Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.4% → 87.4%

Time: 22.0s

Alternatives: 19

Speedup: 24.7×

• 27.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -32000000 \lor \neg \left(t \leq 5.2 \cdot 10^{-37}\right):\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{t}{\frac{\sqrt[3]{\ell}}{\frac{\sqrt[3]{\sin k}}{\sqrt[3]{\ell}}}}\right)}^{3} \cdot \tan k}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{t}\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -32000000.0) (not (<= t 5.2e-37)))
   (/
    (/
     2.0
     (* (pow (/ t (/ (cbrt l) (/ (cbrt (sin k)) (cbrt l)))) 3.0) (tan k)))
    (+ 1.0 (+ 1.0 (pow (/ k t) 2.0))))
   (* 2.0 (* (* (/ (/ l k) k) (/ l t)) (/ (cos k) (pow (sin k) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -32000000.0) || !(t <= 5.2e-37)) {
		tmp = (2.0 / (pow((t / (cbrt(l) / (cbrt(sin(k)) / cbrt(l)))), 3.0) * tan(k))) / (1.0 + (1.0 + pow((k / t), 2.0)));
	} else {
		tmp = 2.0 * ((((l / k) / k) * (l / t)) * (cos(k) / pow(sin(k), 2.0)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -32000000.0) || !(t <= 5.2e-37)) {
		tmp = (2.0 / (Math.pow((t / (Math.cbrt(l) / (Math.cbrt(Math.sin(k)) / Math.cbrt(l)))), 3.0) * Math.tan(k))) / (1.0 + (1.0 + Math.pow((k / t), 2.0)));
	} else {
		tmp = 2.0 * ((((l / k) / k) * (l / t)) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -32000000.0) || !(t <= 5.2e-37))
		tmp = Float64(Float64(2.0 / Float64((Float64(t / Float64(cbrt(l) / Float64(cbrt(sin(k)) / cbrt(l)))) ^ 3.0) * tan(k))) / Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(l / k) / k) * Float64(l / t)) * Float64(cos(k) / (sin(k) ^ 2.0))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -32000000.0], N[Not[LessEqual[t, 5.2e-37]], $MachinePrecision]], N[(N[(2.0 / N[(N[Power[N[(t / N[(N[Power[l, 1/3], $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.2e7 or 5.19999999999999959e-37 < t

   1. Initial program 71.3%

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

      1. associate-/r* 71.3%

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

      2. +-commutative 71.3%

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

   3. Simplified 71.3%

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

      1. associate-/r/ 73.6%

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

      2. add-cube-cbrt 73.4%

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

      3. pow3 73.4%

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

      4. cbrt-div 73.4%

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

      5. rem-cbrt-cube 75.3%

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

      6. associate-/l* 77.3%

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

   5. Applied egg-rr 77.3%

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

      1. cbrt-div 90.0%

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

   7. Applied egg-rr 90.0%

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

      1. cbrt-div 92.7%

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

   9. Applied egg-rr 92.7%

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

   if -3.2e7 < t < 5.19999999999999959e-37

   1. Initial program 40.5%

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

      1. associate-/r* 40.5%

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

      2. associate-*l* 40.5%

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

      3. sqr-neg 40.5%

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

      4. associate-*l* 40.5%

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

      5. *-commutative 40.5%

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

      6. sqr-neg 40.5%

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

      7. associate-*l/ 40.5%

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

      8. associate-*r/ 40.4%

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

      9. associate-/r/ 39.6%

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

   3. Simplified 39.6%

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

   4. Taylor expanded in k around inf 72.0%

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

      1. times-frac 70.2%

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

      2. unpow2 70.2%

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

      3. unpow2 70.2%

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

   6. Simplified 70.2%

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

      1. associate-*l/ 68.8%

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

   8. Applied egg-rr 68.8%

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

   9. Taylor expanded in l around 0 72.0%

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

       1. associate-*r* 72.0%

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

       2. times-frac 72.5%

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

       3. unpow2 72.5%

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

       4. times-frac 90.1%

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

       5. unpow2 90.1%

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

       6. associate-/r* 91.8%

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

   11. Simplified 91.8%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -32000000 \lor \neg \left(t \leq 5.2 \cdot 10^{-37}\right):\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{t}{\frac{\sqrt[3]{\ell}}{\frac{\sqrt[3]{\sin k}}{\sqrt[3]{\ell}}}}\right)}^{3} \cdot \tan k}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{t}\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 2: 83.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;\frac{2}{\left(1 + \left(1 + t_1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)} \leq 5 \cdot 10^{+257}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{3}}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{t}\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<=
        (/
         2.0
         (*
          (+ 1.0 (+ 1.0 t_1))
          (* (tan k) (* (sin k) (/ (pow t 3.0) (* l l))))))
        5e+257)
     (/
      (pow (/ (cbrt (/ 2.0 (tan k))) (/ t (cbrt (/ l (/ (sin k) l))))) 3.0)
      (+ 2.0 t_1))
     (* 2.0 (* (* (/ (/ l k) k) (/ l t)) (/ (cos k) (pow (sin k) 2.0)))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((1.0 + (1.0 + t_1)) * (tan(k) * (sin(k) * (pow(t, 3.0) / (l * l)))))) <= 5e+257) {
		tmp = pow((cbrt((2.0 / tan(k))) / (t / cbrt((l / (sin(k) / l))))), 3.0) / (2.0 + t_1);
	} else {
		tmp = 2.0 * ((((l / k) / k) * (l / t)) * (cos(k) / pow(sin(k), 2.0)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((1.0 + (1.0 + t_1)) * (Math.tan(k) * (Math.sin(k) * (Math.pow(t, 3.0) / (l * l)))))) <= 5e+257) {
		tmp = Math.pow((Math.cbrt((2.0 / Math.tan(k))) / (t / Math.cbrt((l / (Math.sin(k) / l))))), 3.0) / (2.0 + t_1);
	} else {
		tmp = 2.0 * ((((l / k) / k) * (l / t)) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + t_1)) * Float64(tan(k) * Float64(sin(k) * Float64((t ^ 3.0) / Float64(l * l)))))) <= 5e+257)
		tmp = Float64((Float64(cbrt(Float64(2.0 / tan(k))) / Float64(t / cbrt(Float64(l / Float64(sin(k) / l))))) ^ 3.0) / Float64(2.0 + t_1));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(l / k) / k) * Float64(l / t)) * Float64(cos(k) / (sin(k) ^ 2.0))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(2.0 / N[(N[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+257], N[(N[Power[N[(N[Power[N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[(t / N[Power[N[(l / N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 5.00000000000000028e257

   1. Initial program 83.3%

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

      1. associate-/r* 83.3%

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

      2. associate-*l* 77.6%

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

      3. sqr-neg 77.6%

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

      4. associate-*l* 83.3%

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

      5. *-commutative 83.3%

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

      6. sqr-neg 83.3%

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

      7. associate-/r* 83.3%

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

   3. Simplified 85.5%

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

      1. associate-/l/ 85.5%

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

      2. associate-/r/ 83.3%

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

      3. add-cube-cbrt 83.2%

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

      4. pow3 83.2%

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

   5. Applied egg-rr 93.1%

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

   if 5.00000000000000028e257 < (/.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 23.1%

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

      1. associate-/r* 23.1%

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

      2. associate-*l* 23.1%

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

      3. sqr-neg 23.1%

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

      4. associate-*l* 23.1%

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

      5. *-commutative 23.1%

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

      6. sqr-neg 23.1%

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

      7. associate-*l/ 23.1%

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

      8. associate-*r/ 23.1%

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

      9. associate-/r/ 23.1%

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

   3. Simplified 23.1%

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

   4. Taylor expanded in k around inf 58.1%

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

      1. times-frac 56.2%

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

      2. unpow2 56.2%

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

      3. unpow2 56.2%

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

   6. Simplified 56.2%

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

      1. associate-*l/ 54.7%

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

   8. Applied egg-rr 54.7%

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

   9. Taylor expanded in l around 0 58.1%

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

       1. associate-*r* 58.1%

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

       2. times-frac 58.6%

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

       3. unpow2 58.6%

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

       4. times-frac 81.0%

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

       5. unpow2 81.0%

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

       6. associate-/r* 83.2%

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

   11. Simplified 83.2%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)} \leq 5 \cdot 10^{+257}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{t}\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 3: 81.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;\frac{2}{\left(1 + \left(1 + t_1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)} \leq 5 \cdot 10^{+257}:\\ \;\;\;\;\left({\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t}\right)}^{3} \cdot \left(\ell \cdot \frac{\ell}{\sin k}\right)\right) \cdot \frac{1}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{t}\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<=
        (/
         2.0
         (*
          (+ 1.0 (+ 1.0 t_1))
          (* (tan k) (* (sin k) (/ (pow t 3.0) (* l l))))))
        5e+257)
     (*
      (* (pow (/ (cbrt (/ 2.0 (tan k))) t) 3.0) (* l (/ l (sin k))))
      (/ 1.0 (+ 2.0 t_1)))
     (* 2.0 (* (* (/ (/ l k) k) (/ l t)) (/ (cos k) (pow (sin k) 2.0)))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((1.0 + (1.0 + t_1)) * (tan(k) * (sin(k) * (pow(t, 3.0) / (l * l)))))) <= 5e+257) {
		tmp = (pow((cbrt((2.0 / tan(k))) / t), 3.0) * (l * (l / sin(k)))) * (1.0 / (2.0 + t_1));
	} else {
		tmp = 2.0 * ((((l / k) / k) * (l / t)) * (cos(k) / pow(sin(k), 2.0)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((1.0 + (1.0 + t_1)) * (Math.tan(k) * (Math.sin(k) * (Math.pow(t, 3.0) / (l * l)))))) <= 5e+257) {
		tmp = (Math.pow((Math.cbrt((2.0 / Math.tan(k))) / t), 3.0) * (l * (l / Math.sin(k)))) * (1.0 / (2.0 + t_1));
	} else {
		tmp = 2.0 * ((((l / k) / k) * (l / t)) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + t_1)) * Float64(tan(k) * Float64(sin(k) * Float64((t ^ 3.0) / Float64(l * l)))))) <= 5e+257)
		tmp = Float64(Float64((Float64(cbrt(Float64(2.0 / tan(k))) / t) ^ 3.0) * Float64(l * Float64(l / sin(k)))) * Float64(1.0 / Float64(2.0 + t_1)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(l / k) / k) * Float64(l / t)) * Float64(cos(k) / (sin(k) ^ 2.0))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(2.0 / N[(N[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+257], N[(N[(N[Power[N[(N[Power[N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision] * N[(l * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 5.00000000000000028e257

   1. Initial program 83.3%

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

      1. associate-/r* 83.3%

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

      2. associate-*l* 77.6%

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

      3. sqr-neg 77.6%

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

      4. associate-*l* 83.3%

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

      5. *-commutative 83.3%

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

      6. sqr-neg 83.3%

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

      7. associate-/r* 83.3%

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

   3. Simplified 85.5%

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

      1. associate-/l/ 85.5%

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

      2. associate-/r/ 83.3%

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

      3. add-cube-cbrt 83.2%

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

   5. Applied egg-rr 93.1%

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

      1. div-inv 93.1%

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

      2. pow-plus 93.1%

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

      3. associate-/r/ 93.2%

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

      4. associate-/r/ 93.1%

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

      5. metadata-eval 93.1%

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

   7. Applied egg-rr 93.1%

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

      1. cube-prod 91.7%

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

      2. rem-cube-cbrt 91.8%

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

      3. *-commutative 91.8%

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

   9. Simplified 91.8%

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

   if 5.00000000000000028e257 < (/.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 23.1%

