alternative 5: 80.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -7.8 \cdot 10^{-16}:\\ \;\;\;\;\frac{\ell \cdot \left(\ell \cdot \frac{2}{{\left(t \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\tan k}\right)\right)}^{3}}\right)}{2 + t_1}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-79}:\\ \;\;\;\;\frac{2}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \left(\ell \cdot \cos k\right)}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}}{1 + \left(t_1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<= t -7.8e-16)
     (/
      (* l (* l (/ 2.0 (pow (* t (* (cbrt (sin k)) (cbrt (tan k)))) 3.0))))
      (+ 2.0 t_1))
     (if (<= t 2e-79)
       (* (/ 2.0 (* k (* t k))) (/ (* l (* l (cos k))) (pow (sin k) 2.0)))
       (/
        (/ 2.0 (* (tan k) (* (sin k) (pow (/ (pow t 1.5) l) 2.0))))
        (+ 1.0 (+ t_1 1.0)))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double tmp;
	if (t <= -7.8e-16) {
		tmp = (l * (l * (2.0 / pow((t * (cbrt(sin(k)) * cbrt(tan(k)))), 3.0)))) / (2.0 + t_1);
	} else if (t <= 2e-79) {
		tmp = (2.0 / (k * (t * k))) * ((l * (l * cos(k))) / pow(sin(k), 2.0));
	} else {
		tmp = (2.0 / (tan(k) * (sin(k) * pow((pow(t, 1.5) / l), 2.0)))) / (1.0 + (t_1 + 1.0));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double tmp;
	if (t <= -7.8e-16) {
		tmp = (l * (l * (2.0 / Math.pow((t * (Math.cbrt(Math.sin(k)) * Math.cbrt(Math.tan(k)))), 3.0)))) / (2.0 + t_1);
	} else if (t <= 2e-79) {
		tmp = (2.0 / (k * (t * k))) * ((l * (l * Math.cos(k))) / Math.pow(Math.sin(k), 2.0));
	} else {
		tmp = (2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow((Math.pow(t, 1.5) / l), 2.0)))) / (1.0 + (t_1 + 1.0));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (t <= -7.8e-16)
		tmp = Float64(Float64(l * Float64(l * Float64(2.0 / (Float64(t * Float64(cbrt(sin(k)) * cbrt(tan(k)))) ^ 3.0)))) / Float64(2.0 + t_1));
	elseif (t <= 2e-79)
		tmp = Float64(Float64(2.0 / Float64(k * Float64(t * k))) * Float64(Float64(l * Float64(l * cos(k))) / (sin(k) ^ 2.0)));
	else
		tmp = Float64(Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (Float64((t ^ 1.5) / l) ^ 2.0)))) / Float64(1.0 + Float64(t_1 + 1.0)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t, -7.8e-16], N[(N[(l * N[(l * N[(2.0 / N[Power[N[(t * N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2e-79], N[(N[(2.0 / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.79999999999999954e-16

   1. Initial program 67.2%

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

      1. associate-/r* 67.6%

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

      2. associate-*l* 64.0%

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

      3. sqr-neg 64.0%

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

      4. associate-*l* 67.6%

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

      5. *-commutative 67.6%

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

      6. sqr-neg 67.6%

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

      7. associate-*l/ 67.6%

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

      8. associate-*r/ 69.0%

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

      9. associate-/r/ 68.9%

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

   3. Simplified 68.9%

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

      1. add-cube-cbrt 68.8%

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

      2. pow3 68.8%

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

      3. *-commutative 68.8%

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

      4. cbrt-prod 68.8%

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

      5. cbrt-prod 68.7%

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

      6. rem-cbrt-cube 76.1%

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

   5. Applied egg-rr 76.1%

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

      1. associate-*l/ 76.1%

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

      2. associate-*l* 76.0%

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

   7. Applied egg-rr 76.0%

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

      1. unpow2 76.0%

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

      2. *-commutative 76.0%

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

      3. associate-*r/ 76.0%

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

      4. unpow2 76.0%

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

      5. associate-*l* 84.7%

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

   9. Simplified 84.7%

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

   if -7.79999999999999954e-16 < t < 2e-79

   1. Initial program 35.5%

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

      1. associate-/r* 35.5%

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

      2. associate-*l* 35.5%

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

      3. sqr-neg 35.5%

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

      4. associate-*l* 35.5%

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

      5. *-commutative 35.5%

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

      6. sqr-neg 35.5%

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

      7. associate-/r* 35.5%

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

   3. Simplified 35.5%

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

      1. associate-/l/ 35.5%

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

      2. associate-/r/ 35.5%

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

      3. add-cube-cbrt 35.5%

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

      4. pow3 35.5%

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

   5. Applied egg-rr 58.0%

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

      1. cbrt-div 54.9%

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

   7. Applied egg-rr 54.9%

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

   8. Taylor expanded in k around inf 76.4%

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

      1. associate-*r/ 76.4%

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

      2. associate-*r* 76.4%

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

      3. times-frac 76.7%

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

      4. unpow2 76.7%

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

      5. associate-*l* 81.4%

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

      6. unpow2 81.4%

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

      7. associate-*l* 81.5%

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

   10. Simplified 81.5%

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

   if 2e-79 < t

   1. Initial program 60.1%

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

      1. associate-/r* 60.1%

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

      2. +-commutative 60.1%

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

   3. Simplified 60.1%

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

      1. add-sqr-sqrt 60.1%

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

      2. pow2 60.1%

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

      3. sqrt-div 60.1%

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

      4. sqrt-pow1 68.4%

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

      5. metadata-eval 68.4%

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

      6. sqrt-prod 46.9%

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

      7. add-sqr-sqrt 80.9%

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

   5. Applied egg-rr 80.9%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.8 \cdot 10^{-16}:\\ \;\;\;\;\frac{\ell \cdot \left(\ell \cdot \frac{2}{{\left(t \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\tan k}\right)\right)}^{3}}\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-79}:\\ \;\;\;\;\frac{2}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \left(\ell \cdot \cos k\right)}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}\\ \end{array} \]

Alternative 6: 76.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \ell \cdot \left(\ell \cdot \cos k\right)\\ t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_3 := \frac{2}{t \cdot {\left(t \cdot \sin k\right)}^{2}} \cdot \frac{t_1}{t_2}\\ \mathbf{if}\;t \leq -2 \cdot 10^{-10}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-80}:\\ \;\;\;\;\frac{2}{k \cdot \left(t \cdot k\right)} \cdot \frac{t_1}{{\sin k}^{2}}\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{+143}:\\ \;\;\;\;\frac{\frac{2}{t_2}}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* l (* l (cos k))))
        (t_2 (+ 2.0 (pow (/ k t) 2.0)))
        (t_3 (* (/ 2.0 (* t (pow (* t (sin k)) 2.0))) (/ t_1 t_2))))
   (if (<= t -2e-10)
     t_3
     (if (<= t 5.2e-80)
       (* (/ 2.0 (* k (* t k))) (/ t_1 (pow (sin k) 2.0)))
       (if (<= t 7.4e+143)
         (/ (/ 2.0 t_2) (* (pow (/ (pow t 1.5) l) 2.0) (* (tan k) (sin k))))
         t_3)))))

C

double code(double t, double l, double k) {
	double t_1 = l * (l * cos(k));
	double t_2 = 2.0 + pow((k / t), 2.0);
	double t_3 = (2.0 / (t * pow((t * sin(k)), 2.0))) * (t_1 / t_2);
	double tmp;
	if (t <= -2e-10) {
		tmp = t_3;
	} else if (t <= 5.2e-80) {
		tmp = (2.0 / (k * (t * k))) * (t_1 / pow(sin(k), 2.0));
	} else if (t <= 7.4e+143) {
		tmp = (2.0 / t_2) / (pow((pow(t, 1.5) / l), 2.0) * (tan(k) * sin(k)));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = l * (l * cos(k))
    t_2 = 2.0d0 + ((k / t) ** 2.0d0)
    t_3 = (2.0d0 / (t * ((t * sin(k)) ** 2.0d0))) * (t_1 / t_2)
    if (t <= (-2d-10)) then
        tmp = t_3
    else if (t <= 5.2d-80) then
        tmp = (2.0d0 / (k * (t * k))) * (t_1 / (sin(k) ** 2.0d0))
    else if (t <= 7.4d+143) then
        tmp = (2.0d0 / t_2) / ((((t ** 1.5d0) / l) ** 2.0d0) * (tan(k) * sin(k)))
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = l * (l * Math.cos(k));
	double t_2 = 2.0 + Math.pow((k / t), 2.0);
	double t_3 = (2.0 / (t * Math.pow((t * Math.sin(k)), 2.0))) * (t_1 / t_2);
	double tmp;
	if (t <= -2e-10) {
		tmp = t_3;
	} else if (t <= 5.2e-80) {
		tmp = (2.0 / (k * (t * k))) * (t_1 / Math.pow(Math.sin(k), 2.0));
	} else if (t <= 7.4e+143) {
		tmp = (2.0 / t_2) / (Math.pow((Math.pow(t, 1.5) / l), 2.0) * (Math.tan(k) * Math.sin(k)));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = l * (l * math.cos(k))
	t_2 = 2.0 + math.pow((k / t), 2.0)
	t_3 = (2.0 / (t * math.pow((t * math.sin(k)), 2.0))) * (t_1 / t_2)
	tmp = 0
	if t <= -2e-10:
		tmp = t_3
	elif t <= 5.2e-80:
		tmp = (2.0 / (k * (t * k))) * (t_1 / math.pow(math.sin(k), 2.0))
	elif t <= 7.4e+143:
		tmp = (2.0 / t_2) / (math.pow((math.pow(t, 1.5) / l), 2.0) * (math.tan(k) * math.sin(k)))
	else:
		tmp = t_3
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(l * Float64(l * cos(k)))
	t_2 = Float64(2.0 + (Float64(k / t) ^ 2.0))
	t_3 = Float64(Float64(2.0 / Float64(t * (Float64(t * sin(k)) ^ 2.0))) * Float64(t_1 / t_2))
	tmp = 0.0
	if (t <= -2e-10)
		tmp = t_3;
	elseif (t <= 5.2e-80)
		tmp = Float64(Float64(2.0 / Float64(k * Float64(t * k))) * Float64(t_1 / (sin(k) ^ 2.0)));
	elseif (t <= 7.4e+143)
		tmp = Float64(Float64(2.0 / t_2) / Float64((Float64((t ^ 1.5) / l) ^ 2.0) * Float64(tan(k) * sin(k))));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = l * (l * cos(k));
	t_2 = 2.0 + ((k / t) ^ 2.0);
	t_3 = (2.0 / (t * ((t * sin(k)) ^ 2.0))) * (t_1 / t_2);
	tmp = 0.0;
	if (t <= -2e-10)
		tmp = t_3;
	elseif (t <= 5.2e-80)
		tmp = (2.0 / (k * (t * k))) * (t_1 / (sin(k) ^ 2.0));
	elseif (t <= 7.4e+143)
		tmp = (2.0 / t_2) / ((((t ^ 1.5) / l) ^ 2.0) * (tan(k) * sin(k)));
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(l * N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 / N[(t * N[Power[N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2e-10], t$95$3, If[LessEqual[t, 5.2e-80], N[(N[(2.0 / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7.4e+143], N[(N[(2.0 / t$95$2), $MachinePrecision] / N[(N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.00000000000000007e-10 or 7.4000000000000003e143 < t

