Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.9% → 89.7%

Time: 34.3s

Alternatives: 23

Speedup: 28.1×

• 27.2% 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: 54.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.35 \cdot 10^{+71} \lor \neg \left(t \leq 31000000000\right):\\ \;\;\;\;\frac{2}{{\left(\frac{t}{\sqrt[3]{\frac{\ell}{\sin k}} \cdot \sqrt[3]{\ell}}\right)}^{3} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t \cdot \frac{k}{\ell}}}{{\sin k}^{2} \cdot \frac{\frac{k}{\ell}}{\cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -4.35e+71) (not (<= t 31000000000.0)))
   (/
    2.0
    (*
     (pow (/ t (* (cbrt (/ l (sin k))) (cbrt l))) 3.0)
     (* (tan k) (+ 2.0 (pow (/ k t) 2.0)))))
   (/ (/ 2.0 (* t (/ k l))) (* (pow (sin k) 2.0) (/ (/ k l) (cos k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -4.35e+71) || !(t <= 31000000000.0)) {
		tmp = 2.0 / (pow((t / (cbrt((l / sin(k))) * cbrt(l))), 3.0) * (tan(k) * (2.0 + pow((k / t), 2.0))));
	} else {
		tmp = (2.0 / (t * (k / l))) / (pow(sin(k), 2.0) * ((k / l) / cos(k)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -4.35e+71) || !(t <= 31000000000.0)) {
		tmp = 2.0 / (Math.pow((t / (Math.cbrt((l / Math.sin(k))) * Math.cbrt(l))), 3.0) * (Math.tan(k) * (2.0 + Math.pow((k / t), 2.0))));
	} else {
		tmp = (2.0 / (t * (k / l))) / (Math.pow(Math.sin(k), 2.0) * ((k / l) / Math.cos(k)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -4.35e+71) || !(t <= 31000000000.0))
		tmp = Float64(2.0 / Float64((Float64(t / Float64(cbrt(Float64(l / sin(k))) * cbrt(l))) ^ 3.0) * Float64(tan(k) * Float64(2.0 + (Float64(k / t) ^ 2.0)))));
	else
		tmp = Float64(Float64(2.0 / Float64(t * Float64(k / l))) / Float64((sin(k) ^ 2.0) * Float64(Float64(k / l) / cos(k))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -4.35e+71], N[Not[LessEqual[t, 31000000000.0]], $MachinePrecision]], N[(2.0 / N[(N[Power[N[(t / N[(N[Power[N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(k / l), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.3499999999999999e71 or 3.1e10 < t

   1. Initial program 56.7%

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

      1. associate-/r/ 55.5%

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

      2. add-cube-cbrt 55.5%

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

      3. pow3 55.5%

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

      4. cbrt-div 55.4%

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

      5. rem-cbrt-cube 68.7%

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

   3. Applied egg-rr 68.7%

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

      1. distribute-lft-in 68.7%

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

      2. associate-/l* 68.7%

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

      3. associate-/l* 68.7%

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

   5. Applied egg-rr 68.7%

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

      1. distribute-lft-out 68.7%

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

      2. +-commutative 68.7%

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

      3. associate-*r* 68.7%

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

      4. associate-/r/ 68.7%

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

      5. associate-+r+ 68.7%

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

      6. metadata-eval 68.7%

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

   7. Simplified 68.7%

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

      1. cbrt-prod 85.2%

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

   9. Applied egg-rr 85.2%

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

   if -4.3499999999999999e71 < t < 3.1e10

   1. Initial program 50.2%

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

   2. Taylor expanded in t around 0 75.5%

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

      1. associate-/l* 74.9%

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

      2. unpow2 74.9%

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

      3. unpow2 74.9%

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

      4. associate-*r* 74.9%

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

      5. times-frac 84.8%

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

   4. Simplified 84.8%

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

   5. Taylor expanded in k around inf 75.5%

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

      1. associate-/l* 74.9%

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

      2. unpow2 74.9%

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

      3. unpow2 74.9%

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

      4. associate-*r* 74.9%

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

      5. times-frac 84.8%

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

      6. times-frac 93.1%

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

      7. associate-/r/ 93.2%

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

      8. *-commutative 93.2%

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

      9. associate-/r/ 95.6%

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

   7. Simplified 95.6%

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

   8. Taylor expanded in k around 0 95.0%

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

      1. div-inv 95.0%

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

      2. *-commutative 95.0%

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

      3. associate-/l* 93.2%

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

   10. Applied egg-rr 93.2%

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

       1. associate-*r/ 93.2%

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

       2. metadata-eval 93.2%

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

       3. associate-/r* 93.2%

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

       4. associate-/r/ 95.7%

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

       5. *-commutative 95.7%

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

       6. associate-/r* 95.7%

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

   12. Simplified 95.7%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.35 \cdot 10^{+71} \lor \neg \left(t \leq 31000000000\right):\\ \;\;\;\;\frac{2}{{\left(\frac{t}{\sqrt[3]{\frac{\ell}{\sin k}} \cdot \sqrt[3]{\ell}}\right)}^{3} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t \cdot \frac{k}{\ell}}}{{\sin k}^{2} \cdot \frac{\frac{k}{\ell}}{\cos k}}\\ \end{array} \]

Alternative 2: 87.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(1 + \left(t_1 + 1\right)\right)} \leq 10^{+296}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + t_1\right)\right) \cdot {\left(\frac{t}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t \cdot \frac{k}{\ell}}}{{\sin k}^{2} \cdot \frac{\frac{k}{\ell}}{\cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<=
        (/
         2.0
         (*
          (* (tan k) (* (sin k) (/ (pow t 3.0) (* l l))))
          (+ 1.0 (+ t_1 1.0))))
        1e+296)
     (/
      2.0
      (* (* (tan k) (+ 2.0 t_1)) (pow (/ t (cbrt (* l (/ l (sin k))))) 3.0)))
     (/ (/ 2.0 (* t (/ k l))) (* (pow (sin k) 2.0) (/ (/ k l) (cos k)))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((tan(k) * (sin(k) * (pow(t, 3.0) / (l * l)))) * (1.0 + (t_1 + 1.0)))) <= 1e+296) {
		tmp = 2.0 / ((tan(k) * (2.0 + t_1)) * pow((t / cbrt((l * (l / sin(k))))), 3.0));
	} else {
		tmp = (2.0 / (t * (k / l))) / (pow(sin(k), 2.0) * ((k / l) / cos(k)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((Math.tan(k) * (Math.sin(k) * (Math.pow(t, 3.0) / (l * l)))) * (1.0 + (t_1 + 1.0)))) <= 1e+296) {
		tmp = 2.0 / ((Math.tan(k) * (2.0 + t_1)) * Math.pow((t / Math.cbrt((l * (l / Math.sin(k))))), 3.0));
	} else {
		tmp = (2.0 / (t * (k / l))) / (Math.pow(Math.sin(k), 2.0) * ((k / l) / Math.cos(k)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) * Float64((t ^ 3.0) / Float64(l * l)))) * Float64(1.0 + Float64(t_1 + 1.0)))) <= 1e+296)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + t_1)) * (Float64(t / cbrt(Float64(l * Float64(l / sin(k))))) ^ 3.0)));
	else
		tmp = Float64(Float64(2.0 / Float64(t * Float64(k / l))) / Float64((sin(k) ^ 2.0) * Float64(Float64(k / l) / cos(k))));
	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[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+296], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision] * N[Power[N[(t / N[Power[N[(l * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(k / l), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 9.99999999999999981e295

   1. Initial program 81.1%

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

      1. associate-/r/ 80.1%

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

      2. add-cube-cbrt 80.0%

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

      3. pow3 80.0%

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

      4. cbrt-div 80.0%

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

      5. rem-cbrt-cube 87.0%

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

   3. Applied egg-rr 87.0%

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

      1. distribute-lft-in 87.0%

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

      2. associate-/l* 87.0%

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

      3. associate-/l* 88.7%

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

   5. Applied egg-rr 88.7%

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

      1. distribute-lft-out 88.7%

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

      2. +-commutative 88.7%

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

      3. associate-*r* 88.7%

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

      4. associate-/r/ 88.7%

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

      5. associate-+r+ 88.7%

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

      6. metadata-eval 88.7%

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

   7. Simplified 88.7%

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

   if 9.99999999999999981e295 < (/.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 20.4%

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

   2. Taylor expanded in t around 0 56.8%

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

      1. associate-/l* 55.3%

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

      2. unpow2 55.3%

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

      3. unpow2 55.3%

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

      4. associate-*r* 55.3%

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

      5. times-frac 67.8%

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

   4. Simplified 67.8%

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

   5. Taylor expanded in k around inf 56.8%

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

      1. associate-/l* 55.3%

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

      2. unpow2 55.3%

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

      3. unpow2 55.3%

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

      4. associate-*r* 55.3%

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

      5. times-frac 67.8%

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

      6. times-frac 80.2%

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

      7. associate-/r/ 80.2%

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

      8. *-commutative 80.2%

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

      9. associate-/r/ 83.2%

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

   7. Simplified 83.2%

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

   8. Taylor expanded in k around 0 81.7%

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

      1. div-inv 81.7%

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

      2. *-commutative 81.7%

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

      3. associate-/l* 80.2%

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

   10. Applied egg-rr 80.2%

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

       1. associate-*r/ 80.2%

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

       2. metadata-eval 80.2%

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

       3. associate-/r* 80.4%

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

       4. associate-/r/ 83.4%

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

       5. *-commutative 83.4%

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

       6. associate-/r* 83.4%

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

   12. Simplified 83.4%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)} \leq 10^{+296}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\frac{t}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t \cdot \frac{k}{\ell}}}{{\sin k}^{2} \cdot \frac{\frac{k}{\ell}}{\cos k}}\\ \end{array} \]

Alternative 3: 83.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \ell \cdot \frac{\ell}{k \cdot \left(k \cdot {t}^{3}\right)}\\ \mathbf{if}\;t \leq -7.5 \cdot 10^{+180}:\\ \;\;\;\;\sqrt[3]{t_1 \cdot \left(t_1 \cdot t_1\right)}\\ \mathbf{elif}\;t \leq 64000000000:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(t \cdot \left({\sin k}^{2} \cdot \frac{\frac{k}{\ell}}{\cos k}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* l (/ l (* k (* k (pow t 3.0)))))))
   (if (<= t -7.5e+180)
     (cbrt (* t_1 (* t_1 t_1)))
     (if (<= t 64000000000.0)
       (/ 2.0 (* (/ k l) (* t (* (pow (sin k) 2.0) (/ (/ k l) (cos k))))))
       (/
        2.0
        (*
         (+ 2.0 (pow (/ k t) 2.0))
         (* (tan k) (* (sin k) (pow (/ (pow t 1.5) l) 2.0)))))))))

C

double code(double t, double l, double k) {
	double t_1 = l * (l / (k * (k * pow(t, 3.0))));
	double tmp;
	if (t <= -7.5e+180) {
		tmp = cbrt((t_1 * (t_1 * t_1)));
	} else if (t <= 64000000000.0) {
		tmp = 2.0 / ((k / l) * (t * (pow(sin(k), 2.0) * ((k / l) / cos(k)))));
	} else {
		tmp = 2.0 / ((2.0 + pow((k / t), 2.0)) * (tan(k) * (sin(k) * pow((pow(t, 1.5) / l), 2.0))));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = l * (l / (k * (k * Math.pow(t, 3.0))));
	double tmp;
	if (t <= -7.5e+180) {
		tmp = Math.cbrt((t_1 * (t_1 * t_1)));
	} else if (t <= 64000000000.0) {
		tmp = 2.0 / ((k / l) * (t * (Math.pow(Math.sin(k), 2.0) * ((k / l) / Math.cos(k)))));
	} else {
		tmp = 2.0 / ((2.0 + Math.pow((k / t), 2.0)) * (Math.tan(k) * (Math.sin(k) * Math.pow((Math.pow(t, 1.5) / l), 2.0))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(l * Float64(l / Float64(k * Float64(k * (t ^ 3.0)))))
	tmp = 0.0
	if (t <= -7.5e+180)
		tmp = cbrt(Float64(t_1 * Float64(t_1 * t_1)));
	elseif (t <= 64000000000.0)
		tmp = Float64(2.0 / Float64(Float64(k / l) * Float64(t * Float64((sin(k) ^ 2.0) * Float64(Float64(k / l) / cos(k))))));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 + (Float64(k / t) ^ 2.0)) * Float64(tan(k) * Float64(sin(k) * (Float64((t ^ 1.5) / l) ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(l * N[(l / N[(k * N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -7.5e+180], N[Power[N[(t$95$1 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], If[LessEqual[t, 64000000000.0], N[(2.0 / N[(N[(k / l), $MachinePrecision] * N[(t * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(k / l), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.5000000000000003e180

   1. Initial program 46.2%

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

      1. associate-/r* 46.2%

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

      2. associate-*l* 40.4%

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

      3. sqr-neg 40.4%

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

      4. associate-*l* 46.2%

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

      5. *-commutative 46.2%

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

      6. sqr-neg 46.2%

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

      7. associate-*l/ 46.2%

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

      8. associate-*r/ 46.2%

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

      9. associate-/r/ 46.2%

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

   3. Simplified 46.2%

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

   4. Taylor expanded in k around 0 40.4%

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

      1. unpow2 40.4%

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

      2. *-commutative 40.4%

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

      3. unpow2 40.4%

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

   6. Simplified 40.4%

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

      1. add-cbrt-cube 40.4%

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

      2. associate-/l* 40.4%

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

      3. *-commutative 40.4%

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

      4. associate-/l* 40.4%

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

      5. *-commutative 40.4%

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

      6. associate-/l* 60.8%

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

      7. *-commutative 60.8%

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

   8. Applied egg-rr 60.8%

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

      1. associate-*l* 60.8%

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

      2. associate-/r/ 60.8%

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

      3. associate-*l* 60.8%

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

      4. associate-/r/ 60.8%

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

      5. associate-*l* 60.8%

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

      6. associate-/r/ 60.8%

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

      7. associate-*l* 66.8%

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

   10. Simplified 66.8%

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

   if -7.5000000000000003e180 < t < 6.4e10

   1. Initial program 49.3%

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

   2. Taylor expanded in t around 0 71.7%

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

      1. associate-/l* 71.2%

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

      2. unpow2 71.2%

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

      3. unpow2 71.2%

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

      4. associate-*r* 71.2%

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

      5. times-frac 79.9%

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

   4. Simplified 79.9%

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

   5. Taylor expanded in k around inf 71.7%

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

      1. associate-/l* 71.2%

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

      2. unpow2 71.2%

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

      3. unpow2 71.2%

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

      4. associate-*r* 71.2%

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

      5. times-frac 79.9%

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

      6. times-frac 87.9%

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

      7. associate-/r/ 88.0%

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

      8. *-commutative 88.0%

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

      9. associate-/r/ 90.0%

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

   7. Simplified 90.0%

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

      1. div-inv 90.0%

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

      2. *-commutative 90.0%

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

   9. Applied egg-rr 90.0%

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

       1. associate-*r/ 90.0%

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

       2. metadata-eval 90.0%

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

       3. associate-*l* 90.1%

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

       4. *-commutative 90.1%

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

       5. associate-/r* 90.0%

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

   11. Simplified 90.0%

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

   if 6.4e10 < t

   1. Initial program 64.8%

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

      1. distribute-lft-in 64.8%

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

      2. associate-*l* 57.1%

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

      3. add-sqr-sqrt 57.1%

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

      4. pow2 57.1%

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

      5. sqrt-div 57.1%

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

      6. sqrt-pow1 57.2%

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

      7. metadata-eval 57.2%

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

      8. sqrt-prod 33.0%

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

      9. add-sqr-sqrt 57.2%

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

   3. Applied egg-rr 70.7%

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

      1. distribute-lft-out 70.7%

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

      2. +-commutative 70.7%

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

      3. associate-+r+ 70.7%

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

      4. metadata-eval 70.7%

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

      5. associate-*l* 70.7%

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

      6. unpow2 70.7%

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

      7. times-frac 57.0%

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

      8. pow-sqr 56.9%

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

      9. metadata-eval 56.9%

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

      10. unpow2 56.9%

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

      11. associate-*r* 56.9%

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

   5. Simplified 84.0%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{+180}:\\ \;\;\;\;\sqrt[3]{\left(\ell \cdot \frac{\ell}{k \cdot \left(k \cdot {t}^{3}\right)}\right) \cdot \left(\left(\ell \cdot \frac{\ell}{k \cdot \left(k \cdot {t}^{3}\right)}\right) \cdot \left(\ell \cdot \frac{\ell}{k \cdot \left(k \cdot {t}^{3}\right)}\right)\right)}\\ \mathbf{elif}\;t \leq 64000000000:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(t \cdot \left({\sin k}^{2} \cdot \frac{\frac{k}{\ell}}{\cos k}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right)}\\ \end{array} \]