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

      1. associate-/r* 23.1%

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

      2. associate-*l* 23.1%

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

      3. sqr-neg 23.1%

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

      4. associate-*l* 23.1%

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

      5. *-commutative 23.1%

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

      6. sqr-neg 23.1%

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

      7. associate-*l/ 23.1%

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

      8. associate-*r/ 23.1%

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

      9. associate-/r/ 23.1%

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

   3. Simplified 23.1%

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

   4. Taylor expanded in k around inf 58.1%

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

      1. times-frac 56.2%

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

      2. unpow2 56.2%

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

      3. unpow2 56.2%

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

   6. Simplified 56.2%

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

      1. associate-*l/ 54.7%

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

   8. Applied egg-rr 54.7%

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

   9. Taylor expanded in l around 0 58.1%

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

       1. associate-*r* 58.1%

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

       2. times-frac 58.6%

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

       3. unpow2 58.6%

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

       4. times-frac 81.0%

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

       5. unpow2 81.0%

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

       6. associate-/r* 83.2%

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

   11. Simplified 83.2%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)} \leq 5 \cdot 10^{+257}:\\ \;\;\;\;\left({\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t}\right)}^{3} \cdot \left(\ell \cdot \frac{\ell}{\sin k}\right)\right) \cdot \frac{1}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{t}\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 4: 80.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;\frac{2}{\left(1 + \left(1 + t_1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)} \leq 5 \cdot 10^{+257}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k} \cdot \frac{2 \cdot \ell}{\tan k \cdot {t}^{3}}}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{t}\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<=
        (/
         2.0
         (*
          (+ 1.0 (+ 1.0 t_1))
          (* (tan k) (* (sin k) (/ (pow t 3.0) (* l l))))))
        5e+257)
     (/ (* (/ l (sin k)) (/ (* 2.0 l) (* (tan k) (pow t 3.0)))) (+ 2.0 t_1))
     (* 2.0 (* (* (/ (/ l k) k) (/ l t)) (/ (cos k) (pow (sin k) 2.0)))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((1.0 + (1.0 + t_1)) * (tan(k) * (sin(k) * (pow(t, 3.0) / (l * l)))))) <= 5e+257) {
		tmp = ((l / sin(k)) * ((2.0 * l) / (tan(k) * pow(t, 3.0)))) / (2.0 + t_1);
	} else {
		tmp = 2.0 * ((((l / k) / k) * (l / t)) * (cos(k) / pow(sin(k), 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (k / t) ** 2.0d0
    if ((2.0d0 / ((1.0d0 + (1.0d0 + t_1)) * (tan(k) * (sin(k) * ((t ** 3.0d0) / (l * l)))))) <= 5d+257) then
        tmp = ((l / sin(k)) * ((2.0d0 * l) / (tan(k) * (t ** 3.0d0)))) / (2.0d0 + t_1)
    else
        tmp = 2.0d0 * ((((l / k) / k) * (l / t)) * (cos(k) / (sin(k) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((1.0 + (1.0 + t_1)) * (Math.tan(k) * (Math.sin(k) * (Math.pow(t, 3.0) / (l * l)))))) <= 5e+257) {
		tmp = ((l / Math.sin(k)) * ((2.0 * l) / (Math.tan(k) * Math.pow(t, 3.0)))) / (2.0 + t_1);
	} else {
		tmp = 2.0 * ((((l / k) / k) * (l / t)) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow((k / t), 2.0)
	tmp = 0
	if (2.0 / ((1.0 + (1.0 + t_1)) * (math.tan(k) * (math.sin(k) * (math.pow(t, 3.0) / (l * l)))))) <= 5e+257:
		tmp = ((l / math.sin(k)) * ((2.0 * l) / (math.tan(k) * math.pow(t, 3.0)))) / (2.0 + t_1)
	else:
		tmp = 2.0 * ((((l / k) / k) * (l / t)) * (math.cos(k) / math.pow(math.sin(k), 2.0)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + t_1)) * Float64(tan(k) * Float64(sin(k) * Float64((t ^ 3.0) / Float64(l * l)))))) <= 5e+257)
		tmp = Float64(Float64(Float64(l / sin(k)) * Float64(Float64(2.0 * l) / Float64(tan(k) * (t ^ 3.0)))) / Float64(2.0 + t_1));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(l / k) / k) * Float64(l / t)) * Float64(cos(k) / (sin(k) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / t) ^ 2.0;
	tmp = 0.0;
	if ((2.0 / ((1.0 + (1.0 + t_1)) * (tan(k) * (sin(k) * ((t ^ 3.0) / (l * l)))))) <= 5e+257)
		tmp = ((l / sin(k)) * ((2.0 * l) / (tan(k) * (t ^ 3.0)))) / (2.0 + t_1);
	else
		tmp = 2.0 * ((((l / k) / k) * (l / t)) * (cos(k) / (sin(k) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(2.0 / N[(N[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+257], N[(N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * l), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 5.00000000000000028e257

   1. Initial program 83.3%

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

      1. associate-/r* 83.3%

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

      2. associate-*l* 77.6%

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

      3. sqr-neg 77.6%

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

      4. associate-*l* 83.3%

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

      5. *-commutative 83.3%

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

      6. sqr-neg 83.3%

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

      7. associate-*l/ 86.2%

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

      8. associate-*r/ 86.0%

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

      9. associate-/r/ 85.3%

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

   3. Simplified 85.3%

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

      1. expm1-log1p-u 61.7%

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

      2. expm1-udef 55.6%

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

      3. associate-*l/ 56.3%

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

      4. associate-*r* 56.3%

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

   5. Applied egg-rr 56.3%

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

      1. expm1-def 62.5%

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

      2. expm1-log1p 86.0%

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

      3. associate-*r* 86.0%

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

      4. times-frac 91.7%

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

      5. *-commutative 91.7%

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

   7. Simplified 91.7%

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

   if 5.00000000000000028e257 < (/.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 23.1%

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

      1. associate-/r* 23.1%

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

      2. associate-*l* 23.1%

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

      3. sqr-neg 23.1%

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

      4. associate-*l* 23.1%

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

      5. *-commutative 23.1%

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

      6. sqr-neg 23.1%

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

      7. associate-*l/ 23.1%

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

      8. associate-*r/ 23.1%

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

      9. associate-/r/ 23.1%

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

   3. Simplified 23.1%

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

   4. Taylor expanded in k around inf 58.1%

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

      1. times-frac 56.2%

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

      2. unpow2 56.2%

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

      3. unpow2 56.2%

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

   6. Simplified 56.2%

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

      1. associate-*l/ 54.7%

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

   8. Applied egg-rr 54.7%

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

   9. Taylor expanded in l around 0 58.1%

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

       1. associate-*r* 58.1%

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

       2. times-frac 58.6%

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

       3. unpow2 58.6%

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

       4. times-frac 81.0%

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

       5. unpow2 81.0%

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

       6. associate-/r* 83.2%

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

   11. Simplified 83.2%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)} \leq 5 \cdot 10^{+257}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k} \cdot \frac{2 \cdot \ell}{\tan k \cdot {t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{t}\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 5: 85.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -27000000 \lor \neg \left(t \leq 4.8 \cdot 10^{-36}\right):\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot {\left(\frac{t}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{\frac{\sin k}{\ell}}}}\right)}^{3}}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{t}\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -27000000.0) (not (<= t 4.8e-36)))
   (/
    (/ 2.0 (* (tan k) (pow (/ t (/ (cbrt l) (cbrt (/ (sin k) l)))) 3.0)))
    (+ 1.0 (+ 1.0 (pow (/ k t) 2.0))))
   (* 2.0 (* (* (/ (/ l k) k) (/ l t)) (/ (cos k) (pow (sin k) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -27000000.0) || !(t <= 4.8e-36)) {
		tmp = (2.0 / (tan(k) * pow((t / (cbrt(l) / cbrt((sin(k) / l)))), 3.0))) / (1.0 + (1.0 + pow((k / t), 2.0)));
	} else {
		tmp = 2.0 * ((((l / k) / k) * (l / t)) * (cos(k) / pow(sin(k), 2.0)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -27000000.0) || !(t <= 4.8e-36)) {
		tmp = (2.0 / (Math.tan(k) * Math.pow((t / (Math.cbrt(l) / Math.cbrt((Math.sin(k) / l)))), 3.0))) / (1.0 + (1.0 + Math.pow((k / t), 2.0)));
	} else {
		tmp = 2.0 * ((((l / k) / k) * (l / t)) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -27000000.0) || !(t <= 4.8e-36))
		tmp = Float64(Float64(2.0 / Float64(tan(k) * (Float64(t / Float64(cbrt(l) / cbrt(Float64(sin(k) / l)))) ^ 3.0))) / Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(l / k) / k) * Float64(l / t)) * Float64(cos(k) / (sin(k) ^ 2.0))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -27000000.0], N[Not[LessEqual[t, 4.8e-36]], $MachinePrecision]], N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[Power[N[(t / N[(N[Power[l, 1/3], $MachinePrecision] / N[Power[N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.7e7 or 4.8e-36 < t

   1. Initial program 71.3%

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

      1. associate-/r* 71.3%

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

      2. +-commutative 71.3%

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

   3. Simplified 71.3%

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

      1. associate-/r/ 73.6%

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

      2. add-cube-cbrt 73.4%

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

      3. pow3 73.4%

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

      4. cbrt-div 73.4%

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

      5. rem-cbrt-cube 75.3%

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

      6. associate-/l* 77.3%

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

   5. Applied egg-rr 77.3%

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

      1. cbrt-div 90.0%

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

   7. Applied egg-rr 90.0%

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

   if -2.7e7 < t < 4.8e-36

   1. Initial program 40.5%

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

      1. associate-/r* 40.5%

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

      2. associate-*l* 40.5%

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

      3. sqr-neg 40.5%

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

      4. associate-*l* 40.5%

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

      5. *-commutative 40.5%

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

      6. sqr-neg 40.5%

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

      7. associate-*l/ 40.5%

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

      8. associate-*r/ 40.4%

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

      9. associate-/r/ 39.6%

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

   3. Simplified 39.6%

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

   4. Taylor expanded in k around inf 72.0%

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

      1. times-frac 70.2%

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

      2. unpow2 70.2%

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

      3. unpow2 70.2%

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

   6. Simplified 70.2%

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

      1. associate-*l/ 68.8%

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

   8. Applied egg-rr 68.8%

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

   9. Taylor expanded in l around 0 72.0%

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

       1. associate-*r* 72.0%

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

       2. times-frac 72.5%

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

       3. unpow2 72.5%

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

       4. times-frac 90.1%

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

       5. unpow2 90.1%

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

       6. associate-/r* 91.8%

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

   11. Simplified 91.8%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -27000000 \lor \neg \left(t \leq 4.8 \cdot 10^{-36}\right):\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot {\left(\frac{t}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{\frac{\sin k}{\ell}}}}\right)}^{3}}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{t}\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 6: 78.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{\tan k}\\ \mathbf{if}\;t \leq -1.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{-39}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{t}\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{t_1}{\frac{{t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}}}}{2 + \frac{k \cdot k}{t \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{t_1}}{t}\right)}^{3}}{\frac{2}{\ell} \cdot \frac{k}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ 2.0 (tan k))))
   (if (<= t -1.5e+45)
     (/ l (/ (* k (* k (pow t 3.0))) l))
     (if (<= t 8.2e-39)
       (* 2.0 (* (* (/ (/ l k) k) (/ l t)) (/ (cos k) (pow (sin k) 2.0))))
       (if (<= t 1.02e+88)
         (/
          (/ t_1 (/ (pow t 3.0) (/ l (/ (sin k) l))))
          (+ 2.0 (/ (* k k) (* t t))))
         (/ (pow (/ (cbrt t_1) t) 3.0) (* (/ 2.0 l) (/ k l))))))))

C

double code(double t, double l, double k) {
	double t_1 = 2.0 / tan(k);
	double tmp;
	if (t <= -1.5e+45) {
		tmp = l / ((k * (k * pow(t, 3.0))) / l);
	} else if (t <= 8.2e-39) {
		tmp = 2.0 * ((((l / k) / k) * (l / t)) * (cos(k) / pow(sin(k), 2.0)));
	} else if (t <= 1.02e+88) {
		tmp = (t_1 / (pow(t, 3.0) / (l / (sin(k) / l)))) / (2.0 + ((k * k) / (t * t)));
	} else {
		tmp = pow((cbrt(t_1) / t), 3.0) / ((2.0 / l) * (k / l));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = 2.0 / Math.tan(k);
	double tmp;
	if (t <= -1.5e+45) {
		tmp = l / ((k * (k * Math.pow(t, 3.0))) / l);
	} else if (t <= 8.2e-39) {
		tmp = 2.0 * ((((l / k) / k) * (l / t)) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	} else if (t <= 1.02e+88) {
		tmp = (t_1 / (Math.pow(t, 3.0) / (l / (Math.sin(k) / l)))) / (2.0 + ((k * k) / (t * t)));
	} else {
		tmp = Math.pow((Math.cbrt(t_1) / t), 3.0) / ((2.0 / l) * (k / l));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(2.0 / tan(k))
	tmp = 0.0
	if (t <= -1.5e+45)
		tmp = Float64(l / Float64(Float64(k * Float64(k * (t ^ 3.0))) / l));
	elseif (t <= 8.2e-39)
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(l / k) / k) * Float64(l / t)) * Float64(cos(k) / (sin(k) ^ 2.0))));
	elseif (t <= 1.02e+88)
		tmp = Float64(Float64(t_1 / Float64((t ^ 3.0) / Float64(l / Float64(sin(k) / l)))) / Float64(2.0 + Float64(Float64(k * k) / Float64(t * t))));
	else
		tmp = Float64((Float64(cbrt(t_1) / t) ^ 3.0) / Float64(Float64(2.0 / l) * Float64(k / l)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.5e+45], N[(l / N[(N[(k * N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8.2e-39], N[(2.0 * N[(N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.02e+88], N[(N[(t$95$1 / N[(N[Power[t, 3.0], $MachinePrecision] / N[(l / N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(k * k), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[Power[t$95$1, 1/3], $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision] / N[(N[(2.0 / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1.50000000000000005e45