   1. Initial program 62.9%

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

      1. associate-/r* 63.2%

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

      2. associate-*l* 57.0%

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

      3. sqr-neg 57.0%

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

      4. associate-*l* 63.2%

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

      5. *-commutative 63.2%

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

      6. sqr-neg 63.2%

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

      7. associate-/r* 63.2%

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

   3. Simplified 62.9%

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

      1. associate-/l/ 62.9%

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

      2. associate-/r/ 63.2%

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

      3. add-cube-cbrt 63.1%

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

      4. pow3 63.1%

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

   5. Applied egg-rr 78.2%

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

      1. cbrt-div 93.0%

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

   7. Applied egg-rr 93.0%

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

   8. Taylor expanded in l around 0 50.3%

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

      1. associate-*r/ 50.3%

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

      2. associate-*r* 50.3%

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

      3. times-frac 50.3%

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

      4. cube-mult 50.3%

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

      5. unpow2 50.3%

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

      6. associate-*l* 53.7%

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

      7. unpow2 53.7%

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

      8. unpow2 53.7%

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

      9. swap-sqr 68.3%

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

      10. unpow1 68.3%

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

      11. pow-plus 68.3%

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

      12. metadata-eval 68.3%

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

   10. Simplified 76.1%

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

   if -2.00000000000000007e-10 < t < 5.2000000000000002e-80

   1. Initial program 35.5%

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

      1. associate-/r* 35.5%

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

      2. associate-*l* 35.5%

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

      3. sqr-neg 35.5%

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

      4. associate-*l* 35.5%

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

      5. *-commutative 35.5%

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

      6. sqr-neg 35.5%

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

      7. associate-/r* 35.5%

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

   3. Simplified 35.5%

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

      1. associate-/l/ 35.5%

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

      2. associate-/r/ 35.5%

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

      3. add-cube-cbrt 35.5%

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

      4. pow3 35.5%

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

   5. Applied egg-rr 58.0%

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

      1. cbrt-div 54.9%

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

   7. Applied egg-rr 54.9%

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

   8. Taylor expanded in k around inf 76.4%

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

      1. associate-*r/ 76.4%

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

      2. associate-*r* 76.4%

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

      3. times-frac 76.7%

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

      4. unpow2 76.7%

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

      5. associate-*l* 81.4%

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

      6. unpow2 81.4%

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

      7. associate-*l* 81.5%

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

   10. Simplified 81.5%

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

   if 5.2000000000000002e-80 < t < 7.4000000000000003e143

   1. Initial program 63.6%

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

      1. associate-/r* 63.6%

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

      2. +-commutative 63.6%

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

   3. Simplified 63.6%

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

      1. add-sqr-sqrt 63.6%

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

      2. pow2 63.6%

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

      3. sqrt-div 63.6%

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

      4. sqrt-pow1 71.9%

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

      5. metadata-eval 71.9%

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

      6. sqrt-prod 48.7%

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

      7. add-sqr-sqrt 86.0%

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

   5. Applied egg-rr 86.0%

      \[\leadsto \frac{\frac{2}{\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \tan k}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
   6. Step-by-step derivation

      1. div-inv 86.0%

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

      2. *-commutative 86.0%

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

      3. *-commutative 86.0%

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

      4. associate-+r+ 86.0%

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

      5. metadata-eval 86.0%

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

   7. Applied egg-rr 86.0%

      \[\leadsto \color{blue}{\frac{2}{\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)} \cdot \frac{1}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   8. Step-by-step derivation

      1. associate-*l/ 86.0%

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

      2. associate-*r/ 86.0%

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

      3. metadata-eval 86.0%

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

      4. associate-*r* 79.7%

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

   9. Simplified 79.7%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-10}:\\ \;\;\;\;\frac{2}{t \cdot {\left(t \cdot \sin k\right)}^{2}} \cdot \frac{\ell \cdot \left(\ell \cdot \cos k\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-80}:\\ \;\;\;\;\frac{2}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \left(\ell \cdot \cos k\right)}{{\sin k}^{2}}\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{+143}:\\ \;\;\;\;\frac{\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot {\left(t \cdot \sin k\right)}^{2}} \cdot \frac{\ell \cdot \left(\ell \cdot \cos k\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]

Alternative 7: 64.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.66 \cdot 10^{-9}:\\ \;\;\;\;\frac{\ell}{\frac{k}{\ell}} \cdot \frac{\cos k}{\sin k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\ell \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.66e-9)
   (* (/ l (/ k l)) (/ (cos k) (* (sin k) (pow t 3.0))))
   (* 2.0 (* (* l (/ l (* k k))) (/ (cos k) (* t (pow (sin k) 2.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.66e-9) {
		tmp = (l / (k / l)) * (cos(k) / (sin(k) * pow(t, 3.0)));
	} else {
		tmp = 2.0 * ((l * (l / (k * k))) * (cos(k) / (t * pow(sin(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) :: tmp
    if (k <= 1.66d-9) then
        tmp = (l / (k / l)) * (cos(k) / (sin(k) * (t ** 3.0d0)))
    else
        tmp = 2.0d0 * ((l * (l / (k * k))) * (cos(k) / (t * (sin(k) ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.66e-9) {
		tmp = (l / (k / l)) * (Math.cos(k) / (Math.sin(k) * Math.pow(t, 3.0)));
	} else {
		tmp = 2.0 * ((l * (l / (k * k))) * (Math.cos(k) / (t * Math.pow(Math.sin(k), 2.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.66e-9:
		tmp = (l / (k / l)) * (math.cos(k) / (math.sin(k) * math.pow(t, 3.0)))
	else:
		tmp = 2.0 * ((l * (l / (k * k))) * (math.cos(k) / (t * math.pow(math.sin(k), 2.0))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.66e-9)
		tmp = Float64(Float64(l / Float64(k / l)) * Float64(cos(k) / Float64(sin(k) * (t ^ 3.0))));
	else
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l / Float64(k * k))) * Float64(cos(k) / Float64(t * (sin(k) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.66e-9)
		tmp = (l / (k / l)) * (cos(k) / (sin(k) * (t ^ 3.0)));
	else
		tmp = 2.0 * ((l * (l / (k * k))) * (cos(k) / (t * (sin(k) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.66e-9], N[(N[(l / N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.6600000000000001e-9