Alternative 4: 74.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq 3.9 \cdot 10^{-39}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}}\right)}^{3} \cdot \frac{2}{\frac{\cos k}{\sin k}}}\\ \mathbf{elif}\;k \leq 5 \cdot 10^{+223}:\\ \;\;\;\;\frac{2}{\left(t_1 \cdot \frac{k}{\ell \cdot \cos k}\right) \cdot \frac{t \cdot k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t \cdot \frac{k}{\ell}}}{t_1 \cdot \frac{\frac{k}{\ell}}{\cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (<= k 3.9e-39)
     (/
      2.0
      (*
       (pow (/ t (cbrt (* l (/ l (sin k))))) 3.0)
       (/ 2.0 (/ (cos k) (sin k)))))
     (if (<= k 5e+223)
       (/ 2.0 (* (* t_1 (/ k (* l (cos k)))) (/ (* t k) l)))
       (/ (/ 2.0 (* t (/ k l))) (* t_1 (/ (/ k l) (cos k))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double tmp;
	if (k <= 3.9e-39) {
		tmp = 2.0 / (pow((t / cbrt((l * (l / sin(k))))), 3.0) * (2.0 / (cos(k) / sin(k))));
	} else if (k <= 5e+223) {
		tmp = 2.0 / ((t_1 * (k / (l * cos(k)))) * ((t * k) / l));
	} else {
		tmp = (2.0 / (t * (k / l))) / (t_1 * ((k / l) / cos(k)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (k <= 3.9e-39) {
		tmp = 2.0 / (Math.pow((t / Math.cbrt((l * (l / Math.sin(k))))), 3.0) * (2.0 / (Math.cos(k) / Math.sin(k))));
	} else if (k <= 5e+223) {
		tmp = 2.0 / ((t_1 * (k / (l * Math.cos(k)))) * ((t * k) / l));
	} else {
		tmp = (2.0 / (t * (k / l))) / (t_1 * ((k / l) / Math.cos(k)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if (k <= 3.9e-39)
		tmp = Float64(2.0 / Float64((Float64(t / cbrt(Float64(l * Float64(l / sin(k))))) ^ 3.0) * Float64(2.0 / Float64(cos(k) / sin(k)))));
	elseif (k <= 5e+223)
		tmp = Float64(2.0 / Float64(Float64(t_1 * Float64(k / Float64(l * cos(k)))) * Float64(Float64(t * k) / l)));
	else
		tmp = Float64(Float64(2.0 / Float64(t * Float64(k / l))) / Float64(t_1 * Float64(Float64(k / l) / cos(k))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k, 3.9e-39], N[(2.0 / N[(N[Power[N[(t / N[Power[N[(l * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 / N[(N[Cos[k], $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 5e+223], N[(2.0 / N[(N[(t$95$1 * N[(k / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(N[(k / l), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 3.9000000000000003e-39

   1. Initial program 52.5%

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

      1. associate-/r/ 51.8%

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

      2. add-cube-cbrt 51.7%

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

      3. pow3 51.7%

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

      4. cbrt-div 51.7%

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

      5. rem-cbrt-cube 61.3%

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

   3. Applied egg-rr 61.3%

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

      1. distribute-lft-in 61.3%

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

      2. associate-/l* 61.3%

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

      3. associate-/l* 66.1%

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

   5. Applied egg-rr 66.1%

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

      1. distribute-lft-out 66.1%

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

      2. +-commutative 66.1%

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

      3. associate-*r* 66.1%

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

      4. associate-/r/ 66.1%

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

      5. associate-+r+ 66.1%

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

      6. metadata-eval 66.1%

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

   7. Simplified 66.1%

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

   8. Taylor expanded in t around inf 63.4%

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

      1. associate-*r/ 63.4%

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

      2. associate-/l* 63.4%

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

   10. Simplified 63.4%

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

   if 3.9000000000000003e-39 < k < 4.99999999999999985e223

   1. Initial program 57.5%

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

   2. Taylor expanded in t around 0 82.3%

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

      1. associate-/l* 80.4%

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

      2. unpow2 80.4%

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

      3. unpow2 80.4%

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

      4. associate-*r* 80.4%

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

      5. times-frac 84.0%

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

   4. Simplified 84.0%

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

   5. Taylor expanded in k around inf 82.3%

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

      1. associate-/l* 80.4%

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

      2. unpow2 80.4%

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

      3. unpow2 80.4%

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

      4. associate-*r* 80.4%

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

      5. times-frac 84.0%

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

      6. times-frac 87.9%

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

      7. associate-/r/ 87.9%

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

      8. *-commutative 87.9%

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

      9. associate-/r/ 89.8%

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

   7. Simplified 89.8%

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

   8. Taylor expanded in k around 0 93.8%

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

   if 4.99999999999999985e223 < k

   1. Initial program 46.2%

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

   2. Taylor expanded in t around 0 62.3%

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

      1. associate-/l* 62.0%

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

      2. unpow2 62.0%

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

      3. unpow2 62.0%

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

      4. associate-*r* 62.0%

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

      5. times-frac 62.4%

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

   4. Simplified 62.4%

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

   5. Taylor expanded in k around inf 62.3%

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

      1. associate-/l* 62.0%

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

      2. unpow2 62.0%

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

      3. unpow2 62.0%

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

      4. associate-*r* 62.0%

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

      5. times-frac 62.4%

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

      6. times-frac 99.1%

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

      7. associate-/r/ 99.4%

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

      8. *-commutative 99.4%

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

      9. associate-/r/ 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in k around 0 77.9%

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

      1. div-inv 77.9%

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

      2. *-commutative 77.9%

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

      3. associate-/l* 99.4%

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

   10. Applied egg-rr 99.4%

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

       1. associate-*r/ 99.4%

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

       2. metadata-eval 99.4%

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

       3. associate-/r* 99.4%

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

       4. associate-/r/ 99.4%

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

       5. *-commutative 99.4%

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

       6. associate-/r* 99.5%

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

   12. Simplified 99.5%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.9 \cdot 10^{-39}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}}\right)}^{3} \cdot \frac{2}{\frac{\cos k}{\sin k}}}\\ \mathbf{elif}\;k \leq 5 \cdot 10^{+223}:\\ \;\;\;\;\frac{2}{\left({\sin k}^{2} \cdot \frac{k}{\ell \cdot \cos k}\right) \cdot \frac{t \cdot k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t \cdot \frac{k}{\ell}}}{{\sin k}^{2} \cdot \frac{\frac{k}{\ell}}{\cos k}}\\ \end{array} \]

Alternative 5: 73.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin k}^{2} \cdot \frac{k}{\ell \cdot \cos k}\\ \mathbf{if}\;k \leq 1.95 \cdot 10^{-40}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 9.5 \cdot 10^{+218}:\\ \;\;\;\;\frac{2}{t_1 \cdot \frac{t \cdot k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \frac{k}{\ell}\right) \cdot t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (pow (sin k) 2.0) (/ k (* l (cos k))))))
   (if (<= k 1.95e-40)
     (/ 2.0 (* (pow (/ t (cbrt (* l (/ l (sin k))))) 3.0) (* 2.0 k)))
     (if (<= k 9.5e+218)
       (/ 2.0 (* t_1 (/ (* t k) l)))
       (/ 2.0 (* (* t (/ k l)) t_1))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0) * (k / (l * cos(k)));
	double tmp;
	if (k <= 1.95e-40) {
		tmp = 2.0 / (pow((t / cbrt((l * (l / sin(k))))), 3.0) * (2.0 * k));
	} else if (k <= 9.5e+218) {
		tmp = 2.0 / (t_1 * ((t * k) / l));
	} else {
		tmp = 2.0 / ((t * (k / l)) * t_1);
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0) * (k / (l * Math.cos(k)));
	double tmp;
	if (k <= 1.95e-40) {
		tmp = 2.0 / (Math.pow((t / Math.cbrt((l * (l / Math.sin(k))))), 3.0) * (2.0 * k));
	} else if (k <= 9.5e+218) {
		tmp = 2.0 / (t_1 * ((t * k) / l));
	} else {
		tmp = 2.0 / ((t * (k / l)) * t_1);
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64((sin(k) ^ 2.0) * Float64(k / Float64(l * cos(k))))
	tmp = 0.0
	if (k <= 1.95e-40)
		tmp = Float64(2.0 / Float64((Float64(t / cbrt(Float64(l * Float64(l / sin(k))))) ^ 3.0) * Float64(2.0 * k)));
	elseif (k <= 9.5e+218)
		tmp = Float64(2.0 / Float64(t_1 * Float64(Float64(t * k) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(t * Float64(k / l)) * t_1));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(k / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.95e-40], N[(2.0 / N[(N[Power[N[(t / N[Power[N[(l * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 9.5e+218], N[(2.0 / N[(t$95$1 * N[(N[(t * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.9499999999999999e-40

   1. Initial program 52.5%

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

      1. associate-/r/ 51.8%

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

      2. add-cube-cbrt 51.7%

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

      3. pow3 51.7%

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

      4. cbrt-div 51.7%

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

      5. rem-cbrt-cube 61.3%

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

   3. Applied egg-rr 61.3%

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

      1. distribute-lft-in 61.3%

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

      2. associate-/l* 61.3%

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

      3. associate-/l* 66.1%

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

   5. Applied egg-rr 66.1%

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

      1. distribute-lft-out 66.1%

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

      2. +-commutative 66.1%

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

      3. associate-*r* 66.1%

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

      4. associate-/r/ 66.1%

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

      5. associate-+r+ 66.1%

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

      6. metadata-eval 66.1%

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

   7. Simplified 66.1%

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

   8. Taylor expanded in k around 0 63.2%

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

      1. *-commutative 63.2%

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

   10. Simplified 63.2%

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

   if 1.9499999999999999e-40 < k < 9.4999999999999999e218

   1. Initial program 57.5%

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

   2. Taylor expanded in t around 0 82.3%

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

      1. associate-/l* 80.4%

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

      2. unpow2 80.4%

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

      3. unpow2 80.4%

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

      4. associate-*r* 80.4%

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

      5. times-frac 84.0%

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

   4. Simplified 84.0%

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

   5. Taylor expanded in k around inf 82.3%

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

      1. associate-/l* 80.4%

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

      2. unpow2 80.4%

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

      3. unpow2 80.4%

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

      4. associate-*r* 80.4%

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

      5. times-frac 84.0%

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

      6. times-frac 87.9%

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

      7. associate-/r/ 87.9%

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

      8. *-commutative 87.9%

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

      9. associate-/r/ 89.8%

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

   7. Simplified 89.8%

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

   8. Taylor expanded in k around 0 93.8%

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

   if 9.4999999999999999e218 < k

   1. Initial program 46.2%

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

   2. Taylor expanded in t around 0 62.3%

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

      1. associate-/l* 62.0%

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

      2. unpow2 62.0%

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

      3. unpow2 62.0%

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

      4. associate-*r* 62.0%

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

      5. times-frac 62.4%

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

   4. Simplified 62.4%

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

   5. Taylor expanded in k around inf 62.3%

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

      1. associate-/l* 62.0%

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

      2. unpow2 62.0%

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

      3. unpow2 62.0%

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

      4. associate-*r* 62.0%

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

      5. times-frac 62.4%

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

      6. times-frac 99.1%

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

      7. associate-/r/ 99.4%

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

      8. *-commutative 99.4%

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

      9. associate-/r/ 99.7%

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

   7. Simplified 99.7%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.95 \cdot 10^{-40}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 9.5 \cdot 10^{+218}:\\ \;\;\;\;\frac{2}{\left({\sin k}^{2} \cdot \frac{k}{\ell \cdot \cos k}\right) \cdot \frac{t \cdot k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \frac{k}{\ell}\right) \cdot \left({\sin k}^{2} \cdot \frac{k}{\ell \cdot \cos k}\right)}\\ \end{array} \]

Alternative 6: 73.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq 1.45 \cdot 10^{-40}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 4 \cdot 10^{+218}:\\ \;\;\;\;\frac{2}{\left(t_1 \cdot \frac{k}{\ell \cdot \cos k}\right) \cdot \frac{t \cdot k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(t \cdot \left(t_1 \cdot \frac{\frac{k}{\ell}}{\cos k}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (<= k 1.45e-40)
     (/ 2.0 (* (pow (/ t (cbrt (* l (/ l (sin k))))) 3.0) (* 2.0 k)))
     (if (<= k 4e+218)
       (/ 2.0 (* (* t_1 (/ k (* l (cos k)))) (/ (* t k) l)))
       (/ 2.0 (* (/ k l) (* t (* t_1 (/ (/ k l) (cos k))))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double tmp;
	if (k <= 1.45e-40) {
		tmp = 2.0 / (pow((t / cbrt((l * (l / sin(k))))), 3.0) * (2.0 * k));
	} else if (k <= 4e+218) {
		tmp = 2.0 / ((t_1 * (k / (l * cos(k)))) * ((t * k) / l));
	} else {
		tmp = 2.0 / ((k / l) * (t * (t_1 * ((k / l) / cos(k)))));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (k <= 1.45e-40) {
		tmp = 2.0 / (Math.pow((t / Math.cbrt((l * (l / Math.sin(k))))), 3.0) * (2.0 * k));
	} else if (k <= 4e+218) {
		tmp = 2.0 / ((t_1 * (k / (l * Math.cos(k)))) * ((t * k) / l));
	} else {
		tmp = 2.0 / ((k / l) * (t * (t_1 * ((k / l) / Math.cos(k)))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if (k <= 1.45e-40)
		tmp = Float64(2.0 / Float64((Float64(t / cbrt(Float64(l * Float64(l / sin(k))))) ^ 3.0) * Float64(2.0 * k)));
	elseif (k <= 4e+218)
		tmp = Float64(2.0 / Float64(Float64(t_1 * Float64(k / Float64(l * cos(k)))) * Float64(Float64(t * k) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(k / l) * Float64(t * Float64(t_1 * Float64(Float64(k / l) / cos(k))))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k, 1.45e-40], N[(2.0 / N[(N[Power[N[(t / N[Power[N[(l * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4e+218], N[(2.0 / N[(N[(t$95$1 * N[(k / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k / l), $MachinePrecision] * N[(t * N[(t$95$1 * N[(N[(k / l), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.4499999999999999e-40