   1. Initial program 65.7%

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

      1. associate-/r* 65.7%

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

      2. +-commutative 65.7%

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

   3. Simplified 65.7%

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

      1. associate-/r/ 65.5%

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

      2. add-cube-cbrt 65.4%

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

      3. pow3 65.4%

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

      4. cbrt-div 65.5%

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

      5. rem-cbrt-cube 68.4%

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

      6. associate-/l* 68.4%

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

   5. Applied egg-rr 68.4%

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

      1. cbrt-div 89.4%

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

   7. Applied egg-rr 89.4%

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

      1. cbrt-div 95.1%

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

   9. Applied egg-rr 95.1%

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

   10. Taylor expanded in k around 0 57.8%

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

       1. unpow2 57.8%

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

       2. associate-/l* 64.5%

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

       3. *-commutative 64.5%

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

       4. unpow2 64.5%

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

       5. associate-*r* 72.3%

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

   12. Simplified 72.3%

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

   if -1.50000000000000005e45 < t < 8.2e-39

   1. Initial program 40.7%

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

      1. associate-/r* 40.7%

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

      2. associate-*l* 40.7%

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

      3. sqr-neg 40.7%

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

      4. associate-*l* 40.7%

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

      5. *-commutative 40.7%

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

      6. sqr-neg 40.7%

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

      7. associate-*l/ 40.9%

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

      8. associate-*r/ 40.9%

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

      9. associate-/r/ 40.1%

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

   3. Simplified 40.1%

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

   4. Taylor expanded in k around inf 69.9%

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

      1. times-frac 68.1%

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

      2. unpow2 68.1%

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

      3. unpow2 68.1%

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

   6. Simplified 68.1%

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

      1. associate-*l/ 66.8%

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

   8. Applied egg-rr 66.8%

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

   9. Taylor expanded in l around 0 69.9%

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

       1. associate-*r* 69.9%

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

       2. times-frac 70.3%

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

       3. unpow2 70.3%

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

       4. times-frac 87.8%

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

       5. unpow2 87.8%

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

       6. associate-/r* 90.1%

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

   11. Simplified 90.1%

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

   if 8.2e-39 < t < 1.01999999999999998e88

   1. Initial program 80.5%

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

      1. associate-/r* 80.6%

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

      2. associate-*l* 77.8%

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

      3. sqr-neg 77.8%

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

      4. associate-*l* 80.6%

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

      5. *-commutative 80.6%

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

      6. sqr-neg 80.6%

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

      7. associate-/r* 80.6%

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

   3. Simplified 89.0%

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

      1. *-un-lft-identity 89.0%

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

      2. associate-/l* 94.2%

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

   5. Applied egg-rr 94.2%

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

   6. Taylor expanded in k around 0 94.3%

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

      1. unpow2 94.3%

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

      2. unpow2 94.3%

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

   8. Simplified 94.3%

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

   if 1.01999999999999998e88 < t

   1. Initial program 74.5%

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

      1. associate-/r* 74.5%

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

      2. associate-*l* 66.9%

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

      3. sqr-neg 66.9%

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

      4. associate-*l* 74.5%

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

      5. *-commutative 74.5%

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

      6. sqr-neg 74.5%

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

      7. associate-/r* 74.5%

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

   3. Simplified 74.4%

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

      1. associate-/l/ 74.4%

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

      2. associate-/r/ 74.5%

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

      3. add-cube-cbrt 74.5%

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

   5. Applied egg-rr 85.9%

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

      1. div-inv 85.9%

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

      2. pow-plus 85.9%

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

      3. associate-/r/ 85.9%

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

      4. associate-/r/ 85.9%

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

      5. metadata-eval 85.9%

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

   7. Applied egg-rr 85.9%

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

      1. associate-*r/ 85.9%

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

      2. associate-*l/ 85.9%

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

      3. *-rgt-identity 85.9%

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

      4. cube-prod 85.3%

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

      5. rem-cube-cbrt 85.4%

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

      6. associate-/l* 85.3%

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

      7. *-commutative 85.3%

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

   9. Simplified 85.3%

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

   10. Taylor expanded in k around 0 84.7%

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

       1. associate-*r/ 84.7%

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

       2. unpow2 84.7%

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

       3. times-frac 90.1%

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

   12. Simplified 90.1%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{-39}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{t}\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}}}}{2 + \frac{k \cdot k}{t \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t}\right)}^{3}}{\frac{2}{\ell} \cdot \frac{k}{\ell}}\\ \end{array} \]

Alternative 7: 74.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{+45}:\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-70}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t}\right)}^{3}}{\frac{2}{\ell} \cdot \frac{k}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -1.15e+45)
   (/ l (/ (* k (* k (pow t 3.0))) l))
   (if (<= t 2.3e-70)
     (* 2.0 (* (/ (cos k) (pow (sin k) 2.0)) (* (/ l t) (/ l (* k k)))))
     (/ (pow (/ (cbrt (/ 2.0 (tan k))) t) 3.0) (* (/ 2.0 l) (/ k l))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.15e+45) {
		tmp = l / ((k * (k * pow(t, 3.0))) / l);
	} else if (t <= 2.3e-70) {
		tmp = 2.0 * ((cos(k) / pow(sin(k), 2.0)) * ((l / t) * (l / (k * k))));
	} else {
		tmp = pow((cbrt((2.0 / tan(k))) / t), 3.0) / ((2.0 / l) * (k / l));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.15e+45) {
		tmp = l / ((k * (k * Math.pow(t, 3.0))) / l);
	} else if (t <= 2.3e-70) {
		tmp = 2.0 * ((Math.cos(k) / Math.pow(Math.sin(k), 2.0)) * ((l / t) * (l / (k * k))));
	} else {
		tmp = Math.pow((Math.cbrt((2.0 / Math.tan(k))) / t), 3.0) / ((2.0 / l) * (k / l));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -1.15e+45)
		tmp = Float64(l / Float64(Float64(k * Float64(k * (t ^ 3.0))) / l));
	elseif (t <= 2.3e-70)
		tmp = Float64(2.0 * Float64(Float64(cos(k) / (sin(k) ^ 2.0)) * Float64(Float64(l / t) * Float64(l / Float64(k * k)))));
	else
		tmp = Float64((Float64(cbrt(Float64(2.0 / tan(k))) / t) ^ 3.0) / Float64(Float64(2.0 / l) * Float64(k / l)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -1.15e+45], N[(l / N[(N[(k * N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.3e-70], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(l / t), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[Power[N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision] / N[(N[(2.0 / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.15000000000000006e45

   1. Initial program 65.7%

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

      1. associate-/r* 65.7%

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

      2. +-commutative 65.7%

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

   3. Simplified 65.7%

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

      1. associate-/r/ 65.5%

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

      2. add-cube-cbrt 65.4%

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

      3. pow3 65.4%

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

      4. cbrt-div 65.5%

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

      5. rem-cbrt-cube 68.4%

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

      6. associate-/l* 68.4%

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

   5. Applied egg-rr 68.4%

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

      1. cbrt-div 89.4%

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

   7. Applied egg-rr 89.4%

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

      1. cbrt-div 95.1%

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

   9. Applied egg-rr 95.1%

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

   10. Taylor expanded in k around 0 57.8%

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

       1. unpow2 57.8%

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

       2. associate-/l* 64.5%

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

       3. *-commutative 64.5%

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

       4. unpow2 64.5%

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

       5. associate-*r* 72.3%

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

   12. Simplified 72.3%

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

   if -1.15000000000000006e45 < t < 2.30000000000000001e-70

   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. Step-by-step derivation

      1. associate-/r* 40.0%

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

      2. associate-*l* 39.9%

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

      3. sqr-neg 39.9%

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

      4. associate-*l* 40.0%

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

      5. *-commutative 40.0%

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

      6. sqr-neg 40.0%

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

      7. associate-/r* 40.0%

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

   3. Simplified 40.2%

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

      1. associate-/l/ 40.2%

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

      2. associate-/r/ 40.0%

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

      3. add-cube-cbrt 39.9%

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

   5. Applied egg-rr 59.1%

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

   6. Taylor expanded in k around inf 70.3%

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

      1. associate-*r* 70.3%

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

      2. times-frac 70.7%

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

      3. unpow2 70.7%

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

      4. times-frac 88.1%

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

      5. unpow2 88.1%

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

   8. Simplified 88.1%

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

   if 2.30000000000000001e-70 < t

   1. Initial program 76.2%

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

      1. associate-/r* 76.2%

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

      2. associate-*l* 71.2%

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

      3. sqr-neg 71.2%

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

      4. associate-*l* 76.2%

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

      5. *-commutative 76.2%

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

      6. sqr-neg 76.2%

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

      7. associate-/r* 76.2%

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

   3. Simplified 79.8%

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

      1. associate-/l/ 79.7%

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

      2. associate-/r/ 76.2%

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

      3. add-cube-cbrt 76.1%

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

   5. Applied egg-rr 88.8%

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

      1. div-inv 88.8%

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

      2. pow-plus 88.8%

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

      3. associate-/r/ 88.9%

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

      4. associate-/r/ 88.8%

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

      5. metadata-eval 88.8%

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

   7. Applied egg-rr 88.8%

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

      1. associate-*r/ 88.8%

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

      2. associate-*l/ 88.8%

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

      3. *-rgt-identity 88.8%

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

      4. cube-prod 88.5%

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

      5. rem-cube-cbrt 88.7%

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

      6. associate-/l* 88.6%

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

      7. *-commutative 88.6%

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

   9. Simplified 88.6%

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

   10. Taylor expanded in k around 0 79.1%

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

       1. associate-*r/ 79.1%

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

       2. unpow2 79.1%

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

       3. times-frac 85.3%

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

   12. Simplified 85.3%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{+45}:\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-70}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t}\right)}^{3}}{\frac{2}{\ell} \cdot \frac{k}{\ell}}\\ \end{array} \]

Alternative 8: 76.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.15 \cdot 10^{+45}:\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-70}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{t}\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t}\right)}^{3}}{\frac{2}{\ell} \cdot \frac{k}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -2.15e+45)
   (/ l (/ (* k (* k (pow t 3.0))) l))
   (if (<= t 2.3e-70)
     (* 2.0 (* (* (/ (/ l k) k) (/ l t)) (/ (cos k) (pow (sin k) 2.0))))
     (/ (pow (/ (cbrt (/ 2.0 (tan k))) t) 3.0) (* (/ 2.0 l) (/ k l))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -2.15e+45) {
		tmp = l / ((k * (k * pow(t, 3.0))) / l);
	} else if (t <= 2.3e-70) {
		tmp = 2.0 * ((((l / k) / k) * (l / t)) * (cos(k) / pow(sin(k), 2.0)));
	} else {
		tmp = pow((cbrt((2.0 / tan(k))) / t), 3.0) / ((2.0 / l) * (k / l));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -2.15e+45) {
		tmp = l / ((k * (k * Math.pow(t, 3.0))) / l);
	} else if (t <= 2.3e-70) {
		tmp = 2.0 * ((((l / k) / k) * (l / t)) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = Math.pow((Math.cbrt((2.0 / Math.tan(k))) / t), 3.0) / ((2.0 / l) * (k / l));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -2.15e+45)
		tmp = Float64(l / Float64(Float64(k * Float64(k * (t ^ 3.0))) / l));
	elseif (t <= 2.3e-70)
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(l / k) / k) * Float64(l / t)) * Float64(cos(k) / (sin(k) ^ 2.0))));
	else
		tmp = Float64((Float64(cbrt(Float64(2.0 / tan(k))) / t) ^ 3.0) / Float64(Float64(2.0 / l) * Float64(k / l)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -2.15e+45], N[(l / N[(N[(k * N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.3e-70], N[(2.0 * N[(N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[Power[N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision] / N[(N[(2.0 / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.1500000000000002e45