   1. Initial program 56.3%

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

      1. associate-/r* 56.5%

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

      2. associate-*l* 52.8%

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

      3. sqr-neg 52.8%

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

      4. associate-*l* 56.5%

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

      5. *-commutative 56.5%

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

      6. sqr-neg 56.5%

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

      7. associate-/r* 56.5%

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

   3. Simplified 56.3%

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

   4. Taylor expanded in k around 0 54.7%

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

      1. unpow2 54.7%

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

   6. Simplified 54.7%

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

   7. Taylor expanded in t around inf 55.3%

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

      1. times-frac 55.5%

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

      2. unpow2 55.5%

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

      3. associate-/l* 60.7%

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

      4. *-commutative 60.7%

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

   9. Simplified 60.7%

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

   if 1.6600000000000001e-9 < k

   1. Initial program 40.7%

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

      1. associate-/r* 40.7%

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

      2. associate-*l* 40.7%

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

      3. sqr-neg 40.7%

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

      4. associate-*l* 40.7%

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

      5. *-commutative 40.7%

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

      6. sqr-neg 40.7%

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

      7. associate-/r* 40.7%

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

   3. Simplified 40.7%

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

      1. associate-/l/ 40.7%

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

      2. associate-/r/ 40.7%

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

      3. add-cube-cbrt 40.7%

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

      4. pow3 40.7%

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

   5. Applied egg-rr 59.0%

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

      1. cbrt-div 61.2%

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

   7. Applied egg-rr 61.2%

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

   8. Taylor expanded in k around inf 72.8%

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

      1. times-frac 67.7%

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

      2. unpow2 67.7%

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

      3. associate-*r/ 68.2%

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

      4. unpow2 68.2%

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

   10. Simplified 68.2%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.66 \cdot 10^{-9}:\\ \;\;\;\;\frac{\ell}{\frac{k}{\ell}} \cdot \frac{\cos k}{\sin k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\ell \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternative 8: 65.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.3 \cdot 10^{-10}:\\ \;\;\;\;\frac{\ell}{\frac{k}{\ell}} \cdot \frac{\cos k}{\sin k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(t \cdot k\right)} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 3.3e-10)
   (* (/ l (/ k l)) (/ (cos k) (* (sin k) (pow t 3.0))))
   (* 2.0 (* (/ (* l l) (* k (* t k))) (/ (cos k) (pow (sin k) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.3e-10) {
		tmp = (l / (k / l)) * (cos(k) / (sin(k) * pow(t, 3.0)));
	} else {
		tmp = 2.0 * (((l * l) / (k * (t * k))) * (cos(k) / pow(sin(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) :: tmp
    if (k <= 3.3d-10) then
        tmp = (l / (k / l)) * (cos(k) / (sin(k) * (t ** 3.0d0)))
    else
        tmp = 2.0d0 * (((l * l) / (k * (t * k))) * (cos(k) / (sin(k) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.3e-10) {
		tmp = (l / (k / l)) * (Math.cos(k) / (Math.sin(k) * Math.pow(t, 3.0)));
	} else {
		tmp = 2.0 * (((l * l) / (k * (t * k))) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 3.3e-10:
		tmp = (l / (k / l)) * (math.cos(k) / (math.sin(k) * math.pow(t, 3.0)))
	else:
		tmp = 2.0 * (((l * l) / (k * (t * k))) * (math.cos(k) / math.pow(math.sin(k), 2.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 3.3e-10)
		tmp = Float64(Float64(l / Float64(k / l)) * Float64(cos(k) / Float64(sin(k) * (t ^ 3.0))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l * l) / Float64(k * Float64(t * k))) * Float64(cos(k) / (sin(k) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 3.3e-10)
		tmp = (l / (k / l)) * (cos(k) / (sin(k) * (t ^ 3.0)));
	else
		tmp = 2.0 * (((l * l) / (k * (t * k))) * (cos(k) / (sin(k) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 3.3e-10], N[(N[(l / N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l * l), $MachinePrecision] / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.3e-10

   1. Initial program 56.1%

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

      1. associate-/r* 56.2%

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

      2. associate-*l* 52.6%

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

      3. sqr-neg 52.6%

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

      4. associate-*l* 56.2%

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

      5. *-commutative 56.2%

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

      6. sqr-neg 56.2%

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

      7. associate-/r* 56.2%

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

   3. Simplified 56.1%

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

   4. Taylor expanded in k around 0 54.4%

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

      1. unpow2 54.4%

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

   6. Simplified 54.4%

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

   7. Taylor expanded in t around inf 55.0%

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

      1. times-frac 55.3%

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

      2. unpow2 55.3%

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

      3. associate-/l* 60.5%

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

      4. *-commutative 60.5%

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

   9. Simplified 60.5%

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

   if 3.3e-10 < k

   1. Initial program 41.5%

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

      1. associate-/r* 41.5%

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

      2. associate-*l* 41.5%

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

      3. sqr-neg 41.5%

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

      4. associate-*l* 41.5%

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

      5. *-commutative 41.5%

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

      6. sqr-neg 41.5%

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

      7. associate-*l/ 41.5%

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

      8. associate-*r/ 41.5%

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

      9. associate-/r/ 41.4%

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

   3. Simplified 41.4%

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

   4. Taylor expanded in k around inf 72.8%

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

      1. associate-*r* 72.9%

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

      2. times-frac 73.2%

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

      3. unpow2 73.2%

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

      4. unpow2 73.2%

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

      5. associate-*l* 74.8%

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

   6. Simplified 74.8%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.3 \cdot 10^{-10}:\\ \;\;\;\;\frac{\ell}{\frac{k}{\ell}} \cdot \frac{\cos k}{\sin k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(t \cdot k\right)} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 9: 65.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.58 \cdot 10^{-9}:\\ \;\;\;\;\frac{\ell}{\frac{k}{\ell}} \cdot \frac{\cos k}{\sin k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \left(\ell \cdot \cos k\right)}{{\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.58e-9)
   (* (/ l (/ k l)) (/ (cos k) (* (sin k) (pow t 3.0))))
   (* (/ 2.0 (* k (* t k))) (/ (* l (* l (cos k))) (pow (sin k) 2.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.58e-9) {
		tmp = (l / (k / l)) * (cos(k) / (sin(k) * pow(t, 3.0)));
	} else {
		tmp = (2.0 / (k * (t * k))) * ((l * (l * cos(k))) / pow(sin(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) :: tmp
    if (k <= 1.58d-9) then
        tmp = (l / (k / l)) * (cos(k) / (sin(k) * (t ** 3.0d0)))
    else
        tmp = (2.0d0 / (k * (t * k))) * ((l * (l * cos(k))) / (sin(k) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.58e-9) {
		tmp = (l / (k / l)) * (Math.cos(k) / (Math.sin(k) * Math.pow(t, 3.0)));
	} else {
		tmp = (2.0 / (k * (t * k))) * ((l * (l * Math.cos(k))) / Math.pow(Math.sin(k), 2.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.58e-9:
		tmp = (l / (k / l)) * (math.cos(k) / (math.sin(k) * math.pow(t, 3.0)))
	else:
		tmp = (2.0 / (k * (t * k))) * ((l * (l * math.cos(k))) / math.pow(math.sin(k), 2.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.58e-9)
		tmp = Float64(Float64(l / Float64(k / l)) * Float64(cos(k) / Float64(sin(k) * (t ^ 3.0))));
	else
		tmp = Float64(Float64(2.0 / Float64(k * Float64(t * k))) * Float64(Float64(l * Float64(l * cos(k))) / (sin(k) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.58e-9)
		tmp = (l / (k / l)) * (cos(k) / (sin(k) * (t ^ 3.0)));
	else
		tmp = (2.0 / (k * (t * k))) * ((l * (l * cos(k))) / (sin(k) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.58e-9], N[(N[(l / N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.5799999999999999e-9

   1. Initial program 56.3%

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

      1. associate-/r* 56.5%

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

      2. associate-*l* 52.8%

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

      3. sqr-neg 52.8%

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

      4. associate-*l* 56.5%

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

      5. *-commutative 56.5%

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

      6. sqr-neg 56.5%

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

      7. associate-/r* 56.5%

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

   3. Simplified 56.3%

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

   4. Taylor expanded in k around 0 54.7%

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

      1. unpow2 54.7%

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

   6. Simplified 54.7%

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

   7. Taylor expanded in t around inf 55.3%

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

      1. times-frac 55.5%

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

      2. unpow2 55.5%

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

      3. associate-/l* 60.7%

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

      4. *-commutative 60.7%

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

   9. Simplified 60.7%

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

   if 1.5799999999999999e-9 < k

   1. Initial program 40.7%

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

      1. associate-/r* 40.7%

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

      2. associate-*l* 40.7%

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

      3. sqr-neg 40.7%

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

      4. associate-*l* 40.7%

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

      5. *-commutative 40.7%

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

      6. sqr-neg 40.7%

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

      7. associate-/r* 40.7%

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

   3. Simplified 40.7%

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

      1. associate-/l/ 40.7%

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

      2. associate-/r/ 40.7%

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

      3. add-cube-cbrt 40.7%

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

      4. pow3 40.7%

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

   5. Applied egg-rr 59.0%

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

      1. cbrt-div 61.2%

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

   7. Applied egg-rr 61.2%

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

   8. Taylor expanded in k around inf 72.8%

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

      1. associate-*r/ 72.8%

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

      2. associate-*r* 72.8%

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

      3. times-frac 72.8%

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

      4. unpow2 72.8%

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

      5. associate-*l* 74.4%

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

      6. unpow2 74.4%

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

      7. associate-*l* 74.4%

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

   10. Simplified 74.4%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.58 \cdot 10^{-9}:\\ \;\;\;\;\frac{\ell}{\frac{k}{\ell}} \cdot \frac{\cos k}{\sin k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \left(\ell \cdot \cos k\right)}{{\sin k}^{2}}\\ \end{array} \]