   1. Initial program 52.5%

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

      1. associate-/r/ 51.8%

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

      2. add-cube-cbrt 51.7%

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

      3. pow3 51.7%

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

      4. cbrt-div 51.7%

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

      5. rem-cbrt-cube 61.3%

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

   3. Applied egg-rr 61.3%

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

      1. distribute-lft-in 61.3%

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

      2. associate-/l* 61.3%

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

      3. associate-/l* 66.1%

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

   5. Applied egg-rr 66.1%

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

      1. distribute-lft-out 66.1%

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

      2. +-commutative 66.1%

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

      3. associate-*r* 66.1%

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

      4. associate-/r/ 66.1%

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

      5. associate-+r+ 66.1%

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

      6. metadata-eval 66.1%

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

   7. Simplified 66.1%

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

   8. Taylor expanded in k around 0 63.2%

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

      1. *-commutative 63.2%

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

   10. Simplified 63.2%

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

   if 1.4499999999999999e-40 < k < 4.00000000000000033e218

   1. Initial program 57.5%

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

   2. Taylor expanded in t around 0 82.3%

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

      1. associate-/l* 80.4%

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

      2. unpow2 80.4%

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

      3. unpow2 80.4%

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

      4. associate-*r* 80.4%

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

      5. times-frac 84.0%

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

   4. Simplified 84.0%

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

   5. Taylor expanded in k around inf 82.3%

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

      1. associate-/l* 80.4%

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

      2. unpow2 80.4%

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

      3. unpow2 80.4%

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

      4. associate-*r* 80.4%

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

      5. times-frac 84.0%

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

      6. times-frac 87.9%

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

      7. associate-/r/ 87.9%

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

      8. *-commutative 87.9%

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

      9. associate-/r/ 89.8%

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

   7. Simplified 89.8%

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

   8. Taylor expanded in k around 0 93.8%

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

   if 4.00000000000000033e218 < k

   1. Initial program 46.2%

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

   2. Taylor expanded in t around 0 62.3%

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

      1. associate-/l* 62.0%

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

      2. unpow2 62.0%

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

      3. unpow2 62.0%

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

      4. associate-*r* 62.0%

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

      5. times-frac 62.4%

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

   4. Simplified 62.4%

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

   5. Taylor expanded in k around inf 62.3%

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

      1. associate-/l* 62.0%

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

      2. unpow2 62.0%

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

      3. unpow2 62.0%

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

      4. associate-*r* 62.0%

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

      5. times-frac 62.4%

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

      6. times-frac 99.1%

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

      7. associate-/r/ 99.4%

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

      8. *-commutative 99.4%

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

      9. associate-/r/ 99.7%

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

   7. Simplified 99.7%

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

      1. div-inv 99.7%

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

      2. *-commutative 99.7%

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

   9. Applied egg-rr 99.7%

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

       1. associate-*r/ 99.7%

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

       2. metadata-eval 99.7%

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

       3. associate-*l* 99.6%

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

       4. *-commutative 99.6%

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

       5. associate-/r* 99.6%

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

   11. Simplified 99.6%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{-40}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 4 \cdot 10^{+218}:\\ \;\;\;\;\frac{2}{\left({\sin k}^{2} \cdot \frac{k}{\ell \cdot \cos k}\right) \cdot \frac{t \cdot k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(t \cdot \left({\sin k}^{2} \cdot \frac{\frac{k}{\ell}}{\cos k}\right)\right)}\\ \end{array} \]

Alternative 7: 73.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \ell \cdot \cos k\\ \mathbf{if}\;k \leq 1.4 \cdot 10^{-40}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 2.45 \cdot 10^{+218}:\\ \;\;\;\;\frac{2}{\left(t_1 \cdot \frac{k}{t_2}\right) \cdot \frac{t \cdot k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \frac{k}{\ell}\right) \cdot \frac{k \cdot t_1}{t_2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)) (t_2 (* l (cos k))))
   (if (<= k 1.4e-40)
     (/ 2.0 (* (pow (/ t (cbrt (* l (/ l (sin k))))) 3.0) (* 2.0 k)))
     (if (<= k 2.45e+218)
       (/ 2.0 (* (* t_1 (/ k t_2)) (/ (* t k) l)))
       (/ 2.0 (* (* t (/ k l)) (/ (* k t_1) t_2)))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double t_2 = l * cos(k);
	double tmp;
	if (k <= 1.4e-40) {
		tmp = 2.0 / (pow((t / cbrt((l * (l / sin(k))))), 3.0) * (2.0 * k));
	} else if (k <= 2.45e+218) {
		tmp = 2.0 / ((t_1 * (k / t_2)) * ((t * k) / l));
	} else {
		tmp = 2.0 / ((t * (k / l)) * ((k * t_1) / t_2));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0);
	double t_2 = l * Math.cos(k);
	double tmp;
	if (k <= 1.4e-40) {
		tmp = 2.0 / (Math.pow((t / Math.cbrt((l * (l / Math.sin(k))))), 3.0) * (2.0 * k));
	} else if (k <= 2.45e+218) {
		tmp = 2.0 / ((t_1 * (k / t_2)) * ((t * k) / l));
	} else {
		tmp = 2.0 / ((t * (k / l)) * ((k * t_1) / t_2));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	t_2 = Float64(l * cos(k))
	tmp = 0.0
	if (k <= 1.4e-40)
		tmp = Float64(2.0 / Float64((Float64(t / cbrt(Float64(l * Float64(l / sin(k))))) ^ 3.0) * Float64(2.0 * k)));
	elseif (k <= 2.45e+218)
		tmp = Float64(2.0 / Float64(Float64(t_1 * Float64(k / t_2)) * Float64(Float64(t * k) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(t * Float64(k / l)) * Float64(Float64(k * t_1) / t_2)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.4e-40], N[(2.0 / N[(N[Power[N[(t / N[Power[N[(l * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.45e+218], N[(2.0 / N[(N[(t$95$1 * N[(k / t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[(t * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[(k * t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.4e-40

   1. Initial program 52.5%

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

      1. associate-/r/ 51.8%

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

      2. add-cube-cbrt 51.7%

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

      3. pow3 51.7%

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

      4. cbrt-div 51.7%

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

      5. rem-cbrt-cube 61.3%

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

   3. Applied egg-rr 61.3%

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

      1. distribute-lft-in 61.3%

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

      2. associate-/l* 61.3%

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

      3. associate-/l* 66.1%

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

   5. Applied egg-rr 66.1%

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

      1. distribute-lft-out 66.1%

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

      2. +-commutative 66.1%

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

      3. associate-*r* 66.1%

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

      4. associate-/r/ 66.1%

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

      5. associate-+r+ 66.1%

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

      6. metadata-eval 66.1%

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

   7. Simplified 66.1%

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

   8. Taylor expanded in k around 0 63.2%

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

      1. *-commutative 63.2%

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

   10. Simplified 63.2%

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

   if 1.4e-40 < k < 2.4499999999999999e218

   1. Initial program 57.5%

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

   2. Taylor expanded in t around 0 82.3%

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

      1. associate-/l* 80.4%

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

      2. unpow2 80.4%

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

      3. unpow2 80.4%

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

      4. associate-*r* 80.4%

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

      5. times-frac 84.0%

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

   4. Simplified 84.0%

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

   5. Taylor expanded in k around inf 82.3%

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

      1. associate-/l* 80.4%

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

      2. unpow2 80.4%

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

      3. unpow2 80.4%

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

      4. associate-*r* 80.4%

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

      5. times-frac 84.0%

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

      6. times-frac 87.9%

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

      7. associate-/r/ 87.9%

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

      8. *-commutative 87.9%

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

      9. associate-/r/ 89.8%

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

   7. Simplified 89.8%

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

   8. Taylor expanded in k around 0 93.8%

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

   if 2.4499999999999999e218 < k

   1. Initial program 46.2%

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

   2. Taylor expanded in t around 0 62.3%

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

      1. associate-/l* 62.0%

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

      2. unpow2 62.0%

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

      3. unpow2 62.0%

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

      4. associate-*r* 62.0%

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

      5. times-frac 62.4%

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

   4. Simplified 62.4%

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

   5. Taylor expanded in k around inf 62.3%

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

      1. associate-/l* 62.0%

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

      2. unpow2 62.0%

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

      3. unpow2 62.0%

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

      4. associate-*r* 62.0%

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

      5. times-frac 62.4%

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

      6. times-frac 99.1%

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

      7. associate-/r/ 99.4%

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

      8. *-commutative 99.4%

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

      9. associate-/r/ 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in k around inf 99.7%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.4 \cdot 10^{-40}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 2.45 \cdot 10^{+218}:\\ \;\;\;\;\frac{2}{\left({\sin k}^{2} \cdot \frac{k}{\ell \cdot \cos k}\right) \cdot \frac{t \cdot k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \frac{k}{\ell}\right) \cdot \frac{k \cdot {\sin k}^{2}}{\ell \cdot \cos k}}\\ \end{array} \]

Alternative 8: 73.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq 1.8 \cdot 10^{-41}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 2.3 \cdot 10^{+218}:\\ \;\;\;\;\frac{2}{\left(t_1 \cdot \frac{k}{\ell \cdot \cos k}\right) \cdot \frac{t \cdot k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t \cdot \frac{k}{\ell}}}{t_1 \cdot \frac{\frac{k}{\ell}}{\cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (<= k 1.8e-41)
     (/ 2.0 (* (pow (/ t (cbrt (* l (/ l (sin k))))) 3.0) (* 2.0 k)))
     (if (<= k 2.3e+218)
       (/ 2.0 (* (* t_1 (/ k (* l (cos k)))) (/ (* t k) l)))
       (/ (/ 2.0 (* t (/ k l))) (* t_1 (/ (/ k l) (cos k))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double tmp;
	if (k <= 1.8e-41) {
		tmp = 2.0 / (pow((t / cbrt((l * (l / sin(k))))), 3.0) * (2.0 * k));
	} else if (k <= 2.3e+218) {
		tmp = 2.0 / ((t_1 * (k / (l * cos(k)))) * ((t * k) / l));
	} else {
		tmp = (2.0 / (t * (k / l))) / (t_1 * ((k / l) / cos(k)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (k <= 1.8e-41) {
		tmp = 2.0 / (Math.pow((t / Math.cbrt((l * (l / Math.sin(k))))), 3.0) * (2.0 * k));
	} else if (k <= 2.3e+218) {
		tmp = 2.0 / ((t_1 * (k / (l * Math.cos(k)))) * ((t * k) / l));
	} else {
		tmp = (2.0 / (t * (k / l))) / (t_1 * ((k / l) / Math.cos(k)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if (k <= 1.8e-41)
		tmp = Float64(2.0 / Float64((Float64(t / cbrt(Float64(l * Float64(l / sin(k))))) ^ 3.0) * Float64(2.0 * k)));
	elseif (k <= 2.3e+218)
		tmp = Float64(2.0 / Float64(Float64(t_1 * Float64(k / Float64(l * cos(k)))) * Float64(Float64(t * k) / l)));
	else
		tmp = Float64(Float64(2.0 / Float64(t * Float64(k / l))) / Float64(t_1 * Float64(Float64(k / l) / cos(k))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k, 1.8e-41], N[(2.0 / N[(N[Power[N[(t / N[Power[N[(l * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.3e+218], N[(2.0 / N[(N[(t$95$1 * N[(k / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(N[(k / l), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.8e-41

   1. Initial program 52.5%

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

      1. associate-/r/ 51.8%

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

      2. add-cube-cbrt 51.7%

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

      3. pow3 51.7%

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

      4. cbrt-div 51.7%

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

      5. rem-cbrt-cube 61.3%

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

   3. Applied egg-rr 61.3%

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

      1. distribute-lft-in 61.3%

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

      2. associate-/l* 61.3%

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

      3. associate-/l* 66.1%

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

   5. Applied egg-rr 66.1%

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

      1. distribute-lft-out 66.1%

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

      2. +-commutative 66.1%

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

      3. associate-*r* 66.1%

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

      4. associate-/r/ 66.1%

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

      5. associate-+r+ 66.1%

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

      6. metadata-eval 66.1%

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

   7. Simplified 66.1%

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

   8. Taylor expanded in k around 0 63.2%

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

      1. *-commutative 63.2%

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

   10. Simplified 63.2%

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

   if 1.8e-41 < k < 2.3000000000000001e218

   1. Initial program 57.5%

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

   2. Taylor expanded in t around 0 82.3%

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

      1. associate-/l* 80.4%

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

      2. unpow2 80.4%

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

      3. unpow2 80.4%

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

      4. associate-*r* 80.4%

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

      5. times-frac 84.0%

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

   4. Simplified 84.0%

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

   5. Taylor expanded in k around inf 82.3%

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

      1. associate-/l* 80.4%

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

      2. unpow2 80.4%

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

      3. unpow2 80.4%

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

      4. associate-*r* 80.4%

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

      5. times-frac 84.0%

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

      6. times-frac 87.9%

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

      7. associate-/r/ 87.9%

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

      8. *-commutative 87.9%

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

      9. associate-/r/ 89.8%

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

   7. Simplified 89.8%

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

   8. Taylor expanded in k around 0 93.8%

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

   if 2.3000000000000001e218 < k

   1. Initial program 46.2%

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

   2. Taylor expanded in t around 0 62.3%

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

      1. associate-/l* 62.0%

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

      2. unpow2 62.0%

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

      3. unpow2 62.0%

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

      4. associate-*r* 62.0%

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

      5. times-frac 62.4%

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

   4. Simplified 62.4%

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

   5. Taylor expanded in k around inf 62.3%

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

      1. associate-/l* 62.0%

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

      2. unpow2 62.0%

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

      3. unpow2 62.0%

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

      4. associate-*r* 62.0%

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

      5. times-frac 62.4%

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

      6. times-frac 99.1%

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

      7. associate-/r/ 99.4%

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

      8. *-commutative 99.4%

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

      9. associate-/r/ 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in k around 0 77.9%