   1. Initial program 65.7%

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

      1. associate-/r* 65.7%

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

      2. +-commutative 65.7%

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

   3. Simplified 65.7%

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

      1. associate-/r/ 65.5%

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

      2. add-cube-cbrt 65.4%

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

      3. pow3 65.4%

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

      4. cbrt-div 65.5%

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

      5. rem-cbrt-cube 68.4%

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

      6. associate-/l* 68.4%

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

   5. Applied egg-rr 68.4%

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

      1. cbrt-div 89.4%

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

   7. Applied egg-rr 89.4%

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

      1. cbrt-div 95.1%

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

   9. Applied egg-rr 95.1%

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

   10. Taylor expanded in k around 0 57.8%

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

       1. unpow2 57.8%

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

       2. associate-/l* 64.5%

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

       3. *-commutative 64.5%

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

       4. unpow2 64.5%

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

       5. associate-*r* 72.3%

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

   12. Simplified 72.3%

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

   if -2.1500000000000002e45 < t < 2.30000000000000001e-70

   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. Step-by-step derivation

      1. associate-/r* 40.0%

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

      2. associate-*l* 39.9%

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

      3. sqr-neg 39.9%

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

      4. associate-*l* 40.0%

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

      5. *-commutative 40.0%

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

      6. sqr-neg 40.0%

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

      7. associate-*l/ 40.2%

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

      8. associate-*r/ 40.1%

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

      9. associate-/r/ 39.3%

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

   3. Simplified 39.3%

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

   4. Taylor expanded in k around inf 70.3%

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

      1. times-frac 68.5%

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

      2. unpow2 68.5%

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

      3. unpow2 68.5%

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

   6. Simplified 68.5%

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

      1. associate-*l/ 67.1%

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

   8. Applied egg-rr 67.1%

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

   9. Taylor expanded in l around 0 70.3%

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

       1. associate-*r* 70.3%

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

       2. times-frac 70.7%

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

       3. unpow2 70.7%

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

       4. times-frac 88.1%

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

       5. unpow2 88.1%

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

       6. associate-/r* 90.4%

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

   11. Simplified 90.4%

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

   if 2.30000000000000001e-70 < t

   1. Initial program 76.2%

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

      1. associate-/r* 76.2%

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

      2. associate-*l* 71.2%

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

      3. sqr-neg 71.2%

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

      4. associate-*l* 76.2%

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

      5. *-commutative 76.2%

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

      6. sqr-neg 76.2%

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

      7. associate-/r* 76.2%

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

   3. Simplified 79.8%

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

      1. associate-/l/ 79.7%

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

      2. associate-/r/ 76.2%

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

      3. add-cube-cbrt 76.1%

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

   5. Applied egg-rr 88.8%

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

      1. div-inv 88.8%

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

      2. pow-plus 88.8%

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

      3. associate-/r/ 88.9%

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

      4. associate-/r/ 88.8%

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

      5. metadata-eval 88.8%

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

   7. Applied egg-rr 88.8%

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

      1. associate-*r/ 88.8%

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

      2. associate-*l/ 88.8%

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

      3. *-rgt-identity 88.8%

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

      4. cube-prod 88.5%

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

      5. rem-cube-cbrt 88.7%

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

      6. associate-/l* 88.6%

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

      7. *-commutative 88.6%

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

   9. Simplified 88.6%

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

   10. Taylor expanded in k around 0 79.1%

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

       1. associate-*r/ 79.1%

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

       2. unpow2 79.1%

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

       3. times-frac 85.3%

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

   12. Simplified 85.3%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.15 \cdot 10^{+45}:\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-70}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{t}\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t}\right)}^{3}}{\frac{2}{\ell} \cdot \frac{k}{\ell}}\\ \end{array} \]

Alternative 9: 68.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-71}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{t}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t}\right)}^{3}}{\frac{2}{\ell} \cdot \frac{k}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -1.5e-18)
   (/ l (/ (* k (* k (pow t 3.0))) l))
   (if (<= t 6.8e-71)
     (* 2.0 (/ (* (/ (/ l k) k) (/ l t)) (* k k)))
     (/ (pow (/ (cbrt (/ 2.0 (tan k))) t) 3.0) (* (/ 2.0 l) (/ k l))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.5e-18) {
		tmp = l / ((k * (k * pow(t, 3.0))) / l);
	} else if (t <= 6.8e-71) {
		tmp = 2.0 * ((((l / k) / k) * (l / t)) / (k * k));
	} else {
		tmp = pow((cbrt((2.0 / tan(k))) / t), 3.0) / ((2.0 / l) * (k / l));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.5e-18) {
		tmp = l / ((k * (k * Math.pow(t, 3.0))) / l);
	} else if (t <= 6.8e-71) {
		tmp = 2.0 * ((((l / k) / k) * (l / t)) / (k * k));
	} else {
		tmp = Math.pow((Math.cbrt((2.0 / Math.tan(k))) / t), 3.0) / ((2.0 / l) * (k / l));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -1.5e-18)
		tmp = Float64(l / Float64(Float64(k * Float64(k * (t ^ 3.0))) / l));
	elseif (t <= 6.8e-71)
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(l / k) / k) * Float64(l / t)) / Float64(k * k)));
	else
		tmp = Float64((Float64(cbrt(Float64(2.0 / tan(k))) / t) ^ 3.0) / Float64(Float64(2.0 / l) * Float64(k / l)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -1.5e-18], N[(l / N[(N[(k * N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.8e-71], N[(2.0 * N[(N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[Power[N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision] / N[(N[(2.0 / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.49999999999999991e-18

   1. Initial program 64.6%

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

      1. associate-/r* 64.6%

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

      2. +-commutative 64.6%

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

   3. Simplified 64.6%

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

      1. associate-/r/ 64.9%

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

      2. add-cube-cbrt 64.8%

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

      3. pow3 64.8%

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

      4. cbrt-div 64.8%

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

      5. rem-cbrt-cube 67.2%

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

      6. associate-/l* 68.2%

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

   5. Applied egg-rr 68.2%

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

      1. cbrt-div 86.7%

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

   7. Applied egg-rr 86.7%

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

      1. cbrt-div 91.3%

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

   9. Applied egg-rr 91.3%

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

   10. Taylor expanded in k around 0 56.4%

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

       1. unpow2 56.4%

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

       2. associate-/l* 61.9%

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

       3. *-commutative 61.9%

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

       4. unpow2 61.9%

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

       5. associate-*r* 69.8%

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

   12. Simplified 69.8%

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

   if -1.49999999999999991e-18 < t < 6.80000000000000007e-71

   1. Initial program 37.8%

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

      1. associate-/r* 37.8%

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

      2. associate-*l* 37.8%

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

      3. sqr-neg 37.8%

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

      4. associate-*l* 37.8%

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

      5. *-commutative 37.8%

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

      6. sqr-neg 37.8%

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

      7. associate-*l/ 37.8%

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

      8. associate-*r/ 37.8%

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

      9. associate-/r/ 36.9%

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

   3. Simplified 36.9%

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

   4. Taylor expanded in k around inf 72.1%

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

      1. times-frac 70.1%

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

      2. unpow2 70.1%

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

      3. unpow2 70.1%

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

   6. Simplified 70.1%

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

      1. associate-*l/ 68.6%

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

   8. Applied egg-rr 68.6%

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

   9. Taylor expanded in k around 0 60.9%

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

       1. unpow2 60.9%

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

       2. times-frac 77.4%

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

       3. unpow2 77.4%

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

       4. associate-/r* 77.4%

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

   11. Simplified 77.4%

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

   if 6.80000000000000007e-71 < t

   1. Initial program 76.2%

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

      1. associate-/r* 76.2%

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

      2. associate-*l* 71.2%

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

      3. sqr-neg 71.2%

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

      4. associate-*l* 76.2%

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

      5. *-commutative 76.2%

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

      6. sqr-neg 76.2%

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

      7. associate-/r* 76.2%

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

   3. Simplified 79.8%

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

      1. associate-/l/ 79.7%

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

      2. associate-/r/ 76.2%

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

      3. add-cube-cbrt 76.1%

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

   5. Applied egg-rr 88.8%

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

      1. div-inv 88.8%

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

      2. pow-plus 88.8%

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

      3. associate-/r/ 88.9%

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

      4. associate-/r/ 88.8%

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

      5. metadata-eval 88.8%

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

   7. Applied egg-rr 88.8%

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

      1. associate-*r/ 88.8%

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

      2. associate-*l/ 88.8%

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

      3. *-rgt-identity 88.8%

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

      4. cube-prod 88.5%

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

      5. rem-cube-cbrt 88.7%

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

      6. associate-/l* 88.6%

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

      7. *-commutative 88.6%

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

   9. Simplified 88.6%

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

   10. Taylor expanded in k around 0 79.1%

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

       1. associate-*r/ 79.1%

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

       2. unpow2 79.1%

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

       3. times-frac 85.3%

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

   12. Simplified 85.3%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-71}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{t}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t}\right)}^{3}}{\frac{2}{\ell} \cdot \frac{k}{\ell}}\\ \end{array} \]

Alternative 10: 66.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1600000000000:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t}\right)}^{3}}{\frac{2}{\ell} \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\ell \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1600000000000.0)
   (/ (pow (/ (cbrt (/ 2.0 (tan k))) t) 3.0) (* (/ 2.0 l) (/ k l)))
   (* 2.0 (* (* l (/ l (* k k))) (/ (cos k) (* t (pow (sin k) 2.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1600000000000.0) {
		tmp = pow((cbrt((2.0 / tan(k))) / t), 3.0) / ((2.0 / l) * (k / l));
	} else {
		tmp = 2.0 * ((l * (l / (k * k))) * (cos(k) / (t * pow(sin(k), 2.0))));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1600000000000.0) {
		tmp = Math.pow((Math.cbrt((2.0 / Math.tan(k))) / t), 3.0) / ((2.0 / l) * (k / l));
	} else {
		tmp = 2.0 * ((l * (l / (k * k))) * (Math.cos(k) / (t * Math.pow(Math.sin(k), 2.0))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1600000000000.0)
		tmp = Float64((Float64(cbrt(Float64(2.0 / tan(k))) / t) ^ 3.0) / Float64(Float64(2.0 / l) * Float64(k / l)));
	else
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l / Float64(k * k))) * Float64(cos(k) / Float64(t * (sin(k) ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1600000000000.0], N[(N[Power[N[(N[Power[N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision] / N[(N[(2.0 / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.6e12