Alternative 10: 61.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.82 \cdot 10^{-9}:\\ \;\;\;\;\frac{\ell}{\frac{k}{\ell}} \cdot \frac{\cos k}{\sin k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \ell}{{k}^{3}} \cdot \frac{\cos k}{t \cdot k}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.82e-9)
   (* (/ l (/ k l)) (/ (cos k) (* (sin k) (pow t 3.0))))
   (* 2.0 (* (/ (* l l) (pow k 3.0)) (/ (cos k) (* t k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.82e-9) {
		tmp = (l / (k / l)) * (cos(k) / (sin(k) * pow(t, 3.0)));
	} else {
		tmp = 2.0 * (((l * l) / pow(k, 3.0)) * (cos(k) / (t * k)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.82d-9) then
        tmp = (l / (k / l)) * (cos(k) / (sin(k) * (t ** 3.0d0)))
    else
        tmp = 2.0d0 * (((l * l) / (k ** 3.0d0)) * (cos(k) / (t * k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.82e-9) {
		tmp = (l / (k / l)) * (Math.cos(k) / (Math.sin(k) * Math.pow(t, 3.0)));
	} else {
		tmp = 2.0 * (((l * l) / Math.pow(k, 3.0)) * (Math.cos(k) / (t * k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.82e-9:
		tmp = (l / (k / l)) * (math.cos(k) / (math.sin(k) * math.pow(t, 3.0)))
	else:
		tmp = 2.0 * (((l * l) / math.pow(k, 3.0)) * (math.cos(k) / (t * k)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.82e-9)
		tmp = Float64(Float64(l / Float64(k / l)) * Float64(cos(k) / Float64(sin(k) * (t ^ 3.0))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l * l) / (k ^ 3.0)) * Float64(cos(k) / Float64(t * k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.82e-9)
		tmp = (l / (k / l)) * (cos(k) / (sin(k) * (t ^ 3.0)));
	else
		tmp = 2.0 * (((l * l) / (k ^ 3.0)) * (cos(k) / (t * k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.82e-9], N[(N[(l / N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l * l), $MachinePrecision] / N[Power[k, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.8199999999999999e-9

   1. Initial program 56.3%

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

      1. associate-/r* 56.5%

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

      2. associate-*l* 52.8%

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

      3. sqr-neg 52.8%

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

      4. associate-*l* 56.5%

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

      5. *-commutative 56.5%

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

      6. sqr-neg 56.5%

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

      7. associate-/r* 56.5%

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

   3. Simplified 56.3%

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

   4. Taylor expanded in k around 0 54.7%

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

      1. unpow2 54.7%

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

   6. Simplified 54.7%

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

   7. Taylor expanded in t around inf 55.3%

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

      1. times-frac 55.5%

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

      2. unpow2 55.5%

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

      3. associate-/l* 60.7%

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

      4. *-commutative 60.7%

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

   9. Simplified 60.7%

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

   if 1.8199999999999999e-9 < k

   1. Initial program 40.7%

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

      1. associate-/r* 40.7%

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

      2. associate-*l* 40.7%

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

      3. sqr-neg 40.7%

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

      4. associate-*l* 40.7%

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

      5. *-commutative 40.7%

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

      6. sqr-neg 40.7%

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

      7. associate-/r* 40.7%

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

   3. Simplified 40.7%

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

   4. Taylor expanded in k around 0 35.4%

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

      1. unpow2 35.4%

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

   6. Simplified 35.4%

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

   7. Taylor expanded in k around inf 59.2%

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

      1. times-frac 59.2%

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

      2. unpow2 59.2%

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

      3. *-commutative 59.2%

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

   9. Simplified 59.2%

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

   10. Taylor expanded in k around 0 62.1%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.82 \cdot 10^{-9}:\\ \;\;\;\;\frac{\ell}{\frac{k}{\ell}} \cdot \frac{\cos k}{\sin k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \ell}{{k}^{3}} \cdot \frac{\cos k}{t \cdot k}\right)\\ \end{array} \]

Alternative 11: 59.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{\ell \cdot \left(2 \cdot \frac{\ell}{k \cdot \left(k \cdot {t}^{3}\right)}\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \ell}{{k}^{3}} \cdot \frac{\cos k}{t \cdot k}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 5.8e-9)
   (/ (* l (* 2.0 (/ l (* k (* k (pow t 3.0)))))) (+ 2.0 (pow (/ k t) 2.0)))
   (* 2.0 (* (/ (* l l) (pow k 3.0)) (/ (cos k) (* t k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.8e-9) {
		tmp = (l * (2.0 * (l / (k * (k * pow(t, 3.0)))))) / (2.0 + pow((k / t), 2.0));
	} else {
		tmp = 2.0 * (((l * l) / pow(k, 3.0)) * (cos(k) / (t * k)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 5.8d-9) then
        tmp = (l * (2.0d0 * (l / (k * (k * (t ** 3.0d0)))))) / (2.0d0 + ((k / t) ** 2.0d0))
    else
        tmp = 2.0d0 * (((l * l) / (k ** 3.0d0)) * (cos(k) / (t * k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.8e-9) {
		tmp = (l * (2.0 * (l / (k * (k * Math.pow(t, 3.0)))))) / (2.0 + Math.pow((k / t), 2.0));
	} else {
		tmp = 2.0 * (((l * l) / Math.pow(k, 3.0)) * (Math.cos(k) / (t * k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 5.8e-9:
		tmp = (l * (2.0 * (l / (k * (k * math.pow(t, 3.0)))))) / (2.0 + math.pow((k / t), 2.0))
	else:
		tmp = 2.0 * (((l * l) / math.pow(k, 3.0)) * (math.cos(k) / (t * k)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 5.8e-9)
		tmp = Float64(Float64(l * Float64(2.0 * Float64(l / Float64(k * Float64(k * (t ^ 3.0)))))) / Float64(2.0 + (Float64(k / t) ^ 2.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l * l) / (k ^ 3.0)) * Float64(cos(k) / Float64(t * k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 5.8e-9)
		tmp = (l * (2.0 * (l / (k * (k * (t ^ 3.0)))))) / (2.0 + ((k / t) ^ 2.0));
	else
		tmp = 2.0 * (((l * l) / (k ^ 3.0)) * (cos(k) / (t * k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 5.8e-9], N[(N[(l * N[(2.0 * N[(l / N[(k * N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l * l), $MachinePrecision] / N[Power[k, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.79999999999999982e-9

   1. Initial program 56.3%

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

      1. associate-/r* 56.5%

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

      2. associate-*l* 52.8%

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

      3. sqr-neg 52.8%

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

      4. associate-*l* 56.5%

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

      5. *-commutative 56.5%

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

      6. sqr-neg 56.5%

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

      7. associate-*l/ 56.5%

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

      8. associate-*r/ 56.6%

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

      9. associate-/r/ 56.5%

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

   3. Simplified 56.5%

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

      1. add-cube-cbrt 56.5%

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

      2. pow3 56.5%

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

      3. *-commutative 56.5%

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

      4. cbrt-prod 56.4%

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

      5. cbrt-prod 56.3%

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

      6. rem-cbrt-cube 64.2%

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

   5. Applied egg-rr 64.2%

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

      1. associate-*l/ 64.3%

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

      2. associate-*l* 64.3%

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

   7. Applied egg-rr 64.3%

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

      1. unpow2 64.3%

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

      2. *-commutative 64.3%

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

      3. associate-*r/ 64.2%

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

      4. unpow2 64.2%

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

      5. associate-*l* 70.3%

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

   9. Simplified 70.3%

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

   10. Taylor expanded in k around 0 54.6%

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

       1. unpow2 54.6%

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

       2. associate-*l* 60.2%

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

   12. Simplified 60.2%

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

   if 5.79999999999999982e-9 < k

   1. Initial program 40.7%

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

      1. associate-/r* 40.7%

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

      2. associate-*l* 40.7%

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

      3. sqr-neg 40.7%

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

      4. associate-*l* 40.7%

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

      5. *-commutative 40.7%

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

      6. sqr-neg 40.7%

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

      7. associate-/r* 40.7%

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

   3. Simplified 40.7%

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

   4. Taylor expanded in k around 0 35.4%

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

      1. unpow2 35.4%

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

   6. Simplified 35.4%

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

   7. Taylor expanded in k around inf 59.2%

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

      1. times-frac 59.2%

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

      2. unpow2 59.2%

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

      3. *-commutative 59.2%

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

   9. Simplified 59.2%

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

   10. Taylor expanded in k around 0 62.1%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{\ell \cdot \left(2 \cdot \frac{\ell}{k \cdot \left(k \cdot {t}^{3}\right)}\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \ell}{{k}^{3}} \cdot \frac{\cos k}{t \cdot k}\right)\\ \end{array} \]