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

      1. div-inv 77.9%

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

      2. *-commutative 77.9%

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

      3. associate-/l* 99.4%

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

   10. Applied egg-rr 99.4%

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

       1. associate-*r/ 99.4%

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

       2. metadata-eval 99.4%

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

       3. associate-/r* 99.4%

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

       4. associate-/r/ 99.4%

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

       5. *-commutative 99.4%

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

       6. associate-/r* 99.5%

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

   12. Simplified 99.5%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{-41}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 2.3 \cdot 10^{+218}:\\ \;\;\;\;\frac{2}{\left({\sin k}^{2} \cdot \frac{k}{\ell \cdot \cos k}\right) \cdot \frac{t \cdot k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t \cdot \frac{k}{\ell}}}{{\sin k}^{2} \cdot \frac{\frac{k}{\ell}}{\cos k}}\\ \end{array} \]

Alternative 9: 67.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{-13}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.35e-13)
   (/ 2.0 (* (pow (/ t (cbrt (* l (/ l (sin k))))) 3.0) (* 2.0 k)))
   (* 2.0 (* (/ (cos k) (* k k)) (/ (* l l) (* t (pow (sin k) 2.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.35e-13) {
		tmp = 2.0 / (pow((t / cbrt((l * (l / sin(k))))), 3.0) * (2.0 * k));
	} else {
		tmp = 2.0 * ((cos(k) / (k * k)) * ((l * l) / (t * pow(sin(k), 2.0))));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.35e-13) {
		tmp = 2.0 / (Math.pow((t / Math.cbrt((l * (l / Math.sin(k))))), 3.0) * (2.0 * k));
	} else {
		tmp = 2.0 * ((Math.cos(k) / (k * k)) * ((l * l) / (t * Math.pow(Math.sin(k), 2.0))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.35e-13)
		tmp = Float64(2.0 / Float64((Float64(t / cbrt(Float64(l * Float64(l / sin(k))))) ^ 3.0) * Float64(2.0 * k)));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * k)) * Float64(Float64(l * l) / Float64(t * (sin(k) ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.35e-13], N[(2.0 / N[(N[Power[N[(t / N[Power[N[(l * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $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 < 1.35000000000000005e-13

   1. Initial program 53.6%

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

      1. associate-/r/ 53.0%

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

      2. add-cube-cbrt 52.9%

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

      3. pow3 52.9%

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

      4. cbrt-div 52.9%

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

      5. rem-cbrt-cube 62.2%

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

   3. Applied egg-rr 62.2%

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

      1. distribute-lft-in 62.2%

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

      2. associate-/l* 62.2%

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

      3. associate-/l* 66.8%

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

   5. Applied egg-rr 66.8%

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

      1. distribute-lft-out 66.8%

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

      2. +-commutative 66.8%

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

      3. associate-*r* 66.8%

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

      4. associate-/r/ 66.8%

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

      5. associate-+r+ 66.8%

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

      6. metadata-eval 66.8%

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

   7. Simplified 66.8%

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

   8. Taylor expanded in k around 0 63.6%

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

      1. *-commutative 63.6%

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

   10. Simplified 63.6%

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

   if 1.35000000000000005e-13 < k

   1. Initial program 51.3%

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

   2. Taylor expanded in t around 0 77.0%

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

      1. associate-/l* 75.3%

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

      2. unpow2 75.3%

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

      3. unpow2 75.3%

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

      4. associate-*r* 75.3%

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

      5. times-frac 76.9%

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

   4. Simplified 76.9%

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

   5. Taylor expanded in k around inf 77.0%

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

      1. times-frac 75.3%

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

      2. unpow2 75.3%

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

      3. unpow2 75.3%

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

      4. *-commutative 75.3%

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

   7. Simplified 75.3%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{-13}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternative 10: 68.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.15 \cdot 10^{-13}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.15e-13)
   (/ 2.0 (* (pow (/ t (cbrt (* l (/ l (sin k))))) 3.0) (* 2.0 k)))
   (* 2.0 (/ (* (cos k) (* l l)) (* (* k k) (* t (pow (sin k) 2.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.15e-13) {
		tmp = 2.0 / (pow((t / cbrt((l * (l / sin(k))))), 3.0) * (2.0 * k));
	} else {
		tmp = 2.0 * ((cos(k) * (l * l)) / ((k * k) * (t * pow(sin(k), 2.0))));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.15e-13) {
		tmp = 2.0 / (Math.pow((t / Math.cbrt((l * (l / Math.sin(k))))), 3.0) * (2.0 * k));
	} else {
		tmp = 2.0 * ((Math.cos(k) * (l * l)) / ((k * k) * (t * Math.pow(Math.sin(k), 2.0))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.15e-13)
		tmp = Float64(2.0 / Float64((Float64(t / cbrt(Float64(l * Float64(l / sin(k))))) ^ 3.0) * Float64(2.0 * k)));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) * Float64(l * l)) / Float64(Float64(k * k) * Float64(t * (sin(k) ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.15e-13], N[(2.0 / N[(N[Power[N[(t / N[Power[N[(l * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(N[(k * 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 < 1.1499999999999999e-13

   1. Initial program 53.6%

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

      1. associate-/r/ 53.0%

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

      2. add-cube-cbrt 52.9%

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

      3. pow3 52.9%

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

      4. cbrt-div 52.9%

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

      5. rem-cbrt-cube 62.2%

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

   3. Applied egg-rr 62.2%

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

      1. distribute-lft-in 62.2%

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

      2. associate-/l* 62.2%

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

      3. associate-/l* 66.8%

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

   5. Applied egg-rr 66.8%

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

      1. distribute-lft-out 66.8%

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

      2. +-commutative 66.8%

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

      3. associate-*r* 66.8%

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

      4. associate-/r/ 66.8%

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

      5. associate-+r+ 66.8%

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

      6. metadata-eval 66.8%

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

   7. Simplified 66.8%

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

   8. Taylor expanded in k around 0 63.6%

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

      1. *-commutative 63.6%

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

   10. Simplified 63.6%

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

   if 1.1499999999999999e-13 < k

   1. Initial program 51.3%

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

      1. associate-/r* 51.3%

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

      2. associate-*l* 51.3%

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

      3. sqr-neg 51.3%

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

      4. associate-*l* 51.3%

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

      5. *-commutative 51.3%

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

      6. sqr-neg 51.3%

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

      7. associate-*l/ 51.3%

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

      8. associate-*r/ 51.3%

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

      9. associate-/r/ 51.4%

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

   3. Simplified 51.4%

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

   4. Taylor expanded in k around inf 77.0%

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

      1. *-commutative 77.0%

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

      2. unpow2 77.0%

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

      3. unpow2 77.0%

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

      4. *-commutative 77.0%

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

   6. Simplified 77.0%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.15 \cdot 10^{-13}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array} \]

Alternative 11: 68.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-40}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot k} \cdot \left(\frac{\ell \cdot \cos k}{{\sin k}^{2}} \cdot \frac{\ell}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.2e-40)
   (/ 2.0 (* (pow (/ t (cbrt (* l (/ l (sin k))))) 3.0) (* 2.0 k)))
   (* (/ 2.0 (* k k)) (* (/ (* l (cos k)) (pow (sin k) 2.0)) (/ l t)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.2e-40) {
		tmp = 2.0 / (pow((t / cbrt((l * (l / sin(k))))), 3.0) * (2.0 * k));
	} else {
		tmp = (2.0 / (k * k)) * (((l * cos(k)) / pow(sin(k), 2.0)) * (l / t));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.2e-40) {
		tmp = 2.0 / (Math.pow((t / Math.cbrt((l * (l / Math.sin(k))))), 3.0) * (2.0 * k));
	} else {
		tmp = (2.0 / (k * k)) * (((l * Math.cos(k)) / Math.pow(Math.sin(k), 2.0)) * (l / t));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.2e-40)
		tmp = Float64(2.0 / Float64((Float64(t / cbrt(Float64(l * Float64(l / sin(k))))) ^ 3.0) * Float64(2.0 * k)));
	else
		tmp = Float64(Float64(2.0 / Float64(k * k)) * Float64(Float64(Float64(l * cos(k)) / (sin(k) ^ 2.0)) * Float64(l / t)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.2e-40], N[(2.0 / N[(N[Power[N[(t / N[Power[N[(l * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.19999999999999996e-40

   1. Initial program 52.5%

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

      1. associate-/r/ 51.8%

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

      2. add-cube-cbrt 51.7%

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

      3. pow3 51.7%

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

      4. cbrt-div 51.7%

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

      5. rem-cbrt-cube 61.3%

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

   3. Applied egg-rr 61.3%

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

      1. distribute-lft-in 61.3%

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

      2. associate-/l* 61.3%

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

      3. associate-/l* 66.1%

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

   5. Applied egg-rr 66.1%

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

      1. distribute-lft-out 66.1%

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

      2. +-commutative 66.1%

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

      3. associate-*r* 66.1%

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

      4. associate-/r/ 66.1%

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

      5. associate-+r+ 66.1%

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

      6. metadata-eval 66.1%

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

   7. Simplified 66.1%

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

   8. Taylor expanded in k around 0 63.2%

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

      1. *-commutative 63.2%

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

   10. Simplified 63.2%

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

   if 1.19999999999999996e-40 < k

   1. Initial program 55.1%

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

      1. associate-/r* 55.1%

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

      2. associate-*l* 55.2%

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

      3. sqr-neg 55.2%

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

      4. associate-*l* 55.1%

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

      5. *-commutative 55.1%

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

      6. sqr-neg 55.1%

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

      7. associate-*l/ 55.2%

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

      8. associate-*r/ 55.2%

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

      9. associate-/r/ 53.6%

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

   3. Simplified 53.6%

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

   4. Taylor expanded in k around inf 78.1%

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

      1. associate-*r/ 78.1%

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

      2. times-frac 76.5%

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

      3. unpow2 76.5%

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

      4. unpow2 76.5%

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

      5. associate-*r* 76.5%

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

      6. times-frac 79.4%

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

   6. Simplified 79.4%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-40}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot k} \cdot \left(\frac{\ell \cdot \cos k}{{\sin k}^{2}} \cdot \frac{\ell}{t}\right)\\ \end{array} \]

Alternative 12: 73.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{-39}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \frac{k}{\ell}\right) \cdot \left({\sin k}^{2} \cdot \frac{k}{\ell \cdot \cos k}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 4.2e-39)
   (/ 2.0 (* (pow (/ t (cbrt (* l (/ l (sin k))))) 3.0) (* 2.0 k)))
   (/ 2.0 (* (* t (/ k l)) (* (pow (sin k) 2.0) (/ k (* l (cos k))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.2e-39) {
		tmp = 2.0 / (pow((t / cbrt((l * (l / sin(k))))), 3.0) * (2.0 * k));
	} else {
		tmp = 2.0 / ((t * (k / l)) * (pow(sin(k), 2.0) * (k / (l * cos(k)))));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.2e-39) {
		tmp = 2.0 / (Math.pow((t / Math.cbrt((l * (l / Math.sin(k))))), 3.0) * (2.0 * k));
	} else {
		tmp = 2.0 / ((t * (k / l)) * (Math.pow(Math.sin(k), 2.0) * (k / (l * Math.cos(k)))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 4.2e-39)
		tmp = Float64(2.0 / Float64((Float64(t / cbrt(Float64(l * Float64(l / sin(k))))) ^ 3.0) * Float64(2.0 * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(t * Float64(k / l)) * Float64((sin(k) ^ 2.0) * Float64(k / Float64(l * cos(k))))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 4.2e-39], N[(2.0 / N[(N[Power[N[(t / N[Power[N[(l * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(k / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.19999999999999987e-39

   1. Initial program 52.5%

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

      1. associate-/r/ 51.8%

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

      2. add-cube-cbrt 51.7%

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

      3. pow3 51.7%

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

      4. cbrt-div 51.7%

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

      5. rem-cbrt-cube 61.3%

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

   3. Applied egg-rr 61.3%

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

      1. distribute-lft-in 61.3%

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

      2. associate-/l* 61.3%

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

      3. associate-/l* 66.1%

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

   5. Applied egg-rr 66.1%

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

      1. distribute-lft-out 66.1%

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

      2. +-commutative 66.1%

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

      3. associate-*r* 66.1%

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

      4. associate-/r/ 66.1%

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

      5. associate-+r+ 66.1%

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

      6. metadata-eval 66.1%

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

   7. Simplified 66.1%

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

   8. Taylor expanded in k around 0 63.2%

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

      1. *-commutative 63.2%

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

   10. Simplified 63.2%

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

   if 4.19999999999999987e-39 < k

   1. Initial program 55.1%

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

   2. Taylor expanded in t around 0 78.1%

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

      1. associate-/l* 76.6%

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

      2. unpow2 76.6%

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

      3. unpow2 76.6%

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

      4. associate-*r* 76.6%

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

      5. times-frac 79.5%

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

   4. Simplified 79.5%

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

   5. Taylor expanded in k around inf 78.1%

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

      1. associate-/l* 76.6%

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

      2. unpow2 76.6%

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

      3. unpow2 76.6%

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

      4. associate-*r* 76.6%

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

      5. times-frac 79.5%

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

      6. times-frac 90.3%

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

      7. associate-/r/ 90.3%

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

      8. *-commutative 90.3%

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

      9. associate-/r/ 91.8%

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

   7. Simplified 91.8%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{-39}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \frac{k}{\ell}\right) \cdot \left({\sin k}^{2} \cdot \frac{k}{\ell \cdot \cos k}\right)}\\ \end{array} \]

Alternative 13: 65.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.3 \cdot 10^{-41}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(k \cdot \frac{k}{\cos k}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 3.3e-41)
   (/ 2.0 (* (pow (/ t (cbrt (* l (/ l (sin k))))) 3.0) (* 2.0 k)))
   (/ 2.0 (* (* (/ t l) (/ (* k k) l)) (* k (/ k (cos k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.3e-41) {
		tmp = 2.0 / (pow((t / cbrt((l * (l / sin(k))))), 3.0) * (2.0 * k));
	} else {
		tmp = 2.0 / (((t / l) * ((k * k) / l)) * (k * (k / cos(k))));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.3e-41) {
		tmp = 2.0 / (Math.pow((t / Math.cbrt((l * (l / Math.sin(k))))), 3.0) * (2.0 * k));
	} else {
		tmp = 2.0 / (((t / l) * ((k * k) / l)) * (k * (k / Math.cos(k))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 3.3e-41)
		tmp = Float64(2.0 / Float64((Float64(t / cbrt(Float64(l * Float64(l / sin(k))))) ^ 3.0) * Float64(2.0 * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) * Float64(Float64(k * k) / l)) * Float64(k * Float64(k / cos(k)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 3.3e-41], N[(2.0 / N[(N[Power[N[(t / N[Power[N[(l * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k * N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.30000000000000024e-41