   1. Initial program 56.1%

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

      1. associate-/r* 56.1%

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

      2. associate-*l* 52.2%

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

      3. sqr-neg 52.2%

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

      4. associate-*l* 56.1%

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

      5. *-commutative 56.1%

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

      6. sqr-neg 56.1%

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

      7. associate-/r* 56.2%

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

   3. Simplified 57.7%

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

      1. associate-/l/ 57.6%

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

      2. associate-/r/ 56.1%

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

      3. add-cube-cbrt 56.1%

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

   5. Applied egg-rr 72.0%

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

      1. div-inv 72.0%

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

      2. pow-plus 72.0%

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

      3. associate-/r/ 72.0%

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

      4. associate-/r/ 72.0%

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

      5. metadata-eval 72.0%

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

   7. Applied egg-rr 72.0%

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

      1. associate-*r/ 72.0%

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

      2. associate-*l/ 72.0%

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

      3. *-rgt-identity 72.0%

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

      4. cube-prod 66.9%

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

      5. rem-cube-cbrt 67.0%

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

      6. associate-/l* 64.9%

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

      7. *-commutative 64.9%

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

   9. Simplified 64.9%

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

   10. Taylor expanded in k around 0 61.0%

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

       1. associate-*r/ 61.1%

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

       2. unpow2 61.1%

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

       3. times-frac 68.8%

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

   12. Simplified 68.8%

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

   if 1.6e12 < k

   1. Initial program 57.9%

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

      1. associate-/r* 57.9%

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

      2. associate-*l* 58.0%

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

      3. sqr-neg 58.0%

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

      4. associate-*l* 57.9%

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

      5. *-commutative 57.9%

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

      6. sqr-neg 57.9%

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

      7. associate-*l/ 57.9%

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

      8. associate-*r/ 57.9%

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

      9. associate-/r/ 57.9%

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

   3. Simplified 57.9%

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

   4. Taylor expanded in k around inf 72.0%

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

      1. times-frac 70.3%

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

      2. unpow2 70.3%

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

      3. unpow2 70.3%

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

   6. Simplified 70.3%

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

   7. Taylor expanded in l around 0 70.3%

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

      1. unpow2 70.3%

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

      2. associate-*r/ 81.7%

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

      3. unpow2 81.7%

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

   9. Simplified 81.7%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1600000000000:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t}\right)}^{3}}{\frac{2}{\ell} \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\ell \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternative 11: 68.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{-18}:\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{-107}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{t}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k}\right)}^{2}}{{t}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -1.3e-18)
   (/ l (/ (* k (* k (pow t 3.0))) l))
   (if (<= t 8.8e-107)
     (* 2.0 (/ (* (/ (/ l k) k) (/ l t)) (* k k)))
     (/ (pow (/ l k) 2.0) (pow t 3.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.3e-18) {
		tmp = l / ((k * (k * pow(t, 3.0))) / l);
	} else if (t <= 8.8e-107) {
		tmp = 2.0 * ((((l / k) / k) * (l / t)) / (k * k));
	} else {
		tmp = pow((l / k), 2.0) / pow(t, 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-1.3d-18)) then
        tmp = l / ((k * (k * (t ** 3.0d0))) / l)
    else if (t <= 8.8d-107) then
        tmp = 2.0d0 * ((((l / k) / k) * (l / t)) / (k * k))
    else
        tmp = ((l / k) ** 2.0d0) / (t ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.3e-18) {
		tmp = l / ((k * (k * Math.pow(t, 3.0))) / l);
	} else if (t <= 8.8e-107) {
		tmp = 2.0 * ((((l / k) / k) * (l / t)) / (k * k));
	} else {
		tmp = Math.pow((l / k), 2.0) / Math.pow(t, 3.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -1.3e-18:
		tmp = l / ((k * (k * math.pow(t, 3.0))) / l)
	elif t <= 8.8e-107:
		tmp = 2.0 * ((((l / k) / k) * (l / t)) / (k * k))
	else:
		tmp = math.pow((l / k), 2.0) / math.pow(t, 3.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -1.3e-18)
		tmp = Float64(l / Float64(Float64(k * Float64(k * (t ^ 3.0))) / l));
	elseif (t <= 8.8e-107)
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(l / k) / k) * Float64(l / t)) / Float64(k * k)));
	else
		tmp = Float64((Float64(l / k) ^ 2.0) / (t ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -1.3e-18)
		tmp = l / ((k * (k * (t ^ 3.0))) / l);
	elseif (t <= 8.8e-107)
		tmp = 2.0 * ((((l / k) / k) * (l / t)) / (k * k));
	else
		tmp = ((l / k) ^ 2.0) / (t ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -1.3e-18], N[(l / N[(N[(k * N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8.8e-107], N[(2.0 * N[(N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.3e-18

   1. Initial program 64.6%

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

      1. associate-/r* 64.6%

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

      2. +-commutative 64.6%

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

   3. Simplified 64.6%

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

      1. associate-/r/ 64.9%

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

      2. add-cube-cbrt 64.8%

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

      3. pow3 64.8%

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

      4. cbrt-div 64.8%

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

      5. rem-cbrt-cube 67.2%

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

      6. associate-/l* 68.2%

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

   5. Applied egg-rr 68.2%

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

      1. cbrt-div 86.7%

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

   7. Applied egg-rr 86.7%

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

      1. cbrt-div 91.3%

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

   9. Applied egg-rr 91.3%

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

   10. Taylor expanded in k around 0 56.4%

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

       1. unpow2 56.4%

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

       2. associate-/l* 61.9%

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

       3. *-commutative 61.9%

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

       4. unpow2 61.9%

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

       5. associate-*r* 69.8%

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

   12. Simplified 69.8%

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

   if -1.3e-18 < t < 8.8000000000000005e-107

   1. Initial program 35.0%

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

      1. associate-/r* 35.0%

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

      2. associate-*l* 35.0%

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

      3. sqr-neg 35.0%

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

      4. associate-*l* 35.0%

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

      5. *-commutative 35.0%

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

      6. sqr-neg 35.0%

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

      7. associate-*l/ 35.0%

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

      8. associate-*r/ 34.9%

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

      9. associate-/r/ 34.0%

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

   3. Simplified 34.0%

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

   4. Taylor expanded in k around inf 71.0%

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

      1. times-frac 68.8%

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

      2. unpow2 68.8%

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

      3. unpow2 68.8%

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

   6. Simplified 68.8%

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

      1. associate-*l/ 67.1%

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

   8. Applied egg-rr 67.1%

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

   9. Taylor expanded in k around 0 59.5%

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

       1. unpow2 59.5%

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

       2. times-frac 77.7%

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

       3. unpow2 77.7%

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

       4. associate-/r* 77.7%

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

   11. Simplified 77.7%

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

   if 8.8000000000000005e-107 < t

   1. Initial program 74.8%

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

      1. associate-/r* 74.8%

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

      2. +-commutative 74.8%

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

   3. Simplified 74.8%

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

      1. associate-/r/ 77.9%

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

      2. add-cube-cbrt 77.7%

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

      3. pow3 77.7%

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

      4. cbrt-div 77.6%

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

      5. rem-cbrt-cube 78.9%

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

      6. associate-/l* 83.0%

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

   5. Applied egg-rr 83.0%

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

      1. cbrt-div 89.2%

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

   7. Applied egg-rr 89.2%

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

   8. Taylor expanded in k around 0 67.2%

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

      1. associate-/r* 67.3%

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

      2. unpow2 67.3%

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

      3. unpow2 67.3%

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

      4. times-frac 78.9%

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

      5. unpow2 78.9%

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

   10. Simplified 78.9%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{-18}:\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{-107}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{t}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k}\right)}^{2}}{{t}^{3}}\\ \end{array} \]

Alternative 12: 63.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{-65} \lor \neg \left(t \leq 8.8 \cdot 10^{-107}\right):\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{t}}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -1.25e-65) (not (<= t 8.8e-107)))
   (* (/ l (* k k)) (/ l (pow t 3.0)))
   (* 2.0 (/ (* (/ (/ l k) k) (/ l t)) (* k k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.25e-65) || !(t <= 8.8e-107)) {
		tmp = (l / (k * k)) * (l / pow(t, 3.0));
	} else {
		tmp = 2.0 * ((((l / k) / k) * (l / t)) / (k * k));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-1.25d-65)) .or. (.not. (t <= 8.8d-107))) then
        tmp = (l / (k * k)) * (l / (t ** 3.0d0))
    else
        tmp = 2.0d0 * ((((l / k) / k) * (l / t)) / (k * k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.25e-65) || !(t <= 8.8e-107)) {
		tmp = (l / (k * k)) * (l / Math.pow(t, 3.0));
	} else {
		tmp = 2.0 * ((((l / k) / k) * (l / t)) / (k * k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -1.25e-65) or not (t <= 8.8e-107):
		tmp = (l / (k * k)) * (l / math.pow(t, 3.0))
	else:
		tmp = 2.0 * ((((l / k) / k) * (l / t)) / (k * k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -1.25e-65) || !(t <= 8.8e-107))
		tmp = Float64(Float64(l / Float64(k * k)) * Float64(l / (t ^ 3.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(l / k) / k) * Float64(l / t)) / Float64(k * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -1.25e-65) || ~((t <= 8.8e-107)))
		tmp = (l / (k * k)) * (l / (t ^ 3.0));
	else
		tmp = 2.0 * ((((l / k) / k) * (l / t)) / (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -1.25e-65], N[Not[LessEqual[t, 8.8e-107]], $MachinePrecision]], N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.24999999999999996e-65 or 8.8000000000000005e-107 < t

   1. Initial program 70.2%

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

      1. associate-/r* 70.2%

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

      2. associate-*l* 65.2%

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

      3. sqr-neg 65.2%

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

      4. associate-*l* 70.2%

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

      5. *-commutative 70.2%

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

      6. sqr-neg 70.2%

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

      7. associate-*l/ 72.7%

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

      8. associate-*r/ 72.6%

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

      9. associate-/r/ 72.6%

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

   3. Simplified 72.6%

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

   4. Taylor expanded in k around 0 62.7%

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

      1. unpow2 62.7%

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

      2. *-commutative 62.7%

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

      3. times-frac 67.1%

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

      4. unpow2 67.1%

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

   6. Simplified 67.1%

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

   if -1.24999999999999996e-65 < t < 8.8000000000000005e-107

   1. Initial program 33.3%

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

      1. associate-/r* 33.3%

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

      2. associate-*l* 33.3%

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

      3. sqr-neg 33.3%

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

      4. associate-*l* 33.3%

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

      5. *-commutative 33.3%

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

      6. sqr-neg 33.3%

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

      7. associate-*l/ 33.3%

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

      8. associate-*r/ 33.3%

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

      9. associate-/r/ 32.2%

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

   3. Simplified 32.2%

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

   4. Taylor expanded in k around inf 72.0%

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

      1. times-frac 69.6%

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

      2. unpow2 69.6%

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

      3. unpow2 69.6%

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

   6. Simplified 69.6%

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

      1. associate-*l/ 67.8%

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

   8. Applied egg-rr 67.8%

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

   9. Taylor expanded in k around 0 59.6%

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

       1. unpow2 59.6%

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

       2. times-frac 79.1%

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

       3. unpow2 79.1%

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

       4. associate-/r* 79.1%

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

   11. Simplified 79.1%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{-65} \lor \neg \left(t \leq 8.8 \cdot 10^{-107}\right):\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{t}}{k \cdot k}\\ \end{array} \]

Alternative 13: 68.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{-18} \lor \neg \left(t \leq 4.7 \cdot 10^{-71}\right):\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{t}}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -1.1e-18) (not (<= t 4.7e-71)))
   (/ l (/ (* k (* k (pow t 3.0))) l))
   (* 2.0 (/ (* (/ (/ l k) k) (/ l t)) (* k k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.1e-18) || !(t <= 4.7e-71)) {
		tmp = l / ((k * (k * pow(t, 3.0))) / l);
	} else {
		tmp = 2.0 * ((((l / k) / k) * (l / t)) / (k * k));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-1.1d-18)) .or. (.not. (t <= 4.7d-71))) then
        tmp = l / ((k * (k * (t ** 3.0d0))) / l)
    else
        tmp = 2.0d0 * ((((l / k) / k) * (l / t)) / (k * k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.1e-18) || !(t <= 4.7e-71)) {
		tmp = l / ((k * (k * Math.pow(t, 3.0))) / l);
	} else {
		tmp = 2.0 * ((((l / k) / k) * (l / t)) / (k * k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -1.1e-18) or not (t <= 4.7e-71):
		tmp = l / ((k * (k * math.pow(t, 3.0))) / l)
	else:
		tmp = 2.0 * ((((l / k) / k) * (l / t)) / (k * k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -1.1e-18) || !(t <= 4.7e-71))
		tmp = Float64(l / Float64(Float64(k * Float64(k * (t ^ 3.0))) / l));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(l / k) / k) * Float64(l / t)) / Float64(k * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -1.1e-18) || ~((t <= 4.7e-71)))
		tmp = l / ((k * (k * (t ^ 3.0))) / l);
	else
		tmp = 2.0 * ((((l / k) / k) * (l / t)) / (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -1.1e-18], N[Not[LessEqual[t, 4.7e-71]], $MachinePrecision]], N[(l / N[(N[(k * N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.0999999999999999e-18 or 4.69999999999999996e-71 < t

   1. Initial program 70.8%

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

      1. associate-/r* 70.8%

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

      2. +-commutative 70.8%

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

   3. Simplified 70.8%

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

      1. associate-/r/ 72.9%

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

      2. add-cube-cbrt 72.7%

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

      3. pow3 72.7%

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

      4. cbrt-div 72.7%

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

      5. rem-cbrt-cube 74.5%

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

      6. associate-/l* 77.0%

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

   5. Applied egg-rr 77.0%

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

      1. cbrt-div 88.7%

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

   7. Applied egg-rr 88.7%

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

      1. cbrt-div 91.2%

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

   9. Applied egg-rr 91.2%

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

   10. Taylor expanded in k around 0 62.3%

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

       1. unpow2 62.3%

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

       2. associate-/l* 67.6%

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

       3. *-commutative 67.6%

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

       4. unpow2 67.6%

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

       5. associate-*r* 74.5%

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

   12. Simplified 74.5%

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

   if -1.0999999999999999e-18 < t < 4.69999999999999996e-71

   1. Initial program 38.2%

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

      1. associate-/r* 38.2%

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

      2. associate-*l* 38.2%

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

      3. sqr-neg 38.2%

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

      4. associate-*l* 38.2%

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

      5. *-commutative 38.2%

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

      6. sqr-neg 38.2%

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

      7. associate-*l/ 38.1%

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

      8. associate-*r/ 38.1%

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

      9. associate-/r/ 37.2%

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

   3. Simplified 37.2%

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

   4. Taylor expanded in k around inf 71.9%

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

      1. times-frac 69.9%

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

      2. unpow2 69.9%

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

      3. unpow2 69.9%

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

   6. Simplified 69.9%

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

      1. associate-*l/ 68.4%

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

   8. Applied egg-rr 68.4%

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

   9. Taylor expanded in k around 0 60.6%

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

       1. unpow2 60.6%

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

       2. times-frac 77.2%

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

       3. unpow2 77.2%

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

       4. associate-/r* 77.2%

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

   11. Simplified 77.2%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{-18} \lor \neg \left(t \leq 4.7 \cdot 10^{-71}\right):\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{t}}{k \cdot k}\\ \end{array} \]