Alternative 12: 59.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-272}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{elif}\;k \leq 4.2 \cdot 10^{-193}:\\ \;\;\;\;\frac{\ell \cdot \ell}{k \cdot \left(k \cdot {t}^{3}\right)}\\ \mathbf{elif}\;k \leq 1.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \ell}{{k}^{3}} \cdot \frac{\cos k}{t \cdot k}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.2e-272)
   (* 2.0 (* (/ l t) (/ l (pow k 4.0))))
   (if (<= k 4.2e-193)
     (/ (* l l) (* k (* k (pow t 3.0))))
     (if (<= k 1.8e-9)
       (* (/ (/ l k) k) (/ l (pow t 3.0)))
       (* 2.0 (* (/ (* l l) (pow k 3.0)) (/ (cos k) (* t k))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.2e-272) {
		tmp = 2.0 * ((l / t) * (l / pow(k, 4.0)));
	} else if (k <= 4.2e-193) {
		tmp = (l * l) / (k * (k * pow(t, 3.0)));
	} else if (k <= 1.8e-9) {
		tmp = ((l / k) / k) * (l / pow(t, 3.0));
	} else {
		tmp = 2.0 * (((l * l) / pow(k, 3.0)) * (cos(k) / (t * k)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.2d-272) then
        tmp = 2.0d0 * ((l / t) * (l / (k ** 4.0d0)))
    else if (k <= 4.2d-193) then
        tmp = (l * l) / (k * (k * (t ** 3.0d0)))
    else if (k <= 1.8d-9) then
        tmp = ((l / k) / k) * (l / (t ** 3.0d0))
    else
        tmp = 2.0d0 * (((l * l) / (k ** 3.0d0)) * (cos(k) / (t * k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.2e-272) {
		tmp = 2.0 * ((l / t) * (l / Math.pow(k, 4.0)));
	} else if (k <= 4.2e-193) {
		tmp = (l * l) / (k * (k * Math.pow(t, 3.0)));
	} else if (k <= 1.8e-9) {
		tmp = ((l / k) / k) * (l / Math.pow(t, 3.0));
	} else {
		tmp = 2.0 * (((l * l) / Math.pow(k, 3.0)) * (Math.cos(k) / (t * k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.2e-272:
		tmp = 2.0 * ((l / t) * (l / math.pow(k, 4.0)))
	elif k <= 4.2e-193:
		tmp = (l * l) / (k * (k * math.pow(t, 3.0)))
	elif k <= 1.8e-9:
		tmp = ((l / k) / k) * (l / math.pow(t, 3.0))
	else:
		tmp = 2.0 * (((l * l) / math.pow(k, 3.0)) * (math.cos(k) / (t * k)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.2e-272)
		tmp = Float64(2.0 * Float64(Float64(l / t) * Float64(l / (k ^ 4.0))));
	elseif (k <= 4.2e-193)
		tmp = Float64(Float64(l * l) / Float64(k * Float64(k * (t ^ 3.0))));
	elseif (k <= 1.8e-9)
		tmp = Float64(Float64(Float64(l / k) / k) * Float64(l / (t ^ 3.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l * l) / (k ^ 3.0)) * Float64(cos(k) / Float64(t * k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.2e-272)
		tmp = 2.0 * ((l / t) * (l / (k ^ 4.0)));
	elseif (k <= 4.2e-193)
		tmp = (l * l) / (k * (k * (t ^ 3.0)));
	elseif (k <= 1.8e-9)
		tmp = ((l / k) / k) * (l / (t ^ 3.0));
	else
		tmp = 2.0 * (((l * l) / (k ^ 3.0)) * (cos(k) / (t * k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.2e-272], N[(2.0 * N[(N[(l / t), $MachinePrecision] * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4.2e-193], N[(N[(l * l), $MachinePrecision] / N[(k * N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.8e-9], N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l * l), $MachinePrecision] / N[Power[k, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if k < 1.19999999999999995e-272

   1. Initial program 50.9%

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

      1. associate-/r* 51.1%

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

      2. associate-*l* 47.2%

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

      3. sqr-neg 47.2%

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

      4. associate-*l* 51.1%

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

      5. *-commutative 51.1%

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

      6. sqr-neg 51.1%

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

      7. associate-/r* 51.1%

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

   3. Simplified 50.9%

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

   4. Taylor expanded in k around 0 48.7%

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

      1. unpow2 48.7%

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

   6. Simplified 48.7%

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

   7. Taylor expanded in k around inf 52.3%

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

      1. times-frac 52.6%

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

      2. unpow2 52.6%

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

      3. *-commutative 52.6%

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

   9. Simplified 52.6%

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

   10. Taylor expanded in k around 0 52.0%

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

       1. unpow2 52.0%

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

       2. *-commutative 52.0%

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

       3. times-frac 53.5%

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

   12. Simplified 53.5%

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

   if 1.19999999999999995e-272 < k < 4.1999999999999999e-193

   1. Initial program 57.2%

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

      1. associate-/r* 57.2%

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

      2. associate-*l* 50.0%

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

      3. sqr-neg 50.0%

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

      4. associate-*l* 57.2%

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

      5. *-commutative 57.2%

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

      6. sqr-neg 57.2%

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

      7. associate-/r* 57.2%

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

   3. Simplified 56.6%

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

      1. associate-/l/ 56.6%

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

      2. associate-/r/ 57.2%

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

      3. add-cube-cbrt 57.2%

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

      4. pow3 57.2%

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

   5. Applied egg-rr 83.3%

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

   6. Taylor expanded in k around 0 50.0%

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

      1. unpow2 50.0%

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

      2. associate-*l* 56.7%

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

      3. unpow2 56.7%

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

   8. Simplified 56.7%

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

   if 4.1999999999999999e-193 < k < 1.8e-9

   1. Initial program 83.8%

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

      1. associate-/r* 83.8%

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

      2. associate-*l* 83.9%

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

      3. sqr-neg 83.9%

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

      4. associate-*l* 83.8%

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

      5. *-commutative 83.8%

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

      6. sqr-neg 83.8%

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

      7. associate-/r* 83.8%

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

   3. Simplified 83.9%

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

      1. associate-/l/ 83.9%

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

      2. associate-/r/ 83.8%

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

      3. add-cube-cbrt 83.5%

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

      4. pow3 83.5%

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

   5. Applied egg-rr 88.7%

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

   6. Taylor expanded in k around 0 85.3%

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

      1. associate-/r* 87.5%

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

      2. unpow2 87.5%

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

      3. associate-*l/ 88.8%

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

      4. unpow2 88.8%

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

      5. associate-*r/ 88.9%

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

   8. Simplified 88.9%

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

      1. add-log-exp 60.0%

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

   10. Applied egg-rr 60.0%

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

       1. add-log-exp 88.9%

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

       2. associate-/r* 89.0%

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

   12. Applied egg-rr 89.0%

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

   if 1.8e-9 < k

   1. Initial program 40.7%

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

      1. associate-/r* 40.7%

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

      2. associate-*l* 40.7%

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

      3. sqr-neg 40.7%

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

      4. associate-*l* 40.7%

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

      5. *-commutative 40.7%

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

      6. sqr-neg 40.7%

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

      7. associate-/r* 40.7%

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

   3. Simplified 40.7%

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

   4. Taylor expanded in k around 0 35.4%

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

      1. unpow2 35.4%

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

   6. Simplified 35.4%

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

   7. Taylor expanded in k around inf 59.2%

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

      1. times-frac 59.2%

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

      2. unpow2 59.2%

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

      3. *-commutative 59.2%

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

   9. Simplified 59.2%

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

   10. Taylor expanded in k around 0 62.1%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-272}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{elif}\;k \leq 4.2 \cdot 10^{-193}:\\ \;\;\;\;\frac{\ell \cdot \ell}{k \cdot \left(k \cdot {t}^{3}\right)}\\ \mathbf{elif}\;k \leq 1.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \ell}{{k}^{3}} \cdot \frac{\cos k}{t \cdot k}\right)\\ \end{array} \]

Alternative 13: 64.4% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{-63}:\\ \;\;\;\;\frac{\ell \cdot \ell}{k \cdot \left(k \cdot {t}^{3}\right)}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-100}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \left(\ell \cdot \frac{1}{{k}^{4}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{{t}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -1.2e-63)
   (/ (* l l) (* k (* k (pow t 3.0))))
   (if (<= t 3.2e-100)
     (* 2.0 (* (/ l t) (* l (/ 1.0 (pow k 4.0)))))
     (* (/ (/ l k) k) (/ l (pow t 3.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.2e-63) {
		tmp = (l * l) / (k * (k * pow(t, 3.0)));
	} else if (t <= 3.2e-100) {
		tmp = 2.0 * ((l / t) * (l * (1.0 / pow(k, 4.0))));
	} else {
		tmp = ((l / k) / k) * (l / pow(t, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-1.2d-63)) then
        tmp = (l * l) / (k * (k * (t ** 3.0d0)))
    else if (t <= 3.2d-100) then
        tmp = 2.0d0 * ((l / t) * (l * (1.0d0 / (k ** 4.0d0))))
    else
        tmp = ((l / k) / k) * (l / (t ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.2e-63) {
		tmp = (l * l) / (k * (k * Math.pow(t, 3.0)));
	} else if (t <= 3.2e-100) {
		tmp = 2.0 * ((l / t) * (l * (1.0 / Math.pow(k, 4.0))));
	} else {
		tmp = ((l / k) / k) * (l / Math.pow(t, 3.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -1.2e-63:
		tmp = (l * l) / (k * (k * math.pow(t, 3.0)))
	elif t <= 3.2e-100:
		tmp = 2.0 * ((l / t) * (l * (1.0 / math.pow(k, 4.0))))
	else:
		tmp = ((l / k) / k) * (l / math.pow(t, 3.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -1.2e-63)
		tmp = Float64(Float64(l * l) / Float64(k * Float64(k * (t ^ 3.0))));
	elseif (t <= 3.2e-100)
		tmp = Float64(2.0 * Float64(Float64(l / t) * Float64(l * Float64(1.0 / (k ^ 4.0)))));
	else
		tmp = Float64(Float64(Float64(l / k) / k) * Float64(l / (t ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -1.2e-63)
		tmp = (l * l) / (k * (k * (t ^ 3.0)));
	elseif (t <= 3.2e-100)
		tmp = 2.0 * ((l / t) * (l * (1.0 / (k ^ 4.0))));
	else
		tmp = ((l / k) / k) * (l / (t ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -1.2e-63], N[(N[(l * l), $MachinePrecision] / N[(k * N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.2e-100], N[(2.0 * N[(N[(l / t), $MachinePrecision] * N[(l * N[(1.0 / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.2e-63