   1. Initial program 52.5%

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

      1. associate-/r/ 51.8%

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

      2. add-cube-cbrt 51.7%

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

      3. pow3 51.7%

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

      4. cbrt-div 51.7%

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

      5. rem-cbrt-cube 61.3%

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

   3. Applied egg-rr 61.3%

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

      1. distribute-lft-in 61.3%

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

      2. associate-/l* 61.3%

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

      3. associate-/l* 66.1%

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

   5. Applied egg-rr 66.1%

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

      1. distribute-lft-out 66.1%

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

      2. +-commutative 66.1%

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

      3. associate-*r* 66.1%

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

      4. associate-/r/ 66.1%

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

      5. associate-+r+ 66.1%

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

      6. metadata-eval 66.1%

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

   7. Simplified 66.1%

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

   8. Taylor expanded in k around 0 63.2%

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

      1. *-commutative 63.2%

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

   10. Simplified 63.2%

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

   if 3.30000000000000024e-41 < k

   1. Initial program 55.1%

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

   2. Taylor expanded in t around 0 78.1%

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

      1. times-frac 76.6%

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

      2. unpow2 76.6%

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

      3. *-commutative 76.6%

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

      4. unpow2 76.6%

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

   4. Simplified 76.6%

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

   5. Taylor expanded in k around 0 63.2%

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

      1. *-commutative 63.2%

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

      2. unpow2 63.2%

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

      3. times-frac 63.5%

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

      4. unpow2 63.5%

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

   7. Simplified 63.5%

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

      1. pow1 63.5%

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

      2. *-commutative 63.5%

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

      3. frac-times 63.2%

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

      4. associate-/l* 63.2%

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

   9. Applied egg-rr 63.2%

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

       1. unpow1 63.2%

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

       2. times-frac 63.5%

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

       3. associate-/r/ 63.5%

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

   11. Simplified 63.5%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.3 \cdot 10^{-41}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(k \cdot \frac{k}{\cos k}\right)}\\ \end{array} \]

Alternative 14: 66.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{+206}:\\ \;\;\;\;\ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot k\right)}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-67}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(k \cdot \frac{k}{\cos k}\right)}\\ \mathbf{elif}\;t \leq 9.6 \cdot 10^{+111}:\\ \;\;\;\;\frac{2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+181}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\log \left(e^{\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(k \cdot \frac{k}{\ell}\right) \cdot \frac{{t}^{3}}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -4.8e+206)
   (* l (/ l (* (pow t 3.0) (* k k))))
   (if (<= t 4e-67)
     (/ 2.0 (* (* (/ t l) (/ (* k k) l)) (* k (/ k (cos k)))))
     (if (<= t 9.6e+111)
       (/
        (* 2.0 (/ (* (/ l k) (/ l k)) (pow t 3.0)))
        (+ 2.0 (pow (/ k t) 2.0)))
       (if (<= t 1.4e+181)
         (/ 2.0 (/ (* k k) (log (exp (* (/ l t) (/ l (* k k)))))))
         (/ 2.0 (* 2.0 (* (* k (/ k l)) (/ (pow t 3.0) l)))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -4.8e+206) {
		tmp = l * (l / (pow(t, 3.0) * (k * k)));
	} else if (t <= 4e-67) {
		tmp = 2.0 / (((t / l) * ((k * k) / l)) * (k * (k / cos(k))));
	} else if (t <= 9.6e+111) {
		tmp = (2.0 * (((l / k) * (l / k)) / pow(t, 3.0))) / (2.0 + pow((k / t), 2.0));
	} else if (t <= 1.4e+181) {
		tmp = 2.0 / ((k * k) / log(exp(((l / t) * (l / (k * k))))));
	} else {
		tmp = 2.0 / (2.0 * ((k * (k / l)) * (pow(t, 3.0) / l)));
	}
	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 <= (-4.8d+206)) then
        tmp = l * (l / ((t ** 3.0d0) * (k * k)))
    else if (t <= 4d-67) then
        tmp = 2.0d0 / (((t / l) * ((k * k) / l)) * (k * (k / cos(k))))
    else if (t <= 9.6d+111) then
        tmp = (2.0d0 * (((l / k) * (l / k)) / (t ** 3.0d0))) / (2.0d0 + ((k / t) ** 2.0d0))
    else if (t <= 1.4d+181) then
        tmp = 2.0d0 / ((k * k) / log(exp(((l / t) * (l / (k * k))))))
    else
        tmp = 2.0d0 / (2.0d0 * ((k * (k / l)) * ((t ** 3.0d0) / l)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -4.8e+206) {
		tmp = l * (l / (Math.pow(t, 3.0) * (k * k)));
	} else if (t <= 4e-67) {
		tmp = 2.0 / (((t / l) * ((k * k) / l)) * (k * (k / Math.cos(k))));
	} else if (t <= 9.6e+111) {
		tmp = (2.0 * (((l / k) * (l / k)) / Math.pow(t, 3.0))) / (2.0 + Math.pow((k / t), 2.0));
	} else if (t <= 1.4e+181) {
		tmp = 2.0 / ((k * k) / Math.log(Math.exp(((l / t) * (l / (k * k))))));
	} else {
		tmp = 2.0 / (2.0 * ((k * (k / l)) * (Math.pow(t, 3.0) / l)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -4.8e+206:
		tmp = l * (l / (math.pow(t, 3.0) * (k * k)))
	elif t <= 4e-67:
		tmp = 2.0 / (((t / l) * ((k * k) / l)) * (k * (k / math.cos(k))))
	elif t <= 9.6e+111:
		tmp = (2.0 * (((l / k) * (l / k)) / math.pow(t, 3.0))) / (2.0 + math.pow((k / t), 2.0))
	elif t <= 1.4e+181:
		tmp = 2.0 / ((k * k) / math.log(math.exp(((l / t) * (l / (k * k))))))
	else:
		tmp = 2.0 / (2.0 * ((k * (k / l)) * (math.pow(t, 3.0) / l)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -4.8e+206)
		tmp = Float64(l * Float64(l / Float64((t ^ 3.0) * Float64(k * k))));
	elseif (t <= 4e-67)
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) * Float64(Float64(k * k) / l)) * Float64(k * Float64(k / cos(k)))));
	elseif (t <= 9.6e+111)
		tmp = Float64(Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) / (t ^ 3.0))) / Float64(2.0 + (Float64(k / t) ^ 2.0)));
	elseif (t <= 1.4e+181)
		tmp = Float64(2.0 / Float64(Float64(k * k) / log(exp(Float64(Float64(l / t) * Float64(l / Float64(k * k)))))));
	else
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(k * Float64(k / l)) * Float64((t ^ 3.0) / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -4.8e+206)
		tmp = l * (l / ((t ^ 3.0) * (k * k)));
	elseif (t <= 4e-67)
		tmp = 2.0 / (((t / l) * ((k * k) / l)) * (k * (k / cos(k))));
	elseif (t <= 9.6e+111)
		tmp = (2.0 * (((l / k) * (l / k)) / (t ^ 3.0))) / (2.0 + ((k / t) ^ 2.0));
	elseif (t <= 1.4e+181)
		tmp = 2.0 / ((k * k) / log(exp(((l / t) * (l / (k * k))))));
	else
		tmp = 2.0 / (2.0 * ((k * (k / l)) * ((t ^ 3.0) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -4.8e+206], N[(l * N[(l / N[(N[Power[t, 3.0], $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4e-67], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k * N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9.6e+111], N[(N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.4e+181], N[(2.0 / N[(N[(k * k), $MachinePrecision] / N[Log[N[Exp[N[(N[(l / t), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * N[(N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -4.7999999999999999e206

   1. Initial program 42.6%

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

      1. associate-/r* 42.6%

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

      2. associate-*l* 35.8%

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

      3. sqr-neg 35.8%

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

      4. associate-*l* 42.6%

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

      5. *-commutative 42.6%

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

      6. sqr-neg 42.6%

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

      7. associate-*l/ 42.6%

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

      8. associate-*r/ 42.6%

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

      9. associate-/r/ 42.6%

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

   3. Simplified 42.6%

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

   4. Taylor expanded in k around 0 35.8%

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

      1. unpow2 35.8%

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

      2. *-commutative 35.8%

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

      3. unpow2 35.8%

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

   6. Simplified 35.8%

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

      1. *-un-lft-identity 35.8%

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

      2. associate-/l* 59.6%

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

      3. *-commutative 59.6%

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

   8. Applied egg-rr 59.6%

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

      1. associate-/r/ 59.6%

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

   10. Applied egg-rr 59.6%

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

   if -4.7999999999999999e206 < t < 3.99999999999999977e-67

   1. Initial program 46.4%

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

   2. Taylor expanded in t around 0 70.3%

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

      1. times-frac 70.4%

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

      2. unpow2 70.4%

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

      3. *-commutative 70.4%

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

      4. unpow2 70.4%

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

   4. Simplified 70.4%

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

   5. Taylor expanded in k around 0 59.1%

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

      1. *-commutative 59.1%

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

      2. unpow2 59.1%

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

      3. times-frac 68.1%

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

      4. unpow2 68.1%

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

   7. Simplified 68.1%

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

      1. pow1 68.1%

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

      2. *-commutative 68.1%

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

      3. frac-times 59.1%

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

      4. associate-/l* 59.1%

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

   9. Applied egg-rr 59.1%

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

       1. unpow1 59.1%

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

       2. times-frac 68.1%

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

       3. associate-/r/ 68.1%

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

   11. Simplified 68.1%

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

   if 3.99999999999999977e-67 < t < 9.60000000000000023e111

   1. Initial program 80.2%

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

      1. associate-/r* 80.3%

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

      2. associate-*l* 73.6%

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

      3. sqr-neg 73.6%

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

      4. associate-*l* 80.3%

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

      5. *-commutative 80.3%

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

      6. sqr-neg 80.3%

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

      7. associate-*l/ 83.5%

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

      8. associate-*r/ 83.7%

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

      9. associate-/r/ 83.7%

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

   3. Simplified 83.7%

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

   4. Taylor expanded in k around 0 68.3%

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

      1. associate-/r* 65.1%

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

      2. unpow2 65.1%

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

      3. unpow2 65.1%

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

      4. times-frac 78.5%

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

   6. Simplified 78.5%

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

   if 9.60000000000000023e111 < t < 1.39999999999999992e181

   1. Initial program 40.0%

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

   2. Taylor expanded in t around 0 57.1%

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

      1. associate-/l* 57.1%

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

      2. unpow2 57.1%

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

      3. unpow2 57.1%

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

      4. associate-*r* 57.1%

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

      5. times-frac 57.4%

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

   4. Simplified 57.4%

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

   5. Taylor expanded in k around 0 57.4%

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

      1. unpow2 57.4%

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

   7. Simplified 57.4%

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

      1. add-log-exp 62.8%

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

   9. Applied egg-rr 62.8%

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

   if 1.39999999999999992e181 < t

   1. Initial program 73.1%

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

      1. associate-/r/ 72.8%

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

      2. add-cube-cbrt 72.8%

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

      3. pow3 72.8%

         \[\leadsto \frac{2}{\left(\color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right)}^{3}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. cbrt-div 72.8%

         \[\leadsto \frac{2}{\left({\color{blue}{\left(\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\frac{\ell \cdot \ell}{\sin k}}}\right)}}^{3} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. rem-cbrt-cube 73.5%

         \[\leadsto \frac{2}{\left({\left(\frac{\color{blue}{t}}{\sqrt[3]{\frac{\ell \cdot \ell}{\sin k}}}\right)}^{3} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied egg-rr 73.5%

      \[\leadsto \frac{2}{\left(\color{blue}{{\left(\frac{t}{\sqrt[3]{\frac{\ell \cdot \ell}{\sin k}}}\right)}^{3}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Taylor expanded in k around 0 63.6%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   5. Step-by-step derivation

      1. unpow2 63.6%

         \[\leadsto \frac{2}{2 \cdot \frac{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}}{{\ell}^{2}}} \]

      2. unpow2 63.6%

         \[\leadsto \frac{2}{2 \cdot \frac{\left(k \cdot k\right) \cdot {t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      3. times-frac 66.9%

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\left(\frac{k \cdot k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)}} \]

      4. associate-*l/ 76.0%

         \[\leadsto \frac{2}{2 \cdot \left(\color{blue}{\left(\frac{k}{\ell} \cdot k\right)} \cdot \frac{{t}^{3}}{\ell}\right)} \]

   6. Simplified 76.0%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \left(\left(\frac{k}{\ell} \cdot k\right) \cdot \frac{{t}^{3}}{\ell}\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{+206}:\\ \;\;\;\;\ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot k\right)}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-67}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(k \cdot \frac{k}{\cos k}\right)}\\ \mathbf{elif}\;t \leq 9.6 \cdot 10^{+111}:\\ \;\;\;\;\frac{2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+181}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\log \left(e^{\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(k \cdot \frac{k}{\ell}\right) \cdot \frac{{t}^{3}}{\ell}\right)}\\ \end{array} \]