Alternative 14: 60.1% accurate, 24.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.6 \cdot 10^{+16}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{t}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{\frac{t \cdot k}{\ell}} \cdot \frac{-0.16666666666666666}{k}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.6e+16)
   (* 2.0 (/ (* (/ (/ l k) k) (/ l t)) (* k k)))
   (* 2.0 (* (/ l (/ (* t k) l)) (/ -0.16666666666666666 k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.6e+16) {
		tmp = 2.0 * ((((l / k) / k) * (l / t)) / (k * k));
	} else {
		tmp = 2.0 * ((l / ((t * k) / l)) * (-0.16666666666666666 / k));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.6d+16) then
        tmp = 2.0d0 * ((((l / k) / k) * (l / t)) / (k * k))
    else
        tmp = 2.0d0 * ((l / ((t * k) / l)) * ((-0.16666666666666666d0) / k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.6e+16) {
		tmp = 2.0 * ((((l / k) / k) * (l / t)) / (k * k));
	} else {
		tmp = 2.0 * ((l / ((t * k) / l)) * (-0.16666666666666666 / k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.6e+16:
		tmp = 2.0 * ((((l / k) / k) * (l / t)) / (k * k))
	else:
		tmp = 2.0 * ((l / ((t * k) / l)) * (-0.16666666666666666 / k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.6e+16)
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(l / k) / k) * Float64(l / t)) / Float64(k * k)));
	else
		tmp = Float64(2.0 * Float64(Float64(l / Float64(Float64(t * k) / l)) * Float64(-0.16666666666666666 / k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.6e+16)
		tmp = 2.0 * ((((l / k) / k) * (l / t)) / (k * k));
	else
		tmp = 2.0 * ((l / ((t * k) / l)) * (-0.16666666666666666 / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.6e+16], N[(2.0 * N[(N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / N[(N[(t * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.6e16

   1. Initial program 56.1%

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

      1. associate-/r* 56.1%

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

      2. associate-*l* 52.1%

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

      3. sqr-neg 52.1%

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

      4. associate-*l* 56.1%

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

      5. *-commutative 56.1%

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

      6. sqr-neg 56.1%

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

      7. associate-*l/ 58.1%

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

      8. associate-*r/ 57.9%

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

      9. associate-/r/ 57.5%

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

   3. Simplified 57.5%

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

   4. Taylor expanded in k around inf 58.4%

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

      1. times-frac 58.3%

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

      2. unpow2 58.3%

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

      3. unpow2 58.3%

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

   6. Simplified 58.3%

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

      1. associate-*l/ 57.0%

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

   8. Applied egg-rr 57.0%

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

   9. Taylor expanded in k around 0 54.5%

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

       1. unpow2 54.5%

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

       2. times-frac 64.7%

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

       3. unpow2 64.7%

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

       4. associate-/r* 64.7%

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

   11. Simplified 64.7%

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

   if 2.6e16 < k

   1. Initial program 58.2%

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

      1. associate-/r* 58.2%

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

      2. associate-*l* 58.3%

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

      3. sqr-neg 58.3%

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

      4. associate-*l* 58.2%

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

      5. *-commutative 58.2%

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

      6. sqr-neg 58.2%

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

      7. associate-*l/ 58.2%

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

      8. associate-*r/ 58.2%

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

      9. associate-/r/ 58.2%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 58.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around inf 70.9%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. times-frac 69.2%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]

      2. unpow2 69.2%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \]

      3. unpow2 69.2%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \]

   6. Simplified 69.2%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 59.1%

      \[\leadsto 2 \cdot \color{blue}{\left(\left(-0.5 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} \]
   8. Step-by-step derivation

      1. associate--l+ 59.1%

         \[\leadsto 2 \cdot \color{blue}{\left(-0.5 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + \left(\frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)\right)} \]

      2. fma-def 59.1%

         \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(-0.5, \frac{{\ell}^{2}}{{k}^{2} \cdot t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} \]

      3. unpow2 59.1%

         \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

      4. times-frac 55.3%

         \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{t}}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

      5. unpow2 55.3%

         \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

      6. *-commutative 55.3%

         \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{{\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

      7. associate-/r* 55.0%

         \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

      8. unpow2 55.0%

         \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\color{blue}{\ell \cdot \ell}}{t}}{{k}^{4}} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

      9. associate-/l* 55.1%

         \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\color{blue}{\frac{\ell}{\frac{t}{\ell}}}}{{k}^{4}} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

      10. associate-*r/ 55.1%

          \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}} - \color{blue}{\frac{-0.3333333333333333 \cdot {\ell}^{2}}{{k}^{2} \cdot t}}\right) \]

      11. times-frac 55.1%

          \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}} - \color{blue}{\frac{-0.3333333333333333}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t}}\right) \]

      12. unpow2 55.1%

          \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}} - \frac{-0.3333333333333333}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{t}\right) \]

      13. unpow2 55.1%

          \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}} - \frac{-0.3333333333333333}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{t}\right) \]

      14. associate-/l* 61.2%

          \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}} - \frac{-0.3333333333333333}{k \cdot k} \cdot \color{blue}{\frac{\ell}{\frac{t}{\ell}}}\right) \]

   9. Simplified 61.2%

      \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}} - \frac{-0.3333333333333333}{k \cdot k} \cdot \frac{\ell}{\frac{t}{\ell}}\right)} \]

   10. Taylor expanded in k around inf 55.1%

       \[\leadsto 2 \cdot \color{blue}{\frac{-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2}}} \]
   11. Step-by-step derivation

       1. distribute-rgt-out-- 59.3%

          \[\leadsto 2 \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{t} \cdot \left(-0.5 - -0.3333333333333333\right)}}{{k}^{2}} \]

       2. unpow2 59.3%

          \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\ell \cdot \ell}}{t} \cdot \left(-0.5 - -0.3333333333333333\right)}{{k}^{2}} \]

       3. associate-/l* 65.4%

          \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\ell}{\frac{t}{\ell}}} \cdot \left(-0.5 - -0.3333333333333333\right)}{{k}^{2}} \]

       4. metadata-eval 65.4%

          \[\leadsto 2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}} \cdot \color{blue}{-0.16666666666666666}}{{k}^{2}} \]

       5. unpow2 65.4%

          \[\leadsto 2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}} \cdot -0.16666666666666666}{\color{blue}{k \cdot k}} \]

       6. times-frac 60.9%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{\frac{\ell}{\frac{t}{\ell}}}{k} \cdot \frac{-0.16666666666666666}{k}\right)} \]

       7. associate-/l/ 61.1%

          \[\leadsto 2 \cdot \left(\color{blue}{\frac{\ell}{k \cdot \frac{t}{\ell}}} \cdot \frac{-0.16666666666666666}{k}\right) \]

   12. Simplified 61.1%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{k \cdot \frac{t}{\ell}} \cdot \frac{-0.16666666666666666}{k}\right)} \]

   13. Taylor expanded in k around 0 69.6%

       \[\leadsto 2 \cdot \left(\frac{\ell}{\color{blue}{\frac{k \cdot t}{\ell}}} \cdot \frac{-0.16666666666666666}{k}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.6 \cdot 10^{+16}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{t}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{\frac{t \cdot k}{\ell}} \cdot \frac{-0.16666666666666666}{k}\right)\\ \end{array} \]

Alternative 15: 33.1% accurate, 32.4× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(-0.16666666666666666 \cdot \frac{\ell \cdot \ell}{k \cdot \left(t \cdot k\right)}\right) \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* 2.0 (* -0.16666666666666666 (/ (* l l) (* k (* t k))))))

C

double code(double t, double l, double k) {
	return 2.0 * (-0.16666666666666666 * ((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 = 2.0d0 * ((-0.16666666666666666d0) * ((l * l) / (k * (t * k))))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * (-0.16666666666666666 * ((l * l) / (k * (t * k))));
}

Python

def code(t, l, k):
	return 2.0 * (-0.16666666666666666 * ((l * l) / (k * (t * k))))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(-0.16666666666666666 * Float64(Float64(l * l) / Float64(k * Float64(t * k)))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * (-0.16666666666666666 * ((l * l) / (k * (t * k))));
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(-0.16666666666666666 * N[(N[(l * l), $MachinePrecision] / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 56.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

   2. associate-*l* 53.3%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. sqr-neg 53.3%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   4. associate-*l* 56.5%

      \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   5. *-commutative 56.5%

      \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   6. sqr-neg 56.5%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   7. associate-*l/ 58.1%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   8. associate-*r/ 58.0%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   9. associate-/r/ 57.6%

      \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

3. Simplified 57.6%

   \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

4. Taylor expanded in k around inf 60.9%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
5. Step-by-step derivation

   1. times-frac 60.4%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]

   2. unpow2 60.4%

      \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \]

   3. unpow2 60.4%

      \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \]

6. Simplified 60.4%

   \[\leadsto \color{blue}{2 \cdot \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]

7. Taylor expanded in k around 0 31.0%

   \[\leadsto 2 \cdot \color{blue}{\left(\left(-0.5 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} \]
8. Step-by-step derivation

   1. associate--l+ 31.0%

      \[\leadsto 2 \cdot \color{blue}{\left(-0.5 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + \left(\frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)\right)} \]

   2. fma-def 31.0%

      \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(-0.5, \frac{{\ell}^{2}}{{k}^{2} \cdot t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} \]

   3. unpow2 31.0%

      \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

   4. times-frac 30.7%

      \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{t}}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

   5. unpow2 30.7%

      \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

   6. *-commutative 30.7%

      \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{{\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

   7. associate-/r* 31.0%

      \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

   8. unpow2 31.0%

      \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\color{blue}{\ell \cdot \ell}}{t}}{{k}^{4}} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

   9. associate-/l* 31.6%

      \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\color{blue}{\frac{\ell}{\frac{t}{\ell}}}}{{k}^{4}} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

   10. associate-*r/ 31.6%

       \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}} - \color{blue}{\frac{-0.3333333333333333 \cdot {\ell}^{2}}{{k}^{2} \cdot t}}\right) \]

   11. times-frac 33.1%

       \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}} - \color{blue}{\frac{-0.3333333333333333}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t}}\right) \]

   12. unpow2 33.1%

       \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}} - \frac{-0.3333333333333333}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{t}\right) \]

   13. unpow2 33.1%

       \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}} - \frac{-0.3333333333333333}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{t}\right) \]

   14. associate-/l* 34.8%

       \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}} - \frac{-0.3333333333333333}{k \cdot k} \cdot \color{blue}{\frac{\ell}{\frac{t}{\ell}}}\right) \]

9. Simplified 34.8%

   \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}} - \frac{-0.3333333333333333}{k \cdot k} \cdot \frac{\ell}{\frac{t}{\ell}}\right)} \]

10. Taylor expanded in k around inf 29.6%

    \[\leadsto 2 \cdot \color{blue}{\frac{-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2}}} \]
11. Step-by-step derivation

    1. distribute-rgt-out-- 31.3%

       \[\leadsto 2 \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{t} \cdot \left(-0.5 - -0.3333333333333333\right)}}{{k}^{2}} \]

    2. unpow2 31.3%

       \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\ell \cdot \ell}}{t} \cdot \left(-0.5 - -0.3333333333333333\right)}{{k}^{2}} \]

    3. associate-/l* 32.9%

       \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\ell}{\frac{t}{\ell}}} \cdot \left(-0.5 - -0.3333333333333333\right)}{{k}^{2}} \]

    4. metadata-eval 32.9%

       \[\leadsto 2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}} \cdot \color{blue}{-0.16666666666666666}}{{k}^{2}} \]