   1. Initial program 66.3%

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

      1. associate-/r* 66.6%

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

      2. associate-*l* 63.5%

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

      3. sqr-neg 63.5%

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

      4. associate-*l* 66.6%

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

      5. *-commutative 66.6%

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

      6. sqr-neg 66.6%

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

      7. associate-/r* 66.6%

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

   3. Simplified 66.4%

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

      1. associate-/l/ 66.4%

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

      2. associate-/r/ 66.6%

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

      3. add-cube-cbrt 66.5%

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

      4. pow3 66.5%

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

   5. Applied egg-rr 78.2%

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

   6. Taylor expanded in k around 0 60.0%

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

      1. unpow2 60.0%

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

      2. associate-*l* 63.1%

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

      3. unpow2 63.1%

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

   8. Simplified 63.1%

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

   if -1.2e-63 < t < 3.20000000000000017e-100

   1. Initial program 32.4%

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

      1. associate-/r* 32.4%

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

      2. associate-*l* 32.4%

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

      3. sqr-neg 32.4%

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

      4. associate-*l* 32.4%

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

      5. *-commutative 32.4%

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

      6. sqr-neg 32.4%

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

      7. associate-/r* 32.4%

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

   3. Simplified 32.4%

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

   4. Taylor expanded in k around 0 32.4%

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

      1. unpow2 32.4%

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

   6. Simplified 32.4%

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

   7. Taylor expanded in k around inf 65.3%

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

      1. times-frac 65.6%

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

      2. unpow2 65.6%

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

      3. *-commutative 65.6%

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

   9. Simplified 65.6%

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

   10. Taylor expanded in k around 0 63.1%

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

       1. unpow2 63.1%

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

       2. *-commutative 63.1%

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

       3. times-frac 66.7%

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

   12. Simplified 66.7%

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

       1. div-inv 66.7%

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

   14. Applied egg-rr 66.7%

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

   if 3.20000000000000017e-100 < t

   1. Initial program 59.3%

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

      1. associate-/r* 59.3%

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

      2. associate-*l* 54.5%

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

      3. sqr-neg 54.5%

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

      4. associate-*l* 59.3%

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

      5. *-commutative 59.3%

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

      6. sqr-neg 59.3%

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

      7. associate-/r* 59.3%

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

   3. Simplified 59.2%

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

      1. associate-/l/ 59.2%

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

      2. associate-/r/ 59.3%

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

      3. add-cube-cbrt 59.3%

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

      4. pow3 59.2%

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

   5. Applied egg-rr 74.6%

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

   6. Taylor expanded in k around 0 52.3%

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

      1. associate-/r* 51.3%

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

      2. unpow2 51.3%

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

      3. associate-*l/ 52.1%

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

      4. unpow2 52.1%

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

      5. associate-*r/ 56.4%

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

   8. Simplified 56.4%

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

      1. add-log-exp 45.3%

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

   10. Applied egg-rr 45.3%

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

       1. add-log-exp 56.4%

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

       2. associate-/r* 58.8%

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

   12. Applied egg-rr 58.8%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{-63}:\\ \;\;\;\;\frac{\ell \cdot \ell}{k \cdot \left(k \cdot {t}^{3}\right)}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-100}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \left(\ell \cdot \frac{1}{{k}^{4}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{{t}^{3}}\\ \end{array} \]

Alternative 14: 58.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-157} \lor \neg \left(k \leq 4.8 \cdot 10^{-9}\right):\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= k 2e-157) (not (<= k 4.8e-9)))
   (* 2.0 (* (/ l t) (/ l (pow k 4.0))))
   (* (/ l (* k k)) (/ l (pow t 3.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((k <= 2e-157) || !(k <= 4.8e-9)) {
		tmp = 2.0 * ((l / t) * (l / pow(k, 4.0)));
	} else {
		tmp = (l / (k * k)) * (l / pow(t, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((k <= 2d-157) .or. (.not. (k <= 4.8d-9))) then
        tmp = 2.0d0 * ((l / t) * (l / (k ** 4.0d0)))
    else
        tmp = (l / (k * k)) * (l / (t ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((k <= 2e-157) || !(k <= 4.8e-9)) {
		tmp = 2.0 * ((l / t) * (l / Math.pow(k, 4.0)));
	} else {
		tmp = (l / (k * k)) * (l / Math.pow(t, 3.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (k <= 2e-157) or not (k <= 4.8e-9):
		tmp = 2.0 * ((l / t) * (l / math.pow(k, 4.0)))
	else:
		tmp = (l / (k * k)) * (l / math.pow(t, 3.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((k <= 2e-157) || !(k <= 4.8e-9))
		tmp = Float64(2.0 * Float64(Float64(l / t) * Float64(l / (k ^ 4.0))));
	else
		tmp = Float64(Float64(l / Float64(k * k)) * Float64(l / (t ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((k <= 2e-157) || ~((k <= 4.8e-9)))
		tmp = 2.0 * ((l / t) * (l / (k ^ 4.0)));
	else
		tmp = (l / (k * k)) * (l / (t ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[k, 2e-157], N[Not[LessEqual[k, 4.8e-9]], $MachinePrecision]], N[(2.0 * N[(N[(l / t), $MachinePrecision] * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.99999999999999989e-157 or 4.8e-9 < k

   1. Initial program 48.6%

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

      1. associate-/r* 48.7%

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

      2. associate-*l* 45.8%

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

      3. sqr-neg 45.8%

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

      4. associate-*l* 48.7%

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

      5. *-commutative 48.7%

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

      6. sqr-neg 48.7%

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

      7. associate-/r* 48.7%

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

   3. Simplified 48.6%

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

   4. Taylor expanded in k around 0 45.6%

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

      1. unpow2 45.6%

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

   6. Simplified 45.6%

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

   7. Taylor expanded in k around inf 54.8%

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

      1. times-frac 54.9%

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

      2. unpow2 54.9%

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

      3. *-commutative 54.9%

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

   9. Simplified 54.9%

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

   10. Taylor expanded in k around 0 54.1%

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

       1. unpow2 54.1%

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

       2. *-commutative 54.1%

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

       3. times-frac 55.5%

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

   12. Simplified 55.5%

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

   if 1.99999999999999989e-157 < k < 4.8e-9

   1. Initial program 82.5%

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

      1. associate-/r* 82.5%

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

      2. associate-*l* 82.6%

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

      3. sqr-neg 82.6%

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

      4. associate-*l* 82.5%

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

      5. *-commutative 82.5%

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

      6. sqr-neg 82.5%

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

      7. associate-/r* 82.5%

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

   3. Simplified 82.6%

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

      1. associate-/l/ 82.6%

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

      2. associate-/r/ 82.5%

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

      3. add-cube-cbrt 82.2%

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

      4. pow3 82.2%

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

   5. Applied egg-rr 87.8%

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

   6. Taylor expanded in k around 0 84.1%

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

      1. associate-/r* 86.5%

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

      2. unpow2 86.5%

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

      3. associate-*l/ 87.9%

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

      4. unpow2 87.9%

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

      5. associate-*r/ 88.0%

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

   8. Simplified 88.0%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-157} \lor \neg \left(k \leq 4.8 \cdot 10^{-9}\right):\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}\\ \end{array} \]

Alternative 15: 65.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{-66} \lor \neg \left(t \leq 1.95 \cdot 10^{-94}\right):\\ \;\;\;\;\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -7e-66) (not (<= t 1.95e-94)))
   (* (/ (/ l k) k) (/ l (pow t 3.0)))
   (* 2.0 (* (/ l t) (/ l (pow k 4.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -7e-66) || !(t <= 1.95e-94)) {
		tmp = ((l / k) / k) * (l / pow(t, 3.0));
	} else {
		tmp = 2.0 * ((l / t) * (l / pow(k, 4.0)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-7d-66)) .or. (.not. (t <= 1.95d-94))) then
        tmp = ((l / k) / k) * (l / (t ** 3.0d0))
    else
        tmp = 2.0d0 * ((l / t) * (l / (k ** 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -7e-66) || !(t <= 1.95e-94)) {
		tmp = ((l / k) / k) * (l / Math.pow(t, 3.0));
	} else {
		tmp = 2.0 * ((l / t) * (l / Math.pow(k, 4.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -7e-66) or not (t <= 1.95e-94):
		tmp = ((l / k) / k) * (l / math.pow(t, 3.0))
	else:
		tmp = 2.0 * ((l / t) * (l / math.pow(k, 4.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -7e-66) || !(t <= 1.95e-94))
		tmp = Float64(Float64(Float64(l / k) / k) * Float64(l / (t ^ 3.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(l / t) * Float64(l / (k ^ 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -7e-66) || ~((t <= 1.95e-94)))
		tmp = ((l / k) / k) * (l / (t ^ 3.0));
	else
		tmp = 2.0 * ((l / t) * (l / (k ^ 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -7e-66], N[Not[LessEqual[t, 1.95e-94]], $MachinePrecision]], N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / t), $MachinePrecision] * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -7.0000000000000001e-66 or 1.9500000000000001e-94 < t

   1. Initial program 62.5%

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

      1. associate-/r* 62.7%

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

      2. associate-*l* 58.7%

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

      3. sqr-neg 58.7%

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

      4. associate-*l* 62.7%

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

      5. *-commutative 62.7%

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

      6. sqr-neg 62.7%

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

      7. associate-/r* 62.7%

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

   3. Simplified 62.5%

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

      1. associate-/l/ 62.5%

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

      2. associate-/r/ 62.7%

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

      3. add-cube-cbrt 62.6%

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

      4. pow3 62.6%

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

   5. Applied egg-rr 76.2%

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

   6. Taylor expanded in k around 0 55.9%

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

      1. associate-/r* 55.7%

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

      2. unpow2 55.7%

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

      3. associate-*l/ 57.1%

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

      4. unpow2 57.1%

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

      5. associate-*r/ 58.8%

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

   8. Simplified 58.8%

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

      1. add-log-exp 49.1%

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

   10. Applied egg-rr 49.1%

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

       1. add-log-exp 58.8%

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

       2. associate-/r* 60.7%

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

   12. Applied egg-rr 60.7%

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

   if -7.0000000000000001e-66 < t < 1.9500000000000001e-94

   1. Initial program 32.4%

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

      1. associate-/r* 32.4%

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

      2. associate-*l* 32.4%

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

      3. sqr-neg 32.4%

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

      4. associate-*l* 32.4%

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

      5. *-commutative 32.4%

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

      6. sqr-neg 32.4%

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

      7. associate-/r* 32.4%

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

   3. Simplified 32.4%

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

   4. Taylor expanded in k around 0 32.4%

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

      1. unpow2 32.4%

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

   6. Simplified 32.4%

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

   7. Taylor expanded in k around inf 65.3%

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

      1. times-frac 65.6%

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

      2. unpow2 65.6%

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

      3. *-commutative 65.6%

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

   9. Simplified 65.6%

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

   10. Taylor expanded in k around 0 63.1%

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

       1. unpow2 63.1%

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

       2. *-commutative 63.1%

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

       3. times-frac 66.7%

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

   12. Simplified 66.7%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{-66} \lor \neg \left(t \leq 1.95 \cdot 10^{-94}\right):\\ \;\;\;\;\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \end{array} \]