Alternative 15: 65.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.76 \cdot 10^{+208}:\\ \;\;\;\;\ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot k\right)}\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{-35}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(k \cdot \frac{k}{\cos k}\right)}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{+112}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+181}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\log \left(e^{\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(k \cdot \frac{k}{\ell}\right) \cdot \frac{{t}^{3}}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -1.76e+208)
   (* l (/ l (* (pow t 3.0) (* k k))))
   (if (<= t 8.6e-35)
     (/ 2.0 (* (* (/ t l) (/ (* k k) l)) (* k (/ k (cos k)))))
     (if (<= t 1.6e+112)
       (/ (* (/ l k) (/ l k)) (pow t 3.0))
       (if (<= t 1.4e+181)
         (/ 2.0 (/ (* k k) (log (exp (* (/ l t) (/ l (* k k)))))))
         (/ 2.0 (* 2.0 (* (* k (/ k l)) (/ (pow t 3.0) l)))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.76e+208) {
		tmp = l * (l / (pow(t, 3.0) * (k * k)));
	} else if (t <= 8.6e-35) {
		tmp = 2.0 / (((t / l) * ((k * k) / l)) * (k * (k / cos(k))));
	} else if (t <= 1.6e+112) {
		tmp = ((l / k) * (l / k)) / pow(t, 3.0);
	} else if (t <= 1.4e+181) {
		tmp = 2.0 / ((k * k) / log(exp(((l / t) * (l / (k * k))))));
	} else {
		tmp = 2.0 / (2.0 * ((k * (k / l)) * (pow(t, 3.0) / l)));
	}
	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 <= (-1.76d+208)) then
        tmp = l * (l / ((t ** 3.0d0) * (k * k)))
    else if (t <= 8.6d-35) then
        tmp = 2.0d0 / (((t / l) * ((k * k) / l)) * (k * (k / cos(k))))
    else if (t <= 1.6d+112) then
        tmp = ((l / k) * (l / k)) / (t ** 3.0d0)
    else if (t <= 1.4d+181) then
        tmp = 2.0d0 / ((k * k) / log(exp(((l / t) * (l / (k * k))))))
    else
        tmp = 2.0d0 / (2.0d0 * ((k * (k / l)) * ((t ** 3.0d0) / l)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.76e+208) {
		tmp = l * (l / (Math.pow(t, 3.0) * (k * k)));
	} else if (t <= 8.6e-35) {
		tmp = 2.0 / (((t / l) * ((k * k) / l)) * (k * (k / Math.cos(k))));
	} else if (t <= 1.6e+112) {
		tmp = ((l / k) * (l / k)) / Math.pow(t, 3.0);
	} else if (t <= 1.4e+181) {
		tmp = 2.0 / ((k * k) / Math.log(Math.exp(((l / t) * (l / (k * k))))));
	} else {
		tmp = 2.0 / (2.0 * ((k * (k / l)) * (Math.pow(t, 3.0) / l)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -1.76e+208:
		tmp = l * (l / (math.pow(t, 3.0) * (k * k)))
	elif t <= 8.6e-35:
		tmp = 2.0 / (((t / l) * ((k * k) / l)) * (k * (k / math.cos(k))))
	elif t <= 1.6e+112:
		tmp = ((l / k) * (l / k)) / math.pow(t, 3.0)
	elif t <= 1.4e+181:
		tmp = 2.0 / ((k * k) / math.log(math.exp(((l / t) * (l / (k * k))))))
	else:
		tmp = 2.0 / (2.0 * ((k * (k / l)) * (math.pow(t, 3.0) / l)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -1.76e+208)
		tmp = Float64(l * Float64(l / Float64((t ^ 3.0) * Float64(k * k))));
	elseif (t <= 8.6e-35)
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) * Float64(Float64(k * k) / l)) * Float64(k * Float64(k / cos(k)))));
	elseif (t <= 1.6e+112)
		tmp = Float64(Float64(Float64(l / k) * Float64(l / k)) / (t ^ 3.0));
	elseif (t <= 1.4e+181)
		tmp = Float64(2.0 / Float64(Float64(k * k) / log(exp(Float64(Float64(l / t) * Float64(l / Float64(k * k)))))));
	else
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(k * Float64(k / l)) * Float64((t ^ 3.0) / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -1.76e+208)
		tmp = l * (l / ((t ^ 3.0) * (k * k)));
	elseif (t <= 8.6e-35)
		tmp = 2.0 / (((t / l) * ((k * k) / l)) * (k * (k / cos(k))));
	elseif (t <= 1.6e+112)
		tmp = ((l / k) * (l / k)) / (t ^ 3.0);
	elseif (t <= 1.4e+181)
		tmp = 2.0 / ((k * k) / log(exp(((l / t) * (l / (k * k))))));
	else
		tmp = 2.0 / (2.0 * ((k * (k / l)) * ((t ^ 3.0) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -1.76e+208], N[(l * N[(l / N[(N[Power[t, 3.0], $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8.6e-35], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k * N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.6e+112], N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.4e+181], N[(2.0 / N[(N[(k * k), $MachinePrecision] / N[Log[N[Exp[N[(N[(l / t), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * N[(N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -1.76000000000000007e208

   1. Initial program 42.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 42.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 35.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 35.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 42.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 42.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 42.6%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 42.6%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 42.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 42.6%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 42.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around 0 35.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 35.8%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 35.8%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. unpow2 35.8%

         \[\leadsto \frac{\ell \cdot \ell}{{t}^{3} \cdot \color{blue}{\left(k \cdot k\right)}} \]

   6. Simplified 35.8%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{{t}^{3} \cdot \left(k \cdot k\right)}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 35.8%

         \[\leadsto \color{blue}{1 \cdot \frac{\ell \cdot \ell}{{t}^{3} \cdot \left(k \cdot k\right)}} \]

      2. associate-/l* 59.6%

         \[\leadsto 1 \cdot \color{blue}{\frac{\ell}{\frac{{t}^{3} \cdot \left(k \cdot k\right)}{\ell}}} \]

      3. *-commutative 59.6%

         \[\leadsto 1 \cdot \frac{\ell}{\frac{\color{blue}{\left(k \cdot k\right) \cdot {t}^{3}}}{\ell}} \]

   8. Applied egg-rr 59.6%

      \[\leadsto \color{blue}{1 \cdot \frac{\ell}{\frac{\left(k \cdot k\right) \cdot {t}^{3}}{\ell}}} \]
   9. Step-by-step derivation

      1. associate-/r/ 59.6%

         \[\leadsto 1 \cdot \color{blue}{\left(\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}} \cdot \ell\right)} \]

   10. Applied egg-rr 59.6%

       \[\leadsto 1 \cdot \color{blue}{\left(\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}} \cdot \ell\right)} \]

   if -1.76000000000000007e208 < t < 8.6000000000000004e-35

   1. Initial program 48.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0 71.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}}} \]
   3. Step-by-step derivation

      1. times-frac 71.5%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}}} \]

      2. unpow2 71.5%

         \[\leadsto \frac{2}{\frac{\color{blue}{k \cdot k}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]

      3. *-commutative 71.5%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2}}} \]

      4. unpow2 71.5%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]

   4. Simplified 71.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot k}{\cos k} \cdot \frac{t \cdot {\sin k}^{2}}{\ell \cdot \ell}}} \]

   5. Taylor expanded in k around 0 60.2%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\cos k} \cdot \color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}}} \]
   6. Step-by-step derivation

      1. *-commutative 60.2%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}}} \]

      2. unpow2 60.2%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]

      3. times-frac 68.8%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\cos k} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)}} \]

      4. unpow2 68.8%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\cos k} \cdot \left(\frac{t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right)} \]

   7. Simplified 68.8%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\cos k} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)}} \]
   8. Step-by-step derivation

      1. pow1 68.8%

         \[\leadsto \frac{2}{\color{blue}{{\left(\frac{k \cdot k}{\cos k} \cdot \left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)\right)}^{1}}} \]

      2. *-commutative 68.8%

         \[\leadsto \frac{2}{{\color{blue}{\left(\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \frac{k \cdot k}{\cos k}\right)}}^{1}} \]

      3. frac-times 60.2%

         \[\leadsto \frac{2}{{\left(\color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}} \cdot \frac{k \cdot k}{\cos k}\right)}^{1}} \]

      4. associate-/l* 60.2%

         \[\leadsto \frac{2}{{\left(\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot \color{blue}{\frac{k}{\frac{\cos k}{k}}}\right)}^{1}} \]

   9. Applied egg-rr 60.2%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot \frac{k}{\frac{\cos k}{k}}\right)}^{1}}} \]
   10. Step-by-step derivation

       1. unpow1 60.2%

          \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot \frac{k}{\frac{\cos k}{k}}}} \]

       2. times-frac 68.8%

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)} \cdot \frac{k}{\frac{\cos k}{k}}} \]

       3. associate-/r/ 68.8%

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \color{blue}{\left(\frac{k}{\cos k} \cdot k\right)}} \]

   11. Simplified 68.8%

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(\frac{k}{\cos k} \cdot k\right)}} \]

   if 8.6000000000000004e-35 < t < 1.59999999999999993e112

   1. Initial program 75.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 75.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 67.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 67.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 75.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 75.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 75.4%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 79.5%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 79.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 79.6%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 79.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around 0 63.5%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. associate-/r* 59.5%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}}} \]

      2. unpow2 59.5%

         \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{{t}^{3}} \]

      3. unpow2 59.5%

         \[\leadsto \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{{t}^{3}} \]

      4. times-frac 76.1%

         \[\leadsto \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{{t}^{3}} \]

   6. Simplified 76.1%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}} \]

   if 1.59999999999999993e112 < t < 1.39999999999999992e181

   1. Initial program 40.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0 57.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}}} \]
   3. Step-by-step derivation

      1. associate-/l* 57.1%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2} \cdot t}}}} \]

      2. unpow2 57.1%

         \[\leadsto \frac{2}{\frac{\color{blue}{k \cdot k}}{\frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2} \cdot t}}} \]

      3. unpow2 57.1%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2} \cdot t}}} \]

      4. associate-*r* 57.1%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}{{\sin k}^{2} \cdot t}}} \]

      5. times-frac 57.4%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\cos k \cdot \ell}{{\sin k}^{2}} \cdot \frac{\ell}{t}}}} \]

   4. Simplified 57.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot k}{\frac{\cos k \cdot \ell}{{\sin k}^{2}} \cdot \frac{\ell}{t}}}} \]

   5. Taylor expanded in k around 0 57.4%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\ell}{{k}^{2}}} \cdot \frac{\ell}{t}}} \]
   6. Step-by-step derivation

      1. unpow2 57.4%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{t}}} \]

   7. Simplified 57.4%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\ell}{k \cdot k}} \cdot \frac{\ell}{t}}} \]
   8. Step-by-step derivation

      1. add-log-exp 62.8%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\log \left(e^{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}}\right)}}} \]

   9. Applied egg-rr 62.8%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\log \left(e^{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}}\right)}}} \]

   if 1.39999999999999992e181 < t

   1. Initial program 73.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r/ 72.8%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. add-cube-cbrt 72.8%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \sqrt[3]{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. pow3 72.8%

         \[\leadsto \frac{2}{\left(\color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right)}^{3}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. cbrt-div 72.8%

         \[\leadsto \frac{2}{\left({\color{blue}{\left(\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\frac{\ell \cdot \ell}{\sin k}}}\right)}}^{3} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. rem-cbrt-cube 73.5%

         \[\leadsto \frac{2}{\left({\left(\frac{\color{blue}{t}}{\sqrt[3]{\frac{\ell \cdot \ell}{\sin k}}}\right)}^{3} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied egg-rr 73.5%

      \[\leadsto \frac{2}{\left(\color{blue}{{\left(\frac{t}{\sqrt[3]{\frac{\ell \cdot \ell}{\sin k}}}\right)}^{3}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Taylor expanded in k around 0 63.6%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   5. Step-by-step derivation

      1. unpow2 63.6%

         \[\leadsto \frac{2}{2 \cdot \frac{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}}{{\ell}^{2}}} \]

      2. unpow2 63.6%

         \[\leadsto \frac{2}{2 \cdot \frac{\left(k \cdot k\right) \cdot {t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      3. times-frac 66.9%

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\left(\frac{k \cdot k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)}} \]

      4. associate-*l/ 76.0%

         \[\leadsto \frac{2}{2 \cdot \left(\color{blue}{\left(\frac{k}{\ell} \cdot k\right)} \cdot \frac{{t}^{3}}{\ell}\right)} \]

   6. Simplified 76.0%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \left(\left(\frac{k}{\ell} \cdot k\right) \cdot \frac{{t}^{3}}{\ell}\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.76 \cdot 10^{+208}:\\ \;\;\;\;\ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot k\right)}\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{-35}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(k \cdot \frac{k}{\cos k}\right)}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{+112}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+181}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\log \left(e^{\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(k \cdot \frac{k}{\ell}\right) \cdot \frac{{t}^{3}}{\ell}\right)}\\ \end{array} \]

Alternative 16: 60.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ \mathbf{if}\;\ell \leq 4.3 \cdot 10^{-50}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\frac{\ell}{t} \cdot \left(t_1 + \ell \cdot -0.16666666666666666\right)}}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+207}:\\ \;\;\;\;t_1 \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot t_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ l (* k k))))
   (if (<= l 4.3e-50)
     (/ 2.0 (/ (* k k) (* (/ l t) (+ t_1 (* l -0.16666666666666666)))))
     (if (<= l 5.8e+207)
       (* t_1 (/ l (pow t 3.0)))
       (* (/ 2.0 (* k k)) (* (/ l t) t_1))))))

C

double code(double t, double l, double k) {
	double t_1 = l / (k * k);
	double tmp;
	if (l <= 4.3e-50) {
		tmp = 2.0 / ((k * k) / ((l / t) * (t_1 + (l * -0.16666666666666666))));
	} else if (l <= 5.8e+207) {
		tmp = t_1 * (l / pow(t, 3.0));
	} else {
		tmp = (2.0 / (k * k)) * ((l / t) * t_1);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = l / (k * k)
    if (l <= 4.3d-50) then
        tmp = 2.0d0 / ((k * k) / ((l / t) * (t_1 + (l * (-0.16666666666666666d0)))))
    else if (l <= 5.8d+207) then
        tmp = t_1 * (l / (t ** 3.0d0))
    else
        tmp = (2.0d0 / (k * k)) * ((l / t) * t_1)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = l / (k * k);
	double tmp;
	if (l <= 4.3e-50) {
		tmp = 2.0 / ((k * k) / ((l / t) * (t_1 + (l * -0.16666666666666666))));
	} else if (l <= 5.8e+207) {
		tmp = t_1 * (l / Math.pow(t, 3.0));
	} else {
		tmp = (2.0 / (k * k)) * ((l / t) * t_1);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = l / (k * k)
	tmp = 0
	if l <= 4.3e-50:
		tmp = 2.0 / ((k * k) / ((l / t) * (t_1 + (l * -0.16666666666666666))))
	elif l <= 5.8e+207:
		tmp = t_1 * (l / math.pow(t, 3.0))
	else:
		tmp = (2.0 / (k * k)) * ((l / t) * t_1)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(l / Float64(k * k))
	tmp = 0.0
	if (l <= 4.3e-50)
		tmp = Float64(2.0 / Float64(Float64(k * k) / Float64(Float64(l / t) * Float64(t_1 + Float64(l * -0.16666666666666666)))));
	elseif (l <= 5.8e+207)
		tmp = Float64(t_1 * Float64(l / (t ^ 3.0)));
	else
		tmp = Float64(Float64(2.0 / Float64(k * k)) * Float64(Float64(l / t) * t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = l / (k * k);
	tmp = 0.0;
	if (l <= 4.3e-50)
		tmp = 2.0 / ((k * k) / ((l / t) * (t_1 + (l * -0.16666666666666666))));
	elseif (l <= 5.8e+207)
		tmp = t_1 * (l / (t ^ 3.0));
	else
		tmp = (2.0 / (k * k)) * ((l / t) * t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 4.3e-50], N[(2.0 / N[(N[(k * k), $MachinePrecision] / N[(N[(l / t), $MachinePrecision] * N[(t$95$1 + N[(l * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.8e+207], N[(t$95$1 * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l / t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 4.29999999999999997e-50

   1. Initial program 55.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0 60.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}}} \]
   3. Step-by-step derivation

      1. associate-/l* 62.1%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2} \cdot t}}}} \]

      2. unpow2 62.1%

         \[\leadsto \frac{2}{\frac{\color{blue}{k \cdot k}}{\frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2} \cdot t}}} \]

      3. unpow2 62.1%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2} \cdot t}}} \]

      4. associate-*r* 62.1%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}{{\sin k}^{2} \cdot t}}} \]

      5. times-frac 70.1%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\cos k \cdot \ell}{{\sin k}^{2}} \cdot \frac{\ell}{t}}}} \]

   4. Simplified 70.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot k}{\frac{\cos k \cdot \ell}{{\sin k}^{2}} \cdot \frac{\ell}{t}}}} \]

   5. Taylor expanded in k around 0 64.3%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\left(\left(\frac{\ell}{{k}^{2}} + -0.5 \cdot \ell\right) - -0.3333333333333333 \cdot \ell\right)} \cdot \frac{\ell}{t}}} \]
   6. Step-by-step derivation

      1. associate--l+ 64.3%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\left(\frac{\ell}{{k}^{2}} + \left(-0.5 \cdot \ell - -0.3333333333333333 \cdot \ell\right)\right)} \cdot \frac{\ell}{t}}} \]