    5. unpow2 32.9%

       \[\leadsto 2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}} \cdot -0.16666666666666666}{\color{blue}{k \cdot k}} \]

    6. times-frac 34.6%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\frac{\ell}{\frac{t}{\ell}}}{k} \cdot \frac{-0.16666666666666666}{k}\right)} \]

    7. associate-/l/ 34.8%

       \[\leadsto 2 \cdot \left(\color{blue}{\frac{\ell}{k \cdot \frac{t}{\ell}}} \cdot \frac{-0.16666666666666666}{k}\right) \]

12. Simplified 34.8%

    \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{k \cdot \frac{t}{\ell}} \cdot \frac{-0.16666666666666666}{k}\right)} \]

13. Taylor expanded in l around 0 32.0%

    \[\leadsto 2 \cdot \color{blue}{\left(-0.16666666666666666 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} \]
14. Step-by-step derivation

    1. unpow2 32.0%

       \[\leadsto 2 \cdot \left(-0.16666666666666666 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t}\right) \]

    2. unpow2 32.0%

       \[\leadsto 2 \cdot \left(-0.16666666666666666 \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot t}\right) \]

    3. associate-*l* 33.3%

       \[\leadsto 2 \cdot \left(-0.16666666666666666 \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot t\right)}}\right) \]

15. Simplified 33.3%

    \[\leadsto 2 \cdot \color{blue}{\left(-0.16666666666666666 \cdot \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}\right)} \]

16. Final simplification 33.3%

    \[\leadsto 2 \cdot \left(-0.16666666666666666 \cdot \frac{\ell \cdot \ell}{k \cdot \left(t \cdot k\right)}\right) \]

Alternative 16: 35.0% accurate, 32.4× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\frac{-0.16666666666666666}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{t}\right)\right) \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* 2.0 (* (/ -0.16666666666666666 k) (* (/ l k) (/ l t)))))

C

double code(double t, double l, double k) {
	return 2.0 * ((-0.16666666666666666 / k) * ((l / 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 * (((-0.16666666666666666d0) / k) * ((l / k) * (l / t)))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * ((-0.16666666666666666 / k) * ((l / k) * (l / t)));
}

Python

def code(t, l, k):
	return 2.0 * ((-0.16666666666666666 / k) * ((l / k) * (l / t)))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64(-0.16666666666666666 / k) * Float64(Float64(l / k) * Float64(l / t))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * ((-0.16666666666666666 / k) * ((l / k) * (l / t)));
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(-0.16666666666666666 / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 56.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

   2. associate-*l* 53.3%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. sqr-neg 53.3%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   4. associate-*l* 56.5%

      \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   5. *-commutative 56.5%

      \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   6. sqr-neg 56.5%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   7. associate-*l/ 58.1%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   8. associate-*r/ 58.0%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   9. associate-/r/ 57.6%

      \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

3. Simplified 57.6%

   \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

4. Taylor expanded in k around inf 60.9%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
5. Step-by-step derivation

   1. times-frac 60.4%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]

   2. unpow2 60.4%

      \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \]

   3. unpow2 60.4%

      \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \]

6. Simplified 60.4%

   \[\leadsto \color{blue}{2 \cdot \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]

7. Taylor expanded in k around 0 31.0%

   \[\leadsto 2 \cdot \color{blue}{\left(\left(-0.5 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} \]
8. Step-by-step derivation

   1. associate--l+ 31.0%

      \[\leadsto 2 \cdot \color{blue}{\left(-0.5 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + \left(\frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)\right)} \]

   2. fma-def 31.0%

      \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(-0.5, \frac{{\ell}^{2}}{{k}^{2} \cdot t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} \]

   3. unpow2 31.0%

      \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

   4. times-frac 30.7%

      \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{t}}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

   5. unpow2 30.7%

      \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

   6. *-commutative 30.7%

      \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{{\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

   7. associate-/r* 31.0%

      \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

   8. unpow2 31.0%

      \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\color{blue}{\ell \cdot \ell}}{t}}{{k}^{4}} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

   9. associate-/l* 31.6%

      \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\color{blue}{\frac{\ell}{\frac{t}{\ell}}}}{{k}^{4}} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

   10. associate-*r/ 31.6%

       \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}} - \color{blue}{\frac{-0.3333333333333333 \cdot {\ell}^{2}}{{k}^{2} \cdot t}}\right) \]

   11. times-frac 33.1%

       \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}} - \color{blue}{\frac{-0.3333333333333333}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t}}\right) \]

   12. unpow2 33.1%

       \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}} - \frac{-0.3333333333333333}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{t}\right) \]

   13. unpow2 33.1%

       \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}} - \frac{-0.3333333333333333}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{t}\right) \]

   14. associate-/l* 34.8%

       \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}} - \frac{-0.3333333333333333}{k \cdot k} \cdot \color{blue}{\frac{\ell}{\frac{t}{\ell}}}\right) \]

9. Simplified 34.8%

   \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}} - \frac{-0.3333333333333333}{k \cdot k} \cdot \frac{\ell}{\frac{t}{\ell}}\right)} \]

10. Taylor expanded in k around inf 29.6%

    \[\leadsto 2 \cdot \color{blue}{\frac{-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2}}} \]
11. Step-by-step derivation

    1. distribute-rgt-out-- 31.3%

       \[\leadsto 2 \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{t} \cdot \left(-0.5 - -0.3333333333333333\right)}}{{k}^{2}} \]

    2. unpow2 31.3%

       \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\ell \cdot \ell}}{t} \cdot \left(-0.5 - -0.3333333333333333\right)}{{k}^{2}} \]

    3. associate-/l* 32.9%

       \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\ell}{\frac{t}{\ell}}} \cdot \left(-0.5 - -0.3333333333333333\right)}{{k}^{2}} \]

    4. metadata-eval 32.9%

       \[\leadsto 2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}} \cdot \color{blue}{-0.16666666666666666}}{{k}^{2}} \]

    5. unpow2 32.9%

       \[\leadsto 2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}} \cdot -0.16666666666666666}{\color{blue}{k \cdot k}} \]

    6. times-frac 34.6%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\frac{\ell}{\frac{t}{\ell}}}{k} \cdot \frac{-0.16666666666666666}{k}\right)} \]

    7. associate-/l/ 34.8%

       \[\leadsto 2 \cdot \left(\color{blue}{\frac{\ell}{k \cdot \frac{t}{\ell}}} \cdot \frac{-0.16666666666666666}{k}\right) \]

12. Simplified 34.8%

    \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{k \cdot \frac{t}{\ell}} \cdot \frac{-0.16666666666666666}{k}\right)} \]

13. Taylor expanded in l around 0 32.0%

    \[\leadsto 2 \cdot \color{blue}{\left(-0.16666666666666666 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} \]
14. Step-by-step derivation

    1. associate-*r/ 32.0%

       \[\leadsto 2 \cdot \color{blue}{\frac{-0.16666666666666666 \cdot {\ell}^{2}}{{k}^{2} \cdot t}} \]

    2. *-commutative 32.0%

       \[\leadsto 2 \cdot \frac{-0.16666666666666666 \cdot {\ell}^{2}}{\color{blue}{t \cdot {k}^{2}}} \]

    3. associate-/r* 31.3%

       \[\leadsto 2 \cdot \color{blue}{\frac{\frac{-0.16666666666666666 \cdot {\ell}^{2}}{t}}{{k}^{2}}} \]

    4. *-commutative 31.3%

       \[\leadsto 2 \cdot \frac{\frac{\color{blue}{{\ell}^{2} \cdot -0.16666666666666666}}{t}}{{k}^{2}} \]

    5. associate-*l/ 31.3%

       \[\leadsto 2 \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{t} \cdot -0.16666666666666666}}{{k}^{2}} \]

    6. unpow2 31.3%

       \[\leadsto 2 \cdot \frac{\frac{{\ell}^{2}}{t} \cdot -0.16666666666666666}{\color{blue}{k \cdot k}} \]

    7. times-frac 34.0%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\frac{{\ell}^{2}}{t}}{k} \cdot \frac{-0.16666666666666666}{k}\right)} \]

    8. associate-/r* 33.5%

       \[\leadsto 2 \cdot \left(\color{blue}{\frac{{\ell}^{2}}{t \cdot k}} \cdot \frac{-0.16666666666666666}{k}\right) \]

    9. unpow2 33.5%

       \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{t \cdot k} \cdot \frac{-0.16666666666666666}{k}\right) \]

    10. times-frac 34.7%

        \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{k}\right)} \cdot \frac{-0.16666666666666666}{k}\right) \]

15. Simplified 34.7%

    \[\leadsto 2 \cdot \color{blue}{\left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k}\right) \cdot \frac{-0.16666666666666666}{k}\right)} \]

16. Final simplification 34.7%

    \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{t}\right)\right) \]

Alternative 17: 35.2% accurate, 32.4× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\frac{-0.16666666666666666}{k} \cdot \frac{\ell}{k \cdot \frac{t}{\ell}}\right) \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* 2.0 (* (/ -0.16666666666666666 k) (/ l (* k (/ t l))))))

C

double code(double t, double l, double k) {
	return 2.0 * ((-0.16666666666666666 / k) * (l / (k * (t / l))));
}

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 * (((-0.16666666666666666d0) / k) * (l / (k * (t / l))))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * ((-0.16666666666666666 / k) * (l / (k * (t / l))));
}

Python

def code(t, l, k):
	return 2.0 * ((-0.16666666666666666 / k) * (l / (k * (t / l))))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64(-0.16666666666666666 / k) * Float64(l / Float64(k * Float64(t / l)))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * ((-0.16666666666666666 / k) * (l / (k * (t / l))));
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(-0.16666666666666666 / k), $MachinePrecision] * N[(l / N[(k * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 56.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

   2. associate-*l* 53.3%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. sqr-neg 53.3%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   4. associate-*l* 56.5%

      \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   5. *-commutative 56.5%

      \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   6. sqr-neg 56.5%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   7. associate-*l/ 58.1%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   8. associate-*r/ 58.0%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   9. associate-/r/ 57.6%

      \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

3. Simplified 57.6%

   \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

4. Taylor expanded in k around inf 60.9%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
5. Step-by-step derivation

   1. times-frac 60.4%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]

   2. unpow2 60.4%

      \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \]

   3. unpow2 60.4%

      \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \]

6. Simplified 60.4%

   \[\leadsto \color{blue}{2 \cdot \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]

7. Taylor expanded in k around 0 31.0%

   \[\leadsto 2 \cdot \color{blue}{\left(\left(-0.5 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} \]
8. Step-by-step derivation

   1. associate--l+ 31.0%

      \[\leadsto 2 \cdot \color{blue}{\left(-0.5 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + \left(\frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)\right)} \]

   2. fma-def 31.0%

      \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(-0.5, \frac{{\ell}^{2}}{{k}^{2} \cdot t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} \]

   3. unpow2 31.0%

      \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

   4. times-frac 30.7%

      \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{t}}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

   5. unpow2 30.7%

      \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

   6. *-commutative 30.7%

      \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{{\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

   7. associate-/r* 31.0%

      \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

   8. unpow2 31.0%

      \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\color{blue}{\ell \cdot \ell}}{t}}{{k}^{4}} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

   9. associate-/l* 31.6%

      \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\color{blue}{\frac{\ell}{\frac{t}{\ell}}}}{{k}^{4}} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

   10. associate-*r/ 31.6%

       \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}} - \color{blue}{\frac{-0.3333333333333333 \cdot {\ell}^{2}}{{k}^{2} \cdot t}}\right) \]

   11. times-frac 33.1%

       \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}} - \color{blue}{\frac{-0.3333333333333333}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t}}\right) \]

   12. unpow2 33.1%

       \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}} - \frac{-0.3333333333333333}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{t}\right) \]

   13. unpow2 33.1%

       \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}} - \frac{-0.3333333333333333}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{t}\right) \]

   14. associate-/l* 34.8%

       \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}} - \frac{-0.3333333333333333}{k \cdot k} \cdot \color{blue}{\frac{\ell}{\frac{t}{\ell}}}\right) \]

9. Simplified 34.8%

   \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}} - \frac{-0.3333333333333333}{k \cdot k} \cdot \frac{\ell}{\frac{t}{\ell}}\right)} \]

10. Taylor expanded in k around inf 29.6%

    \[\leadsto 2 \cdot \color{blue}{\frac{-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2}}} \]
11. Step-by-step derivation