Alternative 16: 64.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6.8 \cdot 10^{-61}:\\ \;\;\;\;\frac{\ell \cdot \ell}{k \cdot \left(k \cdot {t}^{3}\right)}\\ \mathbf{elif}\;t \leq 1.16 \cdot 10^{-92}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{{t}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -6.8e-61)
   (/ (* l l) (* k (* k (pow t 3.0))))
   (if (<= t 1.16e-92)
     (* 2.0 (* (/ l t) (/ l (pow k 4.0))))
     (* (/ (/ l k) k) (/ l (pow t 3.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -6.8e-61) {
		tmp = (l * l) / (k * (k * pow(t, 3.0)));
	} else if (t <= 1.16e-92) {
		tmp = 2.0 * ((l / t) * (l / pow(k, 4.0)));
	} else {
		tmp = ((l / k) / k) * (l / pow(t, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-6.8d-61)) then
        tmp = (l * l) / (k * (k * (t ** 3.0d0)))
    else if (t <= 1.16d-92) then
        tmp = 2.0d0 * ((l / t) * (l / (k ** 4.0d0)))
    else
        tmp = ((l / k) / k) * (l / (t ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -6.8e-61) {
		tmp = (l * l) / (k * (k * Math.pow(t, 3.0)));
	} else if (t <= 1.16e-92) {
		tmp = 2.0 * ((l / t) * (l / Math.pow(k, 4.0)));
	} else {
		tmp = ((l / k) / k) * (l / Math.pow(t, 3.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -6.8e-61:
		tmp = (l * l) / (k * (k * math.pow(t, 3.0)))
	elif t <= 1.16e-92:
		tmp = 2.0 * ((l / t) * (l / math.pow(k, 4.0)))
	else:
		tmp = ((l / k) / k) * (l / math.pow(t, 3.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -6.8e-61)
		tmp = Float64(Float64(l * l) / Float64(k * Float64(k * (t ^ 3.0))));
	elseif (t <= 1.16e-92)
		tmp = Float64(2.0 * Float64(Float64(l / t) * Float64(l / (k ^ 4.0))));
	else
		tmp = Float64(Float64(Float64(l / k) / k) * Float64(l / (t ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -6.8e-61)
		tmp = (l * l) / (k * (k * (t ^ 3.0)));
	elseif (t <= 1.16e-92)
		tmp = 2.0 * ((l / t) * (l / (k ^ 4.0)));
	else
		tmp = ((l / k) / k) * (l / (t ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -6.8e-61], N[(N[(l * l), $MachinePrecision] / N[(k * N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.16e-92], N[(2.0 * N[(N[(l / t), $MachinePrecision] * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -6.7999999999999996e-61

   1. Initial program 66.3%

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

      1. associate-/r* 66.6%

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

      2. associate-*l* 63.5%

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

      3. sqr-neg 63.5%

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

      4. associate-*l* 66.6%

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

      5. *-commutative 66.6%

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

      6. sqr-neg 66.6%

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

      7. associate-/r* 66.6%

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

   3. Simplified 66.4%

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

      1. associate-/l/ 66.4%

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. associate-/r/ 66.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. add-cube-cbrt 66.5%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}\right) \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. pow3 66.5%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 78.2%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   6. Taylor expanded in k around 0 60.0%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   7. Step-by-step derivation

      1. unpow2 60.0%

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      2. associate-*l* 63.1%

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      3. unpow2 63.1%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{k \cdot \left(k \cdot {t}^{3}\right)} \]

   8. Simplified 63.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

   if -6.7999999999999996e-61 < t < 1.1599999999999999e-92

   1. Initial program 32.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 32.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 32.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 32.4%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 32.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 32.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 32.4%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-/r* 32.4%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 32.4%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around 0 32.4%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\color{blue}{\frac{{\ell}^{2}}{k}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 32.4%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\color{blue}{\ell \cdot \ell}}{k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   6. Simplified 32.4%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\color{blue}{\frac{\ell \cdot \ell}{k}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   7. Taylor expanded in k around inf 65.3%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{3} \cdot \left(t \cdot \sin k\right)}} \]
   8. Step-by-step derivation

      1. times-frac 65.6%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{3}} \cdot \frac{\cos k}{t \cdot \sin k}\right)} \]

      2. unpow2 65.6%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{3}} \cdot \frac{\cos k}{t \cdot \sin k}\right) \]

      3. *-commutative 65.6%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{{k}^{3}} \cdot \frac{\cos k}{\color{blue}{\sin k \cdot t}}\right) \]

   9. Simplified 65.6%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\ell \cdot \ell}{{k}^{3}} \cdot \frac{\cos k}{\sin k \cdot t}\right)} \]

   10. Taylor expanded in k around 0 63.1%

       \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   11. Step-by-step derivation

       1. unpow2 63.1%

          \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

       2. *-commutative 63.1%

          \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

       3. times-frac 66.7%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

   12. Simplified 66.7%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

   if 1.1599999999999999e-92 < t

   1. Initial program 59.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 59.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 54.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 54.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 59.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 59.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 59.3%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-/r* 59.3%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 59.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 59.2%

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. associate-/r/ 59.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. add-cube-cbrt 59.3%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}\right) \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. pow3 59.2%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 74.6%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   6. Taylor expanded in k around 0 52.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   7. Step-by-step derivation

      1. associate-/r* 51.3%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}}} \]

      2. unpow2 51.3%

         \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{{t}^{3}} \]

      3. associate-*l/ 52.1%

         \[\leadsto \frac{\color{blue}{\frac{\ell}{{k}^{2}} \cdot \ell}}{{t}^{3}} \]

      4. unpow2 52.1%

         \[\leadsto \frac{\frac{\ell}{\color{blue}{k \cdot k}} \cdot \ell}{{t}^{3}} \]

      5. associate-*r/ 56.4%

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}} \]

   8. Simplified 56.4%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}} \]
   9. Step-by-step derivation

      1. add-log-exp 45.3%

         \[\leadsto \color{blue}{\log \left(e^{\frac{\ell}{k \cdot k}}\right)} \cdot \frac{\ell}{{t}^{3}} \]

   10. Applied egg-rr 45.3%

       \[\leadsto \color{blue}{\log \left(e^{\frac{\ell}{k \cdot k}}\right)} \cdot \frac{\ell}{{t}^{3}} \]
   11. Step-by-step derivation

       1. add-log-exp 56.4%

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot k}} \cdot \frac{\ell}{{t}^{3}} \]

       2. associate-/r* 58.8%

          \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{k}} \cdot \frac{\ell}{{t}^{3}} \]

   12. Applied egg-rr 58.8%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{k}} \cdot \frac{\ell}{{t}^{3}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.8 \cdot 10^{-61}:\\ \;\;\;\;\frac{\ell \cdot \ell}{k \cdot \left(k \cdot {t}^{3}\right)}\\ \mathbf{elif}\;t \leq 1.16 \cdot 10^{-92}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{{t}^{3}}\\ \end{array} \]

Alternative 17: 58.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{+108}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot -0.5}{{t}^{5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 9.5e+108)
   (* 2.0 (* (/ l t) (/ l (pow k 4.0))))
   (/ (* (* l l) -0.5) (pow t 5.0))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 9.5e+108) {
		tmp = 2.0 * ((l / t) * (l / pow(k, 4.0)));
	} else {
		tmp = ((l * l) * -0.5) / pow(t, 5.0);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 9.5d+108) then
        tmp = 2.0d0 * ((l / t) * (l / (k ** 4.0d0)))
    else
        tmp = ((l * l) * (-0.5d0)) / (t ** 5.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 9.5e+108) {
		tmp = 2.0 * ((l / t) * (l / Math.pow(k, 4.0)));
	} else {
		tmp = ((l * l) * -0.5) / Math.pow(t, 5.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 9.5e+108:
		tmp = 2.0 * ((l / t) * (l / math.pow(k, 4.0)))
	else:
		tmp = ((l * l) * -0.5) / math.pow(t, 5.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 9.5e+108)
		tmp = Float64(2.0 * Float64(Float64(l / t) * Float64(l / (k ^ 4.0))));
	else
		tmp = Float64(Float64(Float64(l * l) * -0.5) / (t ^ 5.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 9.5e+108)
		tmp = 2.0 * ((l / t) * (l / (k ^ 4.0)));
	else
		tmp = ((l * l) * -0.5) / (t ^ 5.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 9.5e+108], N[(2.0 * N[(N[(l / t), $MachinePrecision] * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] * -0.5), $MachinePrecision] / N[Power[t, 5.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 9.50000000000000097e108