      2. unpow2 64.3%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\left(\frac{\ell}{\color{blue}{k \cdot k}} + \left(-0.5 \cdot \ell - -0.3333333333333333 \cdot \ell\right)\right) \cdot \frac{\ell}{t}}} \]

      3. distribute-rgt-out-- 64.3%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\left(\frac{\ell}{k \cdot k} + \color{blue}{\ell \cdot \left(-0.5 - -0.3333333333333333\right)}\right) \cdot \frac{\ell}{t}}} \]

      4. metadata-eval 64.3%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\left(\frac{\ell}{k \cdot k} + \ell \cdot \color{blue}{-0.16666666666666666}\right) \cdot \frac{\ell}{t}}} \]

   7. Simplified 64.3%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\left(\frac{\ell}{k \cdot k} + \ell \cdot -0.16666666666666666\right)} \cdot \frac{\ell}{t}}} \]

   if 4.29999999999999997e-50 < l < 5.79999999999999994e207

   1. Initial program 52.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 50.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 50.2%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 50.2%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 50.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 50.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 50.7%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 50.7%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 50.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 50.8%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 50.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around 0 47.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 47.3%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 47.3%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. unpow2 47.3%

         \[\leadsto \frac{\ell \cdot \ell}{{t}^{3} \cdot \color{blue}{\left(k \cdot k\right)}} \]

   6. Simplified 47.3%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{{t}^{3} \cdot \left(k \cdot k\right)}} \]
   7. Step-by-step derivation

      1. times-frac 47.5%

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]

   8. Applied egg-rr 47.5%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]

   if 5.79999999999999994e207 < l

   1. Initial program 34.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0 67.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}}} \]
   3. Step-by-step derivation

      1. associate-/l* 67.8%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2} \cdot t}}}} \]

      2. unpow2 67.8%

         \[\leadsto \frac{2}{\frac{\color{blue}{k \cdot k}}{\frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2} \cdot t}}} \]

      3. unpow2 67.8%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2} \cdot t}}} \]

      4. associate-*r* 67.8%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}{{\sin k}^{2} \cdot t}}} \]

      5. times-frac 72.8%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\cos k \cdot \ell}{{\sin k}^{2}} \cdot \frac{\ell}{t}}}} \]

   4. Simplified 72.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot k}{\frac{\cos k \cdot \ell}{{\sin k}^{2}} \cdot \frac{\ell}{t}}}} \]

   5. Taylor expanded in k around 0 72.6%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\ell}{{k}^{2}}} \cdot \frac{\ell}{t}}} \]
   6. Step-by-step derivation

      1. unpow2 72.6%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{t}}} \]

   7. Simplified 72.6%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\ell}{k \cdot k}} \cdot \frac{\ell}{t}}} \]
   8. Step-by-step derivation

      1. associate-/r/ 72.6%

         \[\leadsto \color{blue}{\frac{2}{k \cdot k} \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}\right)} \]

   9. Applied egg-rr 72.6%

      \[\leadsto \color{blue}{\frac{2}{k \cdot k} \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.3 \cdot 10^{-50}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\frac{\ell}{t} \cdot \left(\frac{\ell}{k \cdot k} + \ell \cdot -0.16666666666666666\right)}}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+207}:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\\ \end{array} \]

Alternative 17: 62.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.15 \cdot 10^{-177}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;k \leq 5.1 \cdot 10^{+95}:\\ \;\;\;\;\ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\ell}{k \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.15e-177)
   (/ (* (/ l k) (/ l k)) (pow t 3.0))
   (if (<= k 5.1e+95)
     (* l (/ l (* (pow t 3.0) (* k k))))
     (/ (- l) (* k (* t (/ k l)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.15e-177) {
		tmp = ((l / k) * (l / k)) / pow(t, 3.0);
	} else if (k <= 5.1e+95) {
		tmp = l * (l / (pow(t, 3.0) * (k * k)));
	} else {
		tmp = -l / (k * (t * (k / l)));
	}
	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 <= 1.15d-177) then
        tmp = ((l / k) * (l / k)) / (t ** 3.0d0)
    else if (k <= 5.1d+95) then
        tmp = l * (l / ((t ** 3.0d0) * (k * k)))
    else
        tmp = -l / (k * (t * (k / l)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.15e-177) {
		tmp = ((l / k) * (l / k)) / Math.pow(t, 3.0);
	} else if (k <= 5.1e+95) {
		tmp = l * (l / (Math.pow(t, 3.0) * (k * k)));
	} else {
		tmp = -l / (k * (t * (k / l)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.15e-177:
		tmp = ((l / k) * (l / k)) / math.pow(t, 3.0)
	elif k <= 5.1e+95:
		tmp = l * (l / (math.pow(t, 3.0) * (k * k)))
	else:
		tmp = -l / (k * (t * (k / l)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.15e-177)
		tmp = Float64(Float64(Float64(l / k) * Float64(l / k)) / (t ^ 3.0));
	elseif (k <= 5.1e+95)
		tmp = Float64(l * Float64(l / Float64((t ^ 3.0) * Float64(k * k))));
	else
		tmp = Float64(Float64(-l) / Float64(k * Float64(t * Float64(k / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.15e-177)
		tmp = ((l / k) * (l / k)) / (t ^ 3.0);
	elseif (k <= 5.1e+95)
		tmp = l * (l / ((t ^ 3.0) * (k * k)));
	else
		tmp = -l / (k * (t * (k / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.15e-177], N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 5.1e+95], N[(l * N[(l / N[(N[Power[t, 3.0], $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-l) / N[(k * N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.15000000000000011e-177

   1. Initial program 52.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 51.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 47.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 47.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 51.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 51.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 51.6%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 52.1%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 52.2%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 52.2%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 52.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around 0 49.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. associate-/r* 49.1%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}}} \]

      2. unpow2 49.1%

         \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{{t}^{3}} \]

      3. unpow2 49.1%

         \[\leadsto \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{{t}^{3}} \]

      4. times-frac 61.6%

         \[\leadsto \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{{t}^{3}} \]

   6. Simplified 61.6%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}} \]

   if 1.15000000000000011e-177 < k < 5.10000000000000003e95

   1. Initial program 59.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 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* 59.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 59.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      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-*l/ 59.3%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 59.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 57.4%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 57.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around 0 61.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 61.2%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 61.2%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. unpow2 61.2%

         \[\leadsto \frac{\ell \cdot \ell}{{t}^{3} \cdot \color{blue}{\left(k \cdot k\right)}} \]

   6. Simplified 61.2%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{{t}^{3} \cdot \left(k \cdot k\right)}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 61.2%

         \[\leadsto \color{blue}{1 \cdot \frac{\ell \cdot \ell}{{t}^{3} \cdot \left(k \cdot k\right)}} \]

      2. associate-/l* 65.6%

         \[\leadsto 1 \cdot \color{blue}{\frac{\ell}{\frac{{t}^{3} \cdot \left(k \cdot k\right)}{\ell}}} \]

      3. *-commutative 65.6%

         \[\leadsto 1 \cdot \frac{\ell}{\frac{\color{blue}{\left(k \cdot k\right) \cdot {t}^{3}}}{\ell}} \]

   8. Applied egg-rr 65.6%

      \[\leadsto \color{blue}{1 \cdot \frac{\ell}{\frac{\left(k \cdot k\right) \cdot {t}^{3}}{\ell}}} \]
   9. Step-by-step derivation

      1. associate-/r/ 65.6%

         \[\leadsto 1 \cdot \color{blue}{\left(\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}} \cdot \ell\right)} \]

   10. Applied egg-rr 65.6%

       \[\leadsto 1 \cdot \color{blue}{\left(\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}} \cdot \ell\right)} \]

   if 5.10000000000000003e95 < k

   1. Initial program 49.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0 78.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}}} \]
   3. Step-by-step derivation

      1. times-frac 78.8%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}}} \]

      2. unpow2 78.8%

         \[\leadsto \frac{2}{\frac{\color{blue}{k \cdot k}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]

      3. *-commutative 78.8%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2}}} \]

      4. unpow2 78.8%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]

   4. Simplified 78.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot k}{\cos k} \cdot \frac{t \cdot {\sin k}^{2}}{\ell \cdot \ell}}} \]

   5. Taylor expanded in k around 0 58.6%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\cos k} \cdot \color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}}} \]
   6. Step-by-step derivation

      1. *-commutative 58.6%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}}} \]

      2. unpow2 58.6%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]

      3. times-frac 58.9%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\cos k} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)}} \]

      4. unpow2 58.9%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\cos k} \cdot \left(\frac{t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right)} \]

   7. Simplified 58.9%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\cos k} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)}} \]

   8. Taylor expanded in k around 0 59.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   9. Step-by-step derivation

      1. fma-def 59.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{{\ell}^{2}}{{k}^{2} \cdot t}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right)} \]

      2. unpow2 59.3%

         \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      3. unpow2 59.3%

         \[\leadsto \mathsf{fma}\left(-1, \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot t}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      4. associate-/l* 59.3%

         \[\leadsto \mathsf{fma}\left(-1, \color{blue}{\frac{\ell}{\frac{\left(k \cdot k\right) \cdot t}{\ell}}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      5. associate-*r/ 59.3%

         \[\leadsto \mathsf{fma}\left(-1, \frac{\ell}{\color{blue}{\left(k \cdot k\right) \cdot \frac{t}{\ell}}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      6. unpow2 59.3%

         \[\leadsto \mathsf{fma}\left(-1, \frac{\ell}{\left(k \cdot k\right) \cdot \frac{t}{\ell}}, 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t}\right) \]

      7. *-commutative 59.3%

         \[\leadsto \mathsf{fma}\left(-1, \frac{\ell}{\left(k \cdot k\right) \cdot \frac{t}{\ell}}, 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}}\right) \]

   10. Simplified 59.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{\ell}{\left(k \cdot k\right) \cdot \frac{t}{\ell}}, 2 \cdot \frac{\ell \cdot \ell}{t \cdot {k}^{4}}\right)} \]

   11. Taylor expanded in k around inf 59.6%

       \[\leadsto \color{blue}{-1 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   12. Step-by-step derivation

       1. associate-*r/ 59.6%

          \[\leadsto \color{blue}{\frac{-1 \cdot {\ell}^{2}}{{k}^{2} \cdot t}} \]

       2. times-frac 59.4%

          \[\leadsto \color{blue}{\frac{-1}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t}} \]

       3. unpow2 59.4%

          \[\leadsto \frac{-1}{{k}^{2}} \cdot \frac{\color{blue}{\ell \cdot \ell}}{t} \]

       4. associate-/l* 59.5%

          \[\leadsto \frac{-1}{{k}^{2}} \cdot \color{blue}{\frac{\ell}{\frac{t}{\ell}}} \]

       5. times-frac 60.3%

          \[\leadsto \color{blue}{\frac{-1 \cdot \ell}{{k}^{2} \cdot \frac{t}{\ell}}} \]

       6. unpow2 60.3%

          \[\leadsto \frac{-1 \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot \frac{t}{\ell}} \]

       7. associate-*r* 60.9%

          \[\leadsto \frac{-1 \cdot \ell}{\color{blue}{k \cdot \left(k \cdot \frac{t}{\ell}\right)}} \]

       8. mul-1-neg 60.9%

          \[\leadsto \frac{\color{blue}{-\ell}}{k \cdot \left(k \cdot \frac{t}{\ell}\right)} \]

       9. associate-*r/ 60.9%

          \[\leadsto \frac{-\ell}{k \cdot \color{blue}{\frac{k \cdot t}{\ell}}} \]

       10. associate-*l/ 60.9%

           \[\leadsto \frac{-\ell}{k \cdot \color{blue}{\left(\frac{k}{\ell} \cdot t\right)}} \]

       11. *-commutative 60.9%

           \[\leadsto \frac{-\ell}{k \cdot \color{blue}{\left(t \cdot \frac{k}{\ell}\right)}} \]

   13. Simplified 60.9%

       \[\leadsto \color{blue}{\frac{-\ell}{k \cdot \left(t \cdot \frac{k}{\ell}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.15 \cdot 10^{-177}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;k \leq 5.1 \cdot 10^{+95}:\\ \;\;\;\;\ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\ell}{k \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 18: 62.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 5.1 \cdot 10^{-36}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{k \cdot \left(k \cdot {t}^{3}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 5.1e-36)
   (/ 2.0 (/ (* k k) (* (/ l t) (/ l (* k k)))))
   (/ (* l l) (* k (* k (pow t 3.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 5.1e-36) {
		tmp = 2.0 / ((k * k) / ((l / t) * (l / (k * k))));
	} else {
		tmp = (l * l) / (k * (k * 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 <= 5.1d-36) then
        tmp = 2.0d0 / ((k * k) / ((l / t) * (l / (k * k))))
    else
        tmp = (l * l) / (k * (k * (t ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 5.1e-36) {
		tmp = 2.0 / ((k * k) / ((l / t) * (l / (k * k))));
	} else {
		tmp = (l * l) / (k * (k * Math.pow(t, 3.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 5.1e-36:
		tmp = 2.0 / ((k * k) / ((l / t) * (l / (k * k))))
	else:
		tmp = (l * l) / (k * (k * math.pow(t, 3.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 5.1e-36)
		tmp = Float64(2.0 / Float64(Float64(k * k) / Float64(Float64(l / t) * Float64(l / Float64(k * k)))));
	else
		tmp = Float64(Float64(l * l) / Float64(k * Float64(k * (t ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 5.1e-36)
		tmp = 2.0 / ((k * k) / ((l / t) * (l / (k * k))));
	else
		tmp = (l * l) / (k * (k * (t ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 5.1e-36], N[(2.0 / N[(N[(k * k), $MachinePrecision] / N[(N[(l / t), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] / N[(k * N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.09999999999999973e-36

   1. Initial program 47.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0 66.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}}} \]
   3. Step-by-step derivation

      1. associate-/l* 67.1%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2} \cdot t}}}} \]

      2. unpow2 67.1%

         \[\leadsto \frac{2}{\frac{\color{blue}{k \cdot k}}{\frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2} \cdot t}}} \]

      3. unpow2 67.1%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2} \cdot t}}} \]

      4. associate-*r* 67.1%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}{{\sin k}^{2} \cdot t}}} \]

      5. times-frac 75.9%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\cos k \cdot \ell}{{\sin k}^{2}} \cdot \frac{\ell}{t}}}} \]

   4. Simplified 75.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot k}{\frac{\cos k \cdot \ell}{{\sin k}^{2}} \cdot \frac{\ell}{t}}}} \]

   5. Taylor expanded in k around 0 64.4%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\ell}{{k}^{2}}} \cdot \frac{\ell}{t}}} \]
   6. Step-by-step derivation

      1. unpow2 64.4%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{t}}} \]

   7. Simplified 64.4%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\ell}{k \cdot k}} \cdot \frac{\ell}{t}}} \]

   if 5.09999999999999973e-36 < t

   1. Initial program 65.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 65.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 58.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 58.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 65.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 65.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 65.9%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 67.2%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 67.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 67.2%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 67.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around 0 57.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 57.7%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 57.7%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. unpow2 57.7%

         \[\leadsto \frac{\ell \cdot \ell}{{t}^{3} \cdot \color{blue}{\left(k \cdot k\right)}} \]