    1. distribute-rgt-out-- 31.3%

       \[\leadsto 2 \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{t} \cdot \left(-0.5 - -0.3333333333333333\right)}}{{k}^{2}} \]

    2. unpow2 31.3%

       \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\ell \cdot \ell}}{t} \cdot \left(-0.5 - -0.3333333333333333\right)}{{k}^{2}} \]

    3. associate-/l* 32.9%

       \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\ell}{\frac{t}{\ell}}} \cdot \left(-0.5 - -0.3333333333333333\right)}{{k}^{2}} \]

    4. metadata-eval 32.9%

       \[\leadsto 2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}} \cdot \color{blue}{-0.16666666666666666}}{{k}^{2}} \]

    5. unpow2 32.9%

       \[\leadsto 2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}} \cdot -0.16666666666666666}{\color{blue}{k \cdot k}} \]

    6. times-frac 34.6%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\frac{\ell}{\frac{t}{\ell}}}{k} \cdot \frac{-0.16666666666666666}{k}\right)} \]

    7. associate-/l/ 34.8%

       \[\leadsto 2 \cdot \left(\color{blue}{\frac{\ell}{k \cdot \frac{t}{\ell}}} \cdot \frac{-0.16666666666666666}{k}\right) \]

12. Simplified 34.8%

    \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{k \cdot \frac{t}{\ell}} \cdot \frac{-0.16666666666666666}{k}\right)} \]

13. Final simplification 34.8%

    \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{k} \cdot \frac{\ell}{k \cdot \frac{t}{\ell}}\right) \]

Alternative 18: 35.2% accurate, 32.4× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\frac{\ell}{\frac{t \cdot k}{\ell}} \cdot \frac{-0.16666666666666666}{k}\right) \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* 2.0 (* (/ l (/ (* t k) l)) (/ -0.16666666666666666 k))))

C

double code(double t, double l, double k) {
	return 2.0 * ((l / ((t * k) / l)) * (-0.16666666666666666 / 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 * ((l / ((t * k) / l)) * ((-0.16666666666666666d0) / k))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * ((l / ((t * k) / l)) * (-0.16666666666666666 / k));
}

Python

def code(t, l, k):
	return 2.0 * ((l / ((t * k) / l)) * (-0.16666666666666666 / k))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64(l / Float64(Float64(t * k) / l)) * Float64(-0.16666666666666666 / k)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * ((l / ((t * k) / l)) * (-0.16666666666666666 / k));
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(l / N[(N[(t * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 56.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

   2. associate-*l* 53.3%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. sqr-neg 53.3%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   4. associate-*l* 56.5%

      \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   5. *-commutative 56.5%

      \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   6. sqr-neg 56.5%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   7. associate-*l/ 58.1%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   8. associate-*r/ 58.0%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   9. associate-/r/ 57.6%

      \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

3. Simplified 57.6%

   \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

4. Taylor expanded in k around inf 60.9%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
5. Step-by-step derivation

   1. times-frac 60.4%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]

   2. unpow2 60.4%

      \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \]

   3. unpow2 60.4%

      \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \]

6. Simplified 60.4%

   \[\leadsto \color{blue}{2 \cdot \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]

7. Taylor expanded in k around 0 31.0%

   \[\leadsto 2 \cdot \color{blue}{\left(\left(-0.5 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} \]
8. Step-by-step derivation

   1. associate--l+ 31.0%

      \[\leadsto 2 \cdot \color{blue}{\left(-0.5 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + \left(\frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)\right)} \]

   2. fma-def 31.0%

      \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(-0.5, \frac{{\ell}^{2}}{{k}^{2} \cdot t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} \]

   3. unpow2 31.0%

      \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

   4. times-frac 30.7%

      \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{t}}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

   5. unpow2 30.7%

      \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

   6. *-commutative 30.7%

      \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{{\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

   7. associate-/r* 31.0%

      \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

   8. unpow2 31.0%

      \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\color{blue}{\ell \cdot \ell}}{t}}{{k}^{4}} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

   9. associate-/l* 31.6%

      \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\color{blue}{\frac{\ell}{\frac{t}{\ell}}}}{{k}^{4}} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

   10. associate-*r/ 31.6%

       \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}} - \color{blue}{\frac{-0.3333333333333333 \cdot {\ell}^{2}}{{k}^{2} \cdot t}}\right) \]

   11. times-frac 33.1%

       \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}} - \color{blue}{\frac{-0.3333333333333333}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t}}\right) \]

   12. unpow2 33.1%

       \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}} - \frac{-0.3333333333333333}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{t}\right) \]

   13. unpow2 33.1%

       \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}} - \frac{-0.3333333333333333}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{t}\right) \]

   14. associate-/l* 34.8%

       \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}} - \frac{-0.3333333333333333}{k \cdot k} \cdot \color{blue}{\frac{\ell}{\frac{t}{\ell}}}\right) \]

9. Simplified 34.8%

   \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}} - \frac{-0.3333333333333333}{k \cdot k} \cdot \frac{\ell}{\frac{t}{\ell}}\right)} \]

10. Taylor expanded in k around inf 29.6%

    \[\leadsto 2 \cdot \color{blue}{\frac{-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2}}} \]
11. Step-by-step derivation

    1. distribute-rgt-out-- 31.3%

       \[\leadsto 2 \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{t} \cdot \left(-0.5 - -0.3333333333333333\right)}}{{k}^{2}} \]

    2. unpow2 31.3%

       \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\ell \cdot \ell}}{t} \cdot \left(-0.5 - -0.3333333333333333\right)}{{k}^{2}} \]

    3. associate-/l* 32.9%

       \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\ell}{\frac{t}{\ell}}} \cdot \left(-0.5 - -0.3333333333333333\right)}{{k}^{2}} \]

    4. metadata-eval 32.9%

       \[\leadsto 2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}} \cdot \color{blue}{-0.16666666666666666}}{{k}^{2}} \]

    5. unpow2 32.9%

       \[\leadsto 2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}} \cdot -0.16666666666666666}{\color{blue}{k \cdot k}} \]

    6. times-frac 34.6%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\frac{\ell}{\frac{t}{\ell}}}{k} \cdot \frac{-0.16666666666666666}{k}\right)} \]

    7. associate-/l/ 34.8%

       \[\leadsto 2 \cdot \left(\color{blue}{\frac{\ell}{k \cdot \frac{t}{\ell}}} \cdot \frac{-0.16666666666666666}{k}\right) \]

12. Simplified 34.8%

    \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{k \cdot \frac{t}{\ell}} \cdot \frac{-0.16666666666666666}{k}\right)} \]

13. Taylor expanded in k around 0 35.3%

    \[\leadsto 2 \cdot \left(\frac{\ell}{\color{blue}{\frac{k \cdot t}{\ell}}} \cdot \frac{-0.16666666666666666}{k}\right) \]

14. Final simplification 35.3%

    \[\leadsto 2 \cdot \left(\frac{\ell}{\frac{t \cdot k}{\ell}} \cdot \frac{-0.16666666666666666}{k}\right) \]

Alternative 19: 35.4% accurate, 32.4× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \frac{\ell \cdot -0.16666666666666666}{k \cdot \left(k \cdot \frac{t}{\ell}\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* 2.0 (/ (* l -0.16666666666666666) (* k (* k (/ t l))))))

C

double code(double t, double l, double k) {
	return 2.0 * ((l * -0.16666666666666666) / (k * (k * (t / l))));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * ((l * (-0.16666666666666666d0)) / (k * (k * (t / l))))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * ((l * -0.16666666666666666) / (k * (k * (t / l))));
}

Python

def code(t, l, k):
	return 2.0 * ((l * -0.16666666666666666) / (k * (k * (t / l))))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64(l * -0.16666666666666666) / Float64(k * Float64(k * Float64(t / l)))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * ((l * -0.16666666666666666) / (k * (k * (t / l))));
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(l * -0.16666666666666666), $MachinePrecision] / N[(k * N[(k * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 56.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

   2. associate-*l* 53.3%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. sqr-neg 53.3%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   4. associate-*l* 56.5%

      \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   5. *-commutative 56.5%

      \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   6. sqr-neg 56.5%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   7. associate-*l/ 58.1%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   8. associate-*r/ 58.0%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   9. associate-/r/ 57.6%

      \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

3. Simplified 57.6%

   \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

4. Taylor expanded in k around inf 60.9%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
5. Step-by-step derivation

   1. times-frac 60.4%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]

   2. unpow2 60.4%

      \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \]

   3. unpow2 60.4%

      \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \]

6. Simplified 60.4%

   \[\leadsto \color{blue}{2 \cdot \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]

7. Taylor expanded in k around 0 31.0%

   \[\leadsto 2 \cdot \color{blue}{\left(\left(-0.5 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} \]
8. Step-by-step derivation

   1. associate--l+ 31.0%

      \[\leadsto 2 \cdot \color{blue}{\left(-0.5 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + \left(\frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)\right)} \]

   2. fma-def 31.0%

      \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(-0.5, \frac{{\ell}^{2}}{{k}^{2} \cdot t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} \]

   3. unpow2 31.0%

      \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

   4. times-frac 30.7%

      \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{t}}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

   5. unpow2 30.7%

      \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

   6. *-commutative 30.7%

      \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{{\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

   7. associate-/r* 31.0%

      \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

   8. unpow2 31.0%

      \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\color{blue}{\ell \cdot \ell}}{t}}{{k}^{4}} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

   9. associate-/l* 31.6%

      \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\color{blue}{\frac{\ell}{\frac{t}{\ell}}}}{{k}^{4}} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

   10. associate-*r/ 31.6%

       \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}} - \color{blue}{\frac{-0.3333333333333333 \cdot {\ell}^{2}}{{k}^{2} \cdot t}}\right) \]

   11. times-frac 33.1%

       \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}} - \color{blue}{\frac{-0.3333333333333333}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t}}\right) \]

   12. unpow2 33.1%

       \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}} - \frac{-0.3333333333333333}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{t}\right) \]

   13. unpow2 33.1%

       \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}} - \frac{-0.3333333333333333}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{t}\right) \]

   14. associate-/l* 34.8%

       \[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}} - \frac{-0.3333333333333333}{k \cdot k} \cdot \color{blue}{\frac{\ell}{\frac{t}{\ell}}}\right) \]

9. Simplified 34.8%

   \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(-0.5, \frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}} - \frac{-0.3333333333333333}{k \cdot k} \cdot \frac{\ell}{\frac{t}{\ell}}\right)} \]

10. Taylor expanded in k around inf 29.6%

    \[\leadsto 2 \cdot \color{blue}{\frac{-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2}}} \]
11. Step-by-step derivation

    1. distribute-rgt-out-- 31.3%

       \[\leadsto 2 \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{t} \cdot \left(-0.5 - -0.3333333333333333\right)}}{{k}^{2}} \]

    2. unpow2 31.3%

       \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\ell \cdot \ell}}{t} \cdot \left(-0.5 - -0.3333333333333333\right)}{{k}^{2}} \]

    3. associate-/l* 32.9%

       \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\ell}{\frac{t}{\ell}}} \cdot \left(-0.5 - -0.3333333333333333\right)}{{k}^{2}} \]

    4. metadata-eval 32.9%

       \[\leadsto 2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}} \cdot \color{blue}{-0.16666666666666666}}{{k}^{2}} \]

    5. unpow2 32.9%

       \[\leadsto 2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}} \cdot -0.16666666666666666}{\color{blue}{k \cdot k}} \]

    6. times-frac 34.6%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\frac{\ell}{\frac{t}{\ell}}}{k} \cdot \frac{-0.16666666666666666}{k}\right)} \]

    7. associate-/l/ 34.8%

       \[\leadsto 2 \cdot \left(\color{blue}{\frac{\ell}{k \cdot \frac{t}{\ell}}} \cdot \frac{-0.16666666666666666}{k}\right) \]

12. Simplified 34.8%

    \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{k \cdot \frac{t}{\ell}} \cdot \frac{-0.16666666666666666}{k}\right)} \]
13. Step-by-step derivation

    1. frac-times 36.1%

       \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot -0.16666666666666666}{\left(k \cdot \frac{t}{\ell}\right) \cdot k}} \]

14. Applied egg-rr 36.1%

    \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot -0.16666666666666666}{\left(k \cdot \frac{t}{\ell}\right) \cdot k}} \]

15. Final simplification 36.1%

    \[\leadsto 2 \cdot \frac{\ell \cdot -0.16666666666666666}{k \cdot \left(k \cdot \frac{t}{\ell}\right)} \]

Reproduce

herbie shell --seed 2023271 
(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))))