   1. Initial program 53.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 53.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 52.2%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 52.2%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 53.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 53.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 53.8%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-/r* 53.8%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 53.7%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around 0 50.6%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\color{blue}{\frac{{\ell}^{2}}{k}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 50.6%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\color{blue}{\ell \cdot \ell}}{k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   6. Simplified 50.6%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\color{blue}{\frac{\ell \cdot \ell}{k}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   7. Taylor expanded in k around inf 58.9%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{3} \cdot \left(t \cdot \sin k\right)}} \]
   8. Step-by-step derivation

      1. times-frac 59.0%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{3}} \cdot \frac{\cos k}{t \cdot \sin k}\right)} \]

      2. unpow2 59.0%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{3}} \cdot \frac{\cos k}{t \cdot \sin k}\right) \]

      3. *-commutative 59.0%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{{k}^{3}} \cdot \frac{\cos k}{\color{blue}{\sin k \cdot t}}\right) \]

   9. Simplified 59.0%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\ell \cdot \ell}{{k}^{3}} \cdot \frac{\cos k}{\sin k \cdot t}\right)} \]

   10. Taylor expanded in k around 0 58.3%

       \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   11. Step-by-step derivation

       1. unpow2 58.3%

          \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

       2. *-commutative 58.3%

          \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

       3. times-frac 60.1%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

   12. Simplified 60.1%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

   if 9.50000000000000097e108 < t

   1. Initial program 41.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 41.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 32.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 32.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 41.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 41.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 41.6%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-/r* 41.6%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 41.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 41.2%

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. associate-/r/ 41.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. add-cube-cbrt 41.6%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}\right) \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. pow3 41.6%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 63.2%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   6. Taylor expanded in k around 0 32.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{{\ell}^{2} \cdot \left(0.3333333333333333 + \frac{1}{{t}^{2}}\right)}{{t}^{3}} + \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   7. Step-by-step derivation

      1. fma-def 32.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{{\ell}^{2} \cdot \left(0.3333333333333333 + \frac{1}{{t}^{2}}\right)}{{t}^{3}}, \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)} \]

      2. *-commutative 32.9%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\color{blue}{\left(0.3333333333333333 + \frac{1}{{t}^{2}}\right) \cdot {\ell}^{2}}}{{t}^{3}}, \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right) \]

      3. associate-/l* 32.9%

         \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\frac{0.3333333333333333 + \frac{1}{{t}^{2}}}{\frac{{t}^{3}}{{\ell}^{2}}}}, \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right) \]

      4. unpow2 32.9%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{0.3333333333333333 + \frac{1}{\color{blue}{t \cdot t}}}{\frac{{t}^{3}}{{\ell}^{2}}}, \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right) \]

      5. unpow2 32.9%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{0.3333333333333333 + \frac{1}{t \cdot t}}{\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}}, \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right) \]

      6. associate-/r* 32.8%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{0.3333333333333333 + \frac{1}{t \cdot t}}{\frac{{t}^{3}}{\ell \cdot \ell}}, \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}}}\right) \]

      7. unpow2 32.8%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{0.3333333333333333 + \frac{1}{t \cdot t}}{\frac{{t}^{3}}{\ell \cdot \ell}}, \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{{t}^{3}}\right) \]

      8. associate-*l/ 32.8%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{0.3333333333333333 + \frac{1}{t \cdot t}}{\frac{{t}^{3}}{\ell \cdot \ell}}, \frac{\color{blue}{\frac{\ell}{{k}^{2}} \cdot \ell}}{{t}^{3}}\right) \]

      9. unpow2 32.8%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{0.3333333333333333 + \frac{1}{t \cdot t}}{\frac{{t}^{3}}{\ell \cdot \ell}}, \frac{\frac{\ell}{\color{blue}{k \cdot k}} \cdot \ell}{{t}^{3}}\right) \]

      10. associate-*r/ 32.9%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{0.3333333333333333 + \frac{1}{t \cdot t}}{\frac{{t}^{3}}{\ell \cdot \ell}}, \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}}\right) \]

   8. Simplified 32.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{0.3333333333333333 + \frac{1}{t \cdot t}}{\frac{{t}^{3}}{\ell \cdot \ell}}, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}\right)} \]

   9. Taylor expanded in t around 0 41.6%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{{\ell}^{2}}{{t}^{5}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 41.6%

          \[\leadsto \color{blue}{\frac{-0.5 \cdot {\ell}^{2}}{{t}^{5}}} \]

       2. unpow2 41.6%

          \[\leadsto \frac{-0.5 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{t}^{5}} \]

   11. Simplified 41.6%

       \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\ell \cdot \ell\right)}{{t}^{5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{+108}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot -0.5}{{t}^{5}}\\ \end{array} \]

Alternative 18: 28.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\ell \cdot \ell\right) \cdot -0.5}{{t}^{5}} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (/ (* (* l l) -0.5) (pow t 5.0)))

C

double code(double t, double l, double k) {
	return ((l * l) * -0.5) / pow(t, 5.0);
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = ((l * l) * (-0.5d0)) / (t ** 5.0d0)
end function

Java

public static double code(double t, double l, double k) {
	return ((l * l) * -0.5) / Math.pow(t, 5.0);
}

Python

def code(t, l, k):
	return ((l * l) * -0.5) / math.pow(t, 5.0)

Julia

function code(t, l, k)
	return Float64(Float64(Float64(l * l) * -0.5) / (t ^ 5.0))
end

MATLAB

function tmp = code(t, l, k)
	tmp = ((l * l) * -0.5) / (t ^ 5.0);
end

Wolfram

code[t_, l_, k_] := N[(N[(N[(l * l), $MachinePrecision] * -0.5), $MachinePrecision] / N[Power[t, 5.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 52.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

   2. associate-*l* 49.4%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. sqr-neg 49.4%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   4. associate-*l* 52.0%

      \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   5. *-commutative 52.0%

      \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   6. sqr-neg 52.0%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   7. associate-/r* 52.0%

      \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

3. Simplified 51.9%

   \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
4. Step-by-step derivation

   1. associate-/l/ 51.9%

      \[\leadsto \frac{\color{blue}{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   2. associate-/r/ 52.0%

      \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   3. add-cube-cbrt 52.0%

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}\right) \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   4. pow3 52.0%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

5. Applied egg-rr 69.8%

   \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

6. Taylor expanded in k around 0 25.8%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{{\ell}^{2} \cdot \left(0.3333333333333333 + \frac{1}{{t}^{2}}\right)}{{t}^{3}} + \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
7. Step-by-step derivation

   1. fma-def 25.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{{\ell}^{2} \cdot \left(0.3333333333333333 + \frac{1}{{t}^{2}}\right)}{{t}^{3}}, \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)} \]

   2. *-commutative 25.8%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{\color{blue}{\left(0.3333333333333333 + \frac{1}{{t}^{2}}\right) \cdot {\ell}^{2}}}{{t}^{3}}, \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right) \]

   3. associate-/l* 25.8%

      \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\frac{0.3333333333333333 + \frac{1}{{t}^{2}}}{\frac{{t}^{3}}{{\ell}^{2}}}}, \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right) \]

   4. unpow2 25.8%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{0.3333333333333333 + \frac{1}{\color{blue}{t \cdot t}}}{\frac{{t}^{3}}{{\ell}^{2}}}, \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right) \]

   5. unpow2 25.8%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{0.3333333333333333 + \frac{1}{t \cdot t}}{\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}}, \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right) \]

   6. associate-/r* 25.6%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{0.3333333333333333 + \frac{1}{t \cdot t}}{\frac{{t}^{3}}{\ell \cdot \ell}}, \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}}}\right) \]

   7. unpow2 25.6%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{0.3333333333333333 + \frac{1}{t \cdot t}}{\frac{{t}^{3}}{\ell \cdot \ell}}, \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{{t}^{3}}\right) \]

   8. associate-*l/ 25.9%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{0.3333333333333333 + \frac{1}{t \cdot t}}{\frac{{t}^{3}}{\ell \cdot \ell}}, \frac{\color{blue}{\frac{\ell}{{k}^{2}} \cdot \ell}}{{t}^{3}}\right) \]

   9. unpow2 25.9%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{0.3333333333333333 + \frac{1}{t \cdot t}}{\frac{{t}^{3}}{\ell \cdot \ell}}, \frac{\frac{\ell}{\color{blue}{k \cdot k}} \cdot \ell}{{t}^{3}}\right) \]

   10. associate-*r/ 25.9%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{0.3333333333333333 + \frac{1}{t \cdot t}}{\frac{{t}^{3}}{\ell \cdot \ell}}, \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}}\right) \]

8. Simplified 25.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{0.3333333333333333 + \frac{1}{t \cdot t}}{\frac{{t}^{3}}{\ell \cdot \ell}}, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}\right)} \]

9. Taylor expanded in t around 0 23.3%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{{\ell}^{2}}{{t}^{5}}} \]
10. Step-by-step derivation

    1. associate-*r/ 23.3%

       \[\leadsto \color{blue}{\frac{-0.5 \cdot {\ell}^{2}}{{t}^{5}}} \]

    2. unpow2 23.3%

       \[\leadsto \frac{-0.5 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{t}^{5}} \]

11. Simplified 23.3%

    \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\ell \cdot \ell\right)}{{t}^{5}}} \]

12. Final simplification 23.3%

    \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.5}{{t}^{5}} \]

Reproduce

herbie shell --seed 2023283 
(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))))