   6. Simplified 57.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{{t}^{3} \cdot \left(k \cdot k\right)}} \]

   7. Taylor expanded in l around 0 57.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   8. Step-by-step derivation

      1. unpow2 57.7%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. unpow2 57.7%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      3. associate-*l* 66.1%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

   9. Simplified 66.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot {t}^{3}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.1 \cdot 10^{-36}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{k \cdot \left(k \cdot {t}^{3}\right)}\\ \end{array} \]

Alternative 19: 59.1% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ \frac{2}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* (/ 2.0 (* k k)) (* (/ l t) (/ l (* k k)))))

C

double code(double t, double l, double k) {
	return (2.0 / (k * k)) * ((l / t) * (l / (k * k)));
}

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 / (k * k)) * ((l / t) * (l / (k * k)))
end function

Java

public static double code(double t, double l, double k) {
	return (2.0 / (k * k)) * ((l / t) * (l / (k * k)));
}

Python

def code(t, l, k):
	return (2.0 / (k * k)) * ((l / t) * (l / (k * k)))

Julia

function code(t, l, k)
	return Float64(Float64(2.0 / Float64(k * k)) * Float64(Float64(l / t) * Float64(l / Float64(k * k))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (2.0 / (k * k)) * ((l / t) * (l / (k * k)));
end

Wolfram

code[t_, l_, k_] := N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l / t), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in t around 0 63.0%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}}} \]
3. Step-by-step derivation

   1. associate-/l* 63.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2} \cdot t}}}} \]

   2. unpow2 63.1%

      \[\leadsto \frac{2}{\frac{\color{blue}{k \cdot k}}{\frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2} \cdot t}}} \]

   3. unpow2 63.1%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2} \cdot t}}} \]

   4. associate-*r* 63.1%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}{{\sin k}^{2} \cdot t}}} \]

   5. times-frac 69.3%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\cos k \cdot \ell}{{\sin k}^{2}} \cdot \frac{\ell}{t}}}} \]

4. Simplified 69.3%

   \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot k}{\frac{\cos k \cdot \ell}{{\sin k}^{2}} \cdot \frac{\ell}{t}}}} \]

5. Taylor expanded in k around 0 60.9%

   \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\ell}{{k}^{2}}} \cdot \frac{\ell}{t}}} \]
6. Step-by-step derivation

   1. unpow2 60.9%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{t}}} \]

7. Simplified 60.9%

   \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\ell}{k \cdot k}} \cdot \frac{\ell}{t}}} \]
8. Step-by-step derivation

   1. associate-/r/ 60.5%

      \[\leadsto \color{blue}{\frac{2}{k \cdot k} \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}\right)} \]

9. Applied egg-rr 60.5%

   \[\leadsto \color{blue}{\frac{2}{k \cdot k} \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}\right)} \]

10. Final simplification 60.5%

    \[\leadsto \frac{2}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \]

Alternative 20: 59.2% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ \frac{\ell}{\left(k \cdot k\right) \cdot \frac{t}{\ell}} \cdot \frac{\frac{2}{k}}{k} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* (/ l (* (* k k) (/ t l))) (/ (/ 2.0 k) k)))

C

double code(double t, double l, double k) {
	return (l / ((k * k) * (t / l))) * ((2.0 / k) / k);
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (l / ((k * k) * (t / l))) * ((2.0d0 / k) / k)
end function

Java

public static double code(double t, double l, double k) {
	return (l / ((k * k) * (t / l))) * ((2.0 / k) / k);
}

Python

def code(t, l, k):
	return (l / ((k * k) * (t / l))) * ((2.0 / k) / k)

Julia

function code(t, l, k)
	return Float64(Float64(l / Float64(Float64(k * k) * Float64(t / l))) * Float64(Float64(2.0 / k) / k))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l / ((k * k) * (t / l))) * ((2.0 / k) / k);
end

Wolfram

code[t_, l_, k_] := N[(N[(l / N[(N[(k * k), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in t around 0 63.0%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}}} \]
3. Step-by-step derivation

   1. associate-/l* 63.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2} \cdot t}}}} \]

   2. unpow2 63.1%

      \[\leadsto \frac{2}{\frac{\color{blue}{k \cdot k}}{\frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2} \cdot t}}} \]

   3. unpow2 63.1%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2} \cdot t}}} \]

   4. associate-*r* 63.1%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}{{\sin k}^{2} \cdot t}}} \]

   5. times-frac 69.3%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\cos k \cdot \ell}{{\sin k}^{2}} \cdot \frac{\ell}{t}}}} \]

4. Simplified 69.3%

   \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot k}{\frac{\cos k \cdot \ell}{{\sin k}^{2}} \cdot \frac{\ell}{t}}}} \]

5. Taylor expanded in k around 0 60.9%

   \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\ell}{{k}^{2}}} \cdot \frac{\ell}{t}}} \]
6. Step-by-step derivation

   1. unpow2 60.9%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{t}}} \]

7. Simplified 60.9%

   \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\ell}{k \cdot k}} \cdot \frac{\ell}{t}}} \]
8. Step-by-step derivation

   1. *-un-lft-identity 60.9%

      \[\leadsto \color{blue}{1 \cdot \frac{2}{\frac{k \cdot k}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}}}} \]

   2. associate-/r/ 60.5%

      \[\leadsto 1 \cdot \color{blue}{\left(\frac{2}{k \cdot k} \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}\right)\right)} \]

9. Applied egg-rr 60.5%

   \[\leadsto \color{blue}{1 \cdot \left(\frac{2}{k \cdot k} \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}\right)\right)} \]
10. Step-by-step derivation

    1. *-lft-identity 60.5%

       \[\leadsto \color{blue}{\frac{2}{k \cdot k} \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}\right)} \]

    2. *-commutative 60.5%

       \[\leadsto \color{blue}{\left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}\right) \cdot \frac{2}{k \cdot k}} \]

    3. associate-/r* 60.5%

       \[\leadsto \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}\right) \cdot \color{blue}{\frac{\frac{2}{k}}{k}} \]

    4. associate-*r/ 59.8%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot k} \cdot \ell}{t}} \cdot \frac{\frac{2}{k}}{k} \]

    5. associate-/l* 60.5%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot k}}{\frac{t}{\ell}}} \cdot \frac{\frac{2}{k}}{k} \]

    6. associate-/r* 60.5%

       \[\leadsto \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \frac{t}{\ell}}} \cdot \frac{\frac{2}{k}}{k} \]

11. Simplified 60.5%

    \[\leadsto \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \frac{t}{\ell}} \cdot \frac{\frac{2}{k}}{k}} \]

12. Final simplification 60.5%

    \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \frac{t}{\ell}} \cdot \frac{\frac{2}{k}}{k} \]

Alternative 21: 58.8% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\frac{k}{\frac{\ell}{k \cdot k}} \cdot \frac{k}{\frac{\ell}{t}}} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/ 2.0 (* (/ k (/ l (* k k))) (/ k (/ l t)))))

C

double code(double t, double l, double k) {
	return 2.0 / ((k / (l / (k * k))) * (k / (l / t)));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / ((k / (l / (k * k))) * (k / (l / t)))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((k / (l / (k * k))) * (k / (l / t)));
}

Python

def code(t, l, k):
	return 2.0 / ((k / (l / (k * k))) * (k / (l / t)))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(k / Float64(l / Float64(k * k))) * Float64(k / Float64(l / t))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / ((k / (l / (k * k))) * (k / (l / t)));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(k / N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in t around 0 63.0%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}}} \]
3. Step-by-step derivation

   1. associate-/l* 63.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2} \cdot t}}}} \]

   2. unpow2 63.1%

      \[\leadsto \frac{2}{\frac{\color{blue}{k \cdot k}}{\frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2} \cdot t}}} \]

   3. unpow2 63.1%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2} \cdot t}}} \]

   4. associate-*r* 63.1%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}{{\sin k}^{2} \cdot t}}} \]

   5. times-frac 69.3%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\cos k \cdot \ell}{{\sin k}^{2}} \cdot \frac{\ell}{t}}}} \]

4. Simplified 69.3%

   \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot k}{\frac{\cos k \cdot \ell}{{\sin k}^{2}} \cdot \frac{\ell}{t}}}} \]

5. Taylor expanded in k around 0 60.9%

   \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\ell}{{k}^{2}}} \cdot \frac{\ell}{t}}} \]
6. Step-by-step derivation

   1. unpow2 60.9%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{t}}} \]

7. Simplified 60.9%

   \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\ell}{k \cdot k}} \cdot \frac{\ell}{t}}} \]
8. Step-by-step derivation

   1. times-frac 60.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{\ell}{k \cdot k}} \cdot \frac{k}{\frac{\ell}{t}}}} \]

9. Applied egg-rr 60.6%

   \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{\ell}{k \cdot k}} \cdot \frac{k}{\frac{\ell}{t}}}} \]

10. Final simplification 60.6%

    \[\leadsto \frac{2}{\frac{k}{\frac{\ell}{k \cdot k}} \cdot \frac{k}{\frac{\ell}{t}}} \]

Alternative 22: 59.3% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\frac{k \cdot k}{\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}}} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/ 2.0 (/ (* k k) (* (/ l t) (/ l (* k k))))))

C

double code(double t, double l, double k) {
	return 2.0 / ((k * k) / ((l / t) * (l / (k * k))));
}

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 / ((k * k) / ((l / t) * (l / (k * k))))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((k * k) / ((l / t) * (l / (k * k))));
}

Python

def code(t, l, k):
	return 2.0 / ((k * k) / ((l / t) * (l / (k * k))))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(k * k) / Float64(Float64(l / t) * Float64(l / Float64(k * k)))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / ((k * k) / ((l / t) * (l / (k * k))));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(k * k), $MachinePrecision] / N[(N[(l / t), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in t around 0 63.0%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}}} \]
3. Step-by-step derivation

   1. associate-/l* 63.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2} \cdot t}}}} \]

   2. unpow2 63.1%

      \[\leadsto \frac{2}{\frac{\color{blue}{k \cdot k}}{\frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2} \cdot t}}} \]

   3. unpow2 63.1%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2} \cdot t}}} \]

   4. associate-*r* 63.1%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}{{\sin k}^{2} \cdot t}}} \]

   5. times-frac 69.3%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\cos k \cdot \ell}{{\sin k}^{2}} \cdot \frac{\ell}{t}}}} \]

4. Simplified 69.3%

   \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot k}{\frac{\cos k \cdot \ell}{{\sin k}^{2}} \cdot \frac{\ell}{t}}}} \]

5. Taylor expanded in k around 0 60.9%

   \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\ell}{{k}^{2}}} \cdot \frac{\ell}{t}}} \]
6. Step-by-step derivation

   1. unpow2 60.9%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{t}}} \]

7. Simplified 60.9%

   \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\ell}{k \cdot k}} \cdot \frac{\ell}{t}}} \]

8. Final simplification 60.9%

   \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}}} \]

Alternative 23: 34.2% accurate, 42.1× speedup

Math

\[\begin{array}{l} \\ \frac{\ell \cdot \left(-\ell\right)}{k \cdot \left(t \cdot k\right)} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (/ (* l (- l)) (* k (* t k))))

C

double code(double t, double l, double k) {
	return (l * -l) / (k * (t * k));
}

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) / (k * (t * k))
end function

Java

public static double code(double t, double l, double k) {
	return (l * -l) / (k * (t * k));
}

Python

def code(t, l, k):
	return (l * -l) / (k * (t * k))

Julia

function code(t, l, k)
	return Float64(Float64(l * Float64(-l)) / Float64(k * Float64(t * k)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l * -l) / (k * (t * k));
end

Wolfram

code[t_, l_, k_] := N[(N[(l * (-l)), $MachinePrecision] / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in t around 0 63.0%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}}} \]
3. Step-by-step derivation

   1. times-frac 63.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}}} \]

   2. unpow2 63.1%

      \[\leadsto \frac{2}{\frac{\color{blue}{k \cdot k}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]

   3. *-commutative 63.1%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2}}} \]

   4. unpow2 63.1%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]

4. Simplified 63.1%

   \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot k}{\cos k} \cdot \frac{t \cdot {\sin k}^{2}}{\ell \cdot \ell}}} \]

5. Taylor expanded in k around 0 56.2%

   \[\leadsto \frac{2}{\frac{k \cdot k}{\cos k} \cdot \color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}}} \]
6. Step-by-step derivation

   1. *-commutative 56.2%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}}} \]

   2. unpow2 56.2%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]

   3. times-frac 61.6%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\cos k} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)}} \]

   4. unpow2 61.6%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\cos k} \cdot \left(\frac{t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right)} \]

7. Simplified 61.6%

   \[\leadsto \frac{2}{\frac{k \cdot k}{\cos k} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)}} \]

8. Taylor expanded in k around 0 30.7%

   \[\leadsto \color{blue}{-1 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
9. Step-by-step derivation

   1. fma-def 30.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{{\ell}^{2}}{{k}^{2} \cdot t}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right)} \]

   2. unpow2 30.7%

      \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   3. unpow2 30.7%

      \[\leadsto \mathsf{fma}\left(-1, \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot t}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   4. associate-/l* 30.7%

      \[\leadsto \mathsf{fma}\left(-1, \color{blue}{\frac{\ell}{\frac{\left(k \cdot k\right) \cdot t}{\ell}}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   5. associate-*r/ 30.7%

      \[\leadsto \mathsf{fma}\left(-1, \frac{\ell}{\color{blue}{\left(k \cdot k\right) \cdot \frac{t}{\ell}}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   6. unpow2 30.7%

      \[\leadsto \mathsf{fma}\left(-1, \frac{\ell}{\left(k \cdot k\right) \cdot \frac{t}{\ell}}, 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t}\right) \]

   7. *-commutative 30.7%

      \[\leadsto \mathsf{fma}\left(-1, \frac{\ell}{\left(k \cdot k\right) \cdot \frac{t}{\ell}}, 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}}\right) \]

10. Simplified 30.7%

    \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{\ell}{\left(k \cdot k\right) \cdot \frac{t}{\ell}}, 2 \cdot \frac{\ell \cdot \ell}{t \cdot {k}^{4}}\right)} \]

11. Taylor expanded in k around inf 31.4%

    \[\leadsto \color{blue}{-1 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
12. Step-by-step derivation

    1. mul-1-neg 31.4%

       \[\leadsto \color{blue}{-\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]

    2. unpow2 31.4%

       \[\leadsto -\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t} \]

    3. unpow2 31.4%

       \[\leadsto -\frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot t} \]

    4. associate-*l* 32.6%

       \[\leadsto -\frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot t\right)}} \]

13. Simplified 32.6%

    \[\leadsto \color{blue}{-\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}} \]

14. Final simplification 32.6%

    \[\leadsto \frac{\ell \cdot \left(-\ell\right)}{k \cdot \left(t \cdot k\right)} \]

Reproduce

herbie shell --seed 2023279 
(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